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"""This module is deprecated. Please use :mod:`airflow.providers.microsoft.azure.operators.adls_list`."""
import warnings
# pylint: disable=unused-import
from airflow.providers.microsoft.azure.operators.adls_list import AzureDataLakeStorageListOperator # noqa
warnings.warn(
"This module is deprecated. Please use `airflow.providers.microsoft.azure.operators.adls_list`.",
DeprecationWarning,
stacklevel=2,
)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,279
|
{"url":"http:\/\/www.mathnet.ru\/php\/archive.phtml?wshow=paper&jrnid=znsl&paperid=2145&option_lang=eng","text":"Zapiski Nauchnykh Seminarov POMI\n RUS\u00a0 ENG JOURNALS \u00a0 PEOPLE \u00a0 ORGANISATIONS \u00a0 CONFERENCES \u00a0 SEMINARS \u00a0 VIDEO LIBRARY \u00a0 PACKAGE AMSBIB\n General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS\n\n Zap. Nauchn. Sem. POMI: Year: Volume: Issue: Page: Find\n\n Zap. Nauchn. Sem. POMI, 2008, Volume\u00a0358, Pages\u00a054\u201376 (Mi znsl2145)\n\nTime hierarchies for cryptographic function inversion with advice\n\nE.\u00a0A.\u00a0Hirscha, D.\u00a0Yu.\u00a0Grigor'evb, K.\u00a0V.\u00a0Pervyshevcd\n\na St. Petersburg Department of V.\u00a0A.\u00a0Steklov Institute of Mathematics, Russian Academy of Sciences\nb Institute of Mathematical Research of Rennes\nc Saint-Petersburg State University\nd University of California, San Diego, Department of Computer Science and Engineering\n\nAbstract: We prove a\u00a0time hierarchy theorem for inverting functions computable in a slightly non-uniform polynomial time. In particular, we prove that if there is a\u00a0strongly one-way function then for any\u00a0$k$ and for any polynomial\u00a0$p$, there is a\u00a0function\u00a0$f$ computable in linear time with one bit of advice such that there is a\u00a0polynomial-time probabilistic adversary that inverts\u00a0$f$ with probability $\\ge1\/p(n)$ on infinitely many lengths of input while all probabilistic $O(n^k)$-time adversaries with logarithmic advice invert\u00a0$f$ with probability less than $1\/p(n)$ on almost all lengths of input.\nWe also prove a\u00a0similar theorem in the worst-case setting, i.e., if $\\mathbf P\\neq\\mathbf{NP}$, then for every $l>k\\ge1$\n$$(\\mathbf{DTime}[n^k]\\cap\\mathbf{NTime}[n])\/1\\subsetneq(\\mathbf{DTime}[n^l]\\cap\\mathbf{NTime}[n])\/1.$$\nBibl.\u00a0\u2013 16\u00a0titles.\n\nFull text: PDF file (310\u00a0kB)\nReferences: PDF file \u00a0 HTML file\n\nEnglish version:\nJournal of Mathematical Sciences (New York), 2009, 158:5, 633\u2013644\n\nUDC: 510.52\n\nCitation: E.\u00a0A.\u00a0Hirsch, D.\u00a0Yu.\u00a0Grigor'ev, K.\u00a0V.\u00a0Pervyshev, \u201cTime hierarchies for cryptographic function inversion with advice\u201d, Studies in constructive mathematics and mathematical logic. Part\u00a0XI, Zap. Nauchn. Sem. POMI, 358, POMI, St.\u00a0Petersburg, 2008, 54\u201376; J. Math. Sci. (N. Y.), 158:5 (2009), 633\u2013644\n\nCitation in format AMSBIB\n\\Bibitem{HirGriPer08} \\by E.~A.~Hirsch, D.~Yu.~Grigor'ev, K.~V.~Pervyshev \\paper Time hierarchies for cryptographic function inversion with advice \\inbook Studies in constructive mathematics and mathematical logic. Part~XI \\serial Zap. Nauchn. Sem. POMI \\yr 2008 \\vol 358 \\pages 54--76 \\publ POMI \\publaddr St.~Petersburg \\mathnet{http:\/\/mi.mathnet.ru\/znsl2145} \\transl \\jour J. Math. Sci. (N. Y.) \\yr 2009 \\vol 158 \\issue 5 \\pages 633--644 \\crossref{https:\/\/doi.org\/10.1007\/s10958-009-9403-5} \\scopus{https:\/\/www.scopus.com\/record\/display.url?origin=inward&eid=2-s2.0-67349115686}","date":"2021-10-17 03:45:09","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\": 1, \"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\": 0.5330825448036194, \"perplexity\": 10642.788158808537}, \"config\": {\"markdown_headings\": false, \"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-2021-43\/segments\/1634323585120.89\/warc\/CC-MAIN-20211017021554-20211017051554-00625.warc.gz\"}"}
| null | null |
The national headquarters of the Maritime Workers Union of Nigeria, (MWUN) has formerly inaugurated two units in Port Harcourt Port.
They are frozen Fish Transport unit and the Rivers Port Haulage unit, who have been re-unionised and merged into Port Harcourt District of Dockworkers Branch of Maritime Workers Union of Nigeria.
Speaking during the event National Organising Secretary, MWUN, Comrade Abdulahi Oroje, said that the inauguration was informed based the report of the committee set up by the federal government to trim down the multiple associations and agencies operating in the ports legally or illegally.
According to him, it was therefore recommended that all associations related to transportation of general goods in the nation's seaports should belong to the Maritime Workers Union, adding that, it was on that premise the national body came to inaugurate those units in the Eastern Ports of Onne and Port Harcourt.
Comrade Oroje, said that the union has made tremendous break through in sending its members to Durban in South Africa for leadership training to improve their leadership qualities and capabilities as part of human capacity building.
He urged the newly inaugurated units to join hands in building the nation's economy by engaging in genuine ventures that are profitable and beneficial to the lives of the people and the country's economic growth.
Speaking to The Tide shortly after chairman of the Frozen fish handler/Haulage transport, Comrade Emma Fynface expressed joy over their inauguration and assured to work assiduously to ensure that they live and operate within the limits of the Maritime constitution, as well as maintain an effective and efficient safe service delivery culture.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 290
|
\section{1.\ Introduction}
\medskip\noindent
Mainly during the last three years progress has been made in showing that
the class of $(2,2)$ supersymmetric string compactifications is only
a small subset of all four--dimensional perturbative heterotic string vacua
featuring $N=1$ space--time supersymmetry
[2--5,11,13--15,21,26,27,29].
It was long believed that the general class of $(0,2)$ strings might not
be solutions of the string equations of motion at all, for these models
could receive destabilizing instanton corrections [10,28].
However, in a paper by E.\ Silverstein and E.\ Witten [27] it was
argued that for the class of linear $\sigma-$models such terms in the
superpotential cannot occur due to the absence of singularities in the
singlet couplings. Shortly afterwards in [4] we constructed
heterotic exactly solvable $(0,2)$ superconformal field theories (SCFTs)
exhibiting all the properties required for $(0,2)$ string vacua. For
instance, the phenomenologically most interesting feature is that the
gauge group is not restricted to $E_6$ as in the $(2,2)$ case [7]
but can also be $SO(10)$ or $SU(5)$ [28]. The main difficulty
turned out to really identify SCFTs with special points in the moduli
space of a Calabi--Yau $\sigma-$model with a choice of a stable,
holomorphic vector bundle for the left moving $\sigma-$model fermions.
In [5] for at least three $N=1$ models such an identification has
been shown to be possible, including the $SO(10)$ model we will focus on
in this paper. Moreover, for the class of $N=2$ space--time
supersymmetric strings it was furthermore possible to identify all
constructed SCFTs with certain bundles on $K_3\times T_2$ [6].
\par\noindent
In this paper we will investigate the moduli space of a concrete $(0,2)$
model by for the first time calculating all three-- and four--point
functions of the gauge singlets at the exactly solvable point. In [14]
using the Landau--Ginzburg description of the quintic in ${\mathchoice{\BCT}{\BCT}{\BCS}{\BCSS}}{\rm P}[4]$
it was already shown that besides the well known $(2,2)$ moduli space
[8] containing the complex and K\"ahler deformations there are flat
directions in the superpotential due to elements of $H^1({\rm End}(T))$,
as well. At least all states coming from untwisted sectors of the LG
orbifold are mutually integrable. In [27] it was argued, and
exemplified again for the case of the quintic, that even more is true.
All deformations related to parameters of the linear $\sigma-$model are
moduli for any value of the K\"ahler class. At the Landau--Ginzburg point
this includes some moduli from twisted sectors. Thus, for the quintic in
${\mathchoice{\BCT}{\BCT}{\BCS}{\BCSS}}{\rm P}[4]$ there is a 326--dimensional $(0,2)$ moduli space
containing a 102--dimensional $(2,2)$ subspace.
\par\noindent
One advantage of the knowledge of an exactly solvable model for a $(0,2)$
string vacuum is that it
allows one to make definite statements about nonvanishing superpotential
couplings, in general at least to finite order in the superpotential.
We study in detail the $(0,2)$ model with gauge group $SO(10)$ which was
first constructed in [4] and then related to the Calabi--Yau manifold
$\,\hbox{\hbox to -0.2pt{\vrule height 6.5pt width .2pt\hss}\rm P}_{1,1,1,1,2,2}[4\ 4]$ with the bundle $V(1,1,1,1,1;5)$ in [5].
After identifying our SCFT with a special point in the Landau--Ginzburg
sector, we calculate the couplings of all singlets to fourth order and
find that they do not all vanish. At the Landau--Ginzburg point there occur
more than the 329 massless singlets expected from the large radius limit.
Thus, not all singlets at the Landau--Ginzburg point correspond to
flat directions of the space--time superpotential. Unlike the $(2,2)$
case [12,17], at small radius there exists no algebraic
distinction among the complex, K\"ahler and bundle moduli. However, in
our special model, requiring certain properties expected for the
K\"ahler modulus and using F--flatness and D--flatness for the special
enhanced gauge symmetry, at least to lowest order the possible
candidates for the deformation of the radius are highly restricted.
\par\noindent
This paper is organized as follows. In section 2.\ we review the
construction of the $(0,2)$ SCFT. In section 3.\ we give evidence that
the SCFT lies in the Landau--Ginzburg phase of the corresponding linear
$\sigma-$model. In section 4.\ we make use of the SCFT to calculate all
holomorphic three-- and four--point functions of the $SO(10)$ singlets.
Finally, in section 5.\ we use these results to restrict the form of the
K\"ahler modulus and to speculate about new flat directions in the
superpotential not belonging to the moduli space of the linear
$\sigma-$model.
\medskip\noindent
\section{2.\ The exactly solvable model}
\smallskip\noindent
To begin with, we briefly review the model we will deal with in this
paper. The details of the construction of general $(0,2)$ SCFTs can be
found in [4,5]. In order to achieve heterotic modular invariant
partition functions we made use of the technique of simple currents
[19,22--25]. In light cone gauge the starting point
of the construction is the tensor product of CFTs as shown in Table 2.1.
\smallskip\noindent
\centerline{\vbox{
\hbox{\vbox{\offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&\cr}
\def\tablespace\noalign{\hrule}\tablespace{height2pt&\omit&&\omit&&\omit&&\omit&\cr\noalign{\hrule}height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule\halign{&\vrule#&\strut\hskip0.2cm\hfil#\hfill\hskip0.2cm\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
& part && $c$ && $\o{c}$ &\cr
\tablespace\noalign{\hrule}\tablespace\tablerule
& $4D$ space--time, $X^{\mu}$ && $2$ && $2$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $N=2$ Virasoro && $9$ && $9$ &\cr
\tablespace\noalign{\hrule}\tablespace
& $U(1)_2$ && $1$ && $1$ &\cr
\tablespace\noalign{\hrule}\tablespace
& gauge group\ $SO(8)\times E_8$ && $12$ && $12$ &\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr}\hrule}}
\hbox{\hskip 0.5cm Table 2.1 \hskip 0.5cm Underlying CFT for $SO(10)$}}}
\smallskip\noindent
Introducing a special set of simple currents guaranteeing
all properties we want to have for $(0,2)$ models like left
moving $N=2$ world sheet supersymmetry, projection onto
even $U(1)$ charges and an extension of the gauge group from
$SO(8)$ to $SO(10)$, we obtain modular invariant partition
functions of the following form:
$$ Z\sim\vec{\chi}(\tau)\,M(J_{GSO_l})\,\prod_j M(\Upsilon_j)\,\,
M(J_{GSO_r})\,\prod_i M(J_i)\,\,M(J_{(SO(8)\to SO(10))})\,
\vec{\chi}(\o\tau).\eqno(2.1)$$
In order to really get $(0,2)$ models one has to choose some
model dependent simple currents $\Upsilon_j$, which prevent
all the symmetries implemented on the right moving side
from acting also on the left moving side. In the model discussed
in [5], the internal $N=2$ part consists of five copies of the
$k=3$ minimal $N=2$ model (N2Vir(k=3)) and $\Upsilon$ is chosen to be
$$ \Upsilon=\Phi^3_{0,-1}\otimes\left(\Phi^0_{0,0}\right)^4
\otimes\Phi^{U(1)_2}_{1,2}\otimes\Phi^{SO(8)}_0,\eqno(2.2) $$
where $\Phi^l_{m,s}$ denotes the highest weight representations of the
$k=3$ minimal model.
Thus $\Upsilon$ acts nontrivially only on the first factor of the
internal tensor product and on the $U(1)_2$ part. The massless
spectrum contains the usual $N=1$ supergravity multiplet, chiral
multiplets in the four possible representations of $SO(10)_1$
and also some vector multiplets. There are precisely $N_{16}=80$
chiral fields in the spinor representation, $N_{10}=74$ chiral
fields in the vector representation and $N_{1}=350$
chiral fields in the singlet representation of $SO(10)$.
Furthermore, besides the gauge bosons of $SO(10)\times E_8$ the
spectrum contains $N_g=7$ further vector multiplets. Since the
simple current $\Upsilon$ itself and its charge conjugate
are two of these seven further gauge bosons, the special
enhanced gauge group cannot simply be $U(1)^7$.
It is easy to see that the three fields of conformal dimension one
$$\eqalignno{J^\pm(z) &= \Phi^3_{\mp 1}(z) \otimes
e^{\mp i \sqrt{5\over 3} \phi_1(z)}\otimes
e^{\pm i {1\over 2} \phi_{U(1)_2}(z)}, &(2.3)\cr
J^3(z) &= {1\over 2}\left( -5\ j_1(z) + 3\ j_{U(1)_2}(z) \right) &\cr }$$
satisfy the $SU(2)$ Kac--Moody algebra at level $k=3$. $\Phi^3_{\mp 1}(z)$
are primary fields of the $k=3$ parafermionic model and
the $U(1)$ currents of the first N2Vir(k=3) model and the $U(1)_2$ model
are $j_1=i\sqrt{3\over 5} \partial\phi_1$ and $j_{U(1)_2}=i \partial\phi_{U(1)_2}$,
respectively. Thus, the complete gauge group of this model is
$G=SO(10)\times E_8\times SU(2)_3\times U(1)^4$ and the massless spectrum
should also fit into the four allowed representations of $SU(2)_3$.
\par\noindent
In [5] we have listed the explicit form of the massless states in the
spinor and vector representation of $SO(10)$ and have given a monomial
representation of these states such that the Yukawa couplings
$\langle 10\ 16\ 16\rangle$ could be written as a monomial ring
$ {\mathchoice{\BCT}{\BCT}{\BCS}{\BCSS}} [x_i,y_j]/I$. The $x_i$ are four coordinates of weight one and the
$y_j$ are two coordinates of weight two. The ideal $I$ is generated by the
relations $x_i^4=0$ and $y_j\, y_k=0$. The same monomial
ring appears as the cohomological ring of the Calabi--Yau manifold
$\,\hbox{\hbox to -0.2pt{\vrule height 6.5pt width .2pt\hss}\rm P}_{1,1,1,1,2,2}[4\ 4]$ with the gauge bundle $V(1,1,1,1,1;5)$.
Thus, we concluded that the exactly solvable SCFT describes a certain point
in the moduli space of this $(0,2)$ model.
However, we have not determined to which point the SCFT corresponds, namely
what the form is of the quasi--homogeneous polynomials $W_{1,2}(x_i,y_j)$
and $F_{1,\ldots,5}(x_i,y_j)$ defining the complete intersection CY and
the vector bundle $V$, respectively. Furthermore, one has to know at which
radius $r$ of the K\"ahler modulus the model lives. In order to answer
these questions we also have to take the singlet fields into account.
\par\noindent
A singlet in the (--1) ghost picture is of the general form
$$ V_{-1}(z,\overline{z})=e^{-\rho(\overline{z})}{\cal{O}}_1(z,\overline{z})\
{\cal{F}}(z)\ e^{ikX(z,\overline{z})},\eqno(2.4)$$
where ${\cal{O}}_1$ is an internal operator of the $N=2$ theory with
$(c,\o c)=(9,9)$ and ${\cal{F}}$ denotes the left moving $U(1)_2$ part. The
product ${\cal{O}}_{1}\,{\cal{F}}$ has overall conformal dimension
$(h,\o{h})=(1,{1\over2})$ and charge $(q,\o{q})=(0,-1)$. In Table 2.2 and
2.3 we list the explicit form of these internal operators for all the 350
singlets occurring in the model. The degeneracy is due to three reasons.
Firstly, there is the $S_4$ permutation symmetry of the last four
N2Vir(k=3) tensor factors. Secondly, we have the four allowed $SU(2)_3$
representations with degenerated ground states of dimension one to four.
In Table 2.2 and 2.3 we always list the state with highest value of the
$U(1)_2$ quantum number. Thus, the other states in the $SU(2)$ multiplet
can be obtained by applying successively $J^-$. Finally, whenever a state
like, for instance\footnote{$^1$}{We use the notation introduced in
[5].\ $\left[l~\matrix{m&s\cr \o{m}&\o{s}\cr}\right]$ denotes a
primary field of N2Vir(k=3) and $[m]$ denotes one of the four primary
fields of $U(1)_2$.}
$$ \left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[3~\matrix{3&2\cr 3&0\cr}\right]
\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[0].\eqno(2.5)$$
occurs there is also a state
$$ \left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[3~\matrix{3&0\cr 3&0\cr}\right]
\left[2~\matrix{2&2\cr 2&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[0]. \eqno(2.6)$$
The left moving $G^i=\left[0~\matrix{0&2\cr 0&0\cr}\right]$ can be
permuted among all nonzero fields in the last four $N=2$ tensor factors.
In accordance to [14] we denote the untwisted fields by $S$ and the
twisted ones by $S'$.
\medskip\noindent
\centerline{\vbox{
\hbox{\vbox{\offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\def\tablespace\noalign{\hrule}\tablespace{height2pt&\omit&&\omit&&\omit&&\omit&\cr\noalign{\hrule}height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule\halign{&\vrule#&\strut\hskip0.2cm\hfil#\hfill\hskip0.2cm\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
&Type&&\hskip3.7cm${\cal{O}}_{1}$\hskip3.5cm${\cal{F}}$&&$SU(2)$ rep.&°.&\cr
\tablespace\noalign{\hrule}\tablespace\tablerule
& $S_a$ && $\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[3~\matrix{3&2\cr 3&0\cr}\right]
\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[0]$
&& $1$ && $24$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_b$ && $\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[3~\matrix{3&2\cr 3&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[0]$
&& $1$ && $36$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_c$ && $\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[2~\matrix{2&2\cr 2&0\cr}\right]
\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[0]$
&& $1$ && $36$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_d$ && $\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[2~\matrix{2&2\cr 2&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
[0]$
&& $1$ && $16$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_e$ && $\left[3~\matrix{3&0\cr 3&0\cr}\right]
\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[2]$
&& $4$ && $16$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_f$ && $\left[3~\matrix{3&0\cr 3&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[2]$
&& $4$ && $24$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_g$ && $\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[3~\matrix{3&2\cr 3&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[0]$
&& $2$ && $8$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_h$ && $\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[2~\matrix{2&2\cr 2&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[0]$
&& $2$ && $48$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_i$ && $\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[1~\matrix{1&2\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[0]$
&& $2$ && $24$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_j$ && $\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[3~\matrix{3&0\cr 3&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[2]$
&& $3$ && $36$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_k$ && $\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[2]$
&& $3$ && $18$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_l$ && $\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[2~\matrix{2&0\cr 2&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[0~\matrix{0&0\cr 0&0\cr}\right]
[2]$
&& $3$ && $36$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S_m$ && $\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
\left[1~\matrix{1&0\cr 1&0\cr}\right]
[2]$
&& $3$ && $3$ & \cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr}\hrule}}
\hbox{\hskip 0.5cm Table 2.2 \hskip 0.5cm Untwisted Singlets}}}
\medskip\noindent
These are exactly 325 of the 350 singlets. The remaining 25 states occur
in twisted sectors of the $GSO_r$ projection.
\medskip\noindent
\centerline{\vbox{
\hbox{\vbox{\offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\def\tablespace\noalign{\hrule}\tablespace{height2pt&\omit&&\omit&&\omit&&\omit&\cr\noalign{\hrule}height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule\halign{&\vrule#&\strut\hskip0.2cm\hfil#\hfill\hskip0.2cm\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
&Type&&\hskip4.5cm${\cal{O}}_{1}$\hskip4.5cm${\cal{F}}$&&rep.&°.&\cr
\tablespace\noalign{\hrule}\tablespace\tablerule
& $T'$ && $\left[0~\matrix{1&1\cr 0&0\cr}\right]
\left[2~\matrix{-2&0\cr 2&0\cr}\right]
\left[1~\matrix{-3&-2\cr 1&0\cr}\right]
\left[1~\matrix{-3&-2\cr 1&0\cr}\right]
\left[1~\matrix{-3&-2\cr 1&0\cr}\right]
[1]$
&& $1$ && $4$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S'_a$ && $\left[3~\matrix{-4 &-1\cr 3&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[0~\matrix{-2&-2\cr 0&0\cr}\right]
\left[0~\matrix{-2&-2\cr 0&0\cr}\right]
[1]$
&& $1$ && $6$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S'_b$ && $\left[2~\matrix{5 & 5\cr 2&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[0~\matrix{-2&-2\cr 0&0\cr}\right]
[1]$
&& $3$ && $12$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $S'_c$ && $\left[1~\matrix{-1 & -2\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
[0]$
&& $3$ && $3$ & \cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr}\hrule}}
\hbox{\hskip 0.5cm Table 2.3 \hskip 0.5cm Twisted Singlets}}}
\medskip\noindent
These numbers of untwisted and twisted singlets have to be compared with
the numbers of singlets in the LG phase of the corresponding linear
$\sigma-$model. It has been shown in [13] that for a generic choice
of the $W_i(x,y)$ and $F_i(x,y)$ the model contains 339 singlets,
318 of which are untwisted. Note that in the Calabi--Yau limit the
number of singlets is only
$$H^1(M,T)+H^1(M,T^*)+H^1(M,{\rm End}(V))=73+1+255=329.\eqno(2.7)$$
\medskip\noindent
\section{3.\ The Landau--Ginzburg phase}
\medskip\noindent
In order to identify the SCFT with the Landau--Ginzburg phase of the
linear $\sigma-$model we have at least to show that the massless
spectra are the same. For our model of interest the $(0,2)$ superpotential
is
$$ S_{\cal W} =\int d^2 z d\theta\ \left( \Sigma_j W_j(X,Y)
+ \Lambda_a F_a(X,Y)\, \right), \eqno(3.1)$$
where $\Sigma_{1,2}$ and $\Lambda_{1,\ldots,5}$ are Fermi superfields and
$X_{1,\ldots,4}$ and $Y_{1,2}$ are chiral superfields. $ W_{1,2}(X,Y)$
and $F_{1,\ldots,5}(X,Y)$ are quasihomogenous polynomials of degree four.
In the Landau--Ginzburg phase there exists a right $U(1)$ R--symmetry with
charges $\o{q}$ and a left $U(1)$ symmetry with charges $q$. In Table 3.1
we list all the charges of the fields involved in the calculation of the
massless spectrum.
\medskip\noindent
\centerline{\vbox{\hbox{
\vbox{
\hbox{\vbox{\offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&\cr}
\def\tablespace\noalign{\hrule}\tablespace{height2pt&\omit&&\omit&&\omit&&\omit&\cr\noalign{\hrule}height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule\halign{&\vrule#&\strut\hskip0.2cm\hfil#\hfill\hskip0.2cm\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
& field && $q$ && $\o{q}$ &\cr
\tablespace\noalign{\hrule}\tablespace\tablerule
& $x_{1,\ldots,4}$ && ${1\over 5}$ && ${1\over 5}$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $y_{1,2}$ && ${2\over 5}$ && ${2\over 5}$ &\cr
\tablespace\noalign{\hrule}\tablespace
& $\sigma_{1,2}$ && $-{4\over 5}$ && ${1\over 5}$ &\cr
\tablespace\noalign{\hrule}\tablespace
& $\lambda_{1,\ldots,5}$ && $-{4\over 5}$ && ${1\over 5}$ &\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr}\hrule}} }
\hskip 1cm
\vbox{
\hbox{\vbox{\offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&\cr}
\def\tablespace\noalign{\hrule}\tablespace{height2pt&\omit&&\omit&&\omit&&\omit&\cr\noalign{\hrule}height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule\halign{&\vrule#&\strut\hskip0.2cm\hfil#\hfill\hskip0.2cm\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
& field && $q$ && $\o{q}$ &\cr
\tablespace\noalign{\hrule}\tablespace\tablerule
& $\o x_{1,\ldots,4}$ && $-{1\over 5}$ && $-{1\over 5}$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $\o y_{1,2}$ && $-{2\over 5}$ && $-{2\over 5}$ &\cr
\tablespace\noalign{\hrule}\tablespace
& $\o\sigma_{1,2}$ && ${4\over 5}$ && $-{1\over 5}$ &\cr
\tablespace\noalign{\hrule}\tablespace
& $\o\lambda_{1,\ldots,5}$ && ${4\over 5}$ && $-{1\over 5}$ &\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr}\hrule}}} }
\hbox{\hskip 0.5cm Table 3.1 \hskip 0.5cm Left and right charges } }}
\medskip\noindent
For generic choices of the polynomials the massless spectrum has been
calculated for the Landau--Ginzburg orbifold in [13]. There are 80
chiral superfields in the spinor representation, 74 superfields in the
vector representation and 339 superfields in the singlet representation
of SO(10). The question is, whether there exists a choice of the
polynomials $W_j$ and $F_a$ such that there occur 350 singlets, 325
untwisted and 25 twisted, accompanied by an enhanced gauge group of
dimension seven. In [5] we have already made the following guess
for the form of the constraints:
$$\eqalignno{W_1(X,Y)&=\sum_{i=1}^4X_i^4\ +\ \sum_{j=1}^2 Y_j^2,
\quad\quad\ \ W_2(X,Y)=\sum_{i=1}^4 i\,X_i^4\ +\ \sum_{j=1}^2 j\ Y_j^2
&\cr F_i(X,Y)&=X_i^4\quad{\rm for}\ i\in\lbrace 1,\ldots,4\rbrace,
\quad F_5(X,Y) = Y_1\,Y_2, &(3.2)\cr }$$
which was motivated by the fact that the exactly solvable model has the
permutation symmetry $S_4$ in the last four $N=2$ tensor factors. In the
Landau--Ginzburg model the $U(1)$ gauge symmetry of the linear
$\sigma-$model is spontaneously broken to a finite group ${\mathchoice{\BZT}{\BZT}{\BZS}{\BZSS}}_5$, so that
one actually deals with an orbifold theory. Furthermore, to get a
heterotic string theory one has to combine the internal Landau--Ginzburg
sector with the linear part of the gauge group, which is $SO(8)$ in our
case. The GSO projection then selects states with $g=1$ for
$$ g={\rm exp}(-i\pi J_0)\times (-1)^\lambda. \eqno(3.3)$$
$J_0$ is the left $U(1)$ charge and $(-1)^\lambda$ denotes the charges of
the different $SO(8)$ representations. The resulting orbifold has sectors
twisted by $g^k$ for $k=0,\ldots,9$. If $k$ is even we call them $(R,R)$
sectors and if $k$ is odd we call them $(NS,R)$ sector. Finally, since one
is only interested in the massless sector of the string model, one can
employ a Born--Oppenheimer approximation and truncate the fields to their
lowest excited modes. The right moving $N=2$ algebra
$$\lbrace\o{Q}_-,\o{Q}_+\rbrace=\o{L}_0,\quad\o{Q}_-^2=\o{Q}_+^2=0
\eqno(3.4)$$
tells us that massless states are given by the cohomology of $\o{Q}_+$.
There is an expression even off--criticality for this operator in terms
of the fundamental fields in the Lagrangian:
$$\o{Q}_+=i\int(i\o\psi^i\partial\phi_i+{\cal W}|_{\theta=0}).\eqno(3.5)$$
By splitting this into $\o{Q}_+=\o{Q}_{+,r} + \o{Q}_{+,l}$ it was shown in
[20] that one can simply calculate the cohomology of $\o{Q}_{+,l}$ in
the cohomology of $\o{Q}_{+,r}$. In order to get more insight into these
methods the interested reader may take a look into [13,20]. By
going through the calculation of the massless spectrum carried out in
[13], one realizes that there are really more states for the choice
of the polynomials in (3.2). Table 8 of [13] shows that the massless
singlets get modified in the way described in Table 3.2.
\medskip\noindent
\centerline{\vbox{
\hbox{\vbox{\offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&\cr}
\def\tablespace\noalign{\hrule}\tablespace{height2pt&\omit&&\omit&&\omit&&\omit&\cr\noalign{\hrule}height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule\halign{&\vrule#&\strut\hskip0.2cm\hfil#\hfill\hskip0.2cm\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
& $SO(8)\times U(1)$ && $k$ && State &\cr
\tablespace\noalign{\hrule}\tablespace\tablerule
& $1_0$ && $1$ && $P_4(\Phi^i_{-{q_i\over 2}})\lambda^a_{-{3\over 5}}|0\rangle,
\ P_4(\Phi^i_{-{q_i\over 2}})\sigma^j_{-{3\over 5}}|0\rangle$ & \cr
\tablespace\noalign{\hrule}\tablespace
& $1_0$ && $3$ && $\lambda^a_{-{1\over 5}}\lambda^b_{-{1\over 5}}|0\rangle,\
\lambda^a_{-{1\over 5}}\sigma^j_{-{1\over 5}}|0\rangle,\
\sigma^1_{-{1\over 5}}\sigma^2_{-{1\over 5}}|0\rangle$ &\cr
\tablespace\noalign{\hrule}\tablespace
& $1_0$ && $5$ && $\o\lambda^a_{0}\,\o\lambda^b_{0}\,\o\lambda^c_{0}|0\rangle\
{\rm for}\ a,b,c\in \lbrace 1,\ldots,4\rbrace$ &\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr}\hrule}}
\hbox{\hskip 0.5cm Table 3.2 \hskip 0.5cm Massless LG singlets}}}
\medskip\noindent
$P_4(\Phi^i_{-{q_i\over 2}})$ denotes a polynomial of weight four in the
six coordinates $x_i, y_j$. The untwisted states for $k=1$ have to be
considered modulo the equivalence relations
$$\eqalignno{ &F_a(\Phi^i_{-{q_i\over 2}})\lambda^a_{-{3\over 5}}|0\rangle \sim
F_a(\Phi^i_{-{q_i\over 2}})\sigma^j_{-{3\over 5}}|0\rangle \sim 0 & \cr
&W_j(\Phi^i_{-{q_i\over 2}})\lambda^a_{-{3\over 5}}|0\rangle \sim
W_j(\Phi^i_{-{q_i\over 2}})\sigma^j_{-{3\over 5}}|0\rangle \sim 0 & \cr
&P_2(\Phi^i_{-{q_i\over 2}})\left[ {\partial F_a\over \partial
y^{1,2}_{-{1\over 5}}} \lambda^a_{-{3\over 5}} + {\partial W_j\over \partial
y^{1,2}_{-{1\over 5}} } \sigma^j_{-{3\over 5}} \right]|0\rangle \sim 0
&(3.6)\cr &P_1(\Phi^i_{-{q_i\over 2}})\left[ {\partial F_a\over \partial
x^{1,\ldots,4}_{-{1\over 10}} } \lambda^a_{-{3\over 5}} +
{\partial W_j\over \partial x^{1,\dots,4}_{-{1\over 10}} }
\sigma^j_{-{3\over 5}} \right]|0\rangle \sim 0.&\cr }$$
For generic $W_j$ and $F_a$ there are 318 such states. However, for the
symmetric choice in (3.2) we obtain 325 states, which are inevitably
accompanied by seven further gauge fields. Furthermore, there occur four
singlet fields from the $k=5$ sector, which are not present at a generic
point in the moduli space. Thus, together with the 21 states from the
$k=3$ twisted sector there are 25 twisted singlets, the same number as for
the exactly solvable model. This brief excursion to $(0,2)$
Landau--Ginzburg models has provided some more evidence that we can
identify the exactly solvable model with a $(0,2)$ Landau--Ginzburg model
naturally appearing in the $r\to-\infty$ limit of a linear $\sigma-$model.
In the following section we will calculate all three-- and four--point
functions of the SCFT.
\medskip\noindent
\section{4.\ Singlet couplings in the superpotential}
\medskip\noindent
Unlike, for instance, the quintic and the corresponding Landau--Ginzburg
model, in our $(0,2)$ case the number of $SO(10)$ singlets in the
large radius limit is different from the number of singlets in the
Landau--Ginzburg phase. Since we are now also equipped with an explicit
SCFT description, it is possible to investigate moduli
in the neighbourhood of the exactly solvable point.
Thus, we are looking for integrable marginal deformations of the SCFT
preserving the right moving $N=2$ world sheet supersymmetry.
From the space--time point of view this is equivalent to searching for
flat directions in the four--dimensional effective low energy $N=1$
space--time supersymmetric field theory. The scalar potential for such
supergravity theories is generally known as [1,9]
$$U=e^{{\cal K}}\left(D^iW\,G_{ij^{*}}^{-1}\,D^{j^*}W^*-3\,W\right)+
{1\over 2}\sum_a(D^a)^2,\eqno(4.1)$$
where ${\cal K}(\phi_i,\phi^*_i)$ is the K\"ahler potential,
$W(\phi_i)$ the holomorphic superpotential and $T^a$ generators of the
gauge group. The covariant derivative is given by
$$D^iW={\partial W\over\partial\phi_i}+{\partial{\cal K}\over\partial\phi_i}W\eqno(4.2)$$
and $D^a$ are auxiliary fields in the vector multiplets of the gauge bosons
$$ D^a=-{g_a\over 2}\,\left({\partial{\cal K}\over\partial\phi_i}\,T^a\phi_i+\phi^*_i
\,T^a\,{\partial{\cal K}\over\partial\phi^*_i}\right).\eqno(4.3)$$
In the lowest order, the renormalizable field theory limit, ${\cal K}$
is flat and the scalar potential takes the form
$$ U=\sum_a {g_a^2\over 2} |D^a|^2 + \sum_i |F_i|^2= \sum_a
{g_a^2\over 2}\, (\phi^*_i T^a \phi_i)^2
+ \sum_i \left| {\partial W\over \partial\phi_i} \right|^2 \eqno(4.4) $$
where $W$ contains only cubic couplings. In order for the scalar potential
to vanish both the D--terms and the F--terms have to be zero.
\par\noindent
For gauge singlets the condition of D--flatness is satisfied
automatically and one has only to check F--flatness for the superpotential.
However, even though we are dealing mostly with $SO(10)$ singlets
we have to cope with the D--terms arising from the enhanced
gauge symmetry $SU(2)\times U(1)^4$.
The superpotential of the low energy effective field theory can
be determined in the SCFT by calculating correlation functions of the
corresponding vertex operators on the $S^2$ world sheet.
For the following discussion one has to have in mind that the
vertex operators in the SCFT geometrically are tangent vectors
along the moduli space. Suppose one finds a set of scalars
$\lbrace S_i\rbrace$, such that all D--terms vanish and all
superpotential couplings of the form $F(S)$ and $F(S)S'$ are zero for
all scalars $S'$ in the model. Then the entire set $\lbrace S_i\rbrace$
are flat directions and define bona fide moduli of the theory.
However, if one has a set of scalars for which not all terms in
the scalar potential vanish, then one has to be very careful in
drawing any conclusions. It does not necessarily mean that there are
no flat directions at all. This can be seen by studying the following
well known example:\ Consider the simple case of a complex boson
with a global $U(1)$ symmetry and the Higgs potential
$$V(\phi)=\lambda\left(|\phi|^2-{m^2\over2\lambda}\right)^2.\eqno(4.5)$$
Now, expanding around one minimum $\langle\phi\rangle=v=\sqrt{m^2\over 2 \lambda}$
in the usual way
$$ \phi=\eta+v +i \chi \eqno(4.6)$$
one finds in the potential
$$V(\phi)=\lambda\left(4v^2\eta^2+4v\eta(\eta^2+\chi^2)+(\eta^2+\chi^2)^2
\right)\eqno(4.7)$$
both $\eta^2$ and $\chi^4$ couplings.
Thus, neither of the two fields satisfy $F(S)=0$. Nevertheless,
we know that there is a flat direction, the circle with radius $v$.
By looking at the lowest order term $\eta^2$ one can read off that
this flat direction locally is $(\eta,\chi)=\varepsilon(0,1)$.
Surely, choosing polar coordinates the angular variable does not appear
in the potential and defines the flat circle. In our case however,
the local coordinates are given by the primary fields in the SCFT and
without further knowledge there is no guarantee that these are
appropriate to capture certain flat directions explicitly. Furthermore,
unlike this model, one generally does not know the superpotential
to all orders, so that extracting definite statements is a difficult
task. In the next section we will exactly be confronted with such
problems, when we try to identify the K\"ahler modulus.
\par\noindent
First, we want to discuss the former sufficient flatness condition in our
example. In general it is hard to prove in CFT that an infinite set of
$n-$point functions vanishes unless one is equipped with some selection
rules which a priori disallow certain couplings to be nonzero.
In our case we have a lot of such selection rules related to the special
enhanced gauge symmetry. On the one hand side both the left and the
right moving $U(1)$ symmetries from each of the N2Vir(k=3) and $U(1)_2$
factors have to be preserved. On the other hand side there is the special
nonabelian $SU(2)$ gauge symmetry, which also constraints the possible
couplings. In particular, the $SU(2)$ spins have to couple to zero spin in
each correlation function and the relative couplings of members of $SU(2)$
multiplets are determined by Clebsch--Gordan coefficients.
\par\noindent
To make the discussion more transparent we introduce something like
an average, relative charge between the left and right moving sectors of
the singlets. All the singlets in the Tables 2.2 and 2.3 share the common
feature that in each of the last four N2Vir(k=3) factors five times the
difference between the left and the right moving $U(1)$ charge
$q_{\rm rel}$ is constant modulo five. For instance, for the field $T'$
$$ \left[0~\matrix{1&1\cr 0&0\cr}\right]
\left[2~\matrix{-2&0\cr 2&0\cr}\right]
\left[1~\matrix{-3&-2\cr 1&0\cr}\right]
\left[1~\matrix{-3&-2\cr 1&0\cr}\right]
\left[1~\matrix{-3&-2\cr 1&0\cr}\right]
[1] \eqno(4.8)$$
one has
$$ Q(T')=5\left(\left({3\over5}-1\right)-\left(-{1\over5}\right)\right)
=5\left(\left({2\over5}\right)-\left(-{2\over5}\right)\right)
=4\ {\rm mod}\ 5.\eqno(4.9)$$
One can also extend this definition to the other massless states in the
${\bf 16}$ and ${\bf 10}$ representation. In order to make it well--defined
one has to take into account the charge of the $SO(8)$ piece, as well.
Considering the decomposition of representations of $SO(10)$ in terms of
$SO(8)\times U(1)$
$$[16]=[8^1_v]\oplus[8^{-1}_c],\quad\quad[10]=[1^{-2}]\oplus[8^{0}_s]\oplus
[1^2]\eqno(4.10)$$
the correct definition is
$$Q=5\,q_{\rm rel}-{1\over2}(-1)^\lambda\quad{\rm mod}\ 5.\eqno(4.11)$$
This definition is very similar to the space--time R--charge $S_Q$
introduced in [14,15] for the massless states in the
Landau--Ginzburg model, for it contains information about the twisted
sector of the $GSO$ projection, in which the massless states occur.
One obtains for all the massless states in the model Table 4.1 of
space--time R--charges:
\medskip\noindent
\centerline{\vbox{
\hbox{\vbox{\offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr}
\def\tablespace\noalign{\hrule}\tablespace{height2pt&\omit&&\omit&&\omit&&\omit&\cr\noalign{\hrule}height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule\halign{&\vrule#&\strut\hskip0.2cm\hfil#\hfill\hskip0.2cm\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
& $S$ && $T'$ && $S'$ && {\bf 16} && {\bf 10} && ${\bf 10'}$ &\cr
\tablespace\noalign{\hrule}\tablespace\tablerule
& $0$ && $4$ && $2$ && $-{1\over 2}$ && $-1$ && $3$ &\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr}\hrule}}
\hbox{\hskip 0.5cm Table 4.1 \hskip 0.5cm R--charges of massless states}}}
\medskip\noindent
Note that this charges are completely analogous to the R--charges of
the different massless fields in the Landau--Ginzburg model, thus
providing further evidence for the identification of these two models.
The general form of a superpotential coupling of order $n$ is expressed
in terms of world sheet operators in the following way:
$$ C^{1,\ldots,n} =\int d^2 z_n\ldots\int d^2 z_1\langle V^n_0(z_n,\overline{z}_n)\ldots
V^4_0(z_4,\overline{z}_4)\,V^3_{-1}(z_3,\overline{z}_3)\,V^2_{-{1\over 2}}(z_2,\overline{z}_2)\,
V^1_{-{1\over 2}}(z_1,\overline{z}_1)\,\rangle.\eqno(4.12) $$
The lower index indicates the ghost picture, in which the vertex operator
has to be taken. Using the $SL(2,{\mathchoice{\BCT}{\BCT}{\BCS}{\BCSS}})$ symmetry one can shift in the usual
way three coordinates to $\lbrace 0,1,\infty\rbrace$ and can get rid of
three integrations by including the correct measure, for instance
$$\int d^2 z_n\ldots\int d^2 z_4|z_1-z_2|^2\,|z_1-z_3|^2\,|z_2-z_3|^2
\ldots.\eqno(4.13)$$
If the vertex operator for a massless state in a general representation of
the gauge group has the form
$$V_{-1}(z,\overline{z})=e^{-\rho(\overline{z})}\ {\cal{O}} _{1}(z,\overline{z})\ {\cal{F}}(z)\ \lambda^a(z)\
e^{ikX(z,\overline{z})}\eqno(4.14)$$
in the (--1) ghost picture, in the $\left(-{1\over 2}\right)$ ghost
picture it will look like
$$V_{-{1\over2}}(z,\overline{z})=e^{-{\rho(\overline{z})\over2}}\ S^\alpha(\overline{z})\
\Sigma_r{\cal{O}}_{1}(z,\overline{z})\ {\cal{F}}(z)\ \lambda^a(z)\ e^{ikX(z,\overline{z})}\eqno(4.15)$$
with $S^\alpha (\overline{z})$ being a four--dimensional spinor and $\Sigma_r$ the
internal right moving part of the space--time supercharge. In our case, it
is simply the primary field
$$ \Sigma_r=\left[0~\matrix{0&0\cr 1&1\cr}\right]
\left[0~\matrix{0&0\cr 1&1\cr}\right]
\left[0~\matrix{0&0\cr 1&1\cr}\right]
\left[0~\matrix{0&0\cr 1&1\cr}\right]
\left[0~\matrix{0&0\cr 1&1\cr}\right]\, [0].\eqno(4.16)$$
In the $(0)$ ghost picture one gets
$$ V_{0}(z,\overline{z})=(G_{r,\rm tot}+k\psi)\ {\cal{O}} _{1}(z,\overline{z})\ {\cal{F}}(z)\
\lambda^a(z)\ e^{ikX(z,\overline{z})},\eqno(4.17)$$
where $G_{r,\rm tot}$ is the total world sheet supercurrent of the $c=9$
right moving $N=2$ part:
$$G_{r,\rm tot}=\sum_{i=1}^5\left[0~\matrix{0&0\cr 0&0\cr}\right]\ldots
\underbrace{\left[0~\matrix{0&0\cr0&2\cr}\right]}_{i\,{\rm th\ factor}}
\ldots\left[0~\matrix{0&0\cr 0&0\cr}\right]\,[0].\eqno(4.18)$$
One can also attach an R--charge to the two operators $\Sigma_r$ and
$G_{r,\rm tot}$ appearing in the $-{1\over 2}$ and $0$ ghost picture,
respectively. Since $Q(\Sigma_r)=-{3\over2}$ and $Q(G_{r,\rm tot})=0$, the
sum of all $Q$ charges of all internal fields ${\cal{O}}\,{\cal{F}}$ must be equal to
$Q=3$ in order have vanishing R--charge for the entire coupling constant.
Thus, formally a term in the superpotential must have $Q=3$ yielding
already severe constraints on the possible terms in the superpotential.
\par\noindent
Surely, Yukawa couplings like $\langle 10\ 16\ 16\rangle$ or $\langle S\ 10\ 10\rangle$
could take nonzero values but couplings like $\langle 10'\ 16\ 16\rangle$ or
$\langle S'\ 10\ 10\rangle$ are forced to be zero by $Q$ conservation. For
instance, among the couplings of twisted fields only $\langle S'\ 10'\ 10'\rangle$
can also be nonzero. We will come back to such couplings later. Since all
untwisted singlets have zero R--charge, it follows directly that
$$ F(S)=0,\quad\quad F(S)\,S'=0.\eqno(4.19)$$
However, one must not forget the seven D--flatness conditions for
$SU(2)\times U(1)^4$, so that only 318 of the 325 singlets survive
as moduli of the SCFT. The remaining seven are ``eaten" by the
super Higgs mechanism. It is not surprising that this is the
same result as already described in [14] for the Landau--Ginzburg
model.
\par\noindent
At a generic point in the Landau--Ginzburg phase all three--point couplings
are zero, for the four additional fields in the $k=5$ twisted sector do not
occur. However, in our exactly solvable model there exists one coupling
which can be nonzero by R--charge conservation, namely
$$ \langle T'\ S'\ S'\rangle. \eqno(4.20)$$
By taking into account that every individual left and right moving $U(1)$
charge has to be conserved one finds that actually only the following
three--point functions have a chance to be nonzero:
$$ \langle T'\ S_b^{\prime\, (i)}\ S_c^{\prime\, (j)}\rangle, \eqno(4.21)$$
where $i,j=1,0,-1$ are indices of the adjoint representation of $SU(2)$.
Using the explicit form of the three--point functions of the N2Vir(k=3)
and $U(1)_2$ models [30], one obtains for this coupling the
nonvanishing value
$$ C_{T'\,S^{\prime\, (i)}_b\,S^{\prime\, (j)}_c}=
\sqrt 3\, \left(\matrix{0 & 1 & 1 \cr 0 & i & j \cr}\right)
\kappa^3,\quad\quad \kappa= \sqrt{\Gamma\left({3\over5}\right)^3
\Gamma\left({1\over5}\right)\over\Gamma\left({2\over5}\right)^3
\Gamma\left({4\over5}\right)},\eqno(4.22)$$
where $\left(\matrix{j_1 & j_2 & j_3 \cr m_1 & m_2 & m_3 \cr}\right)$
denotes Wigner's $3j$ symbols. Before discussing the resulting obstruction
we move forward and calculate all nonvanishing four--point couplings of
the gauge singlets. R--charge conservation tells us that there
are only three possible types of such couplings
$$\langle T'\ T'\ S\ S\rangle,\quad\quad\langle T'\ S'\ S'\ S\rangle,\quad\quad
\langle S'\ S'\ S'\ S'\rangle.\eqno(4.23)$$
The detailed analysis of all $U(1)$ charges shows that only the following
three couplings of the third type in (4.23) satisfy all selection rules:
$$\langle S_a^{\prime}\ S_a^{\prime}\ S_c^{\prime\,i}\ S_c^{\prime\,j}\rangle,\quad
\langle S_a^{\prime}\ S_b^{\prime\,i}\ S_b^{\prime\,j}\ S_c^{\prime\,k}\rangle,
\quad \langle S_b^{\prime\,i}\ S_b^{\prime\,j}\ S_b^{\prime\,k}\
S_b^{\prime\,l}\rangle.\eqno(4.24)$$
As explicitly shown by E.\ Silverstein in [26], the mere fact that a
four--point function does satisfy all selection rules does not guarantee
that the superpotential coupling is nonzero. In [26] it was shown
that the fourth order coupling of certain twisted singlets for the quintic
in ${\mathchoice{\BCT}{\BCT}{\BCS}{\BCSS}}{\rm P}[4]$ miraculously vanishes even though the conformal field
theoretic four--point function is apparently nonzero. We can follow the
calculation carried out in [26] for the three fourth order couplings
above (4.24), from which we discuss the first one in more detail. First,
we want to calculate the four--point function
$$\langle V^c_{-1}(z_4,\overline{z}_4)\,V^c_{0}(z_3,\overline{z}_3)\,
V^a_{-{1\over2}}(z_2,\overline{z}_2)\,V^a_{-{1\over2}}(z_1,\overline{z}_1)\,\rangle.\eqno(4.25)$$
Since in this case no contact terms can arise, we can
take the zero momentum limit just from the beginning.
The four vertex operators at zero momentum are
$$\eqalignno{
V^a_{-{1\over2}}(z_1,\overline{z}_1)&=e^{-{\rho(\overline{z}_1)\over 2}}\ S^\alpha(\overline{z}_1)
\left[3~\matrix{-4 &-1\cr 4&1\cr}\right]
\left[1~\matrix{-1&0\cr 2&1\cr}\right]^2
\left[0~\matrix{-2&-2\cr 1&1\cr}\right]^2
[1]\ (z_1,\overline{z}_1), &\cr
V^a_{-{1\over2}}(z_2,\overline{z}_2)&=e^{-{\rho(\overline{z}_2)\over 2}}\ S^\beta (\overline{z}_2)
\left[3~\matrix{-4 &-1\cr 4&1\cr}\right]
\left[0~\matrix{-2&-2\cr 1&1\cr}\right]^2
\left[1~\matrix{-1&0\cr 2&1\cr}\right]^2
[1]\ (z_2,\overline{z}_2), &\cr
V^c_0(z_3,\overline{z}_3)&=
\left[1~\matrix{-1 & 0\cr 1&2\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]^4
[-2]\ (z_3,\overline{z}_3), &(4.26)\cr
V^c_{-1}(z_4,\overline{z}_4)&=e^{-{\rho(\overline{z}_4)}}
\left[1~\matrix{-1 & -2\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]^4
[0]\ (z_4,\overline{z}_4).&\cr }$$
Using N2Vir(k)$={SU(2)_k\over U(1)}\times U(1)$ we split the primary
fields of N2Vir(k=3) into parafermi\-onic primaries and vertex
operators of the free boson $\phi$:
$$\left[l~\matrix{q&s\cr\o q&\o s\cr}\right](z,\overline{z})=
\phi^l_{q-s,\o{q}-\o{s}}(z,\overline{z})\ e^{i\alpha_{q,s}\phi(z)}\,
e^{i\alpha_{\o{q},\o{s}}\phi(\overline{z})} \eqno(4.27)$$
with $\alpha_{q,s}={1\over\sqrt{15}}(-q+{5\over2}s)$.
The correlation functions of the four--dimensional space--time fields and
the ghost system are quite simple:
$$\eqalignno{
\langle S^\beta(\overline{z}_2)\ S^\alpha(\overline{z}_1)\rangle&={\delta_{\alpha\beta}\over
(\overline{z}_2-\overline{z}_1)^{1\over 2} } &(4.28)\cr\langle e^{-\rho (\overline{z}_4)}\
e^{-{\rho (\overline{z}_2)\over 2}}\ e^{-{\rho (\overline{z}_1)\over 2}} \rangle &=
{1\over (\overline{z}_2-\overline{z}_1)^{1\over 4}\, (\overline{z}_4-\overline{z}_1)^{1\over 2}\,
(\overline{z}_4-\overline{z}_2)^{1\over 2}}.&\cr } $$
Now, by using $SL(2,{\mathchoice{\BCT}{\BCT}{\BCS}{\BCSS}})$ we set $z_4=0$, $z_2=1$ and $z_1=\infty$ and
realize that the correlation functions (4.28) and the measure in (4.13)
are independent of the variable $z_3=:x$. The correlation functions for the
vertex operators in (4.27) and the $U(1)_2$ piece can be expressed in terms
of $x$ as
$$\langle\ldots\rangle_{U(1)}=|x|^{-{4\over3}}\,|1-x|^{-{4\over3}}.\eqno(4.29)$$
Thus, it only remains to determine five four--point functions for the
parafermionic piece:
$$\eqalignno{P_1&=\langle \phi^0_{0,0}(z_4,\overline{z}_4)\ \phi^0_{0,0}(z_3,\overline{z}_3)\
\phi^1_{-1,-1}(z_2,\overline{z}_2)\ \phi^1_{1,1}(z_1,\overline{z}_1)\rangle &\cr
P_2&=P_3=\langle \phi^1_{-1,1}(z_4,\overline{z}_4)\ \phi^0_{0,0}(z_3,\overline{z}_3)\
\phi^1_{-1,1}(z_2,\overline{z}_2)\ \phi^1_{-1,1}(z_1,\overline{z}_1)\rangle &(4.30)\cr
P_4&=P_5=\langle \phi^0_{0,0}(z_4,\overline{z}_4)\ \phi^1_{-1,1}(z_3,\overline{z}_3)\
\phi^1_{-1,1}(z_2,\overline{z}_2)\ \phi^1_{-1,1}(z_1,\overline{z}_1)\rangle. &\cr}$$
In contrast to the four--point function in [26], here all
parafermionic amplitudes can be expressed in terms of two-- and
three--point functions. These are well known [30], so that we arrive
for the parafermionic correlation function at
$$\prod_{i=1}^5 P_i=\kappa^4\ |x|^{-{4\over15}}\,|1-x|^{-{4\over15}}.
\eqno(4.31)$$
Inserting (4.29,4.31) into the superpotential coupling (4.25), one
finally obtains
$$\eqalignno{
\langle S_c^{\prime\,(i)}\ S_c^{\prime\,(j)}\ S_a^{\prime}\ S_a^{\prime}\rangle&=
\sqrt 3\,\left(\matrix{0 & 1 & 1 \cr 0 & i & j \cr}\right)\
\int d^2 x\ \kappa^4\ |x|^{-{8\over 5}}\, |1-x|^{-{8\over 5}} &\cr
&= \sqrt 3\, \left(\matrix{0 & 1 & 1 \cr 0 & i & j \cr}\right)\
\kappa^4\ B\left({1\over5},{1\over5},{3\over5}\right) &(4.32)\cr }$$
with
$$ B(a,b,c)=\pi\,{\Gamma(a)\Gamma(b)\Gamma(c)\over
\Gamma(a+b)\Gamma(b+c)\Gamma(c+a)}.\eqno(4.33) $$
This coupling is finite and nonzero. In the same way, one
obtains for the second coupling in (4.24)
$$\eqalignno{
\langle S_a^{\prime}\ S_b^{\prime\,(i)}\ S_b^{\prime\,(j)}\ S_c^{\prime\,(k)}
\rangle&=-{3\over\sqrt 2} \left(\matrix{1&1&1\cr-(i+j)&i&j\cr}\right)\,
\left(\matrix{0 & 1 & 1 \cr 0 & i+j & k \cr}\right)\
\int d^2 x \kappa^4\ |x|^{-{6\over 5}}\, |1-x|^{-{8\over 5}} &\cr
&= -{3\over\sqrt 2} \left(\matrix{1&1&1\cr-(i+j)&i&j\cr}\right)\,
\left(\matrix{0 & 1 & 1 \cr 0 & i+j & k \cr}\right)\
\kappa^4\ B\left({2\over 5}, {1\over 5},{2\over 5}\right). &(4.34)\cr }$$
The calculation of the most complicated third coupling fortunately is
exactly the same as for the twisted fourth order coupling in (4.24)
implying that it vanishes after performing the integral over
the complex plane,
$$\langle S_b^{\prime}\ S_b^{\prime}\ S_b^{\prime}\ S_b^{\prime}\rangle =0.
\eqno(4.35)$$
From the conformal field theory point of view we have yet no understanding
why this happens, in particular the arguments of [27] suggest
that the corresponding twisted fields for the quintic are truly moduli,
so that all couplings involving this field should vanish.
\medskip\noindent
\section{5.\ Consequences for the (0,2) moduli space}
\medskip\noindent
The few nonzero superpotential couplings calculated so far do have already
interesting consequences for the moduli space. However, first we want to
discuss the four singlets $T'$. Since singlets of this kind do not occur at
a general point in the Landau--Ginzburg model, it is tempting to identify
them with those appearing in the $k=5$ twisted sector for our special
choice of the polynomials (3.2). Hence, one would expect these fields to
get a mass, when one generically deforms the complex and bundle structure.
Indeed, even though we cannot calculate them exactly, there exist, for
instance, sixth order couplings like
$$\langle T^{\prime}\ T^{\prime}\ S_h\ S_h\ S_h\ S_h\rangle\eqno(5.1)$$
which can create a mass for the singlets $T'$. Since already the
three--point coupling (4.22) containing the singlet $T'$ is finite, one
does not expect miraculous cancellations for other couplings.
\par\noindent
Besides the 318 untwisted moduli one expects at least 11 further flat
directions from the twisted sector. Do the first two orders of the
superpotential allow so many moduli? As the simple Higgs potential
has taught us,
this question is hard to answer, for surely we do not know the entire
superpotential. However, let us mention the following observation:\
Suppose, one can find some untwisted singlets so that giving a
VEV to the field $S_c^{\prime\, (0) }$ and these untwisted singlets
satisfies D--flatness. Then the three--point coupling gives masses to
the four fields $T'$ and four of the twisted fields $S'_b$.
Furthermore, the four--point coupling (4.32) generates mass terms
for the six fields $S'_a$. Thus, including the Higgs effect we are
left with exactly $329$ massless fields, which is the number expected
from the linear $\sigma-$model. As we will show below, the field
$S_c^{\prime\, (0) }$ is not exactly the K\"ahler modulus, but a linear
combination of the fields $S'$. But we are confident that the surviving
number of $329$ massless states is stable under ``rotation" of
$S_c^{\prime\, (0)}$ to $R$. The complexified K\"ahler moduli space of
the linear $\sigma-$model can be sketched like in Figure 1.
\par\noindent
\centerline{\epsfxsize=13.9cm\epsfbox{bild1.ps}}
\par\noindent
In $(2,2)$ compactifications complex and K\"ahler moduli are related to
the $27$ and $\o{27}$ matter fields by the action of the
left moving supercurrent. Thus, there is an algebraic distinction
among complex, K\"ahler and gauge bundle moduli even for
small radius. In the $(0,2)$ case there exist no left moving supercurrent,
so that a priori there is no way to decide to which class of moduli
a given singlet belongs. In the following, we will show how
to use the calculated couplings to determine the form of the K\"ahler
modulus at least to lowest order.
\par\noindent
The following properties are expected from a modulus leading away from
the Landau--Ginzburg radius $r=-\infty$:
\par\noindent
\item{a.)} We completely know the scalar potential in the renormalizable
limit. Thus, in order to determine the local flat direction
we require D--flatness (4.4) and F--flatness up to cubic
couplings.
\par\noindent
\item{b.)} By deforming the radius we expect to obtain the massless
spectrum of the large radius CY limit. In particular, the two
twisted chiral multiplets in the vector representation of
$SO(10)$ should gain a mass.
\par\noindent
\item{c.)} The special enhanced gauge group $SU(2)\times U(1)^4$ is a pure
stringy effect and therefore not present in the CY phase.
\par\noindent
\item{d.)} The massless spectrum features the permutation symmetry $S_4$
in the last four tensor factors. We require that the unique
K\"ahler modulus $R$ also has this $S_4$ symmetry.
\par\noindent
\item{e.)} The untwisted $SO(10)$ singlets are given by polynomials of
degree four modulo some relations. This is analogous to the
complex and bundle deformations of the Calabi--Yau manifold.
Consequently, we expect $R$ to have contributions only from the
$S'$ twisted sector.
\medskip\noindent
We start with point b.)\ and calculate the Yukawa coupling
$\langle 1\ 10\ 10\rangle$ for the two extra ${\bf 10'}$s from the twisted
sector. They are of the form listed in Table 5.1 in the $[1^{-2}]$
sector of $SO(8)$.
\medskip\noindent
\centerline{\vbox{
\hbox{\vbox{\offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\def\tablespace\noalign{\hrule}\tablespace{height2pt&\omit&&\omit&&\omit&&\omit&\cr\noalign{\hrule}height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule\halign{&\vrule#&\strut\hskip0.2cm\hfil#\hfill\hskip0.2cm\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
&Type&&\hskip4.4cm${\cal{O}}_1$\hskip4.3cm${\cal{F}}$&&rep.&°.&\cr
\tablespace\noalign{\hrule}\tablespace\tablerule
& $G$ && $\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right]
\left[1~\matrix{-1&0\cr 1&0\cr}\right] [2]$
&& $2$ && $2$ & \cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr}\hrule}}
\hbox{\hskip 0.5cm Table 5.1 \hskip 0.5cm Twisted ${\bf 10'}$s}}}
\medskip\noindent
Now, one can look for Yukawa couplings containing two
twisted ${\bf 10'}$s and one twisted singlet $S'$. The selection
rules allow only one such coupling, which can be
calculated in the usual way:
$$ \langle S^{\prime\, (i)}_c\ G^{(j)}\ G^{(k)}\rangle ={1\over 3}
\left(\matrix{1&{1\over2}&{1\over2}\cr-(j+k)&j&k\cr}\right)\,
\left(\matrix{0&1&1\cr0&i& j+k\cr}\right)\,\kappa^3.\eqno(5.2)$$
Thus, deforming in the direction of the singlet $S'_c$ gives $G$ a mass
and one is left with the large radius limit for the number of ${\bf 10}$s.
However, since the $SU(2)$ triplet $S'_c$ alone can not break the $SU(2)$
gauge group completely, $R$ must contain contributions from other twisted
states. The most general ansatz compatible with b.)$\,-\,$e.) is
$$ R=\sum_{j=-1}^1 \gamma_j\ S_c^{\prime\,(j)} +
\sum_{j=-1}^1 \sum_{m=1}^4 \beta_j\ S_{b,m}^{\prime\,(j)} +
\sum_{n=1}^6 \alpha\ S_{a,n}^{\prime}.\eqno(5.3) $$
Plugging this ansatz into the flatness conditions (4.4) allows up to
gauge transformations only the following two parameter solution:
$$ R = \gamma\, S_c^{\prime\,(0)}+
\sum_{m=1}^4 \beta\, \left( \ S_{b,m}^{\prime\, (-1)} +
\ S_{b,m}^{\prime\, (+1)}\right) +
\sum_{n=1}^6 {\gamma\over\sqrt 6}\ S_{a,n}^{\prime}. \eqno(5.4)$$
Using the four--point functions (4.32--4.34), one can show that
$R^4\ne 0$ for all $\gamma,\beta\in{\mathchoice{\BCT}{\BCT}{\BCS}{\BCSS}},\gamma\ne 0$. Since we know that
there must exist a flat solution and up to first order the solution is
highly restricted, we conclude that the nonvanishing four--point coupling
indicates merely that the conformal fields are an unappropriate basis, in
which $R$ is curved. The solution (5.4) merely gives the tangent vector at
the SCFT point along this curve. In order to determine the next order
corrections to $R$ one also has to take into account the nonflat
$K(\phi,\phi^*)$. The picture so far obtained for the K\"ahler modulus
is shown in Figure 2.
\par\noindent
\centerline{\epsfxsize=13.9cm\epsfbox{bild2.ps}}
\par\noindent
The sphere visualizes the parameter space of the entire model. The
superpotential is a function (or better a section) over this space.
In this parameter space there is a flat direction $R$, of which our
first order calculation only determines the tangent vector at the SCFT
point. If the SCFT would yield appropriate coordinates (like the
polar coordinates in the Mexican hat example), then the flat direction
would be a geodesic line on the sphere.
\par\noindent
Knowing the singlet associated with the K\"ahler modulus allows one in
principle to investigate couplings of the form
$$\langle R^n\ 10\ 16\ 16\rangle,\quad\quad\quad\langle R^n\ 1\ 10\ 10\rangle.\eqno(5.5)$$
A nonvanishing coupling would detect a radius dependence of the Yukawa
couplings, which perturbatively was argued to be absent [12,18].
Unfortunately, the selection rules do not forbid couplings
of the form (5.5) and their exact calculation is hard to come by.
\par\noindent
Since at the Landau--Ginzburg point there generically occur ten more
$SO(10)$ singlets than in the Calabi--Yau phase, it is tempting
to speculate about new flat directions of the superpotential
leading perhaps to completely new phases of $(0,2)$ models or
even to a different linear $\sigma-$model [16].
However, the data achieved so far neither rule out nor seem to prove such a
possibility. We have tried to find a flat direction with a contribution
from the $T'$ field and only other twisted singlets. Since $T'$
only appears at the very special SCFT point such a flat direction
could not be part of the linear $\sigma-$model moduli space.
However, for such an ansatz no solution to first order exist.
If one also allows contributions from the untwisted singlets
to first order there exist plenty of solutions, as for instance
$$N=\alpha\left(\ \sum_{m=1}^4\left(S_{e,m}^{({1\over2})}-
S_{e,m}^{(-{1\over2})}\right)+\sum_{n=1}^4{\sqrt 2}\left(
S_{g,n}^{({1\over2})}+S_{g,n}^{(-{1\over2})}\right)+
\sum_{p=1}^4T'_p\right).\eqno(5.6)$$
Finally, we want to mention a coincidence, which might perhaps lead to a
better understanding of the $(0,2)$ moduli space. The number of scalars in
the vector representation of $SO(10)$ is $N_{10}=74$, which is the same as
the sum of complex moduli ($b_{21}=73$) and K\"ahler moduli ($b_{11}=1$) of
the underlying Calabi--Yau manifold.
\medskip\noindent
\section{6.\ Summary}
\medskip\noindent
In this paper we have provided further convincing arguments for the
identification of an exactly solvable $(0,2)$ string
model with a special point in the Landau--Ginzburg phase of a $(0,2)$
linear $\sigma-$model. Then, for the first time, we calculated exactly all
three-- and four--point couplings in the space--time superpotential
yielding obstructions against the deformation in all 350 directions
simultaneously. Similarly to the Landau--Ginzburg analysis the nontwisted
moduli could be derived simply by selection rules. Furthermore, to lowest
order we have up to two parameters identified the singlet corresponding to
the K\"ahler modulus. Unfortunately, the available data did not allow us to
find a unique solution for the K\"ahler modulus showing again the
difficulty in making $(0,2)$ models technically as well treatable as
$(2,2)$ models. Since we did not know the entire superpotential, we could
only speculate about the possibility of further flat directions leading
perhaps to another linear $\sigma-$model.
\medskip\noindent
{\bf Acknowledgements}
\smallskip\noindent
It is a pleasure to thank L.\ Dolan, S.\ Kachru, W.\ Nahm,
R.\ Schimmrigk, E.\ Silverstein and E.\ Witten for discussion. This work
is supported by U.S.\ DOE grant No.\ DE--FG05--85ER--40219.
\medskip\noindent
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\vfill
\end
|
{
"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/dirkmittler.homeip.net\/blog\/archives\/tag\/otg-adapter","text":"## One way in which technology appears to be moving forward.\n\nOne of the facts which I only posted about a few years ago, was the existence of external sound devices, which effectively acted as an external, USB-connected sound card, and, whether they could be made to work with certain Android software. That particular sound device had as main feature, studio-quality sound (96kHz, 24-bit).\n\nWell, there is a more recent way to accomplish approximately the same thing:\n\nI should mention in what context this later technology presents itself to the users of mobile devices:\n\nMuch as Apple rolled out smart-phones with no traditional, 3.5mm stereo headphone jacks, Samsung has rolled out similar tablets, where the universal connector-type of the latter, is a USB-C port. The \u2018Tab S6\u2032 is an example of that. Thus, because some users do want to connect \u2018wired\u2019 headphones to this tablet, it\u2019s suggested that users buy a so-called \u201cdongle\u201d, that adapts the USB-C port on the tablet, to a female, 3.5mm stereo phone jack, even the microphone feature of which seems to work. This one cost me CAD 22, including 1-day delivery.\n\nA simple question which some people might have, especially if they are deeply mired in the analog days, and in the technology which existed in the 1970s and 1980s, could be: \u2018Does such a dongle just connect the analog pins of the headphone socket, directly to the pins of the USB socket? If not, what exactly does it do?\u2019\n\nThe correct answer to that sort of question would be the fact that, as small as that end of the dongle is, on the USB-C side, there is a tiny chip. With that tiny chip, the manufacturers have added a completely unpredictable amount of complexity, to how the dongle might work. Chips exist that have 100,000 transistors. And chips also exist that have 1,000,000 transistors, although that last type of chip is less common, and exists in spectacular cases, such as CPUs, GPUs, etc..\n\nWhat this means is that, in theory, the chip in this adapter could do everything that the \u2018Focusrite 2i2\u2032 external sound card was able to do. But, that\u2019s in theory. There are two important ways, in which it will fail to do so, at least at the time I\u2019m writing this:\n\n1. The accuracy of that chip is in doubt, And\n2. The protocol with which the adapter communicates with the USB-C port of the mobile device, which is actually referred to as its USB Profile, has not been made backwards-compatible with the older generation of external sound cards\u2026\n\n(Updated 6\/28\/2020, 12h45\u2026 )\n\n## Testing the Focusrite Scarlett 2i2 external sound device, with my Samsung Tab S Tablet\n\nI have tested, whether this external USB recording tool, works with my Samsung Galaxy Tab S Tablet, using a \u2018StarTech.com\u2019 OTG adapter. The results were resoundingly affirmative.\n\nIn This Earlier Posting, I had tested the same USB Sound Card, with my Samsung Galaxy S6 Smart-Phone. At that time, an attempt also to use it with my Tab S tablet had failed. In order to get the Scarlett 2i2 to work with the Tab S, the following two conditions need to be fulfilled:\n\n1. The amount of current that the USB Slave Device may draw, needs to be reinforced, in principle, with a self-powered OTG adapter, or with a similar arrangement. The \u2018StarTech.com\u2019 is Not a self-powered OTG adapter, and with it, the Scarlett 2i2 is bound to draw too much current, for the likes of the Tab S. It was after all meant as an audio workstation workhorse, and not as a replacement for a simple USB Microphone.\n2. The Master \/ Host Device, the Tab S, needs to have the correct drivers.\n\nCondition (1) is something I was able to fulfill for now, in a roundabout way. I bought a \u2018j5create USB 3.0 4-Ports Mini HUB\u2019, with the part number \u2018JUH340\u2032. This is a self-powered hub by default, with its own power cord, and has Type A USB connectors up-stream and down-stream. Granted, it has a special up-stream cable, that connects to the hub with a special connector, just so that the user does not get this socket confused with the down-stream sockets. But then, the far side of that cable has a standard Type A USB jack.\n\nThis USB jack can be plugged, into the far side of the OTG adapter. Since the hub is self-powered, the current requirements of the Scarlett 2i2 are met by it, and not by the OTG adapter, and thus not by the micro-USB port on the Tab S, the latter of which now faces a minimum current load.\n\n## Retailers find it hard to Define what a USB Y Cable is supposed to be.\n\nIn This Posting, I wrote that I was looking for an OTG cable, with a separate power jack, preferably also a Male USB A. One possible alternative to that, would be a Female USB A to 2x Male USB A, with one of the Male USB jacks distinctly for Power, and the other distinctly for Data.\n\nThe problem with this, is that although both products exist, I don\u2019t own one of either.\n\nFurther, the potential sellers of these cables do not understand what they are selling. They tend to give jumbled descriptions, thinking that if the wording of the description follows a certain trick, this is the correct cable.\n\nOf course, some Electrical understanding allows certain people, to define each type of cable, and thus to name them in a meaningful way. However, often vague descriptions are given, that sound like they might be the correct product, but closer inspection reveals a next-to-useless cable. The most common error, is a Type A Female, connected to 2 Type A Males, in a completely symmetrical way between the Males. The cable is already useless to me, if they are asymmetric in fact, but not labeled in any way that tells me which Male Type A goes into my power source, and which goes into my OTG cable.\n\nAnother type of USB cable which does not suit my needs, is a kind of wiring-bus between several connector-types, but which connects every wire, to each connector\u2026 In theory, some other user might use that as a converter, between any two connectors.\n\nThe correct USB Y Cable costs C$40, while a power-spliced OTG Adapter costs C$ 8.\n\nNow, I have just ordered both types of cable, so that if the OTG is somehow hamstrung software-wise, the USB Y is likely to offer a solution that will still work. Also, it might always come in handy, just to have a USB Y on-hand.\n\nDirk\n\n## Testing the Focusrite Scarlett 2i2 external sound device, with my Samsung S6 Smart-Phone\n\nI have tested, whether this external USB recording tool, works with my Samsung Galaxy S6 Smart-Phone, using an \u2018StarTech.com\u2019 OTG adapter. The results were mixed. In An Earlier Posting, I had tested whether this external USB Sound Card, works under Linux. And the answer to that question was a resounding Yes.\n\nWhen we plug an OTG adapter into a smart-phone or tablet, this puts the mobile device into Master \/ Host Mode, that would otherwise normally work in Slave Mode. Thus, we can then plug in a USB storage device, and hopefully have that recognized, while by default, we can only plug our mobile device into a computer, and have the computer recognize this mobile device, as the storage device.\n\nBut it is also plausible to connect other external devices to our mobile device, when using an OTG adapter. All this happens because the OTG adapter itself contains an additional chip, that gives it the ability to act as a USB Host. Whether such external devices will work or not, generally depends on two factors:\n\n1. Whether the micro-USB port on the mobile device can output enough current, to supply the external \/ Slave device, and\n2. Whether the mobile device possesses the drivers needed, for the USB device in question. Under Linux, this last question is more likely to be answered in the affirmative.\n\nThe OTG adapter I was using, uses its micro-USB side as the only power-supply. This means that if the connected device draws a full 500mA of supply current, we are pushing the limit, that is generally set for USB 2.0\u00a0 PC ports.","date":"2021-01-23 06:20:20","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\": 0.4941699206829071, \"perplexity\": 2337.811819610889}, \"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-2021-04\/segments\/1610703533863.67\/warc\/CC-MAIN-20210123032629-20210123062629-00131.warc.gz\"}"}
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\section{Introduction}
Stellar rotation is a key ingredient for the generation of magnetic fields and magnetic cycles in the Sun and other solar-type stars \citep[e.g.][]{Brun2017}. Stars are observed to spin down as they evolve and lose angular momentum \citep[e.g.][]{Wilson1963,Wilson1964}. Therefore, rotation can also be used as a diagnostic for stellar age, in what we call gyrochronology \citep[ e.g.][]{Skumanich1972,Barnes2003,Barnes2007,Mamajek2008,Garcia2014,Davies2015,Metcalfe2019}. To calibrate the empirical gyrochronology relations, rotation-period estimates for large samples stars are needed as well as precise ages estimates.
Thanks to the NASA mission {\it Kepler}, almost 200,000 stars were observed almost continuously for up to four years. These long duration photometric observations obtained with high precision allow us to measure rotation periods through the modulation of the stellar brightness caused by the passage of spots on the stellar disk. This has been done on a large number of stars observed by {\it Kepler} using different techniques, such as periodogram analysis \citep[e.g.][]{Nielsen2013, Reinhold2013a}, autocorrelation function \citep[e.g.][]{McQuillan2013a, McQuillan2014}, and time-frequency analysis with wavelets \citep[e.g.][]{Garcia2014}. Based on simulated data, the comparison of different pipelines developed to retrieve surface rotation periods using photometric data showed that a combination of different techniques such as done in \citet{Garcia2014} and \citet{Ceillier2016, Ceillier2017} provides the most complete and reliable set of rotation-period estimates \citep[see details in][]{Aigrain2015}.
Ages can be constrained from gyrochronology relations. However, these relations are calibrated, requiring targets with known ages from independent methods. This is the reason why stars belonging to clusters have been used in the past \citep[e.g.][]{Meibom2011a,Meibom2011b,Meibom2015}. Asteroseismology has proven to be a powerful tool to provide precise stellar ages \citep[e.g.][]{Mathur2012,Metcalfe2014,SilvaAguirre2015,Creevey2017,Serenelli2017} and we can now test and improve those relations with a large number of field stars. This led to the results of \citet{vanSaders2016} who found that solar-like stars older than the Sun rotate faster than predicted by the classical gyrochronology relations \citep[see also][]{Angus2015}. The authors suggested that this could be the result of the weakening of the magnetic braking when the Rossby number of the star (ratio between the rotation period and the convective turnover time) reaches a given value. Thus, gyrochronology may not be a uniformly suitable technique for all main-sequence stars, and the apparent weakened braking has implications for the dynamo theory. \citet{vansaders2018} suggested that signatures of this weakened braking might be visible in large samples of field stars with measured rotation periods. However, these relations are still valid for young main-sequence stars. Note that gyrochronology is also not suitable for pre-main-sequence and early spectral type stars \citep[e.g.][]{Kraft1967,Gallet2013,Epstein2014,Amard2016}.
One of the largest analyses of the surface rotation of main-sequence stars observed by {\it Kepler}\ was performed by \citet{McQuillan2014}. They looked for rotation periods in a sample of 133,030 targets using Quarters 3 to 14 and obtained reliable values for 34,030 stars. They estimated ages by comparing their field population to empirical gyrochrones and found that the most slowly rotating stars were consistent with a gyrochronological age of 4.5 Gyr.
In this work we perform a similar analysis focusing on 26,521 M and K dwarfs observed by {\it Kepler}. We use the longest time-series available: up to {\it Kepler}\ Quarter 17, while \citet{McQuillan2014} used only 11 {\it Kepler}\ Quarters. We calibrate the light curves using our own independent software, which high-pass filters the data using filters of 20, 55, and 80 days, preventing us from measuring a harmonic of the real rotation period. In addition, we analyze the PDC-MAP \citep[Presearch Data Conditioning - Maximum A Posteriori; e.g][]{Jenkins2010,Smith2012,Stumpe2012} light curves to be sure that the rotational modulation detected does not result from photometric pollution by nearby stars. We are particularly careful to remove possible polluters such as red giants, classical pulsators, and eclisping binary systems, which can result in the detection of a spurious periodicity. The sample selection and data calibration are described in Sect.~\ref{sec:datasample}. We then apply our rotation pipeline (Sect.~\ref{sec:method}) that consists of the combination of the auto-correlation function, the wavelet analysis, and the composite spectrum (that is a combination of the two former methods) to derive the most reliable periods. Our analysis allows us to retrieve rotation periods for more than 4,000 additional targets in comparison with the analysis in \citet{McQuillan2014}. In particular, we are able to retrieve rotation periods for both fainter and cooler stars. We then measure the photometric magnetic activity proxy $S_{\rm ph}$ (Sect.~\ref{sec:sph}). Finally we interpret the results in Sect.~\ref{sec:res} in terms of rotation, activity, mass, and temperature relations and conclude
in Sect.~\ref{sec:conclusion}.
\section{Data preparation and sample selection}\label{sec:datasample}
\subsection{Data preparation}\label{sec:data}
The light curves are obtained from {\it Kepler}\ pixel-data files using large custom apertures that produce stable light curves. For each pixel in the pixel-data file, a reference flux value is computed as the 99.9\textsuperscript{th} percentile of the flux. Starting from the center of the point-spread function of the target-pixel mask, new pixels are added in one direction of the mask if two conditions are fulfilled: 1) the reference flux of the pixel is higher than a threshold of $100 \text{e}^-\!/\text{s}$; 2) the flux is smaller than the value of the previous pixel within a small tolerance. In most cases, this second condition allows us to remove the pixels corresponding to a second star present in the aperture.
The resulting light curve is processed through the implementation of the {\it Kepler}\ Asteroseismic Data Analysis and Calibration Software \citep[KADACS;][]{Garcia2011}. KADACS corrects for outliers, jumps, and drifts, and it properly concatenates the independent {\it Kepler}\ Quarters on a star-by-star basis. It also fills the gaps shorter than 20 days in long-cadence data following in-painting techniques based on a multi-scale cosine transform \citep{Garcia2014a,Pires2015}. The resulting light curves are high-pass filtered at 20, 55 days (quarter by quarter) and 80 days (using the entire light curve) yielding three different light curves for each target. For light curves longer than one month, KADACS corrects for discontinuities at the edges of the {\it Kepler}\ Quarters. Hereafter, we will refer to KADACS data products, which are optimized for seismic studies, as KEPSEISMIC\footnote{KEPSEISMIC time-series are available at MAST via \dataset[https://doi.org/10.17909/t9-cfke-ps60]{https://doi.org/10.17909/t9-mrpw-gc07}.}.
To correctly prepare the KOI ({\it Kepler}\ Objects of Interest) light curves for seismic analysis, we used the published ephemeris of each star from the MAST (Mikulski Archive for Space Telescopes) to remove the transits and interpolate the resultant gaps using the same in-painting techniques mentioned above.
In the analysis below we compare the results for the different high-pass filters to determine the final stellar rotation period. However, we note that the longer period filters are also less effective at removing long periodicity instrumental trends from the light curves, including the {\it Kepler}\ yearly modulation.
In our analysis, to ensure that the correct rotation period is retrieved, we also use PDC-MAP light curves for Data Release 25. The comparison between KEPSEISMIC and PDC-MAP light curves also helps to identify light curves with photometric pollution by nearby stars. We typically construct the KEPSEISMIC light curves using larger apertures than those of the PDC-MAP time-series, which leads to an increase in the number of polluted light curves. Nevertheless, a significant number of light curves show evidence of pollution or multiple signals in both KEPSEISMIC and PDC-MAP data sets (see Sect.~\ref{sec:sample}, and Tables~4 and 5). To properly understand the source of pollution and/or multiple signals (e.g. possible binary or just nearby star in the field of view) further analyses are needed and currently beyond the scope of this study. Note that, in addition to the difference in the aperture sizes, PDC-MAP light curves are calibrated quarter-by-quarter while the KEPSEISMIC light curves are calibrated using all the quarters at once. Furthermore, PDC-MAP light curves are often high-pass filtered at a period of 20 days which can lead to biased results (see Appendix \ref{app2}).
\subsection{Sample selection}\label{sec:sample}
We analyze long-cadence ($\Delta t=29.42\,\rm min$) data of M and K main-sequence stars observed during the main mission of the {\it Kepler}\ satellite \citep{Borucki2010}. The targets were selected according to the {\it Kepler}\ Stellar Properties Catalog for Data Release 25 \citep[KSPC DR25;][]{Mathur2017}, where M~dwarfs have effective temperatures $T_\text{eff}$\ smaller than 3700 K and K~dwarfs have temperatures within 3700 and 5200 K.
The initial sample is composed by 26,521 targets (24,171 K stars and 2350 M stars) shown in Fig.~\ref{fig:hrsample}.
\begin{figure}[h]
\includegraphics[width=\hsize]{hr_sample.png}
\caption{Surface gravity-effective temperature diagram of the 26,521 M and K dwarfs according to KSPC DR25 \citep{Mathur2017}, color coded by number of stars in each bin. The size of $T_\text{eff}$\ and $\log\, g$\ bins is $\sim16$ K and $\sim 7\times10^{-3}$ dex, respectively. For context, other stars in KSPC DR25 are plotted in gray. Effective temperature ($T_\text{eff}$) and surface gravity ($\log\, g$) values are adopted from KSPC DR25.}\label{fig:hrsample}
\end{figure}
We expect a number of different polluters in the sample of main-sequence M and K stars. These polluters will display stellar variability due to pulsations, eclipses or other astrophysical variability not related to spot-modulation. Therefore, we search and identify such polluters.
We start by removing the known eclipsing binaries \citep[total of 272 stars in the Villanova {\it Kepler}\ Eclipsing Binary Catalog;][]{Kirk2016,Adbul-Masih2016} and known RR Lyrae (3 stars in this sample; Szab\'o et al. in prep). These are listed in Table~4. For rotational analysis of eclipsing binaries see \citet{Lurie2017}.
We also remove possible misclassified red giants (listed in Table~4; Garc\'ia et al. in prep). A significant fraction of those were identified by \citet{Berger2018} using astrometric data from \textit{Gaia} \citep[\textit{Gaia} Data Release~2;][]{Gaia_DR2}. The remainder of the misclassified targets were identified using the hallmark signature of red-giant stars in light curves: the presence of red-giant oscillations. We use both neural network and machine learning techniques that automatically identify power spectra consistent with red-giant stars \citep[see][]{Hon2018,Bugnet2018} and/or by visual examination for red-giant pulsations. In total, 1,221 misclassified red giants were removed from the subsequent rotation analysis. 30 of those targets are also identified as eclipsing binaries (flagged accordingly in Table~4), which may suggest that one of the components of the binary is a red giant.
Another group of potential polluters in the sample corresponds to light curves exhibiting evidence of photometric pollution possibly from nearby stars in the field of view. We consider light curves to be photometrically polluted when the signal is only present in some {\it Kepler}\ Quarters, namely every four Quarters\footnote{Every $\sim90$ days, i.e. every {\it Kepler}\ Quarter, the spacecraft was rotated over $90^\circ$, meaning that the targets are observed by the same modules/channels every four {\it Kepler}\ Quarters.}. We also identify targets as photometrically polluted when their light curves contain a signal or multiple signals only in the KEPSEISMIC time-series. As mentioned previously, the apertures used for the KEPSEISMIC data sets are typically larger than those of the PDC-MAP data sets, and thus more likely to be affected by the contribution of background stars in the field of view. In total, we have flagged the light curves of 255 targets as photometrically polluted (Table~4).
Targets with multiple signals in both the PDC-MAP and KEPSEISMIC light curves are likely to be associated with different unresolved sources. Although determining whether these targets are true binary systems or merely polluted by background stars is beyond the scope of this work, we perform the rotation analysis of these light curves (Table~5). In total, we have identified 270 targets with multiple signals in both KEPSEISMIC and PDC-MAP data sets.
Another concern is pollution by possible classical pulsators (CP) that were not previously identified. We start by flagging the targets that exhibit multiple high-amplitude peaks at relatively high frequencies (higher than $3.5 \mu\text{Hz}$) in the power density spectrum, which are typical of classical pulsators. Then we visually check those targets and also all the other targets for which the rotation estimate (from Sect.~\ref{sec:a2z}) is shorter than 10 days. We only flag stars as CP candidates when there are more than three associated peaks in the power spectra.
We also distinguish between three types of CP candidates. Type~1 candidates (left panels of Fig.~\ref{fig:cp}) show a behaviour somewhat similar to RR Lyrae and Cepheids \citep[see e.g. Szab\'o et al. in prep;][]{Kolenberg2010,Moskalik2015}: high-amplitude and stable flux variations, beating patterns, and a large number of harmonics. Interestingly, a significant fraction of these targets were identified as \textit{Gaia} binary candidates in \citet{Berger2018} and \citet{Simonian2019}. In particular, \citet{Simonian2019} focused on tidally synchronized binary systems. Of the 74 Type~1 candidates we identify common to their analysis, 51 are found to be possible synchronized binaries. Therefore, it is possible that these targets are not classical pulsators but close-in binaries (CB). If that is the case, the signal may still be related to rotation, but may be distinct from the rotational behavior of single stars. For the remainder of this paper, we refer to these targets (350) as Type~1 CP/CB candidates. Type~1 CP/CB candidates are listed and flagged in Table~3. Targets marked as Type~2 CP/CB candidates (9 stars; middle panel of Fig.~\ref{fig:cp}) exhibit a large number of harmonics in the power spectrum, similarly to Type~1 CP/CB candidates. However, these targets differ from Type~1, in particular, the highest peak in the periodogram is the second harmonic associated with the signal instead of the first harmonic (period of the signal). This signature may also be consistent with contact binary systems \citep[see e.g.][]{Lee2016,Colman2017}. Therefore, similarly to Type~1, these targets are flagged as CP/CB candidates. The power spectrum of Type~3 CP candidates (9 stars; right-hand panel of Fig.~\ref{fig:cp}) resembles those of $\gamma$ Doradus or $\delta$ Scuti, depending on characteristic frequencies and nature of the modes \citep[see e.g.][]{Bradley2015,vanReeth2015,BarceloForteza2017}. A proper analysis of these targets is however beyond the scope of this work. Type~2 and 3 candidates are listed in Table~4.
\begin{figure*}[htp]
\includegraphics[width=\hsize]{cpcb_candidates.png}
\caption{Light curve and power density spectrum for an example of the three classical pulsator or close-in binary candidates. {\it Left-hand}: KIC~2996903, Type~1 CP/CB candidate, which exhibit high-amplitude flux variations and high-amplitude peaks with a large number of harmonics in the power density spectrum. {\it Middle}: KIC~5522761, Type~2 CP/CB candidate, which exhibit high-amplitude flux variations and a large number of high-amplitude peaks, with the highest peak being the second harmonic of the signal period. {\it Right-hand}: KIC~5429117, Type~3 CP candidate, which is possibly a $\gamma$ Doradus. Note that targets marked as Type~3 CP candidates may be $\gamma$ Doradus or $\delta$ Scuti depending on the nature of the modes (and characteristic frequencies).}\label{fig:cp}
\end{figure*}
We do not provide rotation periods for confirmed RR Lyrae, misclassified red giants, eclipsing binaries, light curves with photometric pollution, and Type~2 and 3 CP/CB candidates. This leaves us with 24,782 stars for the rotational analysis. Table~\ref{tab0} summarizes the number of polluters and targets used in the subsequent analysis.
\begin{table}[h]
\centering
\begin{tabular}{rl}
\hline\hline
M dwarfs & 2,156\\
K dwarfs & 22,006 \\
Type~1 CP/CB candidates & 350\\
Multiple signals & 270\\\hline
Eclipsing Binaries (EB)& 242\\
Red giants (RG) & 1,191\\
EB \& RG & 30\\
RR Lyrae & 3\\
Photometric pollution & 255\\
Type 2 and 3 CP/CB candidates & 18\\\hline\hline
\end{tabular}
\caption{Summary of the targets classified as M and K~dwarfs in KSPC DR25 \citep{Mathur2017}. The top part of the table corresponds to the targets for which we perform the rotational analysis, while the polluters summarized in the bottom part are not used for rotational analysis.}
\label{tab0}
\end{table}
Finally, possible additional non-single non-main-sequence M and K stars are flagged in Tables~3-5 but not removed from the analysis. We add subgiant and binary flags from \citet[][\textit{Gaia} DR2]{Berger2018}, synchronized binary flag from \citet{Simonian2019}, and FliPer$_\text{Class}$ flag \citep[see][]{Bugnet2019} which indicates solar-type stars, classical pulsators, and binary/photometric pollution. We do not remove these targets from the analysis, but we alert for the possible pollution.
\section{Surface rotation detection}\label{sec:method}
In Sect.~\ref{sec:a2z}, we present the methodology implemented to estimate the surface rotation period. Sections~\ref{sec:auto} and \ref{sec:vis} summarize the results from the automatic selection and visual examination, respectively.
\subsection{Methodology to retrieve rotation periods}\label{sec:a2z}
To extract the rotation-period estimates, we implement the methodology described in \citet{Ceillier2016,Ceillier2017}. It combines a time-frequency analysis and the autocorrelation function (ACF). This methodology was found by \citet{Aigrain2015} to have the best performance in terms of completeness and reliability compared to the periodogram analysis alone, ACF alone, or a combination between the two and spot modeling.
Despite the fact that our KEPSEISMIC light curves have been corrected for instrumental effects \citep[see Sect.~\ref{sec:data};][]{Garcia2011}, calibrated light curves may still exhibit instrumental modulations. We therefore remove {\it Kepler}\ Quarters with anomalously high variance compared with their neighbours from the rotation analysis \citep[see][]{Garcia2014}.
First, we estimate periods from a time-period analysis using the wavelet decomposition \citep{Torrence1998} adapted by \citet{Mathur2010b} using the correction by \citet{Liu2007}. The wavelet analysis assesses the correlation between the mother wavelet and the rebinned data (to decrease the computing time) by sliding the wavelet in time for a given period of the wavelet. The range of periods is probed through an iterative process. For the mother wavelet, we use the Morlet wavelet, which is the convolution between a sinusoidal and a Gaussian function. This analysis provides the wavelet power spectrum (WPS). An example is given in panel b) of Fig.~\ref{fig:a2ztool}, where red and black colors indicate high power, while blue indicates low power. The visual inspection of the WPS also helps us to determined whether the signal is present along the time-series or an artifact resulting from instrumental noise at a particular time. The black hashed area indicates the cone of influence that marks the limit on observing at least four rotations in the light curve. Rotation signals found inside the cone have a lower confidence level. Finally, we obtain the global wavelet power spectrum (GWPS) by computing the sum of the WPS along time for each period of the wavelet (panel c) of Fig.~\ref{fig:a2ztool}. We then fit the GWPS, through a least-squares minimization, with multiple Gaussian functions. The rotation estimate from the wavelet analysis corresponds to the central period of the highest fitted period peak, while the uncertainty corresponds to the half width at half maximum (HWHM) of the corresponding Gaussian profile. Computed in this way, the inferred uncertainty also accounts for possible differential rotation.
Our second method for measuring periods consists of the autocorrelation function of light curves \citep[ACF; following the procedure in ][]{McQuillan2013}, which was combined with wavelet analysis for the first time in \citet{Garcia2014}. The ACF is smoothed using a Gaussian function whose width is a tenth of the most significant period selected from the Lomb-Scargle periodogram \citep{Lomb1976,Scargle1982} of the ACF. We identify the significant peaks and take the highest peak as the rotation-period estimate from the ACF. We also examine the ACF for evidence of double-peaked features resulting from active regions in anti-phase. Panel d) of Fig.~\ref{fig:a2ztool} shows the ACF for a given target in the sample.
Finally, the third method of estimating rotation period utilizes the composite spectrum (CS), which combines the GWPS and the ACF as described by \citet{Ceillier2016,Ceillier2017}. The composite spectrum corresponds to the product of the normalized GWPS and the normalized ACF resampled in the same period of the GWPS. Periods present in both methods, GWPS and ACF, are enhanced by the CS, allowing for a better identification of the intrinsic rotation periods of the star. We also fit the CS with multiple Gaussian functions; the central period and HWHM of the profile corresponding to the highest peak are taken as its period estimate and uncertainty. Panel e) of Fig.~\ref{fig:a2ztool} shows an example of the CS.
For the rotation-period estimate provided in Tables~3 and 5, we prioritize the value returned by the wavelet analysis. When the wavelet power spectrum does not allow us to successfully recover the rotation period, we provide the value recovered from the composite spectrum. If both wavelet power spectrum and composite spectrum fail to infer the rotation period, the rotation period provided is that found by the autocorrelation function without uncertainty. Note that the primary goal of the autocorrelation function and composite spectrum is to validate the rotation period and better identify the reliable results.
\begin{figure}[h!]
\includegraphics[width=\hsize]{4918333_paper.pdf}
\caption{Rotation analysis for KIC~4918333. a) KEPSEISMIC light curve obtained with the 55-day filter. b) Wavelet power spectrum (WPS) where red and black correspond to high power and blue to low power. The cone of influence is shown by the black crossed area. c) Global wavelet power spectrum (GWPS; black) and corresponding best fit with multiple Gaussian functions (red). d) Autocorrelation function (ACF; black) of the light curve and smoothing ACF (red). e) Composite spectrum (black) and respective fit with multiple Gaussian profiles (red). For the GWPS, ACF, and CS, the black dotted lines mark the respective rotation-period estimates.}\label{fig:a2ztool}
\end{figure}
\subsubsection{Automatic selection}\label{sec:auto}
Having the rotation-period estimates from the GWPS, ACF, and CS for the three sets of KEPSEISMIC light curves, we start by selecting the targets with the most reliable rotation estimates.
For the automatic selection, the appropriate filter is chosen according to the rotation period. We note that it is still possible to recover periods longer than the cut-off period of the filter. The transfer function is unity below the cutoff period. Above that, it varies sinusoidally and slowly approaches zero at twice of the cut-off period. Therefore, the amplitude of rotation periods slightly longer than the cut-off period would be only slightly reduced, while rotation periods close to 1.5 times the cut-off period would have roughly half of the original amplitude. It is therefore still possible to extract a high signal-to-noise ratio, reliable peak of a period 1.5 times the cut-off period using our rotation pipeline. However, the 80-day filtered light curves are the least stable often exhibiting instrumental modulations and, thus, we only use them in the automatic selection for rotation periods longer than 60 days. For rotation periods shorter than 23 days, priority is given to the period estimate obtained from the 20-day filter. For rotation periods between 23 and 60 days, the primary filter is the 55-day filter, while for longer periods priority is given to the 80-day filter.
The targets with reliable rotation-period estimates are automatically selected if:
\begin{enumerate}
\item for a given filter, the rotation-period estimates from GWPS, ACF, and CS agree within $2\sigma$ where $\sigma$ is chosen to be the period uncertainty from GWPS;
\item the rotation-period estimates agree within $20\%$ between different filters
\begin{enumerate}
\item for $P_\text{rot}$$<60$ days, the rotation estimates agree between the 20-day and 55-day filters;
\item or for $P_\text{rot}$$\ge60$ days, the rotation estimates agree between the 55-day and 80-day filters;
\end{enumerate}
\item for the appropriate filter, the peak height in the ACF and CS are larger than a given threshold. We adopt the thresholds imposed by \citet{Ceillier2017}:
\begin{enumerate}
\item $G_\text{ACF}\ge0.2$, where $G_\text{ACF}$ is the height of the ACF peak which corresponds to $P_\text{rot}$;
\item $H_\text{ACF}\ge0.3$, where $H_\text{ACF}$ is the mean difference between the height of the ACF peak and the values of the two local minima on either side of the peak;
\item and $H_\text{CS}\ge0.15$, where $H_\text{CS}$ is calculated in the same manner as $H_\text{ACF}$ but for the CS.
\end{enumerate}
\end{enumerate}
Following the steps above, 9,586 targets were automatically selected (this number includes Type~1 CP/CB candidates), which corresponds to $\sim60\%$ of the total number of targets for which we provide rotation-period estimates (Table~3). Targets whose retrieved period is consistent with the reported orbital periods for confirmed and candidate planet hosts (data from the Exoplanet Archive) are reported as targets with no spot modulation (Table~4).
\subsubsection{Visual Check}\label{sec:vis}
For stars that were not automatically selected we proceed to visually check their KEPSEISMIC (three filters) and PDC-MAP light curves, the respective power density spectra, and the rotation diagnostics. We also visually check the light curves of the targets for which the rotation results for the PDC-MAP light curves are not consistent with those for the KEPSEISMIC light curves. Often, half of the rotation period is recovered from the PDC-MAP time-series (see Appendix \ref{app2}). This is probably due to the fact that PDC-MAP applies a 20-day filter, but not systematically in all quarters or to all stars. The comparison with PDC-MAP also helps to identify KEPSEISMIC light curves polluted by nearby stars, as the latter use larger apertures (see Sect.~\ref{sec:data}). Targets showing evidence for photometric pollution are listed in Table~4.
Although multiple signals present in both the KEPSEISMIC and PDC-MAP light curves may still be the result of photometric pollution by background stars, we determine and report the periods of the observed multiple signals (Table~5). Note that these multiple signals are most likely not related to differential rotation as the detected periods are well separated. For most of these targets, the periods of the different signals have to be determined through visual inspection and manually, for example, by limiting the range of period to be searched. For some of the targets, one of the multiple signals is consistent with one of the CP/CB candidates described above. Thus, we also provide the respective flag in Table~5. For signals consistent with Type~2 and 3 we do not provide a period. Finally, we note that some of the signatures can be the result of eclipses or transits. Although some of the targets with multiple signals are KOIs reported as false negatives, none of these targets is a confirmed eclipsing binary.
The rotation-period estimate in Tables~3 and 5 is provided as described in Sect. \ref{sec:a2z}, prioritizing the results from the wavelet analysis.
From the visual inspection, the rotation periods for 6,324 additional targets were determined. In total, we provide rotation-period estimates for 15,910 targets (Tables~3-5; including Type~1 CP/CB candidates and light curves with multiple signals).
Although a significant number of targets exhibit evidence for rotational modulation (3,562), we are not able to confidently recover rotation periods. Generally, their light curves exhibit instrumental effects, which hamper the detection of the true rotation period. We mark these in Table~4 as targets with possible spot modulation.
From this analysis, we find that 5,310 targets (also listed in Table~4) show no evidence for spot modulation. This could be due to the combination of small amplitude spot modulation and noise, or due to the spot visibility, which depends on the stellar inclination angle and spot latitudinal distribution.
\begin{table}[h]
\centering
\begin{tabular}{R{1.5cm}cc}
\hline\hline
\multicolumn{3}{c}{\bf With $\pmb{P}_\text{\!rot}$ estimate}\\\hline
& Auto. selected & Visually selected \\
M dwarfs & 918 & 612 \\
K dwarfs & 8,380 & 5,380 \\\hline
\multicolumn{3}{c}{Type~1 CP/CB candidates\hspace{0.2cm}350} \\\hline
\multicolumn{3}{c}{Multiple signals\hspace{0.2cm}270} \\\hline\\
\multicolumn{3}{c}{\bf Without $\pmb{P}_\text{\!rot}$ estimate}\\\hline
& No rotation & Possible rotation\\
M dwarfs & 494 & 132\\
K dwarfs & 4,816 & 3,430\\\hline\hline
\end{tabular}
\caption{Summary of the results from the rotational analysis of $24,782$ targets. The top part of the table corresponds to the targets for which we provide $P_\text{rot}$\ estimate (automatically and visually selected). The bottom part of the table corresponds to the targets for which we do not provide $P_\text{rot}$\ estimate. Part of those targets do not exhibit spot modulation, while others show possible spot modulation but we are unable to confidently provide a $P_\text{rot}$\ value.}
\label{tab01}
\end{table}\pagebreak
\section{Photometric magnetic activity proxy}\label{sec:sph}
Using CoRot \citep[Convection, Rotation, and planetary Transits;][]{Baglin2006} data for the solar-type star HD~49933, \citet{Garcia2010} showed that the light curve variability due to the presence of magnetic features on the stellar surface --- including starspots --- provides a proxy of stellar magnetic activity. However, brightness variations may include contributions from different phenomena, such as active regions, granulation, oscillations, stellar companions, or instrumental effects. Different phenomena affect the light curve at different timescales. Therefore, to properly estimate a photometric magnetic activity proxy, the stellar rotation period must be taken into account. \citet{Mathur2014} determined that the activity proxy $S_\text{\!ph}$\ computed as the standard deviation of sub-series of length $5\times P_\text{rot}$ provides a reasonable measure of activity and is primarily related to magnetism and minimizes the contributions from other sources of variability. Furthermore, using VIRGO \citep[Variability of Solar Irradiance and Gravity Oscillations][]{Frohlich1995} and GOLF \citep[Global Oscillations at Low Frequency][]{Gabriel1995} data, the photometric activity proxy $S_\text{\!ph}$\ was shown to recover the variation associated with the solar activity cycle at both 11-year and quasi-biennial timescales \citep{Salabert2017}.
For seismic solar-analog stars observed by {\it Kepler}\ and by the ground-based, high-resolution \textsc{Hermes} spectrograph \citep{Raskin2011}, \citet{Salabert2016a} demonstrated that $S_\text{\!ph}$\ measurements are consistent with the chromospheric activity index measured from the Ca K-line emission \citep{Wilson1978}.
Thanks to the {\it Kepler}\ space mission, the photometric activity can be easily estimated through $S_\text{\!ph}$\ for a large number of stars with known rotation periods (which we estimate here). This is a clear advantage in relation to chromospheric activity indexes, which require a large amount of ground-based telescope time and are only possible to measure for bright targets. However, the photometric variability depends on the visibility of active regions. For example, assuming a similar latitudinal distribution of active regions in the Sun for other solar-type stars (note that it may not be true for late-type M dwarfs), $S_\text{\!ph}$\ will correspond to a lower limit of the true photometric activity level for stars with small inclination angle, i.e the angle between the rotation axis and the line of sight, which is unknown for most targets.
In this work, we compute the photometric activity index $S_\text{\!ph}$\ for the M and K~dwarfs with period estimates obtained in Sect.~\ref{sec:method}. In Tables~3 and 5, the $S_\text{\!ph}$\ value and respective uncertainty are provided as the mean value and standard deviation of the $S_\text{\!ph}$\ computed over sub-series of length $5\times P_\text{rot}$. The $S_\text{\!ph}$\ index is corrected for the photon noise following the approach by \citet{Jenkins2010}. However, for $1\%$ of the targets with $P_\text{rot}$\ estimate, the correction from \citet{Jenkins2010} leads to negative $S_\text{\!ph}$\ values. For such targets, the correction to the photon noise is instead computed from the flat component in the power density spectra. In Table~5 (light curves with multiple signals), the multiple $S_\text{\!ph}$\ values for a given target may change significantly as they are computed at different timescales depending on the respective period.
\section{Results}\label{sec:res}
Following the methodology described in Sect.~\ref{sec:method}, surface rotation periods were successfully measured for 15,290 stars ($\sim62\%$ of the targets for which we perform the rotational analysis; 1,530 M and 13,760 K~dwarfs), and for additional 350 Type 1 CP/CB candidates and 270 targets whose light curves show multiple signals. The photometric activity proxy $S_\text{\!ph}$\ was also measured for the same targets.
Tables~3 and 5 summarize the properties and results for the stars with rotation-period estimates, including Type 1 CP/CB candidates and those lightcurves with multiple signals in both KEPSEISMIC and PDC-MAP data sets. Table~4 lists the remainder of the target sample.
Figure~\ref{fig:kp} compares the distribution of {\it Kepler}\ magnitudes (Kp) for targets with rotation-period estimate with those for CP/CB candidates (Type 1, 2, and 3), targets with possible spot modulation, and targets without evidence for spot modulation. The magnitude of stars with possible rotation modulation and of CP/CB candidates is consistent with that of stars with successful rotation measurement. For CP/CB candidates, there is, however, a slight excess of brighter targets. The distribution of targets that do not exhibit spot modulation extends to fainter magnitudes than that of targets with rotation estimates. Faint targets often show high levels of noise which hampers detection of rotational signatures.
\begin{figure}[h]
\includegraphics[width=\hsize]{Kp_Prot2.pdf}\vspace{-0.1cm}
\caption{Comparison between the magnitude distribution for stars with $P_\text{rot}$\ estimate (excluding CP/CB candidates; black solid line) and that for: CP/CB candidates (left; red), stars with possible spot modulation (middle; blue), and stars without spot modulation (right; green). Dashed lines indicate the median magnitude for stars with $P_\text{rot}$\ estimate, while the dotted lines correspond to the median value of the distributions shown in color.}\label{fig:kp}
\end{figure}
Type~1 CP/CB candidates and targets whose light curves show multiple signals are neglected in Figs.~\ref{fig:hist}-\ref{fig:protsph} as well as all the possible non-single non-main-sequence stars flagged by \citet{Berger2018}, \citet{Simonian2019}, and FliPer$_\text{Class}$ \citep{Bugnet2019}. In Appendix \ref{app}, we present the same figures where all targets in Tables~3 and 5 are considered.
Figure~\ref{fig:hist} summarizes the results for the targets with period estimate. M~dwarfs have on average longer rotation periods and larger $S_\text{\!ph}$\ values than K~dwarfs, which is consistent with the results in \citet{McQuillan2014}.
\begin{figure}[h]
\includegraphics[width=\hsize]{ProtSph_hist_type.pdf}\vspace{-0.1cm}
\caption{Distribution of rotation periods (left) and $S_\text{\!ph}$\ values (right) for M (top) and K~dwarfs (bottom) is shown in red. The respective median values are marked by the dotted lines. The distributions and corresponding median value for the full subsample of M anf K~dwarfs with $P_\text{rot}$\ estimate are shown by the black solid and dashed lines, respectively.}\label{fig:hist}
\end{figure}
In the following sections, we take a more detailed look at the dependency of the surface rotation and photometric activity on the stellar effective temperature and mass.
\subsection{Rotation - Mass/Temperature relation}
The left-hand panel of Fig.~\ref{fig:protTeffMass} shows the rotation period as a function of stellar effective temperature (from KSPC DR25). As the effective temperature increases the average rotation period is found to decrease, meaning that hotter stars are generally faster rotators than cooler stars. Our results exhibit two sequences in the $P_\text{rot}$-$T_\text{eff}$\ relation which are consistent with the bimodal $P_\text{rot}$\ distribution previously reported by \citet{McQuillan2013,McQuillan2014}. The vertical features and gaps are the result of artifacts in the {\it Kepler}\ Stellar Properties Catalog temperature scale.
\begin{figure*}[htp]
\includegraphics[width=0.503\hsize]{Prot_vs_Teff02.pdf}
\includegraphics[width=0.5\hsize]{Prot_vs_Mass02.pdf}
\caption{Rotation period as a function of effective temperature (left) and mass (right) color coded by number of stars in a given parameter range. Brighter colors indicate higher density regions than darker colors. Stellar effective temperature and mass are taken from KSPC DR25. } \label{fig:protTeffMass}
\end{figure*}
\begin{figure*}[htp]
\includegraphics[width=0.503\hsize]{Sph_vs_Teff02.pdf}
\includegraphics[width=0.5\hsize]{Sph_vs_Mass02.pdf}
\caption{Photometric activity index $S_\text{\!ph}$\ as a function of effective temperature (left) and mass (right) color coded by number of stars in a given parameter range. Stellar effective temperature and mass are taken from KSPC DR25.} \label{fig:sphTeffMass}
\end{figure*}
The right-hand side of Fig.~\ref{fig:protTeffMass} shows the rotation period as a function of stellar mass \citep[from KSPC DR25;][]{Mathur2017}. Rotation period decreases slightly with increasing mass. In this case, the bimodal rotation-period distribution is not as obvious as that in the $P_\text{rot}$-$T_\text{eff}$\ relation and, in particular, not as clear as in the $P_\text{rot}$-mass relation found by \citet{McQuillan2014}. We note that the stellar masses used in this work are different from those in \citet{McQuillan2014}, as different stellar evolution codes with different physics and observables were used. As shown in Sect.~\ref{sec:mcq}, for the common targets, the period estimates obtained in this study and in \citet{McQuillan2014} are in very good agreement. Thus, the stellar masses are the source for the discrepancy. See Fig.~\ref{fig:mass} for the comparison between the masses from \citet{McQuillan2014} and those from \citet{Mathur2017}.
\begin{figure}\includegraphics[width=\hsize]{mass_vs_mass.png}
\caption{Comparison between the stellar masses from \citet[][$\text{Mass}_\text{McQ}$]{McQuillan2014} and \citet[][$\text{Mass}_\text{KSPC DR25}$]{Mathur2017}.} \label{fig:mass}
\end{figure}
\subsection{Photometric activity - Mass/Temperature relation}
The left-hand panel of Fig.~\ref{fig:sphTeffMass} shows the photometric activity proxy $S_\text{\!ph}$\ as a function of the effective temperature. For the parameter space considered in this work, the photometric activity proxy takes on a wider range of values with increasing effective temperature. The upper envelope of $S_\text{\!ph}$\ values increases with increasing temperature, while the lower envelope decreases. A similar behaviour is found for the $S_\text{\!ph}$\ as a function of mass (right-hand panel in Fig.~\ref{fig:sphTeffMass}). Our results are consistent with those of \citet{McQuillan2014}.
The transition between fully convective stars and stars with a radiative core is expected to take place at $0.35\text{M}_\odot$ \citep[e.g.][]{Chabrier1997}. If the tachocline (transition between a differentially rotating convective envelope and a uniformly rotating radiative core) played an important role in the dynamo mechanism for M~stars, one might expect to observe a transition in the rotation period and photometric activity proxy distributions. However, due to the small number of targets with lower masses, there is no sufficient evidence to support or reject that hypothesis.
\subsection{Photometric activity - rotation relation}
Faster rotators are expected to be more active than slower rotators at fixed effective temperature \citep[e.g.][]{Vaughan1981,Baliunas1983,Noyes1984b}. Therefore, one should expect the photometric activity proxy $S_\text{\!ph}$\ to be related to the rotation period. Figure~\ref{fig:protsph} shows the $S_\text{\!ph}$\ as a function of the rotation period. For M~dwarfs there is no clear relationship. Nevertheless, for K~dwarfs we find a negative correlation: photometric activity increases with increasing rotation rate. The bimodality in the rotation-period distribution is also obvious in the $S_\text{\!ph}$-$P_\text{rot}$\ relation, which exhibits two distinct sequences for faster and slower rotators. Although we have made an effort to identify classical pulsator candidates, we note that there is still the possibility for additional polluters, namely Type~1 CP/CB candidates. We advise caution in particular when dealing with fast rotators with very large $S_\text{\!ph}$\ values. Despite their similarity with targets flagged as Type~1 CP/CB candidates, these targets show three or less harmonics in the power spectra and thus do not obey the criteria imposed in Sect. \ref{sec:sample} to discriminate the CP/CB candidates.
\begin{figure}[h]
\includegraphics[width=\hsize]{Prot_vs_Sph_all.pdf}
\caption{Photometric activity proxy as a function of the rotation period color coded by number of stars in a given parameter range for: all M and K~dwarfs (top), M~dwarfs (middle), and K~dwarfs (bottom). For comparison, the $S_\text{\!ph}$\ values at solar activity maximum (314.5 ppm) and minimum (67.4 ppm) are marked by the dashed green lines. Solar values from \citet{Mathur2014}.}\label{fig:protsph}
\end{figure}\pagebreak
\subsection{Comparison with McQuillan et al. (2013, 2014)}\label{sec:mcq}
In this section, we compare our results with those from \citet{McQuillan2013a,McQuillan2014}. The periods from those works were estimated from the autocorrelation function of the PDC-MAP light curves for {\it Kepler}\ Quarters 3-14 (3 years of data).
Only 11,209 of the targets for which we provide a rotation-period estimate in Table~3 (15,640 targets in total) are common to the detection by \citet{McQuillan2013a,McQuillan2014}.
Figure~\ref{fig:McQ} shows the comparison between the rotation-period estimates, which agree within $2\sigma$ for $\sim99.4\%$ of the common targets ($\sim99.1\%$ within $1\sigma$). The most common cases outside of $2\sigma$ ($\sim0.3\%$ of the common targets) correspond to targets for which \citet{McQuillan2013a,McQuillan2014} measured double the rotation period found in our analysis. For a smaller number of stars ($\sim0.1\%$), \citet{McQuillan2013a,McQuillan2014} recovered half of the rotation period.
\begin{figure}[h]
\includegraphics[width=\hsize]{Prot_vs_McQ.png}
\caption{Comparison between the rotation estimates from this work ($P_\text{rot,\,This\, work}$) and those from \citet[][$P_\text{rot,\,McQ}$]{McQuillan2013a,McQuillan2014}. The dashed lines indicate the two-to-one, one-to-one, and one-to-two lines.}\label{fig:McQ}
\end{figure}
We provide rotation-period estimates for 4,431 targets (3,831 K stars; 618 M stars) that were not reported by \citet{McQuillan2013a,McQuillan2014}. From those only 558 targets were not identified as M and K main-sequence stars by \citet{Brown2011} and \citet{Dressing2013}, which provided the {\it Kepler}\ properties adopted by \citet{McQuillan2013a,McQuillan2014}. Therefore, most of the additional estimates are rotation periods that \citet{McQuillan2013a,McQuillan2014} could not detect with their data and methodology.
\citet{McQuillan2014} reported rotation-period estimates for 465 targets in Table~4, including misclassified red giants (184), eclipsing binaries (5), RR Lyrae (3), Type~2 CP/CB candidates (2), and Type~3 CP candidates~(3). \citet{McQuillan2014} reported rotation periods for 26 targets that we have identified as not showing rotational modulation and 180 targets with possible rotational modulation. The time-series of these targets show significant instrumental effects which would lead to incorrect period estimates. The time-series of 62 targets for which \citet{McQuillan2014} reported $P_\text{rot}$\ exhibit photometric pollution in both PDC-MAP and/or KEPSEISMIC data sets.
We also note that 286 targets with $P_\text{rot}$\ estimate in \citet{McQuillan2014} are flagged as Type~1 CP/CB candidates in Table~3, while 179 show multiple signals (Table~5) which can be related to multiple systems or photometric pollution by background stars. For the latter, an automatic rotation estimate will be biased towards the signal with largest amplitude.
The rotation analysis we perform combines the wavelet analysis with the autocorrelation function of the light curves \citep[e.g.][]{Garcia2014,Ceillier2016,Ceillier2017}. This methodology performs better than the autocorrelation function alone \citep[e.g.][]{McQuillan2013,McQuillan2013a,McQuillan2014} in terms of completeness and reliability \citep{Aigrain2015}. Furthermore, we used both the longest time-series available and our own calibrated light curves (KEPSEISMIC), which may have contributed to the significant improvement in the fraction of rotational signals we detect (see also Appendix~\ref{app2}). Figure \ref{fig:histMcQ} shows the comparison between the number of estimates in this work and in \citet{McQuillan2014}. As mentioned previously, we provide rotation periods for a larger number of stars --- in particular, the fraction of stars cooler than 4200 K with measured rotation periods is larger than that in \citet{McQuillan2014}. Moreover, our methodology is also able to retrieve rotation periods for fainter stars than the analysis by \citet{McQuillan2014}, which only retrieved $P_\text{rot}$\ for 57 targets (M and K dwarfs) fainter than 16 mag.
\begin{figure}[h]
\centering
\includegraphics[width=\hsize]{hist_teff_kp.pdf}
\caption{Distribution of the number of $P_\text{rot}$\ estimates from this work (red) and from the analysis in \citet[][black]{McQuillan2014} as a function of effective temperature (left) and {\it Kepler}\ magnitude (right).}\label{fig:histMcQ}
\end{figure}
\subsection{\textit{Gaia} binary candidates}
In this section, we compare our target sample with the binary candidates proposed in \citet{Simonian2019} and \citet{Berger2018}, both of which used information from \textit{Gaia} DR2. Only 198 targets are in common with the \citet{Simonian2019} sample, which is focused only on the fast rotators ($P_\text{rot}<7$ d), while 22,378 targets are common to the \citet{Berger2018} sample. Appendix~\ref{app} describes these targets in $\log\, g$-$T_\text{eff}$, $P_\text{rot}$-$T_\text{eff}$, $S_\text{\!ph}$-$T_\text{eff}$, and $S_\text{\!ph}$-$P_\text{rot}$ diagrams.
Using \textit{Gaia} DR2, \citet{Simonian2019} found that faster rotators are often systematically offset in luminosity from the single-star main-sequence in comparison to slower rotators. This was interpreted as a signature of tidally-synchronized binaries, for which tidal interactions synchronize the rotation and orbital periods. Both because the fast rotator population in \citet{Simonian2019} was dominated by binary systems, and because our rapid rotators do not behave like typical active spotted stars, we advise caution in the interpretation of measurements of rapidly rotating stars.
The left-hand piechart of Fig.~\ref{fig:chartbin} summarizes the comparison between targets that are both in our sample and that of \citet{Simonian2019}. The size of the slices indicate the percentage of possible tidally-synchronized binaries of each sub-category distinguished in this work. The fractions denoted along the chart indicate the number of possible binaries over the total number of common targets between the two analyses for each sub-category. For example: seven misclassified red giants were analyzed by \citet{Simonian2019}, six of which have luminosity excess consistent with binarity, representing $\sim4\%$ of the targets in this study that are identified as synchronized binaries by \citet{Simonian2019}. $\Delta M_\text{Ks}$ indicates the luminosity excess correction (also listed in Tables~3-~5) which corresponds to the difference between the observed luminosity of a star based on the absolute magnitude in the Ks-band and the expected luminosity for a single star with a given temperature, metallicity, and age inferred from models \citep[for details see][]{Simonian2019}. We adopted the inclusive binary threshold $\Delta M_\text{Ks}<-0.2$ defined by \citet{Simonian2019}.
Interestingly, a significant fraction of possible tidally-synchronized binaries show multiple signals in their KEPSEISMIC and PDC-MAP light curves, and most of the Type~1 CP/CB candidates are identified as possible binaries in \citet{Simonian2019}. Also, four of the misclassified red giants identified as possible binaries show signatures consistent with the Type~1 CP/CB candidates. 72 targets for which we provide $P_\text{rot}$\ estimate (none of which are Type~1 CP/CB candidates or target with multiple signals) are likely to be tidally synchronized binaries according to \citet{Simonian2019}. Note that Figs.~\ref{fig:hist}-\ref{fig:protsph} do not include binary candidates. Moreover, we do not find any particular $P_\text{rot}$\ or $S_\text{\!ph}$\ trend as a function of the luminosity excess correction. However, most of the targets that are possibly tidally-synchronized binaries ($\sim 70\%$) have $S_\text{\!ph}$\ larger than $10^4$ ppm.
\begin{figure}[h]\centering
\includegraphics[trim=0 0 0 29mm,clip,width=\hsize]{piechart.pdf}
\caption{Summary of our results for targets identified as binary candidates by \citet[][left]{Simonian2019} and \citet[][right]{Berger2018}. The size of the slices only concern the targets that are binary candidates. The annotations indicate the fraction of targets flagged as binary candidate over the total number of common targets in each category. Asterisks mark categories with sub-categories (see caption of Tables~3 and~4). Note that three RR Lyrae are analyzed and identified as single main-sequence stars by \citet{Berger2018}.}\label{fig:chartbin}
\end{figure}
Using \textit{Gaia} DR2, \citet{Berger2018} revised the Radii of the {\it Kepler}\ targets and identified misclassified targets (possible subgiants and red giants) and possible binary systems. Flags are added to Tables~3-5. Most of the targets with $P_\text{rot}$\ estimates that were not CP/CB candidates analyzed by \citet[][11,400 out of 13,072]{Berger2018} are found to be likely single stars. 113 CP/CB candidates are identified as possible binaries with 112 of those being Type~1 CP/CB candidates. 69 of the binary candidates show multiple signals in the PDC-MAP and KEPSEISMIC time-series. 101 eclipsing binaries (four are also flagged as misclassified red giants) are found to be \textit{Gaia} binary candidates. Note that misclassified red giants can be in binary systems and, thus, it is reasonable having misclassified red giants with more than one flag. The three RR Lyrae in the sample are common to the analysis of \citet{Berger2018} which identified them as single main-sequence stars.
Finally, for targets in common between \citet{Berger2018} and \citet{Simonian2019}, their results agree reasonably well. All the common targets found to be likely binary systems by \citet{Berger2018} are also identified as possible tidally-synchronized binaries by \citet{Simonian2019}. However, some of the single stars from \citet{Berger2018} are below the threshold imposed by \citet{Simonian2019} for targets to be flagged as binaries.
As mentioned in Sect.~\ref{sec:sample}, 368 presumably fast rotators do not behave as typical active stars. While we have identified three types of possible classical pulsators, it is not clear whether they are indeed classical pulsators. The results from \citet{Simonian2019} and \citet{Berger2018} suggest the interesting possibility of these Type~1 targets being close-in binary systems. A detailed analysis of these targets is beyond the scope of the current work. Nevertheless, we consider that one should be careful drawing conclusions based on the rotation estimate for fast rotators. Note that flags with the results of \citet{Simonian2019} and \citet{Berger2018} are added to Tables~3-5.
Finally, using \textit{Gaia} DR2, \citet{Berger2018} also identified the evolutionary stage of {\it Kepler}\ targets. We have removed the misclassified red-giant candidates from the rotation analysis (Sect.~\ref{sec:sample}; Table~4; Garc\'ia et al. in prep). However, we did not remove the subgiant candidates from the analysis. 61 targets in Table~3 were flagged as subgiants by \citet{Berger2018}. These targets were neglected in Figs.~\ref{fig:hist}-\ref{fig:protsph} and the \textit{Gaia} subgiant flag is provided in Tables~3 and 5.
\section{Summary and conclusions}\label{sec:conclusion}
One can learn about surface rotation and magnetic activity by studying the brightness variations due to dark spots rotating across the visible stellar disk. In this work, we analyze {\it Kepler}\ long-cadence data of 26,521 M and K main-sequence stars. The main goal of this work was to determine the average surface rotation and photometric activity level of the targets using the longest time-series available.
Rotation estimates are obtained by combining wavelet analysis, autocorrelation function, and composite spectrum of light curves \citep[e.g.][]{Mathur2010b,Garcia2014,Ceillier2016,Ceillier2017}. This methodology was found to be the best in terms of completeness and reliability \citep{Aigrain2015}. We compared the results for three KEPSEISMIC time-series (obtained with 20-day, 55-day, and 80-day filters) and PDC-MAP time-series to determine reliable rotation periods.
Given the rotation period, we also calculated the photometric activity proxy $S_\text{\!ph}$\, which corresponds to the average standard deviation computed over subseries of length $5\times P_\text{rot}$ \citep{Mathur2014}. $S_\text{\!ph}$\ is sensitive to the spot visibility and, thus, to their latitudinal distribution and stellar inclination angle. For this reason, $S_\text{\!ph}$\ is likely to be a lower limit of the true photometric activity level. Also, in cases where spots are approximately in anti-phase (approximately $180^\circ$ apart in longitude), $S_\text{\!ph}$\ will underestimate the true activity level. Nevertheless, $S_\text{\!ph}$\ was demonstrated to be consistent with other solar activity proxies \citep{Salabert2017} and complementary to the chromospheric activity $S$ index for solar analogs \citep{Salabert2016a}
We successfully recovered the surface rotation periods and respective photometric activity proxy for 15,290 stars ($\sim 62\%$ of the targets analyzed in Sect. \ref{sec:method}). We provide period estimates for targets whose KEPSEISMIC and PDC-MAP light curves show multiple signals (270 targets). We also provide period estimates for another 350 stars that we flagged as possible classical pulsators or close-in binary systems. Their behaviour is not consistent with that of single active stars, resembling that of RR Lyrae or Cepheids. We also have identified $\gamma$-Doradus or $\delta$-Scuti candidates (18 in total). We note, however, that further analysis is needed to properly classify these 368 targets and determine the source of the multiple signals in the light curves of the 270 targets.
Another 3,562 targets ($\sim14\%$ of the sample) show spot modulation in their light curves, but we are unable recover reliable rotation periods. 5,310 targets ($\sim20\%$ of the sample) do not exhibit any apparent spot modulation. The magnitude distribution of these targets is slightly shifted towards fainter values in comparison with stars with spot modulation in the light curves. We do not provide rotation estimates for confirmed RR Lyrae (3 stars; Szab\'o et al. in prep), known eclipsing binaries \citep[272 stars;][]{Kirk2016,Adbul-Masih2016}, targets identified as misclassified red giants (1221; Garc\'ia et al. in prep), and targets whose light curves show evidence for photometric pollution (255 targets). We consider a light curve photometrically polluted when only particular {\it Kepler}\ Quarters show modulation signals or the signal is only present in the KEPSEISMIC light curves. These targets are listed in Table~4.
\citet{Berger2018} and \citet{Simonian2019} identified possible binary systems and we have crossed-checked our sample with their results. In terms of rotation and photometric activity proxy, we did not find any particular difference between binaries and single stars. Nevertheless, it is interesting to note that a significant number of targets show evidence of photometric pollution by nearby stars. Also, most of the classical pulsator candidates flagged as binary candidates show stable, high-amplitude variations and beating patterns, and we therefore treat them as classical pulsator/close-in binary candidates. Note that we did not remove binary candidates from the analysis, but did include the respective flags from \citet{Berger2018} and \citet{Simonian2019} in our tables
Only $\sim72\%$ of the targets with rotation period estimates are also detected in spot modulation in \citet{McQuillan2013a,McQuillan2014}. For the common targets, the agreement on the $P_\text{rot}$\ estimate is about $99.4\%$ at $2\sigma$. We also show that our methodology is able to recover rotation periods for a larger number of stars (4,431 additional $P_\text{rot}$) than the analysis by \citet{McQuillan2014}. In particular, we provide $P_\text{rot}$\ for a higher fraction of cool and faint stars.
For the parameter range studied here (M and K~dwarfs), we find that the mean rotation period to decrease with increasing stellar effective temperature and mass, i.e. K~dwarfs are on average faster rotators than M~dwarfs. This is consistent with previous findings \citep[e.g][]{McQuillan2014,Garcia2014}. As in \citet{McQuillan2014}, we also found two sequences in the $P_\text{rot}$-$T_\text{eff}$\ relation: a wider and more populated sequence for slower rotators and a narrower and less populated sequence for faster rotators. The bimodality is clear in the rotation-period distribution for M~dwarfs. Due to the wider range of effective temperatures of K~dwarfs in comparison with the M~dwarfs in the sample, the bimodality is not clear in the $P_\text{rot}$\ distribution for K~dwarfs. However, we verified that the bimodality is present while splitting the K~dwarfs in smaller sub-samples according to their temperature. Furthermore, the bimodality is also visible in the density plot of rotation period as a function of effective temperature.
For M and K~dwarfs, we found that the photometric activity proxy takes on a wider range of values as effective temperature and mass increase, and the extremes of the distribution extend to both higher and lower $S_\text{\!ph}$\ values.
The photometric activity proxy $S_\text{\!ph}$\ increases as rotation period decreases. This is consistent with faster rotators being more active than slower rotators \citep[e.g.][]{Vaughan1981,Baliunas1983,Noyes1984b}. The bimodal rotation-period distribution is also visible through the two branches in the $S_\text{\!ph}$-$P_\text{rot}$\ relation. A similar behaviour was also found by \citet{McQuillan2013,McQuillan2014} while using a different measure of photometric variability, $R_\text{var}$ \citep[see Sect. \ref{sec:sph};][]{Basri2011,Basri2013}.
Based on the evidence of two distinct proper motion distributions, \citet{McQuillan2013} interpreted the bimodal rotation-period distribution as evidence for two stellar populations with different ages associated to different star-formation episodes. Using \textit{Gaia} data, the results by \citet{Davenport2017} and \citet{Davenport2018} are consistent with the bimodal rotation-period distribution being associated to a bimodal age distribution. In particular, the authors found that the bimodality is more pronounced at low Galactic scale height which is assumed to be an age indicator. \citet{Montet2017} and \citet{Reinhold2019} found that the fast rotating, more active sequence corresponds to spot-dominated stars, while the slowly rotating less active stars are faculae-dominated. These studies support the idea that solar-type stars transition from spot-dominated to faculae-dominated as stars evolve. \citet{Reinhold2019} suggested that the observed period bimodality is actually a dearth of detections at intermediate rotation periods due to the cancellation between dark spots and bright faculae.
In this work, we found that the photometric activity proxy $S_\text{\!ph}$\ varies approximately within the same range for M~dwarfs in both fast and slow rotator branches. For K~dwarfs, although most of the targets in both branches have $S_\text{\!ph}$\ values smaller than $\sim 7000$~ppm, $S_\text{\!ph}$\ values for slow rotators extend to significantly smaller values ($\sim 200$~ppm), while the fast rotators are mostly within $\sim 600-7000$~ppm.
The methodology followed in this work will be extended to G and F main-sequence stars and subgiants in a future paper. See \citet{Santos2018b} for a brief summary where the analysis is also applied to G main-sequence stars cooler than 5500 K and subgiants cooler than 5500 K and with surface gravities larger than $\log\,g=3.5$ dex.
\acknowledgments
The authors thank R\'obert Szab\'o, Paul G. Beck, Katrien Kolenberg, and Isabel L. Colman for helping on the classification of stars. This paper includes data collected by the {\it Kepler}\, mission and obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the {\it Kepler}\ mission is provided by the National Aeronautics and Space Administration (NASA) Science Mission Directorate. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. ARGS acknowledges the support from NASA under Grant NNX17AF27G. RAG and LB acknowledge the support from PLATO and GOLF CNES grants. SM acknowledges the support from the Ramon y Cajal fellowship number RYC-2015-17697. TSM acknowledges support from a Visiting Fellowship at the Max Planck Institute for Solar System Research. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
\software{KADACS \citep{Garcia2011}, NumPy \citep{numpy}, SciPy \citep{scipy}, Matplotlib \citep{matplotlib}}
\facility{MAST, {\it Kepler}\ Eclipsing Binary Catalog, Exoplanet Archive}
\bibliographystyle{aasjournal}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 14
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Day Centres
SERVICES FOR OLDER PEOPLE
Bathing Service
Trolley shops
Blood Service
Older Peoples' Services
THE GUERNSEY VOLUNTARY SERVICE
The Women's Voluntary Service came into existence in the UK in June 1938 as part of war preparations. The first organiser was Lady Reading who had been asked by the UK government to set up an organisation for women that would work with the ARP (Air Raid Precautions). Their work during the war was invaluable, and in all spheres: support for the troops, organisation of the evacuation of children, soup kitchens and canteens, distribution of clothing for refugees, help for evacuees and for those who had lost all in the bombings. The WVS members were known as the 'Ladies in Green' owing to their distinctive uniforms. For further information see the WRVS Archive Section. In 1966 in recognition of the service that the WVS and its volunteers had given to their country it was granted the honour of adding 'Royal' to the title by Her Majesty the Queen and so became the Women's Royal Voluntary Service.
A branch of the WVS was formed in Guernsey in 1949 with a membership of 6 under the leadership of Miss Doreen Smith-Ainsley. Right from inception it was recognised that the local branch should respond to needs of the island. The initial philosophy was to give practical support to isolated and lonely people, to assist them to maintain their independence and to help them remain in their own homes -particularly in later life. During the first few years the main activities were the Darby and Joan Clubs - one in every parish, the rest tents at the Agricultural Shows and meeting the lifeboat with refreshments after every call out. The scope of the work soon took in other areas of island life and membership gradually increased. In 1960 the Meals on Wheels service was launched, growing in size over the years and in 2013 over 30,000 meals were delivered in Guernsey alone using four dedicated vehicles based at the Princess Elizabeth Hospital.
Alderney's involvement with the WVS began in 1946 soon after the islanders returned after the Second World War. Two ladies from the WVS in London assisted the local people by working in the Food Office and helping with the distribution of clothing. A local branch of the WVS linked with Guernsey was set up in 1951. The longest serving local organiser on Alderney was Pat Hart MBE. WVS Alderney became an independent branch of the WVS in 1957 and the following year permission was granted by the States for the use of a room at the police station in St Annes as the WVS Centre. The Alderney WRVS was awarded the Queens Jubilee Award in 2003 and the medal is on display in the Alderney Museum.
In Guernsey in October 1977 2 Newington Place, Grandes Maisons Road was purchased and named Jubilee House, a name chosen by the members. It acted as a centre for the organisation and a day centre for the elderly. Then in 1993, after significant fundraising, the purpose built Jubilee Day Centre was built in Grandes Maisons Road, St Sampson.
The Guernsey Voluntary Service, otherwise known as the GVS was launched as a charity independent of the WRVS in 2011, the Alderney branch was also incorporated into this re-organisation. No changes were made to the services provided and there has been a great deal of positive feedback regarding the new name and branding. It is felt that the change has had a positive effect in attracting more volunteers- particularly men. There are currently about 500 active volunteers on the islands.
GVS Privacy Policy
privacy_policy_july_2020.pdf
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,447
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Second Update: The Post & Email Speaks with Douglas Vogt, Typesetting Expert, About His Letters to Congress Regarding Birth Certificate Forgery
Friday, July 22, 2011 Saturday, July 23, 2011
WHAT EXCUSE CAN ANYONE MAKE FOR OBAMA NOW?
This "document" lacks the certifying wording present on other birth and marriage certificates issued by the Hawaii Department of Health. Why?
(Jul. 22, 2011) — Mr. Douglas Vogt, owner of Vector Associates and Archive Index Systems, has sent letters to officials in Hawaii and will be sending them to all members of Congress with his report on why he believes the image released by the White House on April 27, 2011 purported to be Barack Obama's long-form birth certificate is a forgery.
The Post & Email previously interviewed Mr. Vogt on June 2, 2011, during which he stated, "I have a unique background for analyzing this document. I owned a typesetting company for 11 years so I know type and form design very well. I currently own Archive Index Systems since 1993, which sells all types of document scanners worldwide and also developed document imaging software…"
We asked Mr. Vogt how many mailings he had completed, and his interview follows below:
MRS. RONDEAU: How many letters have you sent to members of Congress with your report as of now?
MR. VOGT: I haven't finished the cover letter to Congress. I mailed off about 70 of them, but that was 3-4 weeks ago, and that was to several committees. Then I started to write all of them down. I still have to create that mailing list. I have about 89 names, and I have more to do to finish off all the Republicans in the House and the Senate. They'll get it first before their Democrat colleagues.
I've been busy with notifying the U.S. Attorney in Hawaii. You saw the letter to the governor and the attorney general of Hawaii. It looks as if it worked with them, because Loretta Fuddy, the Health Department Director, split! She supposedly "left the island."
MRS. RONDEAU: Have you been able to find out any more specifics on that?
MR. VOGT: Yes, I mailed the thing on July 11th; it was accepted on the 14th, and on the 15th they said that the deputy attorney in the office who was handling the subpoena had left and won't be back until this week sometime and that Fuddy had left the island. It doesn't matter whether the director is there or not; it's a public office, and people come and go all the time. The government office remains open; it's just the next person in charge or even an employee who serves the public, and that's it; you're done. Actually, it's the State of Hawaii which gets served and has to produce it.
The letter to the AG and governor formally notified them that it's a forgery. I sent them both copies of the report: the 28-pager and also the latest affidavit on the news conference.
The news conference analysis was very important, and I think the reason that it shook up the attorney general and governor was that I had correctly identified plausible deniability in that news conference on the 27th of April. It was clear-cut. I'm surprised no one had spotted this. The lawyers hadn't; other people at least didn't say anything; but Bauer and Pfeiffer – Bauer is the former White House attorney and Pfeiffer is communications director – basically set up plausible deniability, and when Obama gave the speech an hour later, he never even mentioned the Certificate of Live Birth that was released an hour before. He only mentioned the Certification of Live Birth of 2008. That's a classic case of plausible deniability where they say, "Well, I never saw it; I never heard it; I didn't order it…" They did it. It's not going to work here. It didn't work during the Nixon administration, and it's not going to work here, because the only one who benefits by that fraud is him. It has nothing to do with money; it has to do with a certificate. That's what we're dealing with.
I think it shook them up. I think it shook up the attorney general in Hawaii and the governor such that they looked at the evidence and said, "He's right." That's plausible deniability; there's no question. They knew they were presenting a fraud. "We're in trouble."
When you read the letter, you'll see it. I said, "This is most likely going to be the end of the Democrat Party." The letter is two pages. The point is that I think this letter scared them, especially when they saw plausible deniability, and I think they realized I'm right. When I defined "plausible deniability" on the affidavit, they said that middle management either is inaccessible or impossible to interrogate because they've beat the scene or they're dead. It's one or the other; that's what happens with criminal organizations to protect the leadership.
[Editor's Note: Vogt issued an original affidavit on May 10, 2011, an expanded 22-page report on May 22, 2011, and a longer report dated June 13, 2011 consisting of 28 pages. A July 4 affidavit containing graphics analyses by Vogt states that "false statements" were made on April 27, 2011, the day on which the image in question was presented to the public as a copy of Obama's original birth record.]
If you remember the Watergate matter, G. Gordon Liddy was volunteered to be assassinated on a Washington, DC street corner if it would save the president. Send Liddy an email; he'll tell you. I'm going to be contacting the U.S. attorney here in Seattle with all the evidence against Robert Bauer to have him prosecuted, because he obviously perpetrated the fraud; he's a principal. I did the same thing with the U.S. attorney in Hawaii to investigate Fuddy and the Hawaii attorney general. They've all been put on notice that "This fraud has happened; these people are most likely involved, and it's your job to investigate."
MRS. RONDEAU: The Post & Email had conducted two Hawaii Petition Campaign mailings, after which we thought we might turn our focus elsewhere. Then Obama released what he said was his birth certificate on April 27 which many have stated is a forgery. As a result, perhaps people such as Fuddy; the attorney general and deputy attorney general, Jill Nagamine; Alvin Onaka, the registrar; the U.S. attorney, as you've mentioned, and the FBI in Hawaii need to hear from as many people as possible.
MR. VOGT: I've notified the U.S. attorney there, Florence Nakakuni; she received hers the same day as the others. She's in the same building as the FBI.
What I want to do is something I learned from my friend, who did something similar with a totally different subject: We have a crook in the White House. We have people who are covering up for him. If the FBI doesn't do anything, I will resort to telling them about not only the plausible deniability, but also the section in the statute about misprision of felony. I tell them very plainly that I'm obligated to tell people, and I've said, "If you do nothing, I'm going to the next group on the list, which is federal judges." I'm going to tell them that I've notified the FBI director and the agent in charge of Hawaii, and they've refused to do anything about it; it's obviously a forgery. I think they're not acting for political reasons. I'm going to ask the federal judges if Mueller can become an accessory after the fact. If you read that law, Section 3, it says that if you hinder the investigation or the prosecution of a guilty party, that is an accessory after the fact. I'm going to ask all the federal judges, including the Supreme Court justices – all the judges who deal with criminal cases, not bankruptcy or probate matters – just those who deal with criminal cases. That's hundreds of judges. If I don't get a response, the next group is all the U.S. attorneys.
MRS. RONDEAU: U.S. attorneys serve at the pleasure of the president, but perhaps we don't have a president right now.
MR. VOGT: We do, but he's a liar. He lied his way to the top; he's a phony. The FBI has to deal with U.S. attorneys and federal judges all the time on a daily basis. So what do you think is going to happen with Mueller and other FBI agents, once they know this stuff, that every U.S. attorney and every federal judge has gotten black-and-white proof that this thing is a forgery? Everybody they work with from whom they need cooperation is going to know the secret. It's no longer going to be a secret. They may not mention it publicly, but they all talk among themselves. This would be the biggest topic among them. Eventually they're going to break, and they're going to break the story to other media. The best thing that could happen for your readers and you is to contact foreign media as well as the smaller- and medium-sized domestic media, both radio and print media, because they aren't so much under the control of the White House, or maybe not even ideologically attuned to them. That's perhaps the best way of doing it. I put up on the secure Birther Summit website blog a suggestion format for a news release.
The only way to force Mueller and the FBI to do something is my plan, which is contacting the federal judges and then the U.S. attorneys, then all of Congress. You tell each and every one, "I've told the FBI; they refuse to do anything; can you please contact them, because the next one on the list is going to be all of these people." They can't do anything to me. I have an absolutely clean record, and they know I'm right. I'm going to try calling Mueller and the FBI agent in Hawaii. I think they'll take my call because I'm the one who wrote the letter and affidavit. A couple of people have called Hawaii and Washington, DC, but they didn't speak with the people to whom I have written. I'm going to see if I can get anything out of them. If I get a chance to talk to them, I'll say, "Listen, if you don't do it, the next one on the list under the law is all the judges." Every single federal judge is going to get a copy of it. Then every single U.S. attorney is going to get it, and they're all going to know that the FBI didn't do anything about it. It'll put the pressure on them.
With Paul Irey's material, my report has a total of ten items. I've offered people who are complaining a $1,000-$2,000 challenge that they couldn't prove all nine of my items wrong, and no one has taken me up on it. They won't. It's just overwhelming.
Anyone who says that the image released by Obama is a forgery has to produce an affidavit that a lawyer can use in court. Some have not come out publicly, so that's just "noise" that the FBI can legally ignore. They have to be formally told that a crime has been committed. They have to make an affidavit, write a letter to the FBI or their congressman, and do the same thing I'm doing. It's basically putting your name and your reputation on the line. I was evidently the first one to do it.
Editor's Note: In an email, Mr. Vogt supplied the following information:
My cover letter will remind them that this scandal is the worst in American history and it is going to hit the domestic as well as foreign media this year. The American voters will want to know why the Republicans did not ask for Congressional hearing or at least ask the FBI to do an investigation. Contact information for Congress, domestic and foreign media is as follows:
Here are links to the names and addresses of all the Congressmen and Senators for you to contact:
Senators: http://www.senate.gov/general/contact_information/senators_cfm.cfm
All elected officials both federal and state:
http://www.congressweb.com/cweb2/index.cfm/siteid/resourcecenter/action/Legislators.Main
All the election officials in the USA :
http://www.congressweb.com/cweb2/index.cfm/siteid/resourcecenter/action/VoterInformation.Main
All the US News Media:
These are the links you can use that will tell you all the newspapers in the USA :
http://www.congressweb.com/cweb2/index.cfm/siteid/resourcecenter/action/Media.Main
For Foreign and Domestic media including TV, Radio, Newspapers all over the world:
http://www.mondotimes.com/
Mailing off news releases to the foreign media might get faster results because they are not under the influence of the Obama administration. If you contact the media while I am notifying the Congress and the US Attorneys and Judges, we might get some action. I will put up a suggested News Release format you should use to notify the news media. Go onto the Birthers Secure Forum: http://birthers.tpath.org/login.aspx
MRS. RONDEAU: What can readers do if they'd like to help defray the costs of postage?
MR. VOGT: It's really simple. The address for Vector is:
12819 S.E. 38th St.
and the phone number is 425-643-1131. Three people called me up the other day and gave me their credit card number, and I actually got to talk to them. I enjoy talking to members of the public. It's the best thing so that somebody doesn't feel as if they're just a number. I also have a chance to tell them to contact the foreign and domestic media. That may be the best way to break the story: drown them, and say, "This is it. It is a fraud. Here are the legal reasons why."
Editor's Note: Mr. Vogt had requested that we send him the reports on the CRS memos written by Joseph DeMaio indicating that Jack Maskell had misled members of Congress regarding Obama's eligibility by representing that simply being born on U.S. soil is enough to be considered a "natural born Citizen." We also sent Vogt our article written about an essay by Attorney Sarah P. Herlihy regarding her advocacy of eliminating the natural born Citizen clause from the U.S. Constitution in which she also hypothesized that a Muslim might become president some day.
After we sent the articles, Vogt reponsded, "Thank you Sharon. Now I can finish my letter to the Congressmen."
Update, July 22, 2011, 7:15 p.m. EDT: The Post & Email has heard from a resident of Hawaii who reported the following of Director of Health Loretta J. Fuddy: "Ms. Loretta Fuddy is here on Island. At a meeting today and booked all next week."
If true, then government officials have blatantly lied to members of the public who have inquired as to whether or not Ms. Fuddy intends to respond to the subpoena issued from the U.S. District Court in Honolulu to make available for inspection the original birth record of Barack Hussein Obama.
Now why would government officials in Hawaii lie about such a thing? Can they be brought up on charges for doing so? They are being paid with taxpayer dollars.
On Tuesday, July 19, 2011, The Post & Email contacted the Hawaii Attorney General's office, specifically asking for Assistant Attorney General Jill T. Nagamine's office. We spoke with an "Adele," who transferred us to a "Jan." We told Jan that as a member of the media, our readers are asking if Ms. Nagamine intends to respond to the subpoena issued to the Hawaii Department of Health by the U.S. District Court requesting that Obama's original birth certificate be made available for inspection. We added that we had just published an interview of someone whose company coworkers in Kenya stated that they were told by Kenyan officials before the 2008 presidential election that Obama's birth record resides in that country. We left an email address and a phone number but received no response.
Update, July 23, 2011: The Hawaii Department of Health's website states that "…Barack Obama posted a certified copy of his original Certificate of Live Birth." If what was posted on April 27, 2011 was a "certified copy of his original," why will Loretta Fuddy not release that "original" contained in the files at the Health Department? Why have the department's employees resorted to lying to continue hiding it?
Tagged: Alvin Onaka, Birther Summit, Congress, Congressional Research Service, CRS memos, Dan Pfeiffer, Douglas Vogt, false statements, FBI, federal judges, forgery, G. Gordon Liddy, Hawaii, Hawaii Attorney General, Hawaii Department of Health, Hawaii Petition Campaign, Jack Maskell, Jill Nagamine, Long-form birth certificate, Loretta J. Fuddy, misprision of felony, natural born citizen, Obama, Obama's eligibility, Paul Irey, plausible deniability, Robert Bauer, Robert S. Mueller III, Sarah P. Herlihy, U.S. Constitution
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,655
|
package procfs
import "testing"
func TestNewFS(t *testing.T) {
if _, err := NewFS("foobar"); err == nil {
t.Error("want NewFS to fail for non-existing mount point")
}
if _, err := NewFS("procfs.go"); err == nil {
t.Error("want NewFS to fail if mount point is not a directory")
}
}
func TestFSXFSStats(t *testing.T) {
stats, err := FS("fixtures").XFSStats()
if err != nil {
t.Fatalf("failed to parse XFS stats: %v", err)
}
// Very lightweight test just to sanity check the path used
// to open XFS stats. Heavier tests in package xfs.
if want, got := uint32(92447), stats.ExtentAllocation.ExtentsAllocated; want != got {
t.Errorf("unexpected extents allocated:\nwant: %d\nhave: %d", want, got)
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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Q: a parallel expectimax Is it possible to make expectimax search parallel? I want to try it on just a basic expectimax algorithm like star1 just to see if there will be significant increase in performance.
What I know right now is to make the alpha-beta pruning parallel similar to the minimax but it's kinda different since chance nodes are involved in expectimax. The problem is I don't know exactly how to do it. I think it's theoretically possible though.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,250
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using System;
using App.Entities.Security;
using App.Security.Infrastructure;
using App.Security.Validation;
using Autofac;
using Microsoft.AspNet.Identity;
using Microsoft.AspNet.Identity.Owin;
using Microsoft.Owin.Security.DataProtection;
namespace App.Security
{
public class SecurityModule : Module
{
protected override void Load(ContainerBuilder builder)
{
builder.Register<PasswordValidator>(x => new CustomPasswordValidator
{
RequiredLength = 6,
RequireNonLetterOrDigit = true,
RequireDigit = false,
RequireLowercase = false,
RequireUppercase = false
});
builder.RegisterType<ApplicationUserManager>().As<UserManager<ApplicationUser, Guid>>().InstancePerLifetimeScope();
builder.RegisterType<IdentityUserStore>().As<IUserStore<ApplicationUser, Guid>>();
builder.RegisterType<IdentityRoleStore>().As<IRoleStore<ApplicationRole, Guid>>().InstancePerLifetimeScope();
builder.RegisterType<ApplicationRoleManager>().As<RoleManager<ApplicationRole, Guid>>()
.InstancePerLifetimeScope();
builder.Register(x =>
new IdentityFactoryOptions<ApplicationUserManager>
{
DataProtectionProvider = new DpapiDataProtectionProvider("App.Api")
}).InstancePerLifetimeScope();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,410
|
\section{Introduction} \label{}
\begin{wrapfigure}{r}{0.4\columnwidth}
\centerline{\includegraphics[width=0.35\columnwidth]{figures/ILD_detector.eps}}
\caption{Layout of the barrel calorimeter system of a LC detector. The
AHCAL is shown in green and the ECAL in blue, while the structure is
surrounded by the magnet.}
\label{fig:barrel}
\end{wrapfigure}
Within the CALICE collaboration~\cite{CALICE} new technologies for
calorimeters for a future linear collider (LC) experiment are
developed and tested. Figure~\ref{fig:barrel} shows a possible design
of the (barrel) calorimeters for a LC detector. The sandwich analog
hadron calorimeter (AHCAL) with 48 layers is a cylindrical structure
with an inner and outer radius of 2.0\,m and 3.1\,m, respectively.
Inside the AHCAL the electromagnetic calorimeter (ECAL) will be
placed, while it is surrounded by the magnet. A major aspect for the
design is the improvement of the jet energy resolution compared to
previous experiments. This can be achieved by measuring the details of
the spatial shower development for a good shower separation and
combining these information with measurements from the tracking
detectors. This approach is known as {\it particle flow} and has been
validated~\cite{PF} with the physics prototype~\cite{PPT} of the
CALICE AHCAL. A very high segmentation of the calorimeters in all
dimensions is mandatory for a good performance of particle flow
algorithms.
A new engineering prototype~\cite{EPT} is currently being developed to
demonstrate that a scalable device can be built that meets the
requirements of an LC experiment. Key requirements for the front-end
electronics are very low power consumption and full integration into
the active calorimeter layers. The prototype is based on scintillating
tiles that are read out by silicon photomultipliers (SiPMs). First
subunits (HCAL base unit, HBU) with 144 detector channels of size
$36\times 36$\,cm$^2$ have been designed and extensively tested in the
laboratory as well as in the DESY test beam facility.
In Sec.~\ref{sec:status} the concept and current status of the
prototype are presented, while results from recent tests are discussed
in Sec.~\ref{sec:results}.
\section{Design and status of the engineering prototype} \label{sec:status}
\begin{wrapfigure}{r}{0.4\columnwidth}
\centerline{\includegraphics[width=0.3\columnwidth]{figures/Proto_Slab.eps}}
\caption{Photo of four assembled HBUs. The detector interface modules
are also shown.}
\label{fig:slab}
\end{wrapfigure}
Figure~\ref{fig:barrel} illustrates that the AHCAL is divided into two
sections along the beam direction and in 16 sectors in
$\phi$-direction. Each sector consists of 48 layers with a total
thickness of 110\,cm and a length of 220\,cm. A single layer consists
of 16\,mm thick stainless steel (or 10\,mm thick tungsten) absorber
plates and an active layer part that is subdivided into several HBUs.
Each layer has about 2500 channels, which adds up to about 4 million
channels for the whole barrel AHCAL. All electronics connections and
interface modules are placed at the two end-faces of the barrel, which
are easily accessible for maintenance and service lines.
The AHCAL layers are subdivided into three parallel slabs, which
consist of six HBUs that are interconnected via ultra-thin flex leads.
Figure~\ref{fig:slab} shows the current setup of four assembled HBUs.
Each HBU features 144 detector channels of $3\times 3$\,cm$^2$ size.
The energy deposited in the calorimeter is sampled by 3\,mm thick
scintillating plastic tiles and the scintillation light is guided with
an integrated wavelength shifting fiber to a SiPM with a size of
1.27\,mm$^2$. The SiPMs comprise 796 pixels operated in Geiger mode
with a gain of $\sim 0.5\cdot 10^6$ - $2.0\cdot 10^6$. The tiles are
connected below the PCB with a nominal distance of 100\,$\mu$m by two
alignment pins that are plugged into holes in the PCB. A photo of 70
assembled tiles on the backside of an HBU is shown in
Fig.~\ref{fig:tiles}.
The 36-channel SPIROC2b ASICs~\cite{ASICs}, that read out the analog
signals from the SiPMs, are mounted on the top side of the PCB and are
lowered into cutouts by $\sim$500\,$\mu$m to reduce the height of the
active layers. They are equipped with 5\,V DACs for channel-wise bias
voltage adjustment. Two gain modes are provided, where the high gain
mode is primarily foreseen for taking calibration data and the low
gain mode measures signals with higher amplitudes up to SiPM
saturation. To avoid the need for an active cooling system inside the
calorimeter layers for the final LC operation, the power consumption
has to be limited to 25\,$\mu$W (40\,$\mu$W) per channel (including
SiPMs). This is only possible when parts of the ASICs are switched off
when they are not needed. This is done according to the bunch train
structure of the LC ({\it power pulsing})~\cite{Peter}. The
on-detector zero suppression with an adjustable threshold is
integrated into the ASICs as well as the digitization step with a
12-bit ADC for charge and a 12-bit TDC for time measurements. The TDC
comprises two ramps with variable lengths between 200\,ns and
5\,$\mu$s, depending on the operation mode (LC or test beam). The
working principle as well as the performance of the TDC in LC and test
beam mode are discussed in Sec.~\ref{sec:TDC}.
\begin{figure
\centering
\subfigure[] {\includegraphics[width=2.5in]{figures/Tiles.eps}\label{fig:tiles}}
\hspace{1.5cm}
\subfigure[] {\includegraphics[width=1.5in]{figures/LED_driver.eps}\label{fig:LED}}
\caption{(a) Photo of 70 tiles that are assembled below an HBU. The
orientation of the wavelength shifting fibers depends on the
position of the holes for the pins. This is determined by the
details of the PCB design. (b) Diagram of the LED driver circuit.}
\end{figure}
Since the response of SiPMs strongly depends on the temperature
($\sim$-1.7\%/K) and the applied bias voltage ($\sim$2.5\%/100\,mV), a
calibration system is needed in order to correct for these effects.
Furthermore, SiPMs saturate at high light intensities due to the
limited number of pixels, which also has to be measured. Therefore,
each channel contains a circuit for pulsing an integrated UV LED,
where the light amplitude can be controlled by an external voltage. A
fast trigger with a pulse width of 40\,ns and an analog bias voltage
of 0\,V - 10\,V is provided by the corresponding detector interface
module (middle interface board in Fig.~\ref{fig:slab}). For the gain
calibration low light intensities are used to extract the gain from
the distances of individual peaks in a single-pixel-spectrum, while
for higher light intensities (corresponding to $\sim$100
minimum-ionizing particles) the SiPMs saturate. Figure~\ref{fig:LED}
shows a circuit diagram of the LED driver. The middle bias capacitor
C$_1=150$\,pF is charged and upon an LED trigger signal, that opens
the transistor, C$_1$ is discharged by a current flowing through the
LED. The resistor R$_1=50\,\Omega$ guarantees a fast fall time of the
optical LED pulse. Pulse lengths of about 10\,ns have been measured
for a large range of amplitudes, which is needed for a good quality of
the single-pixel-spectra. In order to minimize the number of
calibration runs in test beam operations, it is important to have a
reasonably uniform light output for a large number of channels. First
tests are ongoing to investigate, if this can be achieved by closing
solder jumpers to add the bias capacitors C$_2=22$\,pF and
C$_3=82$\,pF to the default capacitor C$_1$. More details of the
driver circuit can be found in~\cite{EPT}. An alternative concept is
also studied that is based on few strong LEDs on special interface
boards, where the light is distributed via notched fibers~\cite{Jiri}.
As shown in Fig.~\ref{fig:slab}, each slab is connected to a data
acquisition system (DAQ). The current setup comprises a Central
Interface Board that hosts the Detector Interface (left), the steering
board for the calibration system (middle) and the power module
(right), that distributes all voltages needed in the slab. More
details about the concept of the DAQ are reported in~\cite{EPT} and
references therein.
\section{Measurements and results} \label{sec:results}
The main goal of the current tests of the prototype modules is the
commissioning of the design concept for a multi-channel prototype. On
one side, this requires tests of the functionality and performance of
all subcomponents in the laboratory as well as in a test beam
environment. This has been done extensively in the past
(see~\cite{EPT} and references therein) and is still ongoing. Results
of recent tests are reported in the following. On the other side,
system aspects have to be considered and the performance of larger
setups as shown in Fig.~\ref{fig:slab} has to be tested. This will be
possible with the next, final stage of the prototype operation, when a
complete layer with up to 18 HBUs (72 ASICs) can be operated and read
out.
\subsection{Time measurement} \label{sec:TDC}
\begin{figure
\centering
\subfigure[] {\includegraphics[width=2.4in, height=1.5in]{figures/TDC_principle.eps}\label{fig:TDC}}
\hspace{1.5cm}
\subfigure[] {\includegraphics[width=2.4in]{figures/LCramp.eps}\label{fig:LCramp}}
\caption{(a) Schematic working principle of the dual slope TDC. Reset
effects are shown in red. (b) TDC ramp in LC mode.}
\end{figure}
In addition to the energy measurement of the shower, it is useful to
measure also the arrival time of a signal in each cell. With this
information one can distinguish between prompt and delayed shower
components, which can be used e.g. for the identification of late
neutrons. This discrimination improves the performance of the particle
flow algorithm. The SPIROC2b ASIC has the capability of measuring the
arrival time of a signal relative to the bunch clock (5\,MHz at a LC)
with a 12-bit Wilkinson TDC. The voltage ramp is started with the
rising edge of the clock and is reset with the next rising clock edge.
Since the reset of the ramp introduces dead time, two ramps have been
implemented. The first ramp is only active on even clock cycles, while
the second ramp is only active on odd clock cycles. A multiplexer
switches between these two ramps (which again may introduce some dead
time). The working principle of the time measurement is depicted in
Fig.~\ref{fig:TDC}.
There are two possible operation modes foreseen in the current HBU
design. In the LC mode the ramp has a length of 200\,ns (to match the
bunch structure of the machine), while in test beam mode the ramp has
a length of about 5\,$\mu$s (in order to reduce dead time due to
multiplexing). These values can be further adapted to the test beam
needs in the DAQ firmware. To exploit the full dynamic range of the
TDC in case of a change of the ramp length, the slope of the ramp can
also be changed by SPIROC2b bias points. The physics goal of the
performance of the time measurement is to achieve a resolution of
about 1\,ns - 3\,ns in test beam mode to be able to distinguish
between prompt and late shower components. In order to test the
performance of the TDC, a signal was injected into one input channel
and the ramp was measured with a stepwise delay of the injected signal
with respect to the SPIROC2b clock. Figure~\ref{fig:LCramp} shows a
typical TDC ramp measured in LC mode. The two ramps have slightly
different peak values, while the dead time between the maximum of one
ramp and the start of the other ramp is caused by the multiplexer.
Both issues will be improved in the next generation of the ASIC. The
resolution is determined to be $\sim$300\,ps. This can be improved in
the future by optimizing the ramp slope and exploiting the full
dynamic range of the TDC. The measurement also shows that the use of
the LC mode in a test beam environment is not ideal, since about 50\%
of a clock cycle would be dead time. Therefore, the clock period is
tuned to about 5\,$\mu$s. This compromises the resolution, since the
slope of the TDC ramp has to be chosen less steep. The measured ramp
in test beam mode provides a resolution of about 3\,ns. This can also
be improved by tuning the TDC ramp as well as optimizing the clock
period, such that a final resolution of 1\,ns will be achievable in
test beam mode.
\subsection{Light amplitude equalization of LED system and SiPM saturation}
\begin{figure
\centering
\subfigure[] {\includegraphics[width=2.3in]{figures/LED_threshold.eps}\label{fig:LED_threshold}}
\hspace{1.5cm}
\subfigure[] {\includegraphics[width=2.3in]{figures/SiPM_saturation.eps}\label{fig:SiPM_saturation}}
\caption{(a) Dependence of the LED bias voltage where the LED starts
to emit light from the bias capacitance. (b) SiPM saturation
measured with the integrated LED calibration system.}
\end{figure}
The main goal of the LED calibration system is to measure the gain of
each SiPM by pulsing LED light of low intensity into the tiles. Since
this has to be achieved in as few calibration runs (or with as few
different LED bias voltages) as possible, the light output of the LEDs
have to be equalized, as discussed in Sec.~\ref{sec:status}. The goal
is to find a suitable combination and optimal values of the capacitors
C$_1$, C$_2$ and C$_3$ for each tile, such that the range of the LED
bias voltage that is needed for the LED to produce photons is as small
as possible. Each possible combination of capacitors has been used to
measure the LED light output as a function of the bias voltage for a
small set of tiles. Figure~\ref{fig:LED_threshold} shows the bias
voltage, where the measured LED light amplitude exceeds a small
threshold (starts flashing), as a function of the total capacitance
(sum of the nominal values of the capacitors). As expected, the
threshold decreases as the capacitance increases. The range of the
bias voltage can already been decreased with this method.
Nevertheless, further optimization of the capacity values is needed
and further studies will be done in order to optimize the calibration
procedure.
The second task of the LED system is to measure the saturation of the
SiPMs. Figure~\ref{fig:SiPM_saturation} shows some typical saturation
curves for different channels measured with LED light in the low gain
mode of the ASIC. Note that the visible saturation is not due to ADC
saturation, since the dynamic range of 12-bit is not fully exploited.
\subsection{MIP and light yield test beam measurement}
\begin{wrapfigure}{r}{0.4\columnwidth}
\centerline{\includegraphics[width=0.35\columnwidth]{figures/Testbeam_Setup.eps}}
\caption{Setup of the light-tight HBU cassette as it is mounted on a
movable stage at the DESY test beam facility. On top of the cassette
the detector interface modules are visible.}
\label{fig:TB}
\end{wrapfigure}
One HBU that is equipped with scintillator tiles is currently under
test at the DESY test beam facility. Electrons with an energy of
2\,GeV are used to investigate the response of the system to MIPs. For
this purpose the HBU is enclosed into a light-tight aluminum cassette
and mounted on a movable stage in order to scan all channels (see
Fig.~\ref{fig:TB}). The MIP signals are measured in auto-trigger mode,
where the threshold is optimized for each channel to measure a full
MIP spectrum, while suppressing most of the SiPM noise. Since in
SPIROC2b the triggers of all channels are connected with a logical OR,
the preamplifiers of all channels that are not used are switched off.
The measurements have been done in high gain mode with a preamplifier
feedback capacitance of 100\,fF and a shaping time of 50\,ns.
Figure~\ref{fig:MIP} shows a typical MIP spectrum. The measured ADC
value from the front-end electronics is converted into a pixel number
by using the measured gain from single-pixel-spectra taken with LED
light. The most probable value is around 15 pixels. Also the trigger
efficiency curve can be observed at the beginning of the spectrum. The
spectrum is fitted with a Landau function convolved with a Gaussian
function to determine the exact position of the most probable value
(light yield). The distribution of the light yield of all investigated
tiles is shown in Fig.~\ref{fig:LY}. The mean value is at 15 pixels,
which is the design value for the used tiles.
\begin{figure
\centering
\subfigure[] {\includegraphics[width=2.3in]{figures/MIP.eps}\label{fig:MIP}}
\hspace{1.5cm}
\subfigure[] {\includegraphics[width=2.3in]{figures/LY.eps}\label{fig:LY}}
\caption{(a) Typical MIP spectrum obtained from a 2\,GeV electron beam
and (b) light yield distribution measured with the latest HBU
version at the DESY test beam.}
\end{figure}
\section{Summary and outlook}
A new engineering prototype for an analog hadron calorimeter is
currently being developed by the CALICE collaboration. The goal is to
show that a realistic LC detector with fully integrated front-end
electronics can be built. The main challenge for the near future is to
construct a full LC detector layer to test the signal integrity and
the concept of power pulsing. Furthermore, a layer for operation in a
hadron test beam environment will be built and used for measuring the
time evolution of hadronic showers.
In order to achieve these goals, the calorimeter base units are tested
in the laboratory as well as in the DESY test beam to characterize
their features and to test the functionality and performance of all
subcomponents as well as overall system aspects. Some of the recent
measurement results are shown in this report, including the time
measurement performance of the ASIC, some aspects of the LED
calibration system and latest test beam measurements of the light
yield of the new scintillator tiles.
The next important steps are the construction of multi-HBU setups to
further investigate the integrated read-out electronics and the
development of a scalable data acquisition system.
\section*{Acknowledgments}
The authors gratefully thank Karsten Gadow, Erika Garutti, Peter
G\"ottlicher, Benjamin Hermberg, Mathias Reinecke, Julian Sauer, Felix
Sefkow and Sebastian Weber for very useful discussions and valuable
contributions to the results presented here.
\begin{footnotesize}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 2,673
|
{"url":"https:\/\/www.semanticscholar.org\/paper\/Fourier-coefficients-of-minimal-and-next-to-minimal-Gourevitch-Gustafsson\/6634dcc03edbbaeb6dcab16a6889d659947b5416","text":"# Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups\n\n@article{Gourevitch2019FourierCO,\ntitle={Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups},\nauthor={D. Gourevitch and H. Gustafsson and A. Kleinschmidt and D. Persson and S. Sahi},\njournal={arXiv: Number Theory},\nyear={2019}\n}\n\u2022 D. Gourevitch, +2 authors S. Sahi\n\u2022 Published 2019\n\u2022 Mathematics, Physics\n\u2022 arXiv: Number Theory\n\u2022 In this paper we analyze Fourier coefficients of automorphic forms on adelic split simply-laced reductive groups $G(\\mathbb{A})$. Let $\\pi$ be a minimal or next-to-minimal automorphic representation of $G(\\mathbb{A})$. We prove that any $\\eta\\in \\pi$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $GL_n$. We also\u2026\u00a0CONTINUE READING","date":"2020-10-29 05:53:33","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\": 0.9016939997673035, \"perplexity\": 1763.790024292159}, \"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-2020-45\/segments\/1603107902745.75\/warc\/CC-MAIN-20201029040021-20201029070021-00052.warc.gz\"}"}
| null | null |
\section{Introduction}
We are given a closed connected $n$-dimensional manifold $M$ with a non-zero de Rham cohomology class $u\in H^1(M;\mathbb{R})$.
This class is represented by closed
differential 1-forms of type Morse, meaning that their zeroes are non-degenerate.
The set of Morse 1-forms in the class $u$ is denoted by $\mathcal{F}_u$\,. For a Morse 1-form $\alpha$, the set of its
zeroes will be denoted $Z(\alpha)$; each zero $p\in Z(\alpha)$ has a Morse index
$i(p)$. Along a path in $\mathcal{F}_u$, the zeroes can be
followed continuously and their respective indices are constant.
As in Morse theory, it is important to equip each $\alpha\in \mathcal{F}_u$ with a {\it descending pseudo-gradient }
$X$ which is said to be {\it adapted} to $\alpha$ (see Definition \ref{df:Adapted}). The set of equipped Morse 1-forms $(\alpha, X)$ with
$\alpha\in \mathcal{F}_u$ is denoted by $\tilde\mathcal{F}_u$\,. The Morse-Novikov theory is devoted to the understanding
of this space.
With each zero $p$ of $\alpha$ in $M$, there are associated a stable and an unstable manifold. These are
respectively denoted by $W^s(p, X)$ and $W^u(p, X)$. Generically,
$X$ has the property
that all these invariant manifolds when
$p$ runs in $Z(\alpha)$ are mutually transverse. In that case, the pair $(\alpha,X)$ will be called {\it Morse-Smale}\footnote{In the literature,
a Morse-Smale vector field fulfills the above condition not only with respect to its set of zeroes but also to its set of periodic orbits.}.
In that case, if an orbit of $X$
is a {\it connecting orbit} going from $p$ to $q$ then $i(p)>i(q)$.
The $\alpha$-{\it length} of a connecting orbit $\ell$, defined by
\begin{equation}\label{eq:LengthCO}
\mathcal{L}(\ell):= -\int_\ell \alpha,
\end{equation}
must be positive. Moreover, a Morse-Smale pair $(\alpha,X)$ yields the so called \emph{Morse-Novikov}
differential $\partial^X$ that counts { the} \emph{incidences} $\scal{p}{q}^X$ between { the}
zeroes of $\alpha$ whose indices verify $i(q)=i(p) -1$ (see Subsection \ref{ssec:Groupoid}). The formulation we are going to use for this complex is due to from Jean-Claude Sikorav in his thesis \cite{sikorav}.\\
In a generic one-parameter family $(X_s)_{s\in I}$ of vector fields adapted to a given $\alpha\in\mathcal{F}_u$, the pair
$(\alpha, X_s)$ is Morse-Smale
for every $s$ outside $ \parent{A\sqcup B}\subset I$, where $A\sqcup B$ is a countable set of {\it bifurcation times}.
To be more precise, for each
$s\in A\sqcup B$ there is only one connecting orbit $\ell$
along which the transversality condition
$W^u(p, X_s)\pitchfork W^s(q,X_s)$ fails
and, for each $L>0$, there are finite subsets
$A_L\subset A$ and $B_L\subset B$ such that, for $s\in A_L\sqcup B_L$,
the non-transverse connecting orbit of $X_s$ has $\alpha$-length less than $L$. The set $A$ -- which is beyond the purposes of the present paper -- consists of \emph{annihilation times} where $i(q)=i(p)-1$
and, at each point of the connecting orbit
$\ell=W^u(p,X_s)\cap W^s(q,X_s)$, there is a two-dimensional plane
tangent to both $W^u(p,X_s)$ and $W^s(q,X_s)$.
We are concerned with \emph{slide times} $s\in B$, where the loss of transversality comes from the fact that $i(p)=i(q)$; in this case, the mentioned connecting orbit $\ell$ is called a {\it slide orbit}.
Remark that
at a slide time, we can have $p=q$ in which case $\ell$ is a {\it self-slide orbit}, classically called {\it homoclinic orbit}.\\
Given a pseudo-gradient $X$ of $\alpha\in \mathcal{F}_u$, a {\it homoclinic orbit}
is a connecting orbit $\ell $ from $p$ to itself, where $p$ is a zero of $\alpha$. This was also considered
by { M. Hutchings}
\cite{hutchings} (see also \cite{moraga}). This orbit makes a loop in $M$
based at $p$ (never smooth there).
In order to avoid dependency on base points, we introduce the \emph{fundamental groupoid} $\Pi$ of $M$ (see Definition \ref{fund_gr}).
Denote by $\Pi_{\lo}\subset \Pi$ the subgroupoid of homotopy classes of based loops in $M$.
This is a topological groupoid and there is a
continuous evaluation map $\Pi_{\lo}\to M$ { which maps each based loop to its base point;
its} fibre is discrete.
The homoclinic orbit $\ell$ based in $p$ has a homotopy class
$g:=[\ell]\in\Pi_{\lo} $ which will be called the {\it $\Pi$-value} of $\ell$.
The canonical isomorphism $H^1(M;\mathbb{R})\cong\Hom\parent{\pi_1(M,p),\mathbb{R}}$ yields
the morphism $g\in \pi_1(M,p)\mapsto u(g)\in\mathbb{R}$ and, if $g$ is the $\Pi$-value of $\ell$,
we have:
\begin{equation}\label{eq:Length}
u(g)=-\mathcal{L}(\ell).
\end{equation}
As a consequence,
$\ell$ is not homotopic to zero.
In what follows, the form $\alpha$ will be kept fixed and we denote by
$\mathcal{F}_\alpha$ the fibre of the forgetful map $\tilde\mathcal{F}_u\to \mathcal{F}_u$.
An element of $\mathcal{F}_\alpha$ will be denoted by $(\alpha,X)$ or by $X$ only if $\alpha$ is implied by the context.
For $g\in \Pi_{\lo}$, we consider the ``stratum'' $\mathcal{S}_g \subset \mathcal{F}_\alpha$ made of elements
$(\alpha, X) $ where $X$ has exactly one homoclinic orbit $\ell$ whose $\Pi$-value is $g$,
the field $X$ being allowed to have another slide orbits.
An element of $\mathcal{S}_g$ will be called a {\it self-slide}.
The stratum $\mathcal{S}_g$ may have several connected components but, by definition of $\Pi_{\lo}$,
the element $g$ indicates which
zero $p$ of $\alpha$ is involved in the self-slide. In particular, with $\mathcal{S}_g$ it
is associated a Morse index which is surely distinct from 0 and $n$ since these indices do not allow
homoclininic orbits.
Generically in $\mathcal{S}_g$, a self-slide possesses a natural {\it character}
which is an element in $\{+,-\}$ and which will be defined later (see Definition
\ref{def:Character}).
We define $\mathcal{S}^+_g$ (resp. $\mathcal{S}^-_g$) to be the open set in $\mathcal{S}_g$ made of elements
having the mentioned character. Finally, $\mathcal{S}^0_g$ denotes the self-slides for which the character
is not defined.
\begin{thm}\label{thm1}
For every $g\in \Pi_{\lo}$ verifying $u(g)<0$, the following holds true:
\begin{enumerate}
\item
The stratum $\mathcal{S}_g$ is a
codimension-one, co-oriented stratum of $\mathcal{F}_\alpha$ of class $C^\infty$.
\item Assume $n>2$.
Then, the stratum $\mathcal{S}_g^0$ is a non-empty
codimension-one, co-oriented sub-stratum of class $C^\infty$ in $\mathcal{S}_g$. Moreover, $\mathcal{S}_g^0$
meets every connected component of $\mathcal{S}_g$.
\end{enumerate}
\end{thm}
\begin{figure}[h]
\includegraphics[width=11cm, height=4cm]{Sg0dansSg-2.pdf}
\caption{Local situation on every connected component of $\mathcal{S}_g\subset\mathcal{F}_\alpha$ near $\mathcal{S}_g^0$.}
\label{fig:Sg0dsSg}
\end{figure}
Since $ \mathcal{S}_g$ is co-oriented, we can study the generic
one-parameter families $(\alpha, X_s)$ which intersect
$\mathcal{S}_g$ \emph{positively} at $(\alpha, X_0)$. Let $q\in Z(\alpha)$ be any zero whose Morse index equals $i(p)-1$.
When passing from $s<0$ to $s>0$,
the incidence $\scal{p}{q}^{X_s}$ gets multiplied by an element
$\lambda$ of the Novikov ring which does not depend on $q$ and
that we call the \emph{self-slide factor}.
It turns out that $\lambda$ depends on the character of $(\alpha, X_0)$.
For clarity, we give a simplified version of Theorem \ref{thm:selfslideAlg} right below. This theorem tells us the precise manner in which the incidence changes. This makes more precise the statement of
\cite[Prop. 2.2.36]{moragaTesis} whose proof was inaccurate.
\begin{thm}\label{thm:selfslideSimplif}
The self-slide factor is given by the following formulas:
\begin{enumerate}
\item $\quad\lambda =1+g \quad$ if $(\alpha, X_0)\in\mathcal{S}_g^-$ (Polynomial type),
\item $\quad\lambda =1+ \sum_{j=1}^{\infty}g^j \ $ if $(\alpha, X_0)\in\mathcal{S}_g^+$ (Series type).
\end{enumerate}
\end{thm}
Of course, in order to keep the squared differential equal to zero there are similar formulas
for the change of the incidence $\scal{q}{p}^{X_s}$ when the Morse indices satisfy $i(q)=i(p)+1$.
The doubling phenomenon that has motivated our article
relates the strata $\mathcal{S}_g$ and $\mathcal{S}_{g^2}$.
If $\ga:\mathbb{S}^1\to \mathcal{F}_\alpha$
denotes a small generic loop going around
the codimension-two stratum $\mathcal{S}_g^0$,
this loop cannot
avoid
crossing $\mathcal{S}_{g^2}$:
\begin{thm}\label{thm:Doubling}
The stratum $\mathcal{S}^0_g$ lies in the closure of $\mathcal{S}_{g^2}$.
Furthermore, there exists a codimension-two stratum $\mathcal{S}_g^{0,0}$ of $\mathcal{S}_g$, contained in $\mathcal{S}_g^0$, such that
$\mathcal{S}_g^0\smallsetminus \mathcal{S}_g^{0,0}$ is a boundary of class $C^1$
of $\mathcal{S}_{g^2}$.
\end{thm}
The precise definition of $\mathcal{S}_g^{0,0}$ yields a decomposition
$\mathcal{S}_g^0\smallsetminus\mathcal{S}_g^{0,0}=\mathcal{S}_g^{0,-}\,\sqcup\,\mathcal{S}_g^{0,+}$ (see Definition \ref{decompositionS^0}).
Locally along $\mathcal{S}_g^0\smallsetminus\mathcal{S}_g^{0,0}$, the stratum $\mathcal{S}_{g^2}$ approaches from one side of
$\mathcal{S}_g$ only. As a matter of fact, $\mathcal{S}_{g^2}$ approaches $\mathcal{S}_g^{0,+}$ (resp. $\mathcal{S}_g^{0,-}$)
from the positive (resp. negative side) of the co-oriented stratum $\mathcal{S}_g$.
Moreover, only $\mathcal{S}_{g^2}^+$ (the positive part of $\mathcal{S}_{g^2}$)
{ approaches $\mathcal{S}_g^0$}.
A precise statement about the latter facts is Theorem \ref{thm:DoublingRefined}, that may be illustrated by the following figure:\\
\begin{figure}[h]
\includegraphics[width=11cm,height=5cm]{Sg0Detail-3.pdf}
\caption{The stratum $\mathcal{S}_g^0\smallsetminus\mathcal{S}_g^{0,0}$ as a boundary of $\mathcal{S}_{g^2}$.}\label{fig:Sg0Detail}
\end{figure}
\begin{remarques}${}$
\smallskip
\noindent 1) It easily seen that the total space of the
fibration $\tilde\mathcal{F}_u\to \mathcal{F}_u$
has also a stratification which is given in each fibre $\mathcal{F}_\alpha$ by the union of
$\mathcal{S}_g^0\subset \mathcal{S}_g\subset \mathcal{F}_\alpha$ where $g$ runs in $\Pi_{\lo}$.
As a stratified fibration, it is locally trivial but it could be globally non-trivial. Indeed, a connected component of a codimension-one stratum could intersect $\mathcal{F}_\alpha$ along some
components of $\mathcal{S}_g$ and $\mathcal{S}_{g'}$ where $g$ and $g'$ are freely homotopic and based in different points.\\
\noindent 2) Our doubling phenomenon evokes
the period doubling bifurcation, also
called Andronov-Hopf's bifurcation. In this aim, it would be good to know that crossing $\mathcal{S}_g$
creates (or destroys) a periodic orbit in the free homotopy class $g$. Shilnikov's theorem \cite{shilnikov}
deals with this question. Unfortunately, it is not applicable here because we are going to use
very non-generic Morse charts (only $+1$ and $-1$ as eigenvalues of the Hessian at critical points).
In counterpart,
such charts offer very nice advantages.\\
\end{remarques}
The motivation of our article comes from {\it pseudo-isotopy} theory
for non-exact closed 1-forms.
Suppose we are given $\alpha_0$ and $ \alpha_1$ in $\mathcal{F}_u$ without zeroes.
Taking a {\it generic path}\footnote{A path of 1-forms in the class $u$ is said to be generic if it avoids $\mathcal{F}_u$ finitely many times only.} $(\alpha_s)$
from $\alpha_0$ to $\alpha_1$ in the cohomology class $u$, that is in the closure of $\mathcal{F}_u$, is it possible
to deform $(\alpha_s)$ -- keeping $\sing{\alpha_0,\alpha_1}$ fixed --
to a path of closed forms with no zeroes? In the latter case, $\alpha_0$ and $\alpha_1$ are said to be {\it isotopic.}
One way to proceed is to equip the path $(\alpha_s)$ with a generic family $(X_s)$
to perform a parametric Morse-Novikov theory. This approach has already been taken
in the same
problem for functions on compact manifolds $N\times [0,1]$ (see \cite{h+w}).
The hope is to produce an algebraic object (such as a group\footnote{We search for the {\it Steinberg group} of $u$ viewed as a representation $\pi_1(M)\to \mathbb{R}$.}) where we could read geometrical
obstructions to isotopy. Bifurcation phenomena visible in 2-parameter families
should correspond to relations holding in this algebraic object.
A first algebraic candidate has already been proposed in \cite{moragaSendai}. Ten algebraic relations appear on \cite[Lemma 2.1]{moragaSendai}.
Theorems \ref{thm:selfslideAlg} and \ref{thm:DoublingRefined} of the present article
correspond to the geometric counterpart of some particular cases of one of the mentioned
relations.
\section{\sc Self-slide stratum, orientation and character}\label{sect:Sg}
We are focusing on self-slides despite some of the statements
hold true for other slides. In the discussion
below, we are given a pair $(\alpha,X)\in\tilde \mathcal{F}_u$ where $\alpha$ is
a closed 1-form on $M$ (in the cohomology class $u$)
equipped with an {\it adapted} vector field $X$ in the sense which is now specified.
\begin{defn}\label{df:Adapted}
For each zero $p$ of $X$ of index $i(p)=i$, Morse coordinates
near $p$ are coordinates where the form
$\alpha$ equals the differential of the standard quadratic form of index $i$
$$Q_i:= -x_1^2+\ldots-x_i^2+x_{i+1}^2+\ldots +x_n^2.
$$
A vector field $X$ is said to be a \emph{(negative) pseudo-gradient
adapted} to $\alpha$ if the two
next conditions are fulfilled:
\begin{itemize}
\item $\alpha(X)<0$ outside $Z(\alpha)$;
\item for every $p\in Z(\alpha)$ there are Morse coordinates about $p$ such that
$X$ coincides with the \emph{standard} descending gradient $X_{i}$
of $Q_i$, where $i=i(p)$, that is:\hfill\break
$X_{i}=2\sum_1^i x_k\partial_{x_k}-2\sum_{i+1}^n x_k\partial_{x_k}.$
\end{itemize}
Given $(\alpha, X)\in\tilde\mathcal{F}_u$ and $p\in Z(\alpha)$, Morse coordinates about $p$
are said to be \emph{adapted} to $(\alpha, X)$ if $X=X_i$ in these coordinates.
It will also said that $X$ is \emph{adapted} to these Morse coordinates.
\end{defn}
\begin{remarque}\label{alexander}
The natural action of $G:=O(i)\times O(n-i)$ on $\mathbb{R}^n$ keeps the pair $(Q_i,X_i)$
invariant and $G$ is the subgroup of $O(i,n-i)$ preserving $X_i$.
Actually, the simplicial group $\mathcal{G}:=\text{Diff}(Q_i,X_i)$ of germs of diffeomorphisms of
$(\mathbb{R}^n,0)$ preserving the pair $(Q_i,X_i)$ retracts by deformation to $G$. Indeed, if
$\varphi\in \mathcal{G}$, the Alexander isotopy $\varphi_t: x\mapsto \frac 1t\varphi(tx)$ is made of elements in $\mathcal{G}$
for every $t\in (0,1]$ and tends to the derivative $\varphi'(0)x$ as $t $ goes to 0.
As a consequence, given $(\alpha, X)\in\tilde\mathcal{F}_u$ and $p\in Z(\alpha)$, the set, modulo G,
of Morse coordinates about $p$ which $X$ is adapted to
is contractible.
\end{remarque}
\subsection{\sc Morse model.} \label{ssec:morse}
Given $p\in Z(\alpha)$ of Morse index $i$,
a {\it Morse model} $\mathcal{M}_p$ with positive parameters $(\delta,\delta^*)$ (which we do not
make explicit in the notation)
is diffeomorphic to the subset of
$\mathbb{R}^i\times\mathbb{R}^{n-i}$ made of pairs $(x^-,x^+)$ such that $Q_i(x^-,x^+)\in [-\delta^*,+\delta^*]$,
$\vert x^-\vert^2\vert x^+\vert^2\leq \delta\de^*$ and $\alpha\vert_{\mathcal{M}_p}= dQ_i$. The bottom of $\mathcal{M}_p$, that is its intersection with $\{Q_i=-\delta^*\}$
is denoted by $\partial^-\mathcal{M}_p$; { similarly,
the top is denoted by} $\partial^+\mathcal{M}_p$.
The rest of the boundary of $\mathcal{M}_p$ is
denoted by $\partial^\ell \mathcal{M}_p$ and $X_i$ is tangent to it. Notice that:
\begin{itemize}
\item the group $G$ preserves $\mathcal{M}_p$ for every parameters $(\delta,\delta^*)$;
\item the set of Morse models, as compact subsets of $\mathbb{R}^n$, is contractible.
\end{itemize}
The flow of $X_i$ is denoted by $(X_i^t)_{t\in \mathbb{R}}$.
The {\it local unstable} (resp. {\it local stable}) manifold is formed by the points
$x\in \mathcal{M}_p$ whose negative (resp. positive) flow line
$X_i^t(x)$ goes to $p$ when $t$ goes to $-\infty$ (resp. $+\infty$) without getting out of $\mathcal{M}_p$.
Denote by $\Sigma^-$ the $(i-1)$-sphere which is formed by the points in the bottom of $\mathcal{M}_p$
which belong to $W_{loc}^u(p,X_i)$; it is called the {\it attaching sphere}.
Similarly, $\Sigma^+$ denotes the $(n-i-1)$-sphere which is contained
both in the top of $\mathcal{M}_p$ and made of points belonging to $W_{loc}^s(p,X_i)$; we call it
the {\it co-sphere}. We will use the the two projections associated with these coordinates:
\begin{equation}
\pi^+:\partial^+\mathcal{M}_p\to \Sigma^+ \quad\text{and} \quad \pi^-:\partial^-\mathcal{M}_p\to \Sigma^-.
\end{equation}
\subsection{\sc Tube and orientation.} \label{ssec:tube}
Let $\alpha$ be a Morse closed 1-form on $M$, let $X$ be an adapted pseudo-gradient
and
$\ell$ be a homoclinic orbit of $X$, based at $p\in Z(\alpha)$.
Denote by $\underline\ell$ the closure of $\ell\smallsetminus \mathcal{M}_p$; it will be named the
{\it restricted} homoclinic orbit. Its end points are denoted respectively $a^-\in \Sigma^-$
and $a^+\in \Sigma^+$. We also introduce a {\it compact tube}
$T$ around $\underline \ell$ made of $X$-trajectories
from $\partial^-\mathcal{M}_p$ to $\partial^+\mathcal{M}_p$.
Up to time rescaling, we may suppose $X^1(a^-)= a^+$. On $T$,
there are coordinates
$(x, y, v, z)\in \mathbb{R}^{i-1}\times \mathbb{R}^{n-i-1}\times[-1,1]\times[0,1]$ with the following properties:
\begin{itemize}
\item $X$ { is positively colinear to } $\partial_z$,
\item $\{z=0\}= T\cap \partial^-\mathcal{M}_p$ and $\{z=1\}= T\cap \partial^+\mathcal{M}_p$;
\item $T\cap\Sigma^-=\{y=0, v=0, z=0\}$ and $T\cap \Sigma^+=\{x=0, v=0, z=1\}$;
\item $\underline\ell=\{x=0,y=0,v=0\}$;
\item $\partial_v$ is tangent to the leaves of $\alpha$.
\end{itemize}
We choose the base $\{z=0\}$ of $T$ to be a closed polydisc.\\
Orient the unstable $W^u(p, X)$.
Thus, the stable manifold is co-oriented. Therefore, we can choose the coordinate $v$ in the tube
{ so that,
for every $z_0\in [0,1]$, the following holds:}
\begin{equation}\label{orientation1}
\partial_v\wedge \text{or}\left(W^u(p, X)\cap\{z=z_0\}\right)= \text{co-or}(W^s(p,X)).
\end{equation}
If the orientation of $W^u(p, X)$ is changed, then the co-orientation of $W^s(p,X)$
is also changed and the above equation shows that the positive direction
of $v$ remains unchanged.
\begin{remarque}\label{rem:holonomie} It is important to notice that (\ref{orientation1})
tells us nothing about the holonomy along $\underline\ell$ of the foliation defined by $X$
(see the next subsection).
Therefore, for a given $\partial_v\in T_{a^+}(\partial^+\mathcal{M}_p)$, the tangent vector
$\partial_v\in T_{a^-}(\partial^-\mathcal{M}_p)$ may have any position not contained in the hyperplane $\mathbb{R}\sing{\partial_x,\partial_y}$,
depending on $X$.\\
\end{remarque}
\subsection{\sc Holonomy and perturbed holonomy.}\label{perturbed}
Since $T$ is compact, there exists some open neighborhood $\mathcal N^-_X\subset \partial^-\mathcal{M}_p $
(depending on $X$)
of $\{z=0\}=T\cap\partial^-\mathcal{M}_p$ on which the {\it holonomy}
along $\underline\ell$ is defined from the transversal $\partial^-\mathcal{M}_p$
to the transversal $\partial^+\mathcal{M}_p$, that is:
if $\ga(u), \ u\in[0,1]$ is a path in $\mathcal N^-_X$ starting from $a^-$,
the solutions $x_u(t)$ of the differential equation $\dot x= X(x)$ with initial condition $x_u(0)=\ga(u)$
cross $\partial^+\mathcal{M}_p$ at a time $\theta(u)$ depending smoothly
on $u$. In particular, $\theta(0)=1$ and $x_0(\theta(0))=a^+$.
In this setting, the map $\ga(1)\mapsto x_1(\theta(1))$ defines a diffeomorphism $H_X$
from $\mathcal N^-_X$ to an open set $\mathcal N^+_X\subset \partial^+\mathcal{M}_p$ which contains $\{z=1\}$.
The existence of such holonomy diffeomorphism is an open property with respect to $X$.
More precisely, if $\widetilde X$ is a close enough approximation of $X$ in the $C^1$-topology,
there is a {\it perturbed holonomy} diffeomorphism
$H_{\widetilde X}$ from an open neighborhood
$ \mathcal N^-_{\widetilde X}$ in $\partial ^-\mathcal{M}_p$ of $\{z=0\}$ in $\partial^-\mathcal{M}_p$
to an open neighborhood $\mathcal N^+_{\widetilde X}$ of $\{z=1\}$ in $\partial^+\mathcal{M}_p$.
\begin{remarque} \label{batons}
According to this property of the perturbed holonomy,
it makes sense to speak of $H_{\widetilde X}^{-1}(\Sigma^+)\cap \{z=0\} $. It is an $(n-i-1)$-disc
close to the $y$-axis in $\{z=0\}$.
Similarly, it makes sense to speak of $H_{\widetilde X}(\Sigma^-)\cap \{z=1\} $. It is an $(i-1)$-disc
close to the $x$-axis in $\{z=1\}$.
\end{remarque}
We now state and prove Item (1) of Theorem \ref {thm1}.
\begin{prop} \label{item1}For every $g\in \Pi_{\rm loop}$ with $u(g)<0$,
the stratum $\mathcal{S}_g$ is a $C^\infty$ codimension-one stratum of $\mathcal{F}_\alpha$.
This stratum has a canonical co-orientation.
\end{prop}
\nd {\bf Proof.\ }
Let $ X_0$ be any point in $\mathcal{S}_g$. Let $\ell$ denote
the homoclinic orbit of $\Pi$-value $g$ and let
$p$ be the involved zero of $\alpha$. We intend to find
a regular real valued equation for $\mathcal{S}_g$ near $ X_0$.
We recall:
\begin{itemize}
\item the local stability
of the adapted vector fields near $p$;
\item the acyclicity of the space of Morse models adapted to $X_0$ near $p$;
near $\{z=0\}\cup \{z=1\}$.
\end{itemize}
Therefore, the action of the group $\text{Diff} (M)$ reduces us to consider a {\it slice}
$S\subset \mathcal{F}_\alpha$ such that every $X\in S$ near $X_0$
satisfies the following: $X=X_0$ in a given Morse model $\mathcal{M}_p$.
We use the tube $T$ and its coordinates as introduced in Subsection \ref{ssec:tube}.
The Implicit Function Theorem allows us to follow continuously, for $X$ close to $X_0$,
a connected component $D(X)$ of $W^u(p,X)\cap T\cap \{z=1\}$ which coincides with
$\{y=0,v=0, z=1\}$ when $X= X_0$. Let
\begin{equation}
p_v:\sing{z=1}\to\sing{ v=0, z=1}
\end{equation}
denote the projection parallel to $\partial_v$ onto the $(x,y)$-space.
The image $p_v(D(X))$
is transverse to $\Sigma^+$.
The intersection is a point $a^+(X)$ which depends $C^\infty $ on $X$. Let $b(X)$
be the point of $D(X)$ which has the same coordinates as $a^+(X)$ except the last coordinate $v$.
Thus, the wanted equation is
\begin{equation}
v(b(X))=0\,.
\end{equation}
This is clearly a $C^\infty $ equation. For proving this equation is regular it is sufficient to exhibit
a one-parameter family $(X_s)$ passing through $X_0$ and satisfying the next inequality:
\begin{equation}\label{trans}
\partial_s\left( v\circ H_s(a^-)\right)_{\vert s=0}> 0\,,
\end{equation}
where
$H_s$ stands for the
perturbed holonomy diffeomorphism of $X_s$ (compare Lemma \ref{velocity}).
This is easy to perform by taking $X_s$ as $X_0$ plus a small vector field
$s\,g(x,y, z,v)\partial_v$
where the function $g$ is non-negative, supported in the interior of the tube $T$ and has a positive integral along $\underline\ell$\,.
After (\ref{orientation1}) we noticed that the positive sense of $v$ does not depend on
the chosen orientation of the unstable manifolds. Therefore,
(\ref{trans}) defines a canonical co-orientation of $\mathcal{S}_g$.
\hfill$\Box$\\
\begin{defn}\label{df:PositTrans}
Let $(X_s)_{s\in \mathcal{O} p(0)}$ be a one-parameter family\,\footnote{Following M. Gromov,
$\mathcal{O} p(0)$ stands for on open interval $(-\ep,\ep)$ as small as wanted.}
of vector fields adapted to $\alpha$ and standard in $\mathcal{M}_p$. It is assumed $X_0\in \mathcal{S}_g$\,.
The family $ X_s$ is said to be
\emph{positively transverse} to the stratum $S_g$ if it satisfies \rm{(\ref{trans})}.
\end{defn}
Consider hence $(X_s)$ being positively transverse. Below, we use the coordinates $(x,y,v)$
of $T\cap\{z=0,1\}$.
According to Subsection \ref{perturbed}, for $s$ close to $0$ there exists an open tube
$T_s\supset T$ made of trajectories of $X_s$ going from $\partial^-\mathcal{M}_p$ to $\partial^+\mathcal{M}_p$. That defines
a {\it holonomy} local diffeomorphism $H_s$ from a neighborhood of $\{z=0\}$ in $\partial^-\mathcal{M}_p$
to a neighborhood of $\{z=1\}$ in $\partial^+\mathcal{M}_p$ and we have:
\begin{equation}\label{positive-trans}
v\left[H_s\bparent{T_s\cap \Sigma^-}\cap \{z=1, x=0\}\right]>0\,,\,\,\,\,\text{when } s>0\,
\end{equation}
A consequence of (\ref{positive-trans}) for the inverse flow is the following:
\begin{equation}\label{negative}
v\left[H_s^{-1}\bparent{T_s\cap\Sigma^+}\cap\{z=0, y=0\} \right]<0\,,\,\,\,\,\text{when } s>0\,
\end{equation}
Actually, the above inequalities are implied by the some relation
between velocities
which is stated below and
will be useful elsewhere. In relation to (\ref{positive-trans}),
we are interested in the local solution $x_s$ of the equation
\begin{equation}\label{x_s}
p_{y,v}H_s(x,0,0)=0
\end{equation}
which equals $0$ when $s=0$; here, $p_{y,v}$ stands for the projection parallel to the span of
$\{\partial_y, \partial_v\}$. And in relation to (\ref{negative}), we consider the solution $y_s$ of
the equation
\begin{equation}\label{s_s}
p_{x,v}H_s^{-1}(0,y,0)=0
\end{equation}
which equals $0$ when $s=0$; here, $p_{x,v}$ stands for the projection parallel to the span of
$\{\partial_x, \partial_v\}$.
\begin{lemme}\label{velocity} With the above data and notations,
the following equality holds:
\begin{equation}
\partial_s \left( v\circ H_s(x_s,0,0)\right)_{\vert s=0}+ \partial_s\left(v\circ H_s^{-1}(0,y_s,0)\right)_{\vert s=0}=0\,.
\end{equation}
In particular, if the first term equals $+1$ the second equals $-1$.
\end{lemme}
\nd {\bf Proof.\ } A Taylor expansion gives
$$
v\circ H_s(x_s,0,0)= v\circ H_s(0,0,0)+O(s^2)\,.
$$
That follows from the fact that the velocity of $x_s$ is a vector which is contained in the kernel
of $dv$. Similarly, we have:
$$
v\circ H_s^{-1}(0,y_s,0)= v\circ H_s^{-1}(0,0,0)+ O(s^2)\,.
$$
Observe that $H_0(x,y,v)= (x,y,v)$. Thus, derivating with respect to $s$ at $s=0$ the composed map $H_s^{-1}\circ H_s= Id$ gives:
$$\partial_s H_s^{-1}(0,0,0)_{\vert s=0}+\partial_s H_s(0,0,0)_{\vert s=0}=0\,.
$$
Altogether, we get the desired formula.\hfill$\Box$\\
\bigskip
\subsection{\sc Equators, signed hemispheres and
latitudes. \label{ssec:Equators}}
We introduce some useful notations. Let $\mathbb{D}^k$, { $ k\geq 1$,}
be the closed Euclidean disc of dimension $k$ and radius $\delta$ { (the first parameter of the Morse model)},
equipped with spherical coordinates $(r, \theta)\in [0,\sqrt\delta]\times \mathbb{S}^{k-1}$. Denote by
\begin{equation}\label{eq:Radial}
\partial_r^{\theta}\in T_0\mathbb{D}^k\,\text{ the unitary vector pointing towards }\theta\in\mathbb{S}^{k-1}.
\end{equation}
Suppose that we are given a preferred co-oriented hyperplane $\Delta\subset T_0\mathbb{D}^k$.
We obtain a \emph{preferred co-oriented equator} $E^\Delta\subset \mathbb{S}^{k-1}$.
The co-orientation of $E^\Delta$ points to an open
\emph{hemisphere} denoted by $\mathcal H^+(\mathbb{S}^{k-1})$;
the opposite
hemisphere is denoted by $\mathcal{H}^-(\mathbb{S}^{k-1})$.
We have two marked points { determined by the oriented normal to $\Delta$}:
a \emph{North pole} $\nu\in \mathcal{H}^+(\mathbb{S}^{k-1})$ and a \emph{South pole} $\sigma\in \mathcal{H}^-(\mathbb{S}^{k-1})$.
Any point $x\in\mathbb{S}^{k-1}$ determines an angle with respect to the North pole $\nu$. The cosinus of this angle defines a \emph{latitude} on $\mathbb{S}^{k-1}$, relative to $\Delta$, that we denote by:
\begin{equation}
\omega_\Delta:\mathbb{S}^{k-1}\to [-1,1].
\end{equation}
Clearly $\omega_\Delta^{-1}(1)=\sing{\nu}, \omega_\Delta^{-1}(-1)=\sing{\sigma}$ and $\omega_\Delta^{-1}(0)=E^\Delta$. Moreover, if we denote by
\begin{equation}
p_\Delta: T_0\mathbb{D}^k\to\mathbb{R}\partial_r^{\nu}
\end{equation}
the projection parallel to $\Delta$, the definition of cosinus reads
\begin{equation}\label{eq:Trigo}
p_\Delta(\partial_r^{\theta})=\omega_\Delta(\theta)\,\partial_r^{\nu}\, .
\end{equation}
\begin{prop}\label{prop:Latitudes}
Every $ X$ in $\mathcal{S}_g$ defines
a preferred latitude on
both the attaching sphere $\Sigma^-$ and the co-sphere $\Sigma^+$.
\end{prop}
For this aim, we use
radial-multispherical coordinates $(\phi,r,\psi)\in S^{i-1}\times [0,\sqrt\delta]\times S^{n-i-1}$ on
each level set of $\mathcal{M}_p$
and we recall the map
\begin{equation}\label{eq:Desc}
Desc:\partial^+\mathcal{M}_p\smallsetminus \Sigma^+\to \partial^-\mathcal{M}_p\smallsetminus \Sigma^-
\end{equation}
obtained by descending the flow lines in $\mathcal{M}_p$. This map reads $Id$ in these coordinates.
The preferred latitude which we are going to define
on $\Sigma^-$ and $\Sigma^+$ will be called respectively the \emph{$\phi$-latitude}
and the \emph{$\psi$-latitude}. We insist that these functions depend on $X\in \mathcal{S}_g$.
We denote them by
\begin{equation}\label{eq:Latitudes}
\omega_\phi^X:\Sigma^-\to [-1,1]\quad\text{and}\quad\omega_\psi^X:\Sigma^+\to [-1,1].
\end{equation}
When the vector field is understood, these functions will be denoted $\omega_\phi$ and $\omega_\psi$.
We shall decorate all the data
related to $\omega_\phi$ or $\omega_\psi$ by using
the letter $\phi$ or $\psi$ respectively;
namely, the preferred hyperplane $\Delta^\phi$, the preferred equator $ E^\phi\subset \Sigma^-$,
the poles $ \nu_\phi\in \mathcal{H}^+(\Sigma^-), \sigma_\phi\in \mathcal{H}^-(\Sigma^-)$ and so on.\\
\nd {\bf Proof.\ }
Take any $X$ in $\mathcal{S}_g$ and denote by $\ell$ its homoclinic
orbit from $p$ to itself. The end point $a^+$ of $\underline\ell$
has coordinates $a^+=( -,0,\psi_0)$; as usual with polar coordinates, when the radius is 0 the spherical coordinate is not defined.
Let
\begin{equation}\label{eq:PsiProj}
\pi^{\psi_0}: \partial^+\mathcal{M}_p \to\{\psi=\psi_0\},\quad \pi^{\psi_0}(\phi, r,\psi)= (\phi, r,\psi_0),
\end{equation}
be the projection onto the meridian $i$-disc.
{ Let $H: \{z=0\}\to \{z=1\}$ denote the holonomy diffeomorphism defined by the vector field $X$
in the tube $T$.
The image of $T\cap\Sigma^- $ through $H$
is a $(i-1)$-disc $D\subset \partial^+\mathcal{M}_p$.} Due to the transversality condition associated with $\ell$,
this disc is a graph over
its projection $D_{\psi_0}:=\pi^{\psi_0}(D)$ { if the tube is small enough around $\underline\ell$.}
Then,
\begin{equation}\label{eq:HypPhi}
\Delta^\phi:=T_{a^+}D_{\psi_0}\subset T_{a^+}\sing{\psi=\psi_0}\text{ is the preferred hyperplane}.
\end{equation}
As we noticed in Remark \ref{rem:holonomie}, the vector $\partial_v\in T_{a^+}\partial^+\mathcal{M}_p$ is neither tangent to $\Sigma^+$ nor to $D$, which implies that
\begin{equation} \label{positive_side}
\parent{d\pi^{\psi_0}}_{a^+}(\partial_v) \text{ defines { a co-orientation of }} \Delta^\phi { \text{ in }
T_{a^+}\sing{\psi=\psi_0} .}
\end{equation}
This provides a preferred latitude on the $\phi$-sphere $\partial\{\psi=\psi_0\}= S^{i-1}\times\{\sqrt\delta\}\times\{\psi_0\}$.
By the canonical isomorphism
\begin{equation}\label{eq:IdentPhi}
S^{i-1}\times \{\sqrt\delta\}\times\{\psi_0\} \cong S^{i-1}\times \{0\}\times\{-\} = \Sigma^-,
\end{equation}
the preferred latitude on $\partial\sing{\psi=\psi_0}$ descends to some $\phi$-latitude
which defines the claimed $\omega^X_{\phi}$
of \eqref{eq:Latitudes}.
For the $\psi$-latitude on the co-sphere $\Sigma^+$, we do the same construction by using
the reversed flow and its holonomy $H^{-1}$. More
precisely,
take the image of $T\cap \Sigma^+$ through
$H^{-1}$; it is a
$(n-i-1)$-disc $D'\subset \partial^-\mathcal{M}_p$ centered in $a^-$ whose spherical coordinates are $ a^- = (\phi_0, 0, -) $. Let
\begin{equation}\label{eq:PhiProj}
\pi^{\phi_0}: \partial^-\mathcal{M}_p \to\{\phi=\phi_0\},\quad \pi^{\phi_0}(\phi, r,\psi)= (\phi_0, r,\psi),
\end{equation}
be the projection onto the meridian $(n-i)$-disc
and let $D_{\phi_0}$ be the image $\pi^{\phi_0}(D')$.
Hence,
\begin{equation}\label{eq:HypPsi}
\Delta^\psi:=T_{a^-}D_{\phi_0}\subset T_{a^-}\sing{\phi=\phi_0} \text{ is the preferred hyperplane.}
\end{equation}
Moreover, $\partial_v\in T_{a^-}\partial^-\mathcal{M}_p$ cannot be neither tangent to $\Sigma^-$ nor to $D'$, which implies that
\begin{equation}\label{eq:PositSi+}
\parent{d\pi^{\phi_0}}_{a^-}(\partial_v)\text{ defines a co-orientation { of} } \Delta^{\psi} {
\text{ in } T_{a^-}\sing{\phi=\phi_0}.}
\end{equation}
This yields a preferred latitude on the sphere $\partial\sing{\phi=\phi_0}$, that can be carried to $\Sigma^+$
by means of
the canonical isomorphism
\begin{equation}\label{eq:IdentPsi}
\sing{\phi_0}\times\{\sqrt\delta\}\times S^{n-i-1}\cong \sing{-}\times\sing{0}\times S^{n-i-1} = \Sigma^+.
\end{equation}
This defines the claimed $\psi$-latitude $\omega_\psi^X$ of \eqref{eq:Latitudes}.
\hfill$\Box$\\
\subsection{\sc Holonomic factor
and character.}
By construction of the $\psi$-latitude,
the tangent space $T_{a^-}\sing{\phi=\phi_0}$ splits as $\mathbb{R}\partial_r^{\nu_{\psi}}\oplus \Delta^\psi$.
This together with the decomposition of $T_{a^+}\sing{\psi=\psi_0}$
{ defined }by the
$\phi$-latitude, we obtain the following decompositions:
\begin{equation}\label{eq:decompTgt}
\begin{cases}
T_{a^-}\sing{z=0} = T_{a^-}\Sigma^-\oplus\parent{\mathbb{R}\partial_r^{\nu_{\psi}}\oplus \Delta^\psi},\\
T_{a^+}\sing{z=1} = \parent{\Delta^\phi\oplus\mathbb{R}\partial_r^{\nu_{\phi}}}\oplus T_{a^+}\Sigma^+\, .
\end{cases}
\end{equation}
Given $(\alpha, X)\in\mathcal{S}_g$, { recall the homoclinic orbit $\ell$ whose $\Pi$-value is $g$ and all
associated objects that we
introduced in Subsection \ref{ssec:tube}: the tube $T$, its coordinates $(x,y,v,z)$
and the holonomy diffeomorphism $H: \{z= 0\} \to \{z= 1\}$. It reads $Id$
in the $(x,y,v)$-coordinates and $H(a^-)=a^+$.}
We are free to choose the coordinates of the tube such that the
unit tangent vector $\partial_v^1:=\partial_v\in T_{a^+}\sing{z=1}$ verifies
\begin{equation}\label{eq:ChoiceVAxis}
\partial_v^1=\partial_r^{\nu_{\phi}}.
\end{equation}
The \emph{linearized holonomy} $T_{a^+}H^{-1}$
sends $\partial_v^1$ onto $\partial_v^0:=\partial_v\in T_{a^-}\sing{z=0}$.
By \eqref{eq:decompTgt}, the latter vector
decomposes as
\begin{equation}\label{eq:HolonDec}
\partial_v^0=v_x+\bar{\eta}\partial_r^{\nu_\psi}+v_y,\text{ where }v_x\in T_{a^-}\Sigma^-, \, v_y\in \Delta^\psi,\, \bar{\eta}\in\mathbb{R}.
\end{equation}
As we pointed in Remark \ref{rem:holonomie}, the only restriction
on the holonomy of $X$ along $\underline{\ell}$ is that $\bar{\eta}\neq 0$.
Moreover, { according to \eqref{eq:PositSi+}, the vector $\partial_v^0$ defines the positive side
of the preferred hyperplane $\Delta^\psi$. As a consequence, }
$\bar{\eta}$ must be positive.
\begin{defn}\label{def:HolFactor}
The holonomic factor associated with $(\alpha, X)$ is the positive real number given by
\begin{equation}
\eta( X):=\,\frac{1}{\bar{\eta}}\,>\,0\, .
\end{equation}
\end{defn}
The next subsets of $\mathcal{S}_g$
are respectively called the \emph{$\phi$-axis} and the \emph{$\psi$-axis} of $\mathcal{S}_g$:
\begin{equation}\label{axis}
\mathcal{S}_g^{\phi}:=\ens{X\in\mathcal{S}_g}{a^+(X)\in E^\psi}\,\text{ and }\quad\mathcal{S}_g^{\psi}:=\ens{X\in\mathcal{S}_g}{a^-(X)\in E^\phi},
\end{equation}
that we also call the \emph{spherical axes}. Here, $a^\pm(X)$
stand for the respective extremities
of the restricted orbit $\underline\ell$, where $\ell$ is the unique homoclinic orbit of $X$ in the homotopy class $g$.
Denote the intersection of the axes:
\begin{equation}
\mathcal{S}_g^{0,0}:=\mathcal{S}_g^\phi\cap\mathcal{S}_g^\psi
\end{equation}
which is empty when one axis is so.
\begin{remarque} When the Morse index of $\mathcal{S}_g$ equals 1, then the $\phi$-equator is empty but there
are still signed poles. In that case, the $\phi$-latitude takes only the values $\{-1,+1\}$ and the $\psi$-axis
is empty. When the Morse index of $\mathcal{S}_g$ equals $n-1$, then the $\psi$-equator and the $\phi$-axis are empty and the $\psi$-latitude is valued in $\{-1,+1\}$. If $n>2$, these two events do not
happen simultaneously. This is the reason for the dimension assumption in Theorem \ref{thm1} (2).
\end{remarque}
We are now ready for defining the important notion of \emph{character}.
\begin{defn}\label{def:Character}
Let $\chi:\mathcal{S}_g\to\mathbb{R} $ be the \emph{character function} given by:
\begin{equation}\label{eq:CharVal}
\chi(X):=\eta( X)\,\omega_\psi^X(a^+(X))+\omega_\phi^X(a^-(X)).
\end{equation}
Let $\mathcal{S}_g^0:=\chi^{-1}\parent{\sing{0}}$. The \emph{character} of $(\alpha,X)$ is then defined as
the map
\begin{equation}
\ep:\mathcal{S}_g\smallsetminus\mathcal{S}_g^0\to\sing{+,-},
\end{equation}
given by the sign of $\chi$. Thus, $\mathcal{S}_g\smallsetminus\mathcal{S}_g^0$ divides into $\mathcal{S}_g^+\sqcup\mathcal{S}_g^-$.
\end{defn}
By the very definition of the latitudes,
it is clear that each axis intersects $\mathcal{S}_g^0$ along $\mathcal{S}_g^{0,0}
, as Figure \ref{fig:Sg0etAxes} suggests (compare to Figure \ref{fig:Sg0dsSg}).
\begin{figure}[h]
\includegraphics[width= 11cm, height=4cm]{Sg0etAxes.pdf}
\caption{The substratum $\mathcal{S}_g^{0,0}\subset \mathcal{S}_g^0$ as the intersection of the $\phi$-axis with the $\psi$-axis.} \label{fig:Sg0etAxes}
\end{figure}
Below, we start giving some information about $\mathcal{S}_g^{0,0}$ and prove Item (2) of Theorem \ref{thm1}.
\begin{prop}\label{item2}${}$
\noindent {\rm 1)} The axes $\mathcal{S}_g^\phi$ and $\mathcal{S}_g^\psi$ are
$C^\infty$ submanifolds of codimension 1 in $\mathcal{S}_g$. Moreover, they intersect transversely.
Hence, their intersection $\mathcal{S}_g^{0,0}$ is a $C^\infty$-submanifold of codimension 2 in $\mathcal{S}_g$.
\noindent {\rm 2)} The stratum $\mathcal{S}_g^0$ is a co-oriented
sub-stratum of codimension 1 of class $C^\infty$
in $\mathcal{S}_g$.
\end{prop}
\nd {\bf Proof.\ } ${}$\\
1) { First assume $1< i < n-1$, where $i$ denotes the Morse index of $\mathcal{S}_g$.
The equation of the { $\psi$-axis} $\mathcal{S}_g^\psi$ in $\mathcal{S}_g$ reads
with the notations introduced in (\ref{eq:Latitudes}) and (\ref{axis}):
$$
\omega_\phi^X(a^-(X))= 0.
$$
Let $X_0\in \mathcal{S}_g^\psi$. In order to prove
that the equation is regular, we have to exhibit a germ
$(X_s)_s$
of path in $\mathcal{S}_g$ passing through $X_0$ such that the $s$-derivative of
$ \omega_\phi^{X_s}(a^-(X_s))$ at $s=0$ is non-zero. For $s$ close to $0$, let $H_s$ be the local
holonomy diffeomorphism
of $X_s$ from a neighborhood of $\sing{z=0}$ in $\partial^-\mathcal{M}_p$ to a neighborhood of $\sing{z=1}$ in $\partial^+\mathcal{M}_p$.
Let $a^-=(\phi_0, 0,-)$ and $a^+=(-,0, \psi_0)$ be the end points of the restricted orbit $\underline\ell$.
We arrange that $H_s$ keeps the $\pi^{\psi_0}$-projection of $H_s(\Sigma^-)$
into the { meridian $\{\psi=\psi_0\}\subset \partial^+\mathcal{M}_p$ }
independent of $s$. Thus, the equator $E^\phi$ is so.
Therefore, we are reduced to
control the $s$-derivative of $\omega_\phi(a^-(X_s))$.
We recall that every germ of isotopy of the holonomy $H_0$ lifts to a deformation of $X_0$
(compare the solution of inequation (\ref{trans})). Then,
we are free to choose the holonomy so that $s\mapsto a^-(X_s)\in\Sigma^-$ crosses $E^\phi$
transversely at time $s=0$ and $a^+(X_s)$ moves in $\Sigma^+$ (in order to guarantee $X_s\in \mathcal{S}_g$).
Thus, we are done.
For a similar reason, the equation $\omega_\psi^X(a^+(X))= 0$
of the { $\phi$-axis} $\mathcal{S}_g^\phi$ is regular.
For showing the transversality { of the two axis}, we consider any $X_{0,0}\in \mathcal{S}_g^{0,0}$.
We choose a (germ of) 2-parameter family $X_{s,u}\in \mathcal{S}_g$ whose holonomy $H_{s,u}$
satisfies the next conditions:
\begin{enumerate}
\item the equator $E^\phi$ is independent of $s$ when $u=0$ and $\partial_s \omega_\phi(a^-(X_{s,0}))>0$;
\item the equator $E^\psi$ is independent of $u$ when $s=0$ and $\partial_u\omega_\psi(a^+(X_{0,u}))>0$:
\item for every $(s,u)$ close to $(0,0)$, we have $a^-(X_{s,u})\in \Sigma^-$ and $a^+(X_{s,u})\in \Sigma^+$.
\end{enumerate}
Condition (3) guarantees that $X_{s,u}$ runs in $\mathcal{S}_g$. Thanks to (1) and (2), the evaluation map
$(s,u)\mapsto (a^-(X_{s,u}), a^+(X_{s,u}))\in \Sigma^-\times\Sigma^+$ is transverse to the submanifold
$E^\phi\times E^\psi$. This proves the independence of $\partial_s X_{s,u}$ and $\partial_u X_{s,u}$
at $ (0,0)$.
Suppose now $1<i=n-1$.
The axis $\mathcal{S}_g^{\phi}$ is empty since $E^\psi=\V$.
We only need to prove the assertion about $\mathcal{S}_g^\psi$. The above proof is still correct by only
noticing that $a^+(X_s)$ remains constantly equal to $a^+\in \sing{(-,0,\nu_\psi),(-,0,\sigma_\psi)}$ and that
$E^\phi$ is a $(n-3)$-sphere inside $\Sigma^-$. The case $i=1<n-1$ is treated analogously.
If $n=2$,
both axes are empty and there is nothing to prove.\\
2) By Definition \ref{def:Character}, the equation of $\mathcal{S}_g^0$ in $\mathcal{S}_g$ reads
$$\chi(X):= \eta(X)\omega_\psi^X(a^+(X))+\omega_\phi^X(a^-(X))=0
$$
For proving that the equation is regular in any $X_0\in \mathcal{S}_g^0$ whe have to exhibit
a germ of path $(X_s)_s$ in $\mathcal{S}_g$ passing through $X_0$ such that
$\partial_s\chi(X_s)>0$.
First, we arrange that the equators $E^\phi $ and $E^\psi$ do not depend on $s$
by requiring that the holonomy $H_s$ along the self-connecting orbit $\ell$ of $X_0$
fulfils the next conditions:
\begin{itemize}
\item for every $s$, there is a self-connecting orbit $\ell_s$ (the end points of the restricted orbit $
\underline\ell_s$ are noted $a^-(X_s)$ and $a^+(X_s)$);
\item the $\pi^{\psi_0}$-projection of $H_s(\Sigma^-)$
into the meridian $\{\psi=\psi_0\}$ is constant;
\item the $\pi^{\phi_0}$-projection of
$(H_s)^{-1}(\Sigma^+)$ into the meridian $\{\phi=\phi_0\}$
is constant.
\end{itemize}
Now, there are two cases depending on whether $\omega_\psi(a^+(X_0))$ equals 0 or not.
If $\omega_\psi(a^+(X_0))\neq 0$,
the germ of $H_s$ at $a^-(X_0)$ is chosen to be a contraction: its
center is $a^+(X_0)$
and its factor is $e^s$ (in the coordinate $(x,y,v)$ of the extremity $\{z=1\}$ of the tube $T$
around $\underline\ell$).
Notice that such a contraction preserves the above requirements for the constancy of the equators.
Then, a calculation shows that the holonomic factor
is multiplied by the same factor, which implies that $\partial_s\chi(X_s)>0$ since $a^\pm(X_s)$ is constant.
Finally, we have to solve the case when $\omega_\psi(a^+(X_0))=0$. Here, we arrange the holonomy
$H_s$ so that $\partial_s\omega_\psi(a^+(X_s))>0$ and $\partial_s\omega_\phi(a^-(X_s))=0$, which again
implies $\partial_s\chi(X_s)>0$ since $\eta( X_s)>0$.
\hfill$\Box$\\
After Proposition \ref{item1} and Proposition \ref{item2} we have a complete proof
of Theorem \ref{thm1}.\hfill$\Box$\\
\subsection{\sc Normalization of crossing path.}
The normalization in question will be used for proving Theorem \ref{thm:selfslideSimplif}
and Theorem \ref{thm:Doubling}. The normalization is achieved by making some group act
on $M$ (actually the product of two groups).
At the end of the subsection it will be proved that the stratification $(\mathcal{S}_g, \mathcal{S}_g^0, \mathcal{S}_g^{0,0})$
is invariant under this action.
In this subsection we use notations as $D_1(0), C_1(0)$ which will be used repeatedly
in Section \ref{section3} (see Notation \ref{not:C1(s)}).
Consider the image $H_0(\Sigma^-)\subset \partial^+\mathcal{M}_p$
by the holonomy map of $X_0$
along its homoclinic orbit $\ell$ in the homotopy
class $g\in \Pi_{\lo}$. Define
$$D_1(0):= H_0(\Sigma^-)\cap\{z=1\}.$$
\begin{defn} \label{normalization} ${}$
\noindent{\rm 1)}The pseudo-gradient $X_0\in \mathcal{S}_g$ is said to be
\emph{normalized} or in \emph{ normal form } if $D_1(0)$ is contained in
the preferred hyperplane $\Delta^\phi$ of the meridian $i$-disc $\{\psi=\psi_0\}$.
\noindent{\rm 2)} A crossing path $(X_s)_s$ of $\mathcal{S}_g$ is said to be normalized if
$X_0$ is so.
\end{defn}
That any $X_0\in \mathcal{S}_g$ can be normalized by conjugation
follows from the next lemma about diffeomorphisms of $\mathcal{M}_p$
whose proof by Taylor expansion is detailed in the Appendix to \cite{flX}.
\begin{lemme} \label{C^1}
Let $K$ be a $C^1$-diffeomorphism of $\partial^+\mathcal{M}_p$ of the form $(\phi, r,\psi)\mapsto
(\phi, r, k(\phi,r,\psi))$. It is assumed that
$k(\phi,0,\psi)=\psi$.
Then, $K$ uniquely extends to $\mathcal{M}_p$ as a $C^1$-diffeomorphism
which is the identity on both stable and unstable local manifolds
and which keeps
the standard vector field $X_i$ invariant. Moreover, the extension ${\overline K}$
is $C^1$-tangent to $Id$ along the attaching sphere $\Sigma^-$.
\end{lemme}
It is worth noticing that the extension cannot be $C^2$ in general, even if $K$ is $C^\infty$.
This lemma can be also used by interchanging the roles of $\partial ^+\mathcal{M}_p$ and $\partial ^-\mathcal{M}_p$
and simultaneously the roles of $\phi$ and $\psi$.
\begin{cor}\label{conjugation} Given a positive crossing path $(X_s)_s$ of the stratum $\mathcal{S}_g$,
there exists a $C^1$-diffeomorphism $\overline K$ of $M$, isotopic to $Id_M$ among the
$C^1$-diffeomorphisms keeping $\alpha$ and $\mathcal{M}_p$ invariant,
such that the crossing path $(\overline K_* X_s)_s$ carried by $\overline K$ is normalized.
Moreover, $\overline K$ may be chosen so that it preserves $\ell$ pointwise.
\end{cor}
Notice that the vector field $(\overline K_* X_s)$ could be $C^0$ only. But it is integrable
and the associate foliation is $C^1$ transversely;
its holonomy is changed by $C^1$-diffeomorphism.\\
\nd {\bf Proof.\ } Recall that $D_1(0)$ is nowhere tangent to the fibres
of the projection $\pi^{\psi_0}$
to the meridian disc $\{\psi=\psi_0\}$. As a consequence,
its projected disc $D_{\psi_0}$ is smooth and there exists a smooth map $\bar k:D_{\psi_0}\to \Sigma^+$
such that $D_1(0)$ reads
$$D_1(0)= \{(\phi,r,\bar k(\phi,r))\mid (\phi,r)\in D_{\psi_0}\}.$$
Since its source is contractible, $\bar k$ is homotopic to the constant map valued in $\psi_0$.
By isotopy extension preserving the fibres of $\pi^{\psi_0}$, there exists some
diffeomorphism $K_1$ of $\mathcal{M}_p$ of the form assumed in
Lemma \ref{C^1} which maps the given $D_1(0)$ to $D_{\psi_0}$.
This $K_1$ extends to $\mathcal{M}_p$ according to
Lemma \ref{C^1}. Since
$K_1$ is isotopic to $Id$ through diffeomorphisms of the same type, its extension
$\overline K_1$ to $\mathcal{M}_p$ also extends to $M$ with the same name. Moreover,
the isotopy of $\overline K_1$ to $Id_M$ is supported in a neighborhood of $\mathcal{M}_p$
and preserves each level set of a local primitive of $\alpha$. Since $\Sigma^\pm$ are fixed by
$\overline K_1$, it is easy to get that $\ell$ is fixed by $\overline K_1$.
After having applied this $\overline K_1$ we are reduced to the case where
$D_1(0)$ is contained in the meridian disc $\{\psi=\psi_0\}$. Decreasing the ``multi-radius''
of the tube $T$ if necessary, the tangent plane
$T_mD_1(0)$ is almost orthogonal to the pole axis
$\mathbb{R}\partial_r^{\nu_\phi}$
in each $m\in D_1(0)$. This implies that, for every $r\in (0,\sqrt\delta)$, the disc $D_1(0)$ is transverse
to the $(i-1)$-sphere of radius $r$ in $\{\psi=\psi_0\}$.
Denote by $C_1(0)$ the image of $D_1(0)$ by $Desc$. It
is diffeomorphic to $S^{i-2}\times(0,1] $
and contained in the spherical annulus $\mathbb{A}_{\psi_0}:=\{(\phi, r, \psi_0)\mid \phi\in \Sigma^-, r\in (0,\sqrt\delta]\}$. By tangency of $D_1(0)$ with the preferred hyperplane $\Delta^\phi$,
the
{\it end} of $C_1(0)$ when $r\to 0$ compactifies as the
$\phi$-equator $E^\phi\subset \Sigma^-$.
Moreover, $C_1(0)$ is transverse (inside $\mathbb{A}_{\psi_0}$) to the sphere $\Sigma^-\times\{(r,\psi_0)\}$ for every $r$, since the corresponding assertion holds in $\partial^+\mathcal{M}_p$.
Thus, there is an annulus $C_{\rm eq}\subset E^\phi\times[0,\sqrt\delta]\times\{\psi_0\} $
such that $C_1(0)$ reads as the graph of some map $\bar\kappa: C_{\rm eq}\to \Sigma^-$ valued in the
complement of the poles. Then, $\bar\kappa$ is homotopic to the map $(\phi,r)\mapsto \phi$
from $C_{\rm eq}$ to the equator $E^\phi$ of $\Sigma^-$.
By isotopy extension preserving each sphere $\Sigma^-\times\{(r,\psi)\}$,
we have some diffeomorphism $\overline K_2$ of $\partial^-\mathcal{M}_p$ of the form
$(\phi,r, \psi)\mapsto (\kappa(\phi,r,\psi),r,\psi)$ which
pushes $C_1(0)$ to its {\it flat} position $C_{\rm eq}$ and satisfies $\kappa(\phi,0,\psi)=\phi$.
By applying Lemma \ref{C^1} ``up side down'', $K_2$ extends to $M$ preserving $\mathcal{M}_p$ with its
standard gradient. On the upper boundary of the Morse model,
this means that $\overline K_2$ pushes $D_1(0)$ to an $(i-1)$-disc in the hyperplane $\Delta^\phi$
by some diffeomorphism tangent to $Id$ in $a_+$. As for $\overline K_1$, this $\overline K_2$
may be chosen so that $\ell$ is fixed pointwise. The composed diffeomorphism
$\overline K_2\circ\overline K_1$ is as desired.
\hfill$\Box$\\
\begin{remarque} Corollary \ref{conjugation} holds true for a finite dimensional family. For instance,
if we are given a two-dimensional germ $\left(X_{s,t}\right)_{s,t}$ adapted to the pair $(\mathcal{S}_g,\mathcal{S}_g^0)$
in $X_{0,0}$ -- in the sense of Definition \ref{decompositionS^0} -- then each crossing path $\ga_t:=(s\mapsto X_{s,t})$ of $\mathcal{S}_g$ has a normalization
by some $C^1$-diffeomorphism $K_t$ depending continuously on $t$ in the $C^1$-topology.
\end{remarque}
\begin{notation}\label{cG}
Let $\mathcal{G}^\pm$ be the group of diffeomorphisms of $M$ isotopic to $Id_M$, fixing the homoclinic orbit
$\ell$ pointwise, preserving the closed one-form $\alpha$ and its standard gradient in $\mathcal{M}_p$,
and having the following form:
\begin{itemize}
\item if $G\in \mathcal{G}^+$, its restriction to $\partial^+\mathcal{M}_p$ reads $\bparent{\phi,r,\psi)\mapsto (\phi, r, k^+(r, \psi)}$
with $k^+(0,\psi)=\psi$;
\item if $G\in \mathcal{G}^-$, its restriction to $\partial^-\mathcal{M}_p$ reads $(\phi,r,\psi)\mapsto \bparent{k^-(\phi, r),r, \psi}$
with $k^-(\phi, 0)=\phi$.
\end{itemize}
\end{notation}
\begin{prop} \label{invariance}
The action of the groups $\mathcal{G}^+$ and $\mathcal{G}^-$ on the space of pseudo-gradients of $\alpha$
preserves the strata $\mathcal{S}_g$, $\mathcal{S}_g^0$ and $\mathcal{S}_g^{0,0}$.
\end{prop}
\nd {\bf Proof.\ } We do it for $\mathcal{G}^+$. Let $G\in \mathcal{G}^+$ and $X_0\in \mathcal{S}_g$. Since
$G$ fixes the homoclinic orbit $\ell$ pointwise, the carried
vector field $G_*(X_0)$
has the same homoclinic orbit. According to the form of the restriction of $G$ to the
upper boundary of $\mathcal{M}_p$, the projection of $D_1(0)$ to the meridian disc is unchanged.
Therefore, the $\phi$-equator is preserved. Looking in the lower boundary, one derives that the
$\phi$-latitude of $a^-(X_0)= \Sigma^-\cap\ell$ is preserved.
Consider the disc $D'_1(0):= H^{-1}_{X_0}(\Sigma^+)$. Recall from Lemma \ref{C^1} that $G\vert_{\partial^-\mathcal{M}_p}$ is tangent to $Id$ at every point of $\Sigma^-$. Therefore, the tangent space $T_{a^-}D'_1(0)$ remains invariant by $G$. It follows that the $\psi$-equator is not changed, and hence,
the $\psi$-latitude of $a^+$ is preserved. Thus we have the invariance of the spherical
axes and of their intersection $\mathcal{S}_g^{0,0}$.
It remains to show that the character function is invariant. We already have seen the invariance of the
latitudes. The last term to control is the holonomic factor $\eta(X_0)$ -- resp.
$\eta(G_*(X_0))$ -- defined in (\ref{eq:HolonDec}), which remains unchanged by the action of $G$ thanks to the invariance of:
\begin{itemize}
\item $\partial_v^1$ by invariance of the $\phi$-latitude,
\item $\partial_v^0$ since
$DG_{a^-}=Id$,
\item the framing in which $\partial_v^0$ decomposes (this framing is preserved by invariance of the $\psi$-latitude). \end{itemize}
\hfill$\Box$\\
\section{ Change in the Morse-Novikov complex}\label{section3}
\subsection{\sc A groupoid approach.} \label{ssec:Groupoid}
A groupoid $\mathscr{G}$ is a small category where every arrow is invertible. For every object $p\in \mathscr{G}^0$, we denote by $1_p\in\Hom(p,p)\subset\mathscr{G}^1$ its \emph{identity} arrow. The maps \emph{source} and \emph{target} $s,t:\mathscr{G}^1\to\mathscr{G}^0$ are the respective components of the map $g\in\Hom(p,q)\mapsto (p,q)\in \mathscr{G}^0\times\mathscr{G}^0$.
\begin{remarque}\label{rem:GroupoidRing}
We denote by $\mathbb{Z}[[\mathscr{G}]]$ the set of formal series of the arrows of $\mathscr{G}$. An element $\lambda\in\mathbb{Z}[[\mathscr{G}]]$
is usually written as $\lambda=\sum_{g\in\mathscr{G}^1}n_g(\lambda)g$, where $ n_g(\lambda)\in\mathbb{Z}$. Define the \emph{support} of $\lambda$ as the set $\supp(\lambda):=\ens{g\in\mathscr{G}^1}{n_g(\lambda)\neq 0}$. Consider the set
\begin{equation}
\mathbb{Z}[\mathscr{G}]:=\ens{\lambda\in\mathbb{Z}[[\mathscr{G}]]}{\supp(\lambda)\text{ is finite}}.
\end{equation}
Given two arrows $g,h$ -- seen as elements of $\mathbb{Z}[\mathscr{G}]$ -- the product $gh\in\mathbb{Z}[\mathscr{G}]$ is defined by their composition in $\mathscr{G}^1$ when $t(g)=s(h)$ and by $0$ otherwise. Extending the previous rule distributively with respect to the sum, we obtain a ring structure for $\mathbb{Z}[\mathscr{G}]$. Moreover, when $\mathscr{G}^0$ is finite, the element $1 := \sum_{p\in\mathscr{G}^0}1_p\in\mathbb{Z}[\mathscr{G}]$ gives an \emph{identity element} for this product. We call $\mathbb{Z}[\mathscr{G}]$ the \emph{groupoid ring} associated with $\mathscr{G}$. The next definition is classical.
\end{remarque}
\begin{defn}\label{fund_gr}
The fundamental groupoid $\Pi$ of the manifold $M$ is defined as follows: its objects are the points of
$M$ and, if $(p,q) $ is a pair of points, $\Hom(p,q)$ is the set of homotopy classes
of paths from $p$ to $q$. If $\ga$ is a such a path, its homotopy class $[\ga]$ will be called the
$\Pi$-value of $\ga$.
\end{defn}
Given any representative $\alpha$ of the fixed cohomology class $u$, we obtain a groupoid morphism
\begin{equation}
u_\alpha:\Pi\to\mathbb{R},
\quad\quad g\mapsto \int_{\gamma}\alpha,
\end{equation}
where $g$ is the $\Pi$-value
of a path $\gamma$ in $M$ and $\mathbb{R}$ is seen as a groupoid with a single object. The restriction of any such $u_\alpha$ to the fundamental group $\pi_1(M,p)$ clearly coincides with the group morphism associated with $u$. The subgroupoid $\Pi_{\lo}\subset \Pi$ is formed with the homotopy classes
of based loops in $M$. \\
We denote by $\Pi_\alpha$ the full subcategory of $\Pi$ whose set of objects coincides with $Z(\alpha)$. By Remark \ref{rem:GroupoidRing}, when $\alpha$ is Morse and $Z(\alpha)$ is non-empty, we may consider the groupoid ring $\mathbb{Z}[\Pi_\alpha]$.\\
A formal series $\lambda\in\mathbb{Z}[[\Pi_\alpha]]$ fulfills the \emph{Novikov condition} if
\begin{equation}\label{eq:NovikovCond}
\text{for every } L\in\mathbb{R}, \text{ the set } \supp(\lambda)\cap u_\alpha^{-1}(L,+\infty) \text{ is finite}.
\end{equation}
Denote by $\Lambda_\alpha\subset \mathbb{Z}[[\Pi_\alpha]] $ the subset of formal series satisfying the Novikov condition. It can be easily checked that the product rule given in Remark \ref{rem:GroupoidRing} also gives a ring structure to $\Lambda_\alpha$, having the same identity element. We call $\Lambda_\alpha$ the \emph{Novikov ring associated with $\alpha$}.
\begin{ex}
Let $ g\in \Pi_{\lo}$ with $u(g)<0$
-- for instance,
the $\Pi$-value of a self slide $\ell$.
The following formal series are elements of the Novikov ring:
\begin{equation}
\sum_{j=1}^\infty g^j\,\quad\text{and}\quad
\sum_{j=1}^\infty (-1)^j g^j\
\end{equation}
Indeed, the Novikov condition \eqref{eq:NovikovCond} is fulfilled
since $u_\alpha(g^j)=j.u_\alpha(g)$ which goes to $-\infty $
when $j\to +\infty$. Thus, they belong to the Novikov ring $\Lambda_\alpha$.
In particular $1+g+g^2+\ldots$ is a
unit whose inverse is $1-g$. \\
\end{ex}
We are going to deduce a chain complex $\bparent{C_*(\alpha),\partial^X}$
{ when $(\alpha, X)$ is a Morse-Smale pair.}\\
For a given index $i$, the left $\Lambda_{\alpha}$-module freely generated by $Z_i(\alpha)$ is denoted by $C_i(\alpha)$.
The map $\partial_* ^X: C_*(\alpha)\to C_{*-1}(\alpha)$ is then defined by:
\begin{equation}\label{eq:NovDiff}
\partial^X_*(p)=\sum_{q\in Z_{*-1}(\alpha)}\scal{p}{q}^X q,
\end{equation}
where the \emph{incidence} $\scal{p}{q}^X$ is { the algebraic count in $\Lambda_\alpha$ which
we are going to define, associated with the set $\Orb^X(p,q)$ of connecting orbits from $p$ to $q$.
First, we define the sign of a connecting orbit $\ell\in\Orb^X(p,q)$. Given a point $x\in\ell$,
the sign $\ep_\ell$ is defined by the following equation:
\begin{equation}
\ep_\ell\, X(x)\wedge \text{co-or}\parent{W^s(q,X)}= \text{or} \parent{W^u(p,X)}.
\end{equation}
This definition is clearly indepedent of $x\in\ell$.}
\begin{defn} { Given a Morse-Smale pair $(\alpha, X)$,
the \emph{Morse-Novikov incidence} associated with the data $(p,q,X)$, $p\in Z_i(\alpha)$,
$q\in Z_{i-1}(\alpha)$,}
is defined by:
\begin{equation}
\scal{p}{q}^X:=\sum_{\ell\in\Orb^X(p,q)}\ep_\ell \,g_\ell\in\Lambda_\alpha,
\end{equation}
where $g_{\ell}$ denotes the $\Pi$-value of the connecting orbit $\ell$.
\end{defn}
The map $\partial^X$ as in \eqref{eq:NovDiff} is indeed a differential; this can be found in \cite{latour}. The resulting $\bparent{C_*(\alpha),\partial^X}$ is known as the Morse-Novikov complex (see \cite{novikov}, \cite{sikorav}).\\
We denote by $\Orb^X_L(p,q)$ the set of connecting orbits from $p$ to $q$ whose $\alpha$-length $ \mathcal{L}(\ell)$ is
less than a fixed $L>0$. Since these orbits verify the inequality $u_{\alpha}(g_\ell)>-L$, we are led to define a
$L$-\emph{truncation} map $\mathcal{T}_L: \Lambda_\alpha\to\mathbb{Z}[\Pi_\alpha]$ by:
\begin{equation}
\mathcal{T}_L(\lambda):=\sum_{u_{\alpha}(g)>-L}n_g(\lambda)g.
\end{equation}
Two elements $\lambda,\mu\in\Lambda_{\alpha}$ are said to be \emph{$L$-equal} if $\mathcal{T}_L(\lambda-\mu)=0$, which is denoted by $\lambda\equiv_L\mu$.\\
Finally, the \emph{$L$-incidence} is defined as follows: \begin{equation}
\scal{p}{q}^X_L:=\sum_{\ell\in\Orb^X_L(p,q)}\ep_\ell \,g_\ell\in\mathbb{Z}[\Pi_\alpha].
\end{equation}
Of course
we have $\mathcal{T}_L\bparent{\scal{p}{q}^X}=\scal{p}{q}^X_L$.
\subsection{\sc Effect of self-slides on the incidence.}
Consider a generic one-parameter family of pseudo-gradients $(X_s)_s$ adapted to the
fixed Morse closed 1-form $\alpha$ such that $ X_0\in\mathcal{S}_g$. By
definition,
$X_0$ has a unique homoclinic orbit $\ell$ connecting $p\in Z(\alpha)$ to itself
whose $\Pi$-value is $g$.
Since $\mathcal{S}_g^0$ has codimension one in $\mathcal{S}_g$ (Proposition \ref{item2}), generically
the character of $X_0$ is defined.
Denote the index of $p$ by $i$.
The next definition specifies some genericity conditions that will be needed to prove the theorem below. The remainder of this section is devoted
to its proof and consequences.
\begin{defn} Let $L>0$.
\noindent{\rm 1)} The pair $(\alpha, X)$ is said to be \emph{Morse-Smale up to} $L$ if, for every pair of
zeroes $p,q\in Z(\alpha)$ and every $X$-orbit $\ell$
from $p$ to $q$ with $\int_\ell \alpha>-L$, the unstable and stable manifolds, $W^u(p,X)$
and $W^s(q,X)$, are transverse along $\ell$.\smallskip
\noindent{\rm 2)} When $X\in \mathcal{S}_g$ and $u_\alpha(g)>-L$, then $(\alpha,X)$ is said to be
\emph{almost Morse-Smale up to }$L$,
if the preceding transversality condition is fulfilled except for the unique homoclinic orbit
whose $\Pi$-value is $g$.
\end{defn}
If $(X_s)_s$ is a path which intersects $\mathcal{S}_g$ transversely in $ X_0$ and if $(\alpha,X_0)$ is almost
Morse-Smale up to $L$, then $(\alpha, X_s)$ is easily checked to be Morse-Smale up to $L$ for every $s\neq 0$ close enough to 0.
\begin{thm}\label{thm:selfslideAlg}
Let $(X_s)_{s}$ be a path intersecting $\mathcal{S}_g$ transversely in $ X_0$ and let $L> - u_\alpha(g)$.
Assume $(\alpha,X_0)$ to be almost Morse-Smale up to $L$, assume that its character is defined and the
$\phi$-latitude is non-zero. Let $q\in Z_{i-1}(\alpha)$. The $L$-incidence $\scal{p}{q}^{X_s}_L$ is defined
when $s=0_+$ and $s=0_-$; it is denoted by $\scal{p}{q}^\pm_L$ respectively. Then, we have the
following.
\smallskip
\noindent When $(X_s)_s$ intersects the stratum $\mathcal{S}_g$ positively,
the next relations hold in $\Lambda_\alpha$:
\begin{enumerate}
\item if $ X_0\in\mathcal{S}_g^+$, then $\scal{p}{q}^+_L\equiv_L\bparent{1+g+g^2+g^3+\ldots}
\cdot \scal{p}{q}^-_L\, ,$
\item if $X_0\in\mathcal{S}_g^-$, then $\scal{p}{q}^+_L\equiv_L\bparent{1+g}\cdot \scal{p}{q}^-_L\, .
$
\end{enumerate}
\smallskip
\noindent
When $(X_s)_s$ intersects the stratum $\mathcal{S}_g$ negatively, we have:
\begin{enumerate}
\item[(1')] if $ X_0\in\mathcal{S}_g^+$, then $\scal{p}{q}^+_L\equiv_L\bparent{1-g}\cdot \scal{p}{q}^-_L\, ,$
\item[(2')] if $ X_0\in\mathcal{S}_g^-$, then $\scal{p}{q}^+_L\equiv_L\bparent{1-g+g^2-g^3+\ldots}\cdot \scal{p}{q}^-_L\, .
$
\end{enumerate}
\end{thm}
\noindent{\bf Proof of $(1)\iff(1')\iff (2)\iff (2')$.} The first equivalence is obvious
since $1- g$ is the inverse of $ 1+\Sigma_{j=1}^\infty g^j$; and similarly for the last equivalence.
We are left with the middle equivalence. It is obtained by changing the vector $\partial_v$
into its opposite in the coordinates of the tube around the homoclinic orbit of $X_0$. This amounts to put a sign $-$ in Formula (\ref{orientation1}). The latter change has three effects:
\begin{enumerate}
\item[i)] It reverses the co-orientation of $\mathcal{S}_g$. Hence, positive and negative crossings are exchanged.
\item[ii)] The character is changed into its opposite since the $\phi$- and $\psi$-latitudes are.
Thus, both sides of $\mathcal{S}^0_g$ are exchanged.
\item[iii)] The homoclinic orbit becomes negative in the following sense: if the $\phi$-sphere
is seen
as the boundary of the meridian disc of $a^+$, the new positive hemisphere projects to the preferred
hyperplane $\Delta^\phi$ by reversing the orientation. This implies that in the algebraic counting
of connecting orbits from $p$ to $q$ (where $i(p)=i(q)+1$) the coefficient
$g$ has to be changed to $-g$ (see
the orientation claim in Lemma \ref{lem:C1(s)}).
\end{enumerate}
We are left to prove the Theorem \ref{thm:selfslideAlg} in case (1). This will be done
in Subsection \ref{continued}.
\hfill$\Box$\\
According to Proposition \ref{invariance}, the statement of Theorem \ref{thm:selfslideAlg}
is invariant by the groups $\mathcal{G}^\pm$
introduced in Notation \ref{cG}. After Corollary \ref{conjugation}, it is sufficient to consider the case
where the crossing path in question is {\it normalized} in the sense of Definition \ref{normalization}.
This assumption is done in what follows.
We need some more notations and lemmas.
The setting of Theorem \ref{thm:selfslideAlg} is still assumed.
\begin{notation}{\rm ${}$\label{not:C1(s)}
\noindent 1) The symbol $\mathcal O_-$ stands for an open interval $(-\epsilon,0)$ whose size
is not specified and which is understood as small
as needed by the statement; and similarly for $\mathcal O_+$. If $A$ is a closed subset of $B$, $\mathcal Op(A)$ stands for an open
neighborhood of $A$ in $B$ which is not specified
\smallskip
\noindent 2) Recall from Subsection \ref{perturbed} that $H_s$ stands for the local holonomy diffeomorphism
along the homoclinic orbit $\ell$ from $\mathcal Op(\{z=0\}) \subset \partial^- \mathcal{M}_p$
to $\mathcal Op(\{z=1\}) \subset \partial^+ \mathcal{M}_p$.
\smallskip
\noindent 3)
For $s\in \mathcal Op(0)$,
let $D_1(s)$ denote the image $H_s(\Sigma^-)\cap\{z=1\}$.
Consider its projection $\pi^{\psi_0}(D_1(s))$
onto the meridian disc $\{\psi=\psi_0\}$
and define
\begin{equation}
a^+(s):= \pi^{\psi_0}(D_1(s))\cap\mathbb{R}\partial_v^1.
\end{equation}
The {\it crossing velocity} of the crossing path is
\begin{equation}\label{initial_velocity}
V_1 := \frac {d }{ds} a^+(s)_{\vert s=0}.
\end{equation}
Up to a reparametrization in $s$, it can be assumed to be equal to 1.
\smallskip
\noindent 4) For $s\in \mathcal O_\pm$, by definition of a crossing path $D_1(s)$ avoids $\Sigma^+$. Therefore we are allowed to define $C_1(s):= Desc(D_1(s))\subset \partial^-\mathcal{M}_p$. It is still
an $(i-1)$-disc.
}
\end{notation}
\begin{lemme}\label{lem:C1(s)}
Recall the natural projection $\pi^-: \partial^- \mathcal{M}_p\to \Sigma^-$.
Let $K$ be any compact disc in the open hemisphere $\mathcal H^-(\Sigma^-)$ {\rm (}as in {\rm Subsection \ref{ssec:Equators})}.
Then, for $s\in \mathcal O_-$, the disc $C_1(s)\cap (\pi^-)^{-1}(K)$ is a graph over $K$
of a section of $\pi^-$
which goes to the zero-section $0_K$ of $\pi^- $
in the $C^1$-topology when $s$ goes to $0$.
Moreover, $\pi^-: C_1(s)\to \Sigma^-$ is orientation reversing.
A similar statement holds when $K\subset \mathcal H^+(\Sigma^-)$ and $s\in\mathcal O_+$,
except that $\pi^-: C_1(s)\to \Sigma^-$ is orientation preserving in that case.
\end{lemme}
\nd {\bf Proof.\ }
The statement about orientation is clear after the claim
about the $C^1$-convergence.
Consider the case $s\in\mathcal O_-$, the other case being similar.
Recall the normalization assumption: the disc $D_1(0)$ is contained in the meridian disc $\{\psi=\psi_0\}$.
Recall the projection $\pi^{\psi_0}$ of $\partial^+\mathcal{M}_p$ to the meridian disc.
The normalization implies that the projected discs $\pi^{\psi_0}(D_1(s))$ tend to $D_1(0)$
in the $C^1$-topology.
Recall the identification $\partial(\{\psi=\psi_0\})\cong \Sigma^-$ of \eqref{eq:IdentPhi}
and think of $K$ as a compact subset of the South hemisphere in the boundary of the meridian disc
$\{\psi=\psi_0\}$.
For every such $K$,
the next property holds:
\begin{itemize}
\item[] {\it For every $s$ close enough to 0 and for every $\phi\in K$,
the disc $\pi^{\psi_0}(D_1(s))$
intersects the ray directed by $\partial_r^\phi$ in one point only and transversely. }
\end{itemize}
This point is denoted by $m_s(\phi)$; it is the image of some
$\tilde m_s(\phi)\in D_1(s)$ through $\pi^{\psi_0}$.
We have $m_0(\phi)=\tilde m_0(\phi)= a^+$, but when $s\neq 0$, the point
$\tilde m_s(\phi)$ has well-defined multi-spherical coordinates $(\phi, r_s(\phi),\psi_s(\phi))$ where
$r_s$ and $\psi_s$ depend smoothly on $s$.
Going back to $\partial^-\mathcal{M}_p$ by the map $Desc$, we see that $C_1(s)$ is the image of a section
of the projection $\pi^-$ over $K$.
When $s\in \mathcal{O}_-$ goes to 0, then $D_1(s)\cap \{(\phi,r,\psi)\in \partial^+\mathcal{M}_p
\mid \phi\in K\} $
goes to $a^+$
in the metric sense. In particular, $\max_{\phi\in K} \{r\,\vert\, (\phi,r,\psi)\in D_1(s)\}$
goes to 0.
Therefore, $C_1(s)\cap (\pi^-)^{-1}(K)$ goes to $0_K$ in the $C^0$-topology
when $s$ goes to $0$ negatively.
For the $C^1$-convergence, we use that $K$ is far from the
$\phi$-equator.
Therefore, the angle in the meridian disc $\{\psi=\psi_0\}$ between the ray $\mathbb{R}\partial_r^\phi$
and the tangent plane to $ \pi^{\psi_0}(D_1(s))$ at $ m_s(\phi)$
is bounded from below. Including the fact that $s\to 0_-$ implies $r\to 0$, it follows
that the smooth functions $r_s(\phi)$ and $\psi_s(\phi)$ satisfy
$$\left\{\begin{array}{c}
\vert dr\vert= O_s(r) \vert d\phi\vert,\\
\vert d\psi\vert =O_s(r) \vert d\phi\vert,
\end{array}
\right.
$$
where $O_s(r)$ stands for a quantity which is uniformly bounded with respect to $r$ when $s$ goes to 0.
This yields the wanted $C^1$-convergence
of the part of $C_1(s)$ over $K\subset \mathcal H^-(\Sigma^-)$ to $0_K = K$.
\hfill$\Box$\\
\subsection{\sc Geometric interpretation of the character function.}\label{interpretation}
We still consider a germ of {\it normalized} positive crossing path $\left(X_s\right)_s$.
Let $D_1'(0) $ be the connected component $W^s(p, X_0)\cap \{z=0\}$ which contains $a^-$.
It is an $(n-i-1)$-disc which is the image of $\Sigma^+$
by the inverse holonomy diffeomorphism $H_0^{-1}$ along $\ell$.
For every $s\in \mathcal Op(0)$, consider now $D_1'(s):= H_s^{-1}(\Sigma^+)\cap \{ z=0\}$.
Recall from Subsection \ref{ssec:tube}
that $\Sigma^-\cap \{z=0\}$ is identified
with the $x$-axis whereas $D_1'(0)$ is identified with the $y$-axis.
Let also $p_v:\sing{z=0}\to\sing{ v=0, z=0}$ denote the projection parallel to $\partial_v$ onto the $(x,y)$-space.
When $s\in \mathcal Op(0)$ goes to 0, the family $D'_1(s)$ accumulates
to the $y$-axis in the $C^1 $-topology. And, according to Lemma \ref{lem:C1(s)},
the family $C_1(s)\cap \{z=0\} $
accumulates to the $x$-axis in the $C^1$-topology if and only if $s\, \omega_\phi(a^-)$ goes to $0_+$.
In particular, when $s\, \omega_\phi(a^-)>0$ the projections $p_v(C_1(s))$ and $p_v(D'_1(s))$
intersect
in a unique point $b_1(s)$ and transversely.
If $s\, \omega_\phi(a^-)$ is negative, then $C_1(s)\cap \{z=0\} $ is empty.\\
Denote by $c_1(s)\text{ and } d_1'(s)$ the only points in $C_1(s)\cap\sing{z=0}$ and in $D_1'(s)$
respectively such that $p_v(c_1(s))=b_1(s)=p_v(d_1'(s))$. Consider the real number
\begin{equation}\label{eq:v1(s)}
v_1(s):=v(c_1(s))-v(d_1'(s))\quad \text{ for every } s \text{ such that }s\, \omega_\phi(a^-)\in \mathcal O_+.
\end{equation}
This function $v_1(s)$ depends smoothly on $s$. Its derivative with respect to $s$
is denoted by $\dot{v}_1(s)$.
\begin{remarque}\label{rem:g2}
By construction, $v_1(s)=0$ implies $c_1(s)=d_1'(s)$ which in turn implies
the existence of an orbit $\ell_s\in\Orb^{X_s}(p, p)$ passing through $c_1(s)$ such that $[\ell_s]=g^2$.
\end{remarque}
Lemma \ref{geometry-character} will show
the geometrical meaning of the character function $\chi$ at $X=X_0$.
For simplicity, in what follows $a^\pm(X_0)$ will be denoted by $a^\pm$.
\begin{lemme}\label{geometry-character} Given a normalized
positive crossing path $(X_s)_s$ whose crossing velocity
defined in \eqref{initial_velocity} equals $ +1$,
the next relation holds: \begin{equation}\label{eq:CharGeom}
\omega_\phi(a^-)\, \dot{v}_1(0)=\chi( X_0).
\end{equation}
\end{lemme}
\nd {\bf Proof.\ }
Let us study the $v$-coordinate of $c_1(s)$ first. We notice that, if $\bar c_1(s)$ is another point
of $C_1(s)$ depending smoothly on $s$ and such that $\bar c_1(0)=a^-=(\phi_0,0,-)$,
we have the same velocity in $s= 0$:
\begin{equation}\label{same-speed}
\frac{d}{ds} v\left(\bar c_1(s)\right)_{\vert s=0} =\frac{d}{ds} v\left(c_1(s)\right)_{\vert s=0}.
\end{equation}
Indeed, $C_1(s)$ accumulates $C^1$ to $\Sigma^-$ (Lemma \ref{lem:C1(s)}), then the difference $\dot c_1(0)-\dot{\bar c}_1(0)$ is a vector in $T_{a^-}\Sigma^-$.
We apply this remark to the point $\bar c_1(s):=C_1(s)\cap\{\phi=\phi_0\}$. We now lift
that point to $d_1(s)\in D_1(s)$ by $Desc^{-1}$.
Since $Desc$ preserves the $(\phi, r, \psi)$-coordinates, both paths
$s\mapsto \bar c_1(s)$ and $s\mapsto d_1(s)$ have the same coordinates
$(\phi(s)=\phi_0, r(s), \psi(s))$ when $s\neq 0$.
Since $\bar c_1(s)\in\{\phi=\phi_0 \}$, the vector $\dot d_1(0)$ belongs to the $(n-i)$-plane
which is span of $\{\partial_r^{\phi_0}, T_{a^+}\Sigma^+\}$.
Let ${\hat d}_1(0)$
be its projection to the line $\mathbb{R}\partial_r ^{\phi_0}$ in the meridian disc $\{\psi=\psi_0\}$.
Then,
\begin{equation}\label{eq:Info1}
\begin{array}{l}
{\hat d}_1(0)=\rho\,\partial_r^{\phi_0}, \text{ where } \rho={\displaystyle\frac{d}{ds}r(s)_{\vert s=0} }.\\
\end{array}
\end{equation}
By definition of the $\phi$-latitude (Proposition \ref{prop:Latitudes} and \eqref{eq:Trigo}) we have:
\begin{equation}
p_{\Delta^\phi}({\hat d}_1(0))=\rho\, p_{\Delta^{\phi}}\parent{\partial_r^{\phi_0}}=\rho\,\omega_\phi (a^-)\partial_r^{\nu_{\phi}}.
\end{equation}
By definition, the hyperplane $\Delta^\phi$ is tangent in $a^+$ to the projection $p_y(D_1(0))$.
Therefore, some calculus of Taylor expansion allows us to deduce that
\begin{equation}
p_{\Delta^\phi}({\hat d}_1(0))=\frac{d}{ds} a^+(s)_{\vert s=0}=+1\,.
\end{equation}
We derive:
\begin{equation}\label{eq:rho}
\rho=\frac{1}{\omega_\phi(a^-)}\ .
\end{equation}
Since $d_1(s) $ goes to $a^+=(-,0,\psi_0)$ when $s$ goes to $0$ and since the radial velocity is
preserved by $Desc$, then we have:
\begin{equation}
\dot{\bar c}_1(0)= \rho\,\partial_r^{\psi_0}\in T_{a^-}\{\phi=\phi_0\}.
\end{equation}
Using again
\eqref{eq:Trigo}, but relatively
to the hyperplane $\Delta^{\psi}$ which defines the $\psi$-latitude
we obtain
\begin{equation}
p_{\Delta^{\psi}}(\dot{\bar c}_1(0))=\rho\,\omega_{\psi}(a^+)\partial_r^{\nu_{\psi}}.
\end{equation}
This together with the decomposition of $T_{a^-}\{z=0\}$ of \eqref{eq:decompTgt} says that
there are $w_x\in T_{a^-}\Sigma^-$ and $ w_y\in \Delta^\psi$ such that
\begin{equation}\label{eq:c1'(0)}
\dot{\bar c}_1(0)=w_x+ \rho\,\omega_{\psi}(a^+)\partial_r^{\nu_{\psi}} + w_y.
\end{equation}
Consider the projection $p_{xy}:T_{a^-}\sing{z=0}\to \mathbb{R}\partial_v^0$ parallel
to the hyperplane $ T_{a^-}\Sigma^-\oplus \Delta^{\psi}$. Thanks to \eqref{eq:c1'(0)}, we have
$v(\dot{\bar c}_1(0))\, \partial_v^0=p_{xy}(\dot{\bar c}_1(0))=\rho\,\omega_{\psi}(a^+)p_{xy}(\partial_r^{\nu_{\psi}})$. On the other hand,
\eqref{eq:HolonDec} tells us that:
\begin{equation}\label{eq:HolomMod}
v(\partial_r^{\nu_{\psi}})=\frac{1}{\bar{\eta}}=\eta\,.
\end{equation}
Putting together (\ref{same-speed}), \eqref{eq:rho}, \eqref{eq:c1'(0)} and \eqref{eq:HolomMod} we obtain:
\begin{equation}\label{eq:c1der}
v(\dot{c}_1(0))= v(\dot{\bar c}_1(0))=\rho\,\omega_{\psi}(a^+)\, v(\partial_r^{\nu_\psi})=\frac{\omega_{\psi}(a^+)}{\omega_{\phi}(a^-)}\,\eta\,.
\end{equation}\\
We come now to estimate the term $v(\dot{d'}_1(0))$. We apply Lemma \ref{velocity}
for comparing velocities associated with the holonomy $H_s$ and its inverse. From the formula (\ref{initial_velocity})
we derive that
$\frac {d}{ds} (v\circ H_s)(a^-)_{\vert s=0}= +1$. Then, the inverse holonomy satisfies
\begin{equation}\label{second}
\frac {d}{ds} (v\circ H_s^{-1})(a^+)_{\vert s=0}= -1
\end{equation}
from which it is easily derived that $v(\dot{d'}_1(0))=-1$. Therefore:
\begin{equation}\label{eq:GrandV2}
\dot v_1(0)=\eta\frac{\omega_{\psi}(a^+)}{\omega_{\phi}(a^-)}+1
\end{equation}
which is a reformulation of the desired formula.\hfill$\Box$\\
Lemma \ref{lem:Passages} right below is the last tool that we need for proving Theorem
\ref{thm:selfslideAlg}. It extracts the geometrical information contained in Equation (\ref{eq:CharGeom}).
The setting is the same as in the previous lemma. We are only looking at normalized paths
$(X_s)_s$ which cross $\mathcal{S}_g$ positively in a point $X_0$ whose character is positive.
A similar statement holds for paths crossing at a point whose character is negative.
\begin{lemme}\label{lem:Passages}${}$
\noindent {\rm 1)} Suppose $\dot{v}_1(0)$ and the $\phi$-latitude
$\omega_\phi(a^-)$ are positive.
Then, for $s\in \mathcal O_+$ there are sequences of non-empty $(i-1)$-discs $(D_k(s))_{k>1}$ and
$(C_k(s))_{k>1}$ inductively defined from the previous $D_1(s)$ and $C_1(s)$ by
\begin{equation}\label{defC_k}
\left\{
\begin{array}{l}
D_k(s):= H_s\left(C_{k-1}(s)\right) \cap \{z=1\}\\
C_k(s) := Desc\left( D_k(s)\right) \subset \partial^- \mathcal{M}_p
\end{array}
\right.
\end{equation}
Moreover, when $s$ goes to 0, the discs $C_k(s)$ tend to the North hemisphere
$\mathcal H^+(\Sigma^-)$
in the $C^1$-topology, uniformly over every compact set of $\mathcal H^+(\Sigma^-)$.
When $s\in \mathcal O_-$, both previous sequences are empty when $k>1$.
\smallskip
\noindent {\rm 2)} If $\dot{v}_1(0)$ and $\omega_\phi(a^-)$ are negative,
then for $s\in \mathcal O_-$ the disc $C_2(s)$ is well defined as in \eqref{defC_k}
and the next ones are empty. Moreover,
$C_2(s)$ tends to $\mathcal H^+(\Sigma^-)$ in the $C^1$-topology with the reversed orientation.
When $s\in \mathcal O_+$, every disc in \eqref{defC_k} is empty when $k>1$.
\end{lemme}
Notice that, according to Lemma \ref{geometry-character} the assumption of 1) reads also:
the character and $\omega_\phi(a^-)$ are positive; and the assumption of 2) reads:
the character is positive and $\omega_\phi(a^-)<0$.\\
\nd {\bf Proof.\ }
1) When $s\in \mathcal O_-$, the disc $C_1(s)$ does not meet the tube $T$ around the
homoclinic orbit $\ell$. Then $D_2(s)$ is empty and, hence, all the further discs are so.
Assume now that $s\in \mathcal O_+$. In that case, $C_1(s) $ goes to $\mathcal H^+(\Sigma^-)$
(Lemma \ref{lem:C1(s)})
and therefore meets the set $\{z=0\}$. Then,
the discs $D_2(s)$ and $C_2(s) $ defined in \eqref{defC_k} are non empty.
We are going to compute the position of $C_2(s)$ with respect to $D'_1(s)$ measured
by some $v_2(s)$ in the direction
of the $v$-coordinate.
We shall check the positivity of $\dot v_2(0)$ which will allow us to pursue the induction.
Recall the projection $\pi^{\psi_0}:\partial^+\mathcal{M}_p\to \{\psi=\psi_0\}$ and define the spherical annulus
$A:= (\pi^{\psi_0})^{-1}(\mathbb{R}\partial_v)$.
Consider the point $\tilde c_1(s)$ which is the transverse intersection
$C_1(s)\cap H_s^{-1}(A)$. By projecting to
the $v$-axis we find a function $v(\tilde c_1(s))$ which satisfies
\begin{equation}
\frac{d}{ds}v(\tilde c_1(s))_{\vert s=0}= \frac{d}{ds}v( c_1(s))_{\vert s=0}
\end{equation}
Recall the definition of $d'_1(s)\in D_1'(s)$ from (\ref{eq:v1(s)}).
Compute the derivative $V_2$ at $s=0$ of $v\left[H_s(\tilde c_1(s))\right]-v\left[H_s(d'_1(s))\right]$,
which is nothing but the velocity of the projection of $H_s(\tilde c_1(s))\in D_2(s)$ onto the $v$-axis of
$\{z=1\}$ at $s=0$.
Using $\tilde c_1(0)= d'_1(0)=a^-$ and $dH_0(a^-)=Id$ in the coordinates of the tube $T$, we find:
\begin{equation}\label{V_2}
V_2= \dot v_1(0)
\end{equation}
which is positive by assumption. This $V_2$ will play the same r\^ole as the crossing velocity.
Since $V_2>0$, Lemma \ref{lem:C1(s)} tells us that $C_2(s)$ meets $\{z=0\}$ when
$s\in \mathcal O_+$. Therefore, we choose points $c_2(s)\in C_2(s)=Desc(D_2(s)) $ and
$d'_2(s)\in D'_1(s)$ which forms the unique pair of points of the respective subsets
which have the same $p_v$-projection. We define
\begin{equation}
v_2(s):= v(c_2(s))-v(d'_2(s))
\end{equation}
The computation of $\dot v_2(0)$ is exactly the same except we have to replace
$V_1=1$ with $V_2$. The result is:
\begin{equation}
\dot v_2(0)= \eta\frac{\omega_\psi(a^+)}{\omega_\phi(a^-)} V_2+1.
\end{equation}
Here, it is appearing some discussion according to the sign of $\omega_\psi(a^+)$:
\begin{itemize}
\item[(i)] if $\omega_\psi(a^+)$ is positive, then $\dot v_2(0)$ is larger than $ V_1=+1$. In that case
the induction goes on with $V_k>V_{k-1}>\ldots> 1$.
\item[(ii)] if $\omega_\psi(a^+)$ is negative, then $0<V_2=\dot v_1(0)<1$, where the last inequality comes from \eqref{eq:GrandV2}. Therefore,
$\dot v_2(0)-1$ is the product of two numbers\footnote{One of them being $V_2$.} of opposite signs
and whose absolute values are smaller than 1. Thus, $\dot v_2(0) $ belongs to $(0,1)$. Such a fact
is preserved at each step of the induction.
\end{itemize}
The induction can be carried on.
\smallskip
\noindent 2) Take $s\in \mathcal O_-$. The calculation yielding the equality (\ref{V_2}) still holds and tells us that
$V_2$ is negative. Remark that $H_s(d'_1(s))\in \Sigma^+$. As $s<0$, one derives:
$$
v\left[H_s(\tilde c_1(s))\right]=v\left[H_s(\tilde c_1(s))\right]-v\left[H_s(d'_1(s))\right]>0.
$$ Thus, Lemma \ref{lem:C1(s)} says that $C_2(s)$ tends to $\mathcal H^+(\Sigma^+)$
in the $C^1$-topology.
As $\omega_\phi(a^-)<0$,
$C_2(s)$ does not meet $\{z=0\}$ and the next discs are empty.
Concerning the orientation, we check that $D_2(s)$ tends to $-D_1(0)$ in $\partial ^+\mathcal{M}_p$. Then,
$C_2(s) $ tends to $-\mathcal H^+(\Sigma^-)$. Finally,
the statement when $s\in \mathcal O_+$ is clear.
\hfill$\Box$\\
\subsection{\sc Proof of Theorem \ref{thm:selfslideAlg} continued.\label{continued}}
We continue the proof which begins just after the statement of that theorem. After a series of
equivalences, we are left to prove the case (1) of a positive crossing of the stratum $\mathcal{S}_g$
at a point $(\alpha,X_0)$ where the character is positive.
We recall that
the statement of Theorem \ref{thm:selfslideAlg} is preseved under the action of the groups
$\mathcal{G}^\pm$. Therefore, we may assume that $X_0\in \mathcal{S}_g$ is normalized.
The element $g\in \Pi_\alpha$ is thought of as an arrow from the set of zeroes
$Z(\alpha)$ into itself. Then $g$ determines its origin $p$ which is also its end point.
Recall that the Morse index of $p$ is $i$.
We look at any zero $q\in Z(\alpha)$ of Morse index $i-1$. We have to compute the change of
$\langle p,q\rangle^X$ when $X$ changes from $X_{0_-}$ to $X_{0_+}$ in a crossing path
$\left(X_s\right)_s$.
Taking into account that we search for an isomorphism of complexes over the Novikov ring $\Lambda_\alpha$,
the first attempt is to look at each connecting orbit from $p$ to $q$
separately. But it does not work.
It is useful
to make
some partition, adapted to $g$, of the set of connecting orbits.\\
\noindent {\sc Partition of the connecting orbits.} We may assume that each connecting orbit is the unique one in its homotopy class. In general, one would take the multiplicity into account. The equivalence relation
defining the partition is the following: $\Gamma_0\sim \Gamma_1$
if and only if the homotopy class of $\Gamma_1$ reads
$[\Gamma_1]= g^k[\Gamma_0] \text{ with } k\in \mathbb{Z}$.
Consider $[\Gamma]_{\sim}$, the $\sim$-class of a fixed connecting orbit $\Gamma$.
Since the $\alpha$-lengths of connecting orbits are positive, we have $u([\Gamma'])<0$ for every $\Gamma'\in [\Gamma]_{\sim}$.
Therefore, as $u(g)<0$, there are only finitely many
connecting orbits $\Gamma'\in [\Gamma]_{\sim}$ verifying
$u([\Gamma'])\geq u([\Gamma])$. Let $\Gamma_0$
be
a connecting orbit in $[\Gamma]_{\sim}$
such that $u([\Gamma_0])$
is maximal. Then, any
element of $[\Gamma]_{\sim}$
reads $g^k[\Gamma_0]$ with
$k\geq 0$.\\
\noindent{\sc End of the proof.} Now, without loss of generality we may assume that the above
partition has one $\sim$-class only and that the maximal element $\Gamma$ is a positive connecting orbit. Let $b:= \Gamma\cap \Sigma^-$ and $\Delta_s$ be the connected component of $W^s(q,X_s)\cap\partial^-\mathcal{M}_p$ containing $b$, which is a $(n-i)$-disc intersecting transversely $\Sigma^-$, only at $b$. Since, we are looking at the change formula up to lenght $L>0$, we may assume that the crossing path $\left(X_s\right)_s$ is defined for $s\in[-1,1]$ and for every $s\neq 0$ the pair $(\alpha,X_s)$ is Morse-Smale up to $L$. Therefore, as the property of being Morse-Smale up to $L$ is open, we are allowed to slightly move this path (in particular, $X_0$)
keeping the end points fixed.
Thus, generically $b$ does not lie on the equator $E^\phi$, and $\Delta_s$ may be supposed to coincide with $\Delta_0$ for every $s$.
There are still four cases to consider
where $a^-$ stands for $a^-(X_0)$ and $\mathcal H^\pm$ stand for $\mathcal H^\pm(\Sigma^-)$:
\begin{enumerate}
\item[(a.1)] The $\phi$-latitude $\omega_\phi(a^-)$ is positive and $b$ belongs to $\mathcal H^+$.
\item[(a.2)] The $\phi$-latitude $\omega_\phi(a^-)$ is positive and $b$ belongs to $\mathcal H^-$.
\item[(b.1)] The $\phi$-latitude $\omega_\phi(a^-)$ is negative and $b$ belongs to
$\mathcal H^+$.
\item[(b.2)] The $\phi$-latitude $\omega_\phi(a^-)$ is negative and $b$ belongs to $\mathcal H^-$.
\end{enumerate}
The proof consists of applying Lemma \ref{lem:Passages}. It is convenient to use the following
definiton. \\
\noindent{\bf Definitions.}${}$\smallskip
\noindent 1) {\it The positive (resp. negative) part of $W^u(p, X_0)$ is the union of the $X_0$-orbits passing
through the positive (resp. negative) hemisphere $\mathcal H^+(\Sigma^-)$. It will be denoted
by $W^u(p, X_0)^\pm$.}\smallskip
\noindent 2) {\it For a given $k>0$, we say that the unstable manifolds $W^u(p, X_s)$
accumulate to $g^k\cdot W^u(p, X_0)^\pm$ when $s$ goes to $0_-$ or $0_+$ if it is true when
lifting to the universal cover, that is:
if $\tilde p$ (resp. $\widetilde X_s$)
is a lift of $p$ (resp. $X_s$), the unstable manifolds $W^u(\tilde p, \widetilde X_s)$
accumulate to $W^u(g^k\tilde p, \widetilde X_0)^\pm$.}\\
Here, it is worth noticing that when a point lies in the accumulation set its whole
{$X_0$-orbit}\break
is accumulated. As a consequence, Lemma \ref{lem:C1(s)} tells us that
$W^u(p, X_s)$ accumulate to \break
$g\cdot W^u(p, X_0)^\pm$ in the $C^1$-topology
when $s$ goes to $0_\pm$. Thanks to this topology,
it makes sense to compare the orientations. The result is the following: when $s\to 0_\pm$, then
$W^u(p, X_s)$ accumulate to $\pm g\cdot W^u(p, X_0)^\pm$. Accumulation to
$g^k\cdot W^u(p, X_0)^\pm$
for some $k>1$ is dictated by Lemma \ref{lem:Passages} depending on the sign of the character
and the $\phi$-latitude.
We are now ready for proving Theorem
\ref{thm:selfslideAlg} in each case.
Let $\lambda_-(\Gamma)$ (resp. $\lambda_+(\Gamma)$) denote the element of the Novikov ring $\Lambda_\alpha$ which is
the contribution of $[\Gamma]_{\sim}$
(up to the given $L>0$) in $\langle p,q\rangle^X_s$ when $s<0$
(resp. $s>0$). We have to check the next formula in each case (a.1) ... (b.2).
\begin{equation}\label{change}
\lambda_+(\Gamma)= (1+g+g^2+\ldots)\cdot \lambda_-(\Gamma)
\end{equation}
\noindent {\bf Case (a.1).} When $s\to 0_-$, the oriented unstable manifolds $W^u(p, X_s)$ accumulate \break to $-g\cdot W^u(p,X_s)^-$ and nothing else. Therefore,
as $b\in \mathcal H^+$, we have
$\lambda_-(\Gamma)= [\Gamma]$.
When $s\to 0_+$, then $W^u(p, X_s)$ accumulate to $+g^k\cdot W^u(p,X_0)^+$ for every $k>0$ and will intersect $g^k\cdot\Delta_0$ transversely at a single point. Thus,
we have
$\lambda_+(\Gamma)=(1+g+g^2+\ldots)\cdot [\Gamma]$.
The change of $\lambda_\pm(\Gamma)$ from $s<0$ to $s>0$
is well given by
Formula \eqref{change}.\\
\noindent {\bf Case (a.2).} As $b\in \mathcal H^-$ and taking into account the accumulation described
right above, we have:
$ \lambda_-(\Gamma)= (1- g)\cdot [\Gamma]$ and $ \lambda_+(\Gamma)= [\Gamma]$.
The wanted formula is still satisfied.\\
\noindent {\bf Case (b.1).} Here, the accumulation of $W^u(p, X_s)$ is
dictated by part 2) of Lemma \ref{lem:Passages} and the reason why formula \eqref{change} holds is more surprising than in the previous cases.
When $s\to 0_-$, the manifolds $W^u(p, X_s)$ accumulate to $-g\cdot W^u(p, X_0)^-$
and to $-g^2\cdot W^u(p, X_0)^+$ and then nothing else.
When $s\to 0_+$, the manifolds $W^u(p, X_s)$ accumulate to
$+g\cdot W^u(p, X_0)^+$ and nothing else.
As $b\in \mathcal H^+$, we have $\lambda_-(\Gamma)= (1-g^2)\cdot [\Gamma]$
and
$\lambda_+(\Gamma)= (1+g)\cdot [\Gamma]$.
Formula \eqref{change} is right since the identity
$(1+g+g^2+\ldots)(1-g^2) = 1+g$ holds in the Novikov ring.\\
\noindent {\bf Case (b.2).} Accumulation is as right above. One derives that
$\lambda_-(\Gamma)= (1-g)\cdot [\Gamma]$ and $\lambda_+(\Gamma)= [\Gamma]$. The wanted formula is still satisfied.
The proof of Theorem \ref{thm:selfslideAlg} is now complete. \hfill$\Box$\\
\vskip 1cm
\section{\sc Proof of Theorem \ref{thm:Doubling}}
\subsection{\sc Notations and statement.}
In this section, we state and prove the refined version of Theorem \ref{thm:Doubling} which is given
right below after specifying some definition and notations. In the whole section
$X_0$ is an element in the codimension-one stratum $\mathcal{S}_g^0\subset \mathcal{S}_g$, the co-oriented
locus where the character function $\chi$ vanishes. It is assumed that $(\alpha,X_0)$
does not belong to $\mathcal{S}_g^{0,0}$, the locus where both $\phi$- and $\psi$-latitudes vanish.
\begin{defn} \label{decompositionS^0}
${}$
\noindent {\rm 1)} Let $\mathbb{R}_+$ {\rm (}resp. $\mathbb{R}_-${\rm )}
be the set of positive {\rm (}resp. negative{\rm )} real numbers.
The open set $\mathcal{S}_g^{0,\pm}\subset \mathcal{S}_g^0$ is defined by the sign of the $\phi$-latitude, that is,
$\omega_\phi(X_0)\in \mathbb{R}_\pm$.\smallskip
\noindent {\rm 2)} Let $X_0\in \mathcal{S}_g^0$.
Let $\bigl(\mathcal{D}(s,t):=X_{s,t}\bigr)$
be a germ in $X_0$ of a two-parameter family of vector fields of pseudo-gradients of $\alpha$.
It is assumed that $\mathcal{D}$ passes through $X_0$ when
$(s,t)=(0,0)$. This germ is said to be \emph{adapted} to the pair $(\mathcal{S}_g,\mathcal{S}_g^0)$
if the next conditions are fulfilled:
\begin{enumerate}
\item The one-parameter family $\bigl(\mathcal{D}(0,t)\bigr)_t$ is contained in $\mathcal{S}_g$, transverse to
$\mathcal{S}_g^0$ and $\displaystyle\frac{\partial \mathcal{D}}{\partial t}(0,0)$ points toward{s} $\mathcal{S}_g^+$.
\item The partial derivative $\displaystyle\frac{\partial \mathcal{D}}{\partial s}(0,0)$ is transverse to $\mathcal{S}_g$ and points towards to its positive side.
\end{enumerate}
\end{defn}
In particular, $\mathcal{D}$ is transverse to $\mathcal{S}_g^0$.
\begin{thm}\label{thm:DoublingRefined}
Let $\mathcal{D}$ be
a germ of $2$-disc transverse to $\mathcal{S}_g^0\smallsetminus \mathcal{S}_g^{0,0}$ and adapted to
the pair $(\mathcal{S}_g,\mathcal{S}_g^0)$. Then $\mathcal{D}$ intersects $\mathcal{S}_{g^2}$ transversely and
the trace on $\mathcal{D}$ of the strata $\bparent{ \mathcal{F}_\alpha\,,\,\mathcal{S}_g\cup \mathcal{S}^+_{g^2}\, ,\, \mathcal{S}_g^{0,\pm}}$
is $C^1$-diffeomorphic to
$$\bparent{\mathbb{R}^2,\, \mathbb{R}\times \{0\}\cup\{0\}\times \mathbb{R}_{\pm}\,,\, \{(0,0) \}}\,.$$
Moreover, the natural co-orientation of $\mathcal{S}_{g^2}$ restricts to
the natural co-orientation of $\mathcal{S}_g^{0}$ in $\mathcal{S}_g$ or to its opposite depending upon
$\mathcal{S}_{g^2}$ approaches $\mathcal{S}_g^{0,+}$ or $\mathcal{S}_g^{0,-}$ respectively.
\end{thm}
We first prove Theorem \ref{thm:DoublingRefined} for particular germs $\mathcal{D}$ which we call
{\it elementary}. These germs consist of one-parameter family of positive normalized crossing paths
of $\mathcal{S}_g$ in the sense of Definition \ref{normalization}.
\subsection{\sc Elementary crossing path.}\hskip -.2cm
\footnote{This terminology is inspired by that which J. Cerf used in his study of the stratification of smooth functions \cite{cerf}.}${}$
Let $(X_s)_s$ be a normalized positive crossing path of $\mathcal{S}_g$.
After the normalization we are allowed
to prescribe more special dynamics of $X_s$;
the perturbed holonomy will be specified near the respective homoclinic orbit. This is the flavor
of what {\it elementary} means.
Consider the spherical annulus $\mathbb{A}_{\psi_0}:=\Sigma^-\times (0,\sqrt\delta)\times \{\psi_0\}\subseteq \partial^-\mathcal{M}_p$.
Assume the $\psi$-latitude of $a^+$ different from zero. Therefore,
the preimage $D'_1(s) $ of $\Sigma^+$ by the
holonomy diffeomorphism $H_s$
is transverse to $\mathbb{A}_{\psi_0}$. Call $b(s)$ the intersection point $D'_1(s)\cap \mathbb{A}_{\psi_0}$
when the intersection is non-empty; it is the case either when $s< 0$ or $s>0$ but not in both cases.
\begin{defn}\label{elementary}
The path $\left(X_s\right)_s$ is said to be \emph{elementary} if the next conditions are fulfilled.
\begin{enumerate}
\item The disc $D_1(s)$ moves in the meridian disc $\{\psi=\psi_0\}$ parallely to
the the hyperplane $\Delta^\phi$.
\item We have ${\displaystyle \frac{da^+(s)}{ds}=1}$ for every $s$.
\item The point $b(s)$ runs on the ray
$\{(\phi_0,r,\psi_0)\mid r\geq 0\}$ for $s$ close to $0$
\emph{positively} or \emph{negatively}
if the
$\psi$-latitude $\omega_\psi(a^+(X_0))$ is \emph{negative} or \emph{positive}
respectively
\item The
velocity
of $b(s)$
is
\begin{equation}
\frac{db(s)}{ds}=-\frac{1}{\eta\,\omega_\psi(a^+)}\ .
\end{equation}
Here, $\eta$ stands for the holonomic factor of $X_0$ {\rm (Definition \ref{def:HolFactor})}.
\end{enumerate}
\end{defn}
This definition makes sense only when the $\psi $-latitude of
$a^+(X_0)$ is not $0$, that is, when $X_0$ does not lie on the $\phi$-axis $\mathcal{S}_g^\phi$, (see \eqref{axis}). This
is always the case when $X_0$ belongs to $\mathcal{S}_g^0\smallsetminus \mathcal{S}_g^{0,0}$.
\begin{lemme}\label{lemme_elementary} Let
$X_{0}\in \mathcal{S}_g\smallsetminus \mathcal{S}_g^\phi$ be a pseudo-gradient of $\alpha$ in normal
form. Then there exists a germ of elementary path $\left(X_s\right)$
passing through $X_0$ and depending smoothly on $s$ in the $C^1$-topology.
\end{lemme}
\nd {\bf Proof.\ } Recall the tube $T$ with coordinates $(x,y,v,z)$ around the restricted homoclinic orbit $\underline{\ell}$ of $X_0$.
The holonomy $H_0$ is defined on a neighborhood of $\{z=0\}$ in $\partial^-\mathcal{M}_p$
to a neighborhood of $\{z=1\}$ in $\partial^+\mathcal{M}_p$.
Up to an isotopy of $M$ fixing $\mathcal{M}_p$, a one-parameter perturbation $X_s$ of $X_0$ is determined
by its perturbed holonomy $H_s$.
This holonomy can be decomposed as $H_s=H_0\circ K_s$ where $K_s$
is a diffeomorphism of $\partial^-\mathcal{M}_p$ supported in the interior of this manifold with boundary
and $C^1$-close to $Id$.
In order to satisfy conditions (1) and (2) of Definition \ref{elementary}, we first choose $a^+(s)$
and $D_1(s)$ before constructing $H_s$.
Denote $a^+(s)$ the point in $\{\psi=\psi_0\}$ moving on the oriented pole axis with velocity $+1$
and such that $a^+(0)=a^+$. Let $D_1(s)$ be the paralell disc to $D_1(0)$ passing through $a^+(s)$.
Take $K_s$, smooth with respect to $s$, such that:
\begin{equation}\label{K_s}
K_s(\Sigma^-\cap\{z=0\})=H_0^{-1}(D_1(s))
\end{equation}
Apart from equality \eqref{K_s}, the restriction $K_s\vert_{\Sigma^-}$ is not imposed.
For instance, we may
choose $K_s(0,0,0,0)=(0,0,s,0)=H_0^{-1}(a^+(s))$. This implies the next constraint on
$K_s^{-1}$, namely:
\begin{equation}
\frac{\partial}{\partial s} (v\circ K_s^{-1})(0,0,0,0)\vert_{s=0}= -1
\end{equation}
Apart from this velocity constraint in $s=0$, we are free for defining $K_s^{-1}\vert_{H_0^{-1}(\Sigma^+)\cap
\{z=0\}}$,
that is the value of $K_s^{-1}(0,y,0,0)$. In particular, we can move $b(s)$
with the radial velocity which is prescribed in
Definition \ref{elementary}.
Indeed, according to the formula \eqref{eq:HolonDec}
relating the $v$-coordinates of the unit vectors $\partial_r^{\psi_0}$ and
$\partial_r^{\nu_\psi}$ and using \eqref{eq:Trigo}, we get $v(\dot b(0))= -1$.
\hfill$\Box$\\
Clearly, this lemma holds with parameters, for instance when the data is a one-parameter family
in $\mathcal{S}_g\smallsetminus \mathcal{S}_g^\phi$.
Then, the next corollary follows.
\begin{cor}\label{cor_elementary} Let
$X_{0,0}\in \mathcal{S}_g^0\smallsetminus\mathcal{S}_g^{0,0}$ and let $\ga(t)=\left(X_{0,t}\right)_t$ be a germ
of path in $\mathcal{S}_g$ passing through $X_{0,0}$ and crossing $\mathcal{S}^0_g$ transversely, such that
$\frac{\partial\gamma}{\partial t}(0)$ points towards $\mathcal{S}_g^+$. Then, there exists a
two-parameter family $\mathcal{D}=\left(X_{s,t}\right)$
of pseudo-gradients of $\alpha$ adapted to $(\mathcal{S}_g,\mathcal{S}^0_g)$ such that,
for every $t$ close to $0$, the path $s\mapsto X_{s,t}$ is elementary.
Moreover, there are such $X_{s,t}$
which are smooth with respect to the parameters
in the $C^1$-topology.\\
\end{cor}
\begin{defn}\label{two-elementary}
Let $\mathcal{D}$ be a $2$-disc transverse to $\mathcal{S}_g^0\smallsetminus \mathcal{S}_g^{0,0}$ and adapted to the pair $(\mathcal{S}_g,\mathcal{S}_g^0)$. We say that $\mathcal{D}$ is \emph{elementary} if it is made of a one-parameter family of elementary crossing paths as in {\rm Corollary \ref{cor_elementary}}.
\end{defn}
\noindent{\bf Proof of Theorem \ref{thm:DoublingRefined}.}
First, we prove the theorem in the
particular case where the transverse disc $\mathcal{D}$
is
elementary.
Even in this particular case the proof is slightly different depending on where the base point
$X_{0,0}$ lies either in $\mathcal{S}_g^{0,-}$ or $\mathcal{S}_g^{0,+}$. In each case, the proof has three items:
\begin{enumerate}
\item What is the trace on $\mathcal{D}$ of $\mathcal{S}_{g^2}$? Is there a non-empty trace of $\mathcal{S}_{g^k}$ for $k\neq 1\text{ or } 2$?
\item Is $\mathcal{D}$ transverse to $\mathcal{S}_{g^2}$? How is the positive co-orientation of $\mathcal{S}_{g^2}$?
\item Which
part $\mathcal{S}_{g^2}^+$ or $\mathcal{S}_{g^2}^-$
is intersected by $\mathcal{D}$?\\
\end{enumerate}
\noindent{\bf Case $X_{0,0}\in \mathcal{S}_g^{0,-}$.} In other words, $a^-(X_{0,0})$ has a negative $\phi$-latitude.
\smallskip
\noindent (1) The pseudo-gradient $X_{0,t}$ has a homoclinic orbit $\ell_t$
based in $p$ and the $\phi$-latitude of
$a^-(X_{0,t})$ lies in $[-1,0)$ for every $t\in \mathcal{O} p(0)$. Denote $\phi_t$ the spherical coordinate of
$a^-(X_{0,t})$. We use the tube $T$ around $\ell_0$ and its extremities: $\{z=0\}\subset \partial^-\mathcal{M}_p$
and $\{z=1\}\subset \partial^+\mathcal{M}_p$.
For simplicity, we more specify the path $\left(X_{0,t}\right)_t$
by adding some assumptions (the discussion is similar with the other cases of latitudes by using
other specifications\,\footnote{If
$\omega_\phi(a^-(X_{0,0}))=-1$, one makes $\eta(X_{0,t})$ vary increasingly with $t$. Since
$\omega_\psi(a^+(X_{0,0}))$ must be positve, $\frac{\partial \chi}{\partial t}(X_{0,t})>0$.}):
\begin{enumerate}
\item[(i)]The $\phi$-latitude $\omega_\phi(a^-(X_{0,0}))$ is not equal to $-1$
and the $\phi$-equator of $X_{0,t}$ is fixed.
\item[(ii)] The point $a^+(X_{0,t})= \Sigma^+\cap\ell_t$ and the $\psi$-equator of $X_{0,t}$
are fixed.
\item[(iii)] The holonomic factor $\eta(X_{0,t})$ remains
constant and is denoted by $\eta$.
\end{enumerate}
Notice
that (i) allows one to take
$(X_{0,t})_t$ positively transverse to $\mathcal{S}_g^0$ satisfying (ii) and (iii). More precisely, one makes the $\phi$-coordinate $\phi_t$ of $a^-(X_{0,t})$ vary on $t$ by increasing the $\phi$-latitude.
Denote the spherical coordinate of $a^+(X_{0,t})$ by $\psi_0$, independent of $t$.
In this setting, as the paths $s\mapsto X_{s,t}$ are elementary the discs $D_1(s,t)\subset \{z=1\}$
depend only on $s$ and are denoted by $D_1(s)$. For every $s\neq 0$, their
images by the descent map are
discs $C_1(s)$ contained in the {\it spherical annulus}
$\mathbb{A}_{\psi_0}:= \{(\phi,r,\psi_0)\}$.
When $s$ goes to $0_-$, according to Lemma \ref{lem:C1(s)}
the discs $C_1(s)$ accumulate to the negative hemisphere
$\mathcal{H}^-(\Sigma^-)$.
Since $s\mapsto X_{s,t}$ is elementary and $\omega_{\psi}(\psi_0)>0$, the disc $D'_1(s,t)$, preimage in $\{z=0\}$ of $\Sigma^+$
by the respective perturbed holonomy, intersects $\mathbb{A}_{\psi_0}$ in one point $b(s,t)$ when $s\leq 0$ and nowhere when $s>0$, according to Definition \ref{elementary}. When $t$ is constant, $b(s,t)$ moves on the ray
$\{(\phi_t, r,\psi_0)\mid r\geq 0\}$ and its velocity
is given by the formula in Definition \ref{elementary}.
According to Remark~\ref{rem:g2}, we have:
\begin{equation}\label{equationSg2}
X_{s,t}\in \mathcal{S}_{g^2}\quad \text{if and only if}\quad b(s,t)\in C_1(s).
\end{equation}
Denote by $c_1(s,t)$ the intersection point of
$C_1(s)$ with the meridian disc $\{\phi= \phi_t\} $. When $t$
is constant, $c_1(s,t)$ also moves on the ray
$\{(\phi_t, r,\psi_0) \mid r\geq 0\}$ and its radial velocity
is the same as the one of its lift through $Desc$ in $D_1(s)\subset \partial M_p^+$. Therefore,
\begin{equation}
\frac{\partial c_1(s,t)}{\partial s}= \frac {1}{\omega_\phi(\phi_t)}
\end{equation}
As $X_{0,0}$ belongs to $\mathcal{S}_g^0$, that is $\chi(X_{0,0})=0$,
the
curves $b(s,0)$ and $c_1(s,0)$, defined for $s<0$,
have the same radial velocities. Since both tend to
$a^-(X_{0,0})$ on the same ray when $s$ goes to $0_-$,
we have $b(s,0)=c_1(s,0)$ for every $s$. Then, Equation (\ref{equationSg2})
tells us that $X_{s,0}\in\mathcal{S}_{g^2}$ for every $s$ close to $0$ negatively.
For $t\neq 0$ and $s<0$, the radial velocities of $c_1(s,t)$ and $b(s,t)$ are distinct while
their limits
when $s$ goes to $0$
coincide. Therefore, Equation (\ref{equationSg2}) is never satisfied. When $s>0$, the discs
$C_1(s) $ accumulate to the positive hemisphere $\mathcal{H}^+(\Sigma^-)$. There is no chance
for $C_1(s)$ to intersect
$D'_1(s,t)$ which is
far from any point in
$\mathcal{H}^+(\Sigma^-)$.
What about $\mathcal{S}_{g^k}$? If $k\leq 0$, we have $u(g^k)\geq 0$ and there is no homoclinic orbit in the homotopy class $g^k$. When $k>2$, we have to discuss the successive
passages of the unstable manifold $W^u(p,X_{s,t})$ in $\partial^+\mathcal{M}_p$, more precisely in
$\{z=0\}$.
According to Lemma \ref{lem:Passages}, if $t>0$, that is $\chi(X_{0,t})>0$, only the discs
$C_2(s,t)$ of the second passage
are non-empty, but they accumulate to the positive hemisphere $\mathcal{H}^+(\Sigma^-)$.
Therefore, no further passage could give rise to a homoclinic orbit.
If $t<0$, one is able to see that there are infinitely many passages in $\{z=0\}$. But, by velocity considerations
$C_k(s,t)$ never meet $D'_1(s,t)$. We do not give more details here because this is similar
to the symmetric case $X_{0,0}\in \mathcal{S}_g^{0,+}$ and $t>0$ where the analysis of velocities
will be completely performed. Thus, the first item of case $X_{0,0}\in \mathcal{S}_g^{0,-}$ is proved.\\
\noindent (2) The reason for transversality to $\mathcal{S}_{g^2}$ relies again on some
computations of velocity.
Define for $s\leq 0$:
$$
\delta(s,t):= v\left(c_1(s,t)\right)-v\left(b_1(s,t) \right) \quad \text{and} \quad
V(t):=\frac{\partial\delta(s,t)}{\partial s} \vert_{s=0}\,.
$$
Despite the points are not the same, this velocity $V(t)$ at $s=0$ is easily checked to be
the same as the velocity computed in Lemma \ref{geometry-character}. Then, for every $t$ close to $0$
we have:
\begin{equation}\label{V}
V(t)= \eta\,\frac{\omega_{\psi}(\psi_0)}{\omega_\phi(\phi_t)} +1\quad \text{which implies} \quad \frac{dV(t)}{dt}<0\,.
\end{equation}
By definition of the character function, we have $V(0)=0$ which implies $V(t)<0$ for $t>0$.
Define
$V(s,t):= \partial_s\delta (s,t)$. By construction of $(X_{s,t})$, we have $V(s,0)=0$ for every $s<0$ close to 0.
According to (\ref{V}), the second partial derivative
$\partial^2_{t s}\delta(s,t)$ is negative for every $(s,t)$ close to $(0,0)$ with $s\leq 0$ (here we use the
smoothness with respect to the parameters\footnote{The vector fields in a normalized crossing path are
not smooth with respect to the space variable. Their holonomy is $C^1$ only. Nevertheless, as the
$C^1$-maps (of degree zero) $\partial^-\mathcal{M}_p\to \partial^+\mathcal{M}_p$ form a Banach manifold it makes sense to
consider a smooth family of such holonomies.}).
By integrating
in the variable $s$ from $s_0<0$ to $0$ and noticing that $\delta(0,t)=0$, we get:
\begin{equation}\label{pt}
\frac{\partial \delta}{\partial t} (s_0,t)= -\frac{\partial}{\partial t} \left(\int_{s_0}^{0}\frac{\partial \delta}{\partial s}(s,t)\, ds\right)
= -\left(\int_{s_0}^{0} \partial_{st}^2\delta(s,t)\, ds\right) >0.
\end{equation}
For $t=0$, this is exactly the transversality of $\mathcal{D}$ to $\mathcal{S}_{g^2}$ at
$X_{s_0,0}$. \\
We are now looking at orientation. Take
$s_0<0$ such that $b(s_0,0)$
lies in $\{z=0\}$. It belongs to a homoclinic orbit $\ell'$ in the homotopy class $g^2$. There is a tube
$T'$ around $\ell'$ with coordinates $(x',y', v', z') $. The $y'$-axis is contained in
$D'_1(s_0,0)$ and is given a co-orientation which follows from the co-orientation
of $D'_1(0,0)$ by continuity.
The $x'$-axis is contained in $C_1(s_0)$. Its projection to the $x$-axis is orientation reversing
(Lemma~\ref{lem:C1(s)}). Therefore:
\begin{equation} \label{w}
v(\partial_{v'}) <0.
\end{equation}
By (\ref{pt}) we have:
${\displaystyle \frac{\partial}{\partial t}[v\left(c_1(s_0,t)\right)-v\left(b_1(s_0,t) \right)]_{\vert_{t=0}} = \frac{\partial \delta}{\partial t} (s_0,0)>0}$.
By replacing $v$ with $v'$ in the last inequality, we get:
$$ \frac{\partial}{\partial t}[v'\left(c_1(s,t)\right)-v'\left(b_1(s,t) \right)]_{\vert_{t=0}}<0\,.$$
This translates the fact that $\partial _t$ points to the negative side of $\mathcal{S}_{g^2}$ for $s<0$
while for $s=0$, $\partial _t$ defines the positive co-orientation of $\mathcal{S}_g^0$ in $\mathcal{S}_g$.\\
\noindent (3) Let $L>0$ be a lenght. Consider a small circle $\ga\subset \mathcal{D}$ centered at the origin
of the coordinates $(s,t)$ and turning positively with respect to the orientation given by these coordinates.
If the radius of $\ga$ is small enough\footnote{More $L$ is large, more this radius has to be small.}
and if $X_{0,0}$ is generic,
$\ga$ avoids all codimension-one strata in $\mathcal F_\alpha$ except:
\begin{itemize}
\item $\mathcal{S}_g$ which is crossed once in $\mathcal{S}_g^-$ positively, and once in $\mathcal{S}_g^+$ negatively,
\item $\mathcal{S}_{g^2}$ which is crossed once positively according to the above discussion.
\end{itemize}
The product of the self-slide factors should equal 1 up to $L$ in the Novikov ring.
The self-slide factor of the a small arc of $\gamma$ crossing
$\mathcal{S}_{g^2}$ is for now unknown; call it $m(g)$.
This (commutative) product
is
$$(1+g)\cdot(1+g+g^2+...)^{-1}\cdot m(g)\,\,\equiv_L 1. $$
Then,
$m(g)\,\equiv_L\, (1-g^2)^{-1} $, that can only happen if
the crossing of $\mathcal{S}_{g^2}$
takes place in $\mathcal{S}_{g^2}^+$. The proof of Theorem \ref{thm:DoublingRefined} is complete
in the case $X_{0,0}\in \mathcal{S}_g^{0,-}$.\\
\noindent{\bf Case $X_{0,0}\in \mathcal{S}_g^{0,+}$.} In other words, $a^-(X_{0,0})$ has a positive $\phi$-latitude.
\smallskip
\noindent (1) The discussion is led in the same manner as in the previous case,
with same notation and simplified setting. We only
mention the main differences. Here, $a^-(X_{0,0})$ belongs to the positive hemisphere of
$\Sigma^-$ while $\psi_0$ belongs to the negative hemisphere of $\Sigma^+$. The discs $C_1(s)$ intersect
$\{z=0\}$ only when $s>0$. Therefore, for $s<0$ there is no chance for meeting $\mathcal{S}_{g^k}$,
for any $k\neq 0$.
According to Lemma \ref{lem:Passages} there are
infinitely many passages
$C_k(s), \ k\geq 1, s>0$ of $W^u(p, X_{s,t})$ in $\partial^-\mathcal{M}_p$ meeting $\{z=0\}$. Recall that
the $(i-1)$-discs $C_k(s)$
do not depend on $t$. The
fact that $X_{s,t}$ belongs to
$\mathcal{S}_{g^2}$ if and only if
$s>0,\ t = 0$ is proved exactly as
in the previous case.
Then, we are left
to show that for every $k>2$, $\mathcal{S}_{g^k}$ does not
intersect $\mathcal{D}$. Here it is important
to think of $\mathcal{D}$ as a germ because for a given representative this result is not true; when $k$
increases, the domain of the representative has to be restricted.
Let $C_k(s,t)$ denote the $(i-1)$-disc in $\partial^-\mathcal{M}_p$
corresponding to the $k$-th passage of the unstable manifold $W^u(p, X_{s,t})$
(see Lemma \ref{lem:Passages}); let $D'_1(s,t)$
denote the $(n-i-1)$-disc
corresponding to the first passage of the stable manifold $W^s(p, X_{s,t})$.
Observe that
$\mathcal{D}$ intersects $\mathcal{S}_{g^{k+1}}$ if and only if, for $(s,t)$ close to $(0,0)$,
$C_k(s,t)$ intersects $D'_1(s,t)$. This translates
in the next equation:
\begin{equation}
c_k(s,t)= d'_k(s,t)
\end{equation}
where $c_k(s,t)$ and $d'_k(s,t)$ are
the only two points of $\{z=0\}$ lying respectively on $C_k(s,t)$ and $D'_1(s,t)$
which have the same $(x,y)$-coordinates. Then, the above equation becomes:
\begin{equation}\label{equality}
v\bigl( c_k(s,t)\bigr)= v\bigl(d'_k(s,t)\bigr)\,.
\end{equation}
When $s$ goes to $0$, these two points go to the same point
$a^-(X_{s,t})\in\Sigma^-$. By computations done in the proof of Lemma \ref{lem:Passages},
we know that:
\begin{equation}
\frac{\partial}{\partial s}\left[v\bigl( c_k(s,t)\bigr)- v\bigl(d'_k(s,t)\bigr)\right]_{\vert_{s=0}}\neq 0\,.
\end{equation}
It follows that, for $s$ close to $0$ (closeness depending on $t$),
the equation (\ref{equality}) cannot be fulfilled. Theorem \ref{thm:DoublingRefined}
is now proved for elementary
2-discs as in Definition \ref{two-elementary}.\\
Let $\mathcal{D}$ be
an elementary 2-disc centered at $X_{0,0}$. Let $X_{0,0}$ run
in a neighborhood
$U\subset \mathcal{S}_g^0\smallsetminus\mathcal{S}_g^{0,0}$. This move extends to a move of $\mathcal{D}$
among
elementary discs. Limit ourselves to the $C^1$-topology in order that
the space of vector fields is a Banach manifold. Then, we have a $C^1$-map
$$F: U\times [-1,+1]^2\to \mathcal{F}_\alpha,\quad (X,s,t) \mapsto F(X,s,t),$$
such that $F(X,0,0)= X$, $F(X,0,t)$ belongs to $ \mathcal{S}_g$
and span$\{ \partial_sF,\partial_tF\}$
is transverse to
$ \mathcal{S}_g\smallsetminus\mathcal{S}_g^{0,0}$.
Here, we notice that the perturbed holonomy $H_{F(X,s,t)}$ depends linearly on $(s,t)$
near $a^-(X_{0,0})\in \partial^-\mathcal{M}_p$
and the extension by partition of unity makes $F$ depend smoothly on $(s,t)$ for the $C^1$-topology of
vector fields.
The Inverse Function Theorem is available and states that after a suitable restriction
(that we do not denote)
$F$ is a diffeomorphism
onto its image $\mathcal N$ which is an open set in $\mathcal{F}_\alpha$ ($C^1$-completed).
Therefore $\mathcal N$ has a product structure and a projection $P: \mathcal N\to [-1,+1]^2$ such that, for every $X\in \mathcal N$, the next equivalences hold:
\begin{equation}\label{equiv}
\left\{
\begin{array}{lcl}
X \in \mathcal{S}_g & \Longleftrightarrow& \bparent{s\circ P}(X)=0\\
X \in \mathcal{S}_g ^0 &\Longleftrightarrow & P(X)=(0,0)\\
X \in \mathcal{S}_{g^2}& \Longleftrightarrow & \bparent{t\circ P}(X)=0 \text{ and }
\bparent{s\circ P}(X)\in \mathbb{R}_\pm
\text{ depending on }X_{0,0}\in \mathcal{S}_g^{0,\pm}.
\end{array}
\right.
\end{equation}
Let $\mathcal{D}'$ be any germ of two-parameter family centered in $X_{0,0}$ transverse to
$\mathcal{S}_g^0\smallsetminus\mathcal{S}_g^{0,0}$ and contained in $\mathcal N$.
Its projection $P\circ \mathcal{D}'$ is submersive. The equivalences (\ref{equiv}) finish the proof of Theorem
\ref{thm:DoublingRefined}.
\hfill$\Box$\\
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"redpajama_set_name": "RedPajamaArXiv"
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Q: Download other page using AJAX Is there a way to download the page using javascript/ajax functions except for jQuery?
It is not an option.
A: Sure, use the ol' XMLHttpRequest.
A: If you are referring to downloading a html page served on another domain using ajax, then the answer is no. Due to security browsers does not allow that. Read more about that here: http://en.wikipedia.org/wiki/Same_origin_policy
You could probably solve this by using a service such as http://pipes.yahoo.com/pipes/ which uses CORS
A: It's possible using XMLHttpRequest if the page you're hitting is on the same domain. Just set the content type in the response to something like application/force-download.
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{"url":"https:\/\/bookdown.org\/huckley\/Physical_Processes_In_Ecosystems\/4-3-physics-forcelaws.html","text":"## 4.3 Basic Force Laws\n\nThe \u201cstage\u201d on which inorganic or organic processes take place was thought before 1905 to be the ordinary three-dimensional space of Euclidean geometry, with change occurring in a medium called \u201ctime.\u201d Einstein, in his theory of relativity, demonstrated the need for us to reshape our notions of space and time, since in different frames of reference (moving at different velocities with respect to each other), lengths and times will differ. Events which seem to occur simultaneously in one frame will not appear to be simultaneous to an observer in another frame. Relativity is of course a rather heady subject, and one which we won\u2019t discuss further, but it points up the difficulties which arise in defining precisely and appreciating the most basic quantities in physics.\n\nA notion of what \u201ctime\u201d is, for example, is so ingrained in our everyday experience, that we hardly think it necessary to define it. (PROBLEM: Try to define time!) The best the dictionary seems able to do is to define time as the \u201cperiod during which an action, process, etc., continues; measured or measurable duration.\u201d (Emphasis added.) Thus \u201ctime\u201d is a \u201cperiod,\u201d or a \u201cduration.\u201d What, pray, is a \u201cperiod\u201d or a \u201cduration?\u201d We may invent some more synonyms, but eventually we find ourselves caught in a circular definition\u2026time is time (and then you wave your hand and say \u201cYou know what I mean!\u201d). We are just going to have to content ourselves with this, provided we can give a fairly precise means of measuring that which we call \u201ctime.\u201d One way is to divide the period of rotation of the earth into 86,400 equal parts and call it the \u201csecond.\u201d This has been found to be insufficiently accurate for many reasons (one is that the earth is not that constant a clock). In 1960, the \u201csecond\u201d was defined in an international agreement to be a certain fraction of the particular year beginning on the vernal equinox of 1900, ending on the vernal equinox of 1901. This being a rather difficult standard to move about from laboratory to laboratory (you can\u2019t even go to Paris for it!), a recent definition has been provisionally accepted based on the number of oscillations of the cesium atom.\n\nWithout attempting to give a verbal definition of length (\u201cYou know what I mean!\u201d), we give you the standard length of one meter, which was formerly defined to be the distance between two scratches on a platinum-iridium bar kept in Paris and measured under standard conditions of temperature and pressure. This standard was based on an old measure of the earth\u2019s circumference, thought to have been 40,000 kilometers on a great circle passing through Greenwich, but the standard is now based on the wavelength of a particular line in the emission spectrum of krypton 86. This new primary standard is a more reproducible standard than the Pt-Ir bar (How wide is the scratch on the Pt-Ir bar? How closely can standard temperature and pressure be held?), and a more convenient measure since it can be maintained in one\u2019s own laboratory for about the cost of three or four round trip fares to Paris.\n\nThe \u201cactors\u201d on our \u201cstage\u201d of three-dimensional space plus time are particles\u2014atoms, molecules, protons, neutrons, electrons, etc. The particles are characterized by several fundamental properties, which when fully understood should, in theory at least, allow us to comprehend the most complicated of life processes. These properties (expressed as the forces which particles exert on each other) are surprisingly few in number; there are four basic force laws, (1) gravitational, (2) electromagnetic, (3,4) weak and strong nuclear, which are used in conjunction with the law of \u201cinertial force\u201d and which embrace all of what the physicist understands of the universe today. (There may be more force laws forthcoming, e.g., a law describing the force holding the constituent parts of a proton together, but it\u2019s not likely that any newly discovered force laws will have a strong bearing on life processes.)\n\nThe atomic theory in conjunction with a knowledge of the force laws will allow us to view a gas as a collection of moving particles, whose pressure is the result of collisions of these particles with the walls of its containing vessel, or perhaps with your eardrums. We will be able to calculate pressure in terms of the \u201cinertial force\u201d (i.e., the change in momentum of the molecules as they bang into the wall and reverse their direction). The drift of the particles, if they\u2019re all moving in one direction, will be called wind. If the motion of the particles is random, we shall call it heat. Ii the motion is in waves of excess density occurring at a regular frequency, we will know it as sound, whose pitch we\u2019ll discover depends on the frequency.\n\nThe understanding of these things based on so few underlying principles, is a remarkable achievement. We shall proceed by discussing the notions of mass and force, and then by describing the force laws.\n\n### 4.3.1 Inertia\n\nIn just the same way that \u201ctime\u201d and \u201clength\u201d were so difficult, in fact impossible, to provide true definitions for (in the mathematical sense of definition), we shall find that \u201cmass\u201d and \u201cforce\u201d must also be defined with some hand-waving. The student should not despair\u2014this is not the fault of physics, nor of this presentation\u2014it\u2019s just the way life is. Even the mathematician must face up to a fundamental imperfection in his otherwise perfect discipline. For mathematics must eventually trace all of its definitions back to some primitive concept, the universally accepted primitive concept being that of the \u201cset.\u201d What is a \u201cset\u201d of objects? Well, it\u2019s a \u201ccollection\u201d of them. What\u2019s a collection? It\u2019s a \u201cgroup.\u201d Et cetera. Eventually we use up all our synonyms for \u201cset,\u201d and return to\u2026\u201cset.\u201d The definition is circular. Thus mathematicians rely on grasping intuitively, without precise definition, the notion of set. Once that is accepted, of course, mathematics is on sound footing.\n\nThus we shall simultaneously introduce the ideas of force and mass (and hence, inertia) by presenting Newton\u2019s first and second laws:\n\nNewton\u2019s first law (law of inertia): Every body will remain in a state of uniform motion unless acted on by external force.\n\nNewton\u2019s second law: The acceleration of a particle is directly proportional to the resultant external force acting on the particle, is inversely proportional to the mass of the particle, and has the same direction as the resultant force.\n\nThe second law is usually written: $$$F=ma \\tag{4.1}$$$ where\n$$a$$ = acceleration (change in velocity per unit time)\n$$m$$ = mass\n$$F$$ = force\n\nIf we know the meaning of position and time, then velocity (time rate of change of position, meters\/sec) and hence acceleration (time rate of change of velocity, meters \/sec2) give us no problem. Let us say for the moment that we have some intuitive sense of what \u201cmass\u201d is. In fact, let us \u201cdefine\u201d one kilogram of mass to be the \u201cquantity of matter\u201d in a certain cylinder of platinum-iridium alloy preserved at the International Bureau of Weights and Measures in Paris, and let us measure unknown masses by balancing them opposite this standard (both masses presumably being acted on by the same \u201cforce\u201d due to gravity).\n\nWe then might be inclined to \u201cdefine\u201d force in such a way that Newton\u2019s second law holds. That is, if we observe that a body is either at rest or moving in a straight line at constant velocity (what\u2019s a straight line?), we will say that no net force is acting on the body. Or contrariwise, that if the body is accelerating, then a net force must be acting on the body. This has the effect of rendering Newton\u2019s second law as a mere definition with no physical content, and hence not an experimentally verifiable law of physics. However, the real content of Newton\u2019s laws is supposed to be this: \u201cthat the force is supposed to have some independent properties, in addition to the law $$F = ma$$; but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law $$F = ma$$ is an incomplete law. It implies that if we study the mass times the acceleration and call the product the force, i.e., if we study the characteristics of force as a program of interest, then we shall find that forces have some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces will be simple\u201d (Feynman 1963, Ch. 12). Furthermore, there is the implication that forces are of material origin, that if a body is observed to accelerate, we will find some physical body nearby which is the source of that force. Thus we must consider simultaneously with Newton\u2019s second law, the force laws associated with the nearby presence of matter.\n\nThis we\u2019ll do, upon noting that if Newton\u2019s second law holds, we may assign the units to force of the right-hand side of the expression, kg m\/s2, and since this is a clumsy unit, we shall call it the \u201cnewton\u201d:\n\n1 newton (nt) $$\\equiv$$ 1 kg m\/s2","date":"2021-05-13 02:56:06","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\": 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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.673226535320282, \"perplexity\": 704.1464602619328}, \"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-2021-21\/segments\/1620243992721.31\/warc\/CC-MAIN-20210513014954-20210513044954-00512.warc.gz\"}"}
| null | null |
Fred Howe, né le à Bredbury (Angleterre), mort en 1984, est un footballeur anglais, qui évolue au poste d'attaquant à Liverpool dans les années 1930.
Carrière
1931-1933 : Stockport County
1933-1935 : Hyde United
1935-1938 : Liverpool
1938 : Manchester City
1938-1939 : Grimsby Town
1946-1947 : Oldham Athletic
Notes et références
Naissance en septembre 1912
Naissance dans le Cheshire
Footballeur anglais
Joueur du Stockport County FC
Joueur du Liverpool FC
Joueur du Manchester City FC
Joueur du Grimsby Town FC
Joueur de l'Oldham Athletic AFC
Décès en 1984
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,767
|
Would you know whether your financial adviser were the next Bernie Madoff? Odds are you wouldn't. Swindlers don't exactly wear name tags as they fleece investors.
"We have witnessed a growing number of older Americans fall victim to financial swindles," says Don Blandin, CEO of the Investor Protection Trusts, which is cosponsoring a one-day hotline for older people.
See also: Financial help for elderly parents.
Check out telemarketing pitches; skepticism is one of the best forms of caution.
"Millions are in danger of being exploited," and one in five people over the age of 65 already has been exploited, according to Blandin's group.
"You can't get access to your money." If your broker-adviser tells you there's a "lock-up" period and you can't get your money out on short notice, stay away from the product. "When presented with an investment proposal, seek the guidance of a third party who has no vested interest in your decision — be very cautious with any investment that limits your ability to access the funds for unexpected personal or medical emergencies," advises Louis Straney, an expert on security fraud and author of the Investor's Guide to Loss Recovery.
"You'd better act now." Some advisers may try to rope you in by pushing the quick sell. They use phrases like "if you act now," or "we have limited space available." Any investment should be carefully considered over time. You shouldn't feel obligated just because you know the broker or if he has bought you a steak dinner.
And ignore a fancy office or bank lobby location. That doesn't mean your investment is any safer.
"Office location is of little importance," adds Straney. "In fact, cosmetic issues are a distraction to the tasks at hand. The critical issues are the training, supervision and motivation of the adviser."
Straney says you should ask about the adviser's background and qualifications, whether investors have access to key managers, and the breadth of products and services the broker is offering.
"If every recommendation is plastered with the company's logo," Straney says, "look elsewhere."
Come-ons tied to the headlines. Many swindles are tied to the news. Is the price of gold or oil soaring? Does the idea of buying cheap properties appeal to you? According to the North American Securities Administrators Association (NASAA), top scams over the past year involved distressed real estate, energy and precious metals investments, promissory notes and securitized life settlement contracts. That last vehicle involves buying a "piece" of somebody else's life insurance policy.
"Con artists follow the news and seek ways to exploit the headlines to the advantage while leaving investors holding an empty bag," said David Massey, NASAA president and North Carolina deputy securities administrator, in a statement.
Targeting of a person with cognitive impairment. Many swindlers target people whose mental sharpness may not be what it once was. Fraud merchants also go after the most vulnerable — women between the ages of 80 and 89 who live alone and need help with activities of daily life, according to the MetLife Institute. The cost to this group is nearly $3 billion annually.
The flimflam artists know that most older investors buy on trust alone. So there's one simple rule for you to follow: If you can't fully understand how an investment works and all of its downsides, then don't invest in it.
Shady marketing. Check out telemarketing, direct mail and "affinity" (social group) pitches. Skepticism is one of the best forms of caution. Keep talking to your loved ones, friends and neighbors. The best form of protection is vigilance.
Investor Protection Trust. It offers a helpful video on investment fraud among older people. Also see the IPT elder investment fraud program.
FINRA. This securities industry regulator has a host of resources that provide investor alerts and broker/adviser background checks.
Financial Planning Association. This group can provide basic financial advice and help you locate a certified financial planner near you.
John F. Wasik is a personal finance columnist for Reuters and the author of The Cul-de-Sac Syndrome (www.johnwasik.com) and 12 other books.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,341
|
@interface GIDSignInPreferencesTest : XCTestCase
@end
@implementation GIDSignInPreferencesTest
- (void)testGIDVersion {
NSString *version = GIDVersion();
XCTAssertTrue([version hasPrefix:@"gid-"]);
}
- (void)testGIDEnvironment {
NSString *environment = GIDEnvironment();
NSString *expectedEnvironment;
#if TARGET_OS_MACCATALYST
expectedEnvironment = @"macos-cat";
#elif TARGET_OS_IOS
#if TARGET_OS_SIMULATOR
expectedEnvironment = @"ios-sim";
#else
expectedEnvironment = @"ios";
#endif // TARGET_OS_SIMULATOR
#elif TARGET_OS_OSX
expectedEnvironment = @"macos";
#endif // TARGET_OS_MACCATALYST
XCTAssertEqualObjects(environment, expectedEnvironment);
}
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,661
|
{"url":"http:\/\/tug.org\/pipermail\/texhax\/2011-July\/017909.html","text":"# [texhax] \\headline in LaTeX document\n\nv_2e at ukr.net v_2e at ukr.net\nTue Jul 19 14:56:21 CEST 2011\n\n Hello!\nI have encountered a problem trying to process a .tex-file containing\nthe command \"\\headline\" with the 'pdflatex' from my TeX Live\ndistribution. The following error message occurs in the log file:\n\n! Undefined control sequence.\nThe control sequence at the end of the top line\nof your error message was never \\def'ed. If you have\nmisspelled it (e.g., \\hobx'), type I' and the correct\nspelling (e.g., I\\hbox'). Otherwise just continue,\nand I'll forget about whatever was undefined.\n\nThe same happens when I try to use \"\\footline\".\nI searched a lot over the Internet, and as far as I understood,\nthese are both the plain TeX commands, but I was given a LaTeX\ntemplate containing such commands which I don't know how to process\nand haven't found a clear answer to this question.\n\nI'll appreciate any suggestions and advices.\n\n-----\n<v_2e at ukr.net>\n`","date":"2017-12-14 16:53:00","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\": 0.9823892712593079, \"perplexity\": 4105.787263008728}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"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-2017-51\/segments\/1512948545526.42\/warc\/CC-MAIN-20171214163759-20171214183759-00102.warc.gz\"}"}
| null | null |
Nick's Flick Picks: The Blog
A film blog under the influence
Picked Flick #92: Alice Adams
Among Katharine Hepburn's most famous and auspicious screen collaborators—including, in my own order of preference, George Cukor, Cary Grant, and Spencer Tracy—director George Stevens is the least fêted member, but his achievements with Hepburn should not be undervalued. Once an established star, she never looked more radiant than she did in 1942's Woman of the Year, where Stevens' generous showcasing of her look and her performance beautifully counteracts the script's rather mean imbalance against her. (Well, maybe until that cooking scene, anyway.) Earlier than that, Stevens gamely ushered her through a spritely and underseen J.M. Barrie adaptation called Quality Street (1937), where Hepburn's comic dual-performance paves the way as none of her previous roles had done for the screwball delights of Howard Hawks' Bringing Up Baby (1938) and the aristocratic wit of Cukor's The Philadelphia Story (1940). But moving back still earlier, it's not clear that any of Hepburn's once and future heights would have been reached without the pretext of her first truly great performance in Alice Adams, which finds her amidst a glorious, once-in-a-lifetime metamorphosis from the queer, coltish ingénue of 1933 (Little Women, Morning Glory, Christopher Strong) into the rounded sophistication of her later work, somehow softer and more confident all at the same time. Alice Adams is the moment where Hepburn becomes a star, and also the moment where she becomes truly lovable.
Adapted from a novel by Booth Tarkington—the writer, too, of The Magnificent Ambersons—Alice Adams tells the story of a lower-middle-class girl, not far past her schooling years, who positively quivers with longing to join the coterie of her more fashionable peers and to find the kind of domestic bliss that presumably once united her parents (the excellent Fred Stone and Ann Shoemaker), whose tacit bond of affection is now sorely tested by illness, monetary need, and other trials of late middle-age. Alice Adams is the kind of girl who would adore Pride and Prejudice, even though in real life she might well have settled for Mr. Collins. One of the major ambitions of the screenplay and, I'm guessing, the novel is to keep Alice so dotingly loyal to her family even as she dreams of something bigger or other than what they have, which often compels a shame of her circumstances and a coy dishonesty about who she is and who they are. That the emblematically patrician Hepburn is so convincing within both this cast and this caste is a complete revelation, even more so in hindsight than it must have been in 1935, but her empathetic connection to this girl's gossamer aspirations couldn't be clearer. Her body and voice are much more relaxed than we're used to seeing and hearing them, and even though she takes center stage in a way she wouldn't truly do again until David Lean's Summertime in 1955, she holds the movie, as she does the character, with graceful, unpugnacious care, as though cupping her hand around the spores of a dandelion, keeping them from blowing away.
Stevens, so intuitive and judicious in realizing his best films, cuts to Hepburn at unexpected moments, lingering on her face longer than other directors would—possibly because, as in Woman of the Year, he's found the right angles and lighting concepts to make Hepburn's proudly intellectual face stay remarkably open and emotive. But more than that, his gift falls in knowing when to cut to Alice, when to understand the debates and dramatic actions surrounding her as essentially her story, rather than that of the bumptious family unit or the town at large. The two centerpiece sequences of the movie, when the guileless and ill-dressed but optimistic Alice takes her Cagneyish brother Walter to a local-society ball, and later when suitor Fred MacMurray arrives chez Adams for an uncomfortably hot and subtly humiliating evening of dinner and conversation, rank among the greatest passages of narrative filmmaking in the American cinema of that decade. The style is elegant and holistic, even as it magically embraces such different elements—MacMurray's somewhat lumpen appeal, the adroit conveyance of stifling temperature, the wholly unexpected elegance of Walter's dancing, a tart cameo from Hattie McDaniel, a romantic proclivity for fades and dissolves on Alice when her spirits flag, and the almost neo-realist shot where she kicks her wilting, homemade bouquet of hand-picked violets under a chair. There is also, of course, the justly famous and encapsulating shot of Hepburn weeping in her bedroom after the ball, filmed through the rivulets of rain running down her window. Moment by moment by moment, Alice Adams reverberates with humble but sure-handed technique and a credible reverence for modesty as a virtue. The last line of the movie is "Gee whiz!", and as dramatically precipitous as it is—the one major miscalculation in the script, I think—the sentiment is fully shared by the audience. (Click here for the full list of Nick's Picked Flicks.)
Labels: 1930s, Best Actress, Favorites, Katharine Hepburn
posted by NicksFlickPicks at 10:45 PM
tim r said...
You make this sound so great. I will be getting the DVD imported I think. Hepburn fever in this household, you know...
NICK DAVIS
VISIT NICK–DAVIS.COM...
Follow @NicksFlickPicks
Hot Off the Presses!
The Desiring-Image: Gilles Deleuze and Contemporary Queer Cinema ($30/pbk). By Nick Davis. Oxford University Press, 2013. The book that earned me tenure at Northwestern. Offers a new theoretical model of queer film, born from Gilles Deleuze's rarely-integrated notions of cinema and desire. Chapter-length readings of Dead Ringers, Naked Lunch, Shortbus, The Watermelon Woman, Brother to Brother, Beau travail, and Velvet Goldmine, plus other films along the way! Written for a scholarly audience but hopefully interesting to anyone curious about recent cinema, ideas about desire, or LGBT aesthetics and politics. "Important and needed work...Deeply original." —D.N. Rodowick, "Seductive in its intellect and humbling in its prose." —Michele Aaron
Reading the Bromance: Homosocial Relationships in Film and Television ($32/pbk). Ed. Michael DeAngelis. Wayne State University Press, 2014. Academic pieces that dig into recent portraits in popular media, comic and dramatic, of intimacies between straight(ish) men. Includes the essay "'I Love You, Hombre': Y tu mamá también as Border-Crossing Bromance" by Nick Davis, as well as chapters on Superbad, Humpday, Jackass, The Wire, and other texts. Written for a mixed audience of scholars, students, and non-campus readers. Forthcoming in June 2014. "Remarkably sophisticated essays." —Janet Staiger, "Essential reading for anyone interested in contemporary models of gender and sexuality." —Harry Benshoff
Fifty Key American Films ($31/pbk). Ed. Sabine Haenni, John White. Routledge, 2009. Includes my essays on The Wild Party, The Incredibles, and Brokeback Mountain. Intended as both a newcomer's guide to the terrain and a series of short, exploratory essays about such influential works as The Birth of a Nation, His Girl Friday, Sweet Sweetback's Baadasssss Song, Taxi Driver, Blade Runner, Daughters of the Dust, and Se7en.
The Cinema of Todd Haynes: All That Heaven Allows ($25/pbk). Ed. James Morrison. Wallflower Press, via Columbia University Press, 2007. Includes the essay "'The Invention of a People': Velvet Goldmine and the Unburying of Queer Desire" by Nick Davis, later expanded and revised in The Desiring-Image. More, too, on Poison, Safe, Far From Heaven, and Haynes's other films by Alexandra Juhasz, Marcia Landy, Todd McGowan, James Morrison, Anat Pick, and other scholars. "A collection as intellectually and emotionally generous as Haynes' films" —Patricia White, Swarthmore College
Film Studies: The Basics ($23/pbk). By Amy Villarejo. Routledge, 2006, 2013. Award-winning film scholar and teacher Amy Villarejo finally gives us the quick, smart, reader-friendly guide to film vocabulary that every teacher, student, and movie enthusiast has been waiting for, as well as a one-stop primer in the past, present, and future of film production, exhibition, circulation, and theory. Great glossary, wide-ranging examples, and utterly unpretentious prose that remains rigorous in its analysis; the book commits itself at every turn to the artistry, politics, and accessibility of cinema.
Most recent screenings in each race;
multiple nominees appear wherever they scored their most prestigious nod... and yes, that means Actress trumps Actor!
* Denotes a recent reappraisal
Picture Noms % Seen:
Friendly Persuasion
Director Noms % Seen:
I Want to Live!
Actress Noms % Seen:
A Star Is Born ('54)
The Letter ('29)
Actor Noms % Seen:
The Affairs of Cellini
Sup Actress Noms % Seen:
Sup Actor Noms % Seen:
The Day of the Locust
The Paper Chase
Cinematography Noms % Seen:
King Kong ('76)
Shanghai Triad
Screenplay Noms % Seen:
The Strange Love of Martha Ivers D
The Bachelor and the Bobby-Soxer* B
Boomerang! C+
This Blog Sponsored by...
Chicagoans! This site doesn't even accept advertising, but I'm making an unsolicited exception for the best, freshest, most affordable meal you can enjoy in the Loop, at any time of the day, whether you're on the go or eager to sit. Cuban and Latin American sandwiches, coffees, pastries, salads, shakes, and other treats. Hand-picked, natural, and slow-cooked ingredients. My friendly neighborhood place, a jewel in my life even before the Reader and Time Out figured it out. Visit!
Picked Flick #93: I ♥ Huckabees
R.I.P. Rosa Parks
Picked Flick #94: Cemetery Man
Picked Flick #95: Possessed
Picked Flick #96: Masked and Anonymous
Picked Flick #97: George Washington
Picked Flick #98: Brother's Keeper
Picked Flicks #99: The Breakfast Club & Pretty in ...
Let's Start Picking Some Flicks
Watch this space! Chicago has a new, exciting, important, and totally accessible cadre of queer film critics who are joining forces to bring screenings, special events, and good, queer-focused movie chats to our fair city. Read our mission! Stay tuned for events! Cruise the website, and help get this great new group off the ground by enrolling as a friend (it's free!) and by asking how you can help.
since 5.27.05
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,047
|
// ***********************************************************************
// <copyright file="FieldDefinition.cs" company="ServiceStack, Inc.">
// Copyright (c) ServiceStack, Inc. All Rights Reserved.
// </copyright>
// <summary>Fork for YetAnotherForum.NET, Licensed under the Apache License, Version 2.0</summary>
// ***********************************************************************
namespace ServiceStack.OrmLite;
using System;
using System.Collections;
using System.Reflection;
using ServiceStack.Text;
/// <summary>
/// Class FieldDefinition.
/// </summary>
public class FieldDefinition
{
/// <summary>
/// Gets or sets the model def.
/// </summary>
public ModelDefinition ModelDef { get; set; }
/// <summary>
/// Gets or sets the name.
/// </summary>
/// <value>The name.</value>
public string Name { get; set; }
/// <summary>
/// Gets or sets the alias.
/// </summary>
/// <value>The alias.</value>
public string Alias { get; set; }
/// <summary>
/// Gets the name of the field.
/// </summary>
/// <value>The name of the field.</value>
public string FieldName => this.Alias ?? this.Name;
/// <summary>
/// Gets or sets the type of the field.
/// </summary>
/// <value>The type of the field.</value>
public Type FieldType { get; set; }
/// <summary>
/// Gets or sets the field type default value.
/// </summary>
/// <value>The field type default value.</value>
public object FieldTypeDefaultValue { get; set; }
/// <summary>
/// Gets or sets the type of the treat as.
/// </summary>
/// <value>The type of the treat as.</value>
public Type TreatAsType { get; set; }
/// <summary>
/// Gets the type of the column.
/// </summary>
/// <value>The type of the column.</value>
public Type ColumnType => TreatAsType ?? FieldType;
/// <summary>
/// Gets or sets the property information.
/// </summary>
/// <value>The property information.</value>
public PropertyInfo PropertyInfo { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is primary key.
/// </summary>
/// <value><c>true</c> if this instance is primary key; otherwise, <c>false</c>.</value>
public bool IsPrimaryKey { get; set; }
/// <summary>
/// Gets or sets a value indicating whether [automatic increment].
/// </summary>
/// <value><c>true</c> if [automatic increment]; otherwise, <c>false</c>.</value>
public bool AutoIncrement { get; set; }
/// <summary>
/// Gets or sets a value indicating whether [automatic identifier].
/// </summary>
/// <value><c>true</c> if [automatic identifier]; otherwise, <c>false</c>.</value>
public bool AutoId { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is nullable.
/// </summary>
/// <value><c>true</c> if this instance is nullable; otherwise, <c>false</c>.</value>
public bool IsNullable { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is indexed.
/// </summary>
/// <value><c>true</c> if this instance is indexed; otherwise, <c>false</c>.</value>
public bool IsIndexed { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is unique index.
/// </summary>
/// <value><c>true</c> if this instance is unique index; otherwise, <c>false</c>.</value>
public bool IsUniqueIndex { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is clustered.
/// </summary>
/// <value><c>true</c> if this instance is clustered; otherwise, <c>false</c>.</value>
public bool IsClustered { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is non clustered.
/// </summary>
/// <value><c>true</c> if this instance is non clustered; otherwise, <c>false</c>.</value>
public bool IsNonClustered { get; set; }
/// <summary>
/// Gets or sets the name of the index.
/// </summary>
/// <value>The name of the index.</value>
public string IndexName { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is row version.
/// </summary>
/// <value><c>true</c> if this instance is row version; otherwise, <c>false</c>.</value>
public bool IsRowVersion { get; set; }
/// <summary>
/// Gets or sets the length of the field.
/// </summary>
/// <value>The length of the field.</value>
public int? FieldLength { get; set; } // Precision for Decimal Type
/// <summary>
/// Gets or sets the scale.
/// </summary>
/// <value>The scale.</value>
public int? Scale { get; set; } // for decimal type
/// <summary>
/// Gets or sets the default value.
/// </summary>
/// <value>The default value.</value>
public string DefaultValue { get; set; }
/// <summary>
/// Gets or sets the check constraint.
/// </summary>
/// <value>The check constraint.</value>
public string CheckConstraint { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is unique constraint.
/// </summary>
/// <value><c>true</c> if this instance is unique constraint; otherwise, <c>false</c>.</value>
public bool IsUniqueConstraint { get; set; }
/// <summary>
/// Gets or sets the order.
/// </summary>
/// <value>The order.</value>
public int Order { get; set; }
/// <summary>
/// Gets or sets the foreign key.
/// </summary>
/// <value>The foreign key.</value>
public ForeignKeyConstraint ForeignKey { get; set; }
/// <summary>
/// Gets or sets the get value function.
/// </summary>
/// <value>The get value function.</value>
public GetMemberDelegate GetValueFn { get; set; }
/// <summary>
/// Gets or sets the set value function.
/// </summary>
/// <value>The set value function.</value>
public SetMemberDelegate SetValueFn { get; set; }
/// <summary>
/// Gets the value.
/// </summary>
/// <param name="instance">The instance.</param>
/// <returns>System.Object.</returns>
public object GetValue(object instance)
{
var type = instance.GetType();
if (PropertyInfo.DeclaringType?.IsAssignableFrom(type) != true)
{
if (instance is IDictionary d)
return d[Name];
var accessor = TypeProperties.Get(type).GetAccessor(Name);
return accessor?.PublicGetter(instance);
}
return this.GetValueFn?.Invoke(instance);
}
/// <summary>
/// Sets the value.
/// </summary>
/// <param name="instance">The instance.</param>
/// <param name="value">The value.</param>
public void SetValue(object instance, object value)
{
if (instance is IDictionary d)
{
d[Name] = value;
return;
}
this.SetValueFn?.Invoke(instance, value);
}
/// <summary>
/// Gets the name of the quoted.
/// </summary>
/// <param name="dialectProvider">The dialect provider.</param>
/// <returns>System.String.</returns>
public string GetQuotedName(IOrmLiteDialectProvider dialectProvider)
{
return IsRowVersion
? dialectProvider.GetRowVersionSelectColumn(this).ToString()
: dialectProvider.GetQuotedColumnName(FieldName);
}
/// <summary>
/// Gets the quoted value.
/// </summary>
/// <param name="fromInstance">From instance.</param>
/// <param name="dialect">The dialect.</param>
/// <returns>System.String.</returns>
public string GetQuotedValue(object fromInstance, IOrmLiteDialectProvider dialect = null)
{
var value = GetValue(fromInstance);
return (dialect ?? OrmLiteConfig.DialectProvider).GetQuotedValue(value, ColumnType);
}
/// <summary>
/// Gets or sets the sequence.
/// </summary>
/// <value>The sequence.</value>
public string Sequence { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is computed.
/// </summary>
/// <value><c>true</c> if this instance is computed; otherwise, <c>false</c>.</value>
public bool IsComputed { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is persisted.
/// </summary>
/// <value><c>true</c> if this instance is persisted; otherwise, <c>false</c>.</value>
public bool IsPersisted { get; set; }
/// <summary>
/// Gets or sets the compute expression.
/// </summary>
/// <value>The compute expression.</value>
public string ComputeExpression { get; set; }
/// <summary>
/// Gets or sets the custom select.
/// </summary>
/// <value>The custom select.</value>
public string CustomSelect { get; set; }
/// <summary>
/// Gets or sets the custom insert.
/// </summary>
/// <value>The custom insert.</value>
public string CustomInsert { get; set; }
/// <summary>
/// Gets or sets the custom update.
/// </summary>
/// <value>The custom update.</value>
public string CustomUpdate { get; set; }
/// <summary>
/// Gets a value indicating whether [requires alias].
/// </summary>
/// <value><c>true</c> if [requires alias]; otherwise, <c>false</c>.</value>
public bool RequiresAlias => Alias != null || CustomSelect != null;
/// <summary>
/// Gets or sets the name of the belong to model.
/// </summary>
/// <value>The name of the belong to model.</value>
public string BelongToModelName { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is reference.
/// </summary>
/// <value><c>true</c> if this instance is reference; otherwise, <c>false</c>.</value>
public bool IsReference { get; set; }
public FieldReference FieldReference { get; set; }
/// <summary>
/// Gets or sets the custom field definition.
/// </summary>
/// <value>The custom field definition.</value>
public string CustomFieldDefinition { get; set; }
/// <summary>
/// Gets or sets a value indicating whether this instance is reference type.
/// </summary>
/// <value><c>true</c> if this instance is reference type; otherwise, <c>false</c>.</value>
public bool IsRefType { get; set; }
/// <summary>
/// Gets or sets a value indicating whether [ignore on update].
/// </summary>
/// <value><c>true</c> if [ignore on update]; otherwise, <c>false</c>.</value>
public bool IgnoreOnUpdate { get; set; }
/// <summary>
/// Gets or sets a value indicating whether [ignore on insert].
/// </summary>
/// <value><c>true</c> if [ignore on insert]; otherwise, <c>false</c>.</value>
public bool IgnoreOnInsert { get; set; }
/// <summary>
/// Gets or sets a value indicating whether [return on insert].
/// </summary>
/// <value><c>true</c> if [return on insert]; otherwise, <c>false</c>.</value>
public bool ReturnOnInsert { get; set; }
/// <summary>
/// Returns a <see cref="System.String" /> that represents this instance.
/// </summary>
/// <returns>A <see cref="System.String" /> that represents this instance.</returns>
public override string ToString() => Name;
/// <summary>
/// Shoulds the skip insert.
/// </summary>
/// <returns><c>true</c> if XXXX, <c>false</c> otherwise.</returns>
public bool ShouldSkipInsert() => IgnoreOnInsert || AutoIncrement || IsComputed && !IsPersisted || IsRowVersion;
/// <summary>
/// Shoulds the skip update.
/// </summary>
/// <returns><c>true</c> if XXXX, <c>false</c> otherwise.</returns>
public bool ShouldSkipUpdate() => IgnoreOnUpdate || IsComputed && !IsPersisted;
/// <summary>
/// Shoulds the skip delete.
/// </summary>
/// <returns><c>true</c> if XXXX, <c>false</c> otherwise.</returns>
public bool ShouldSkipDelete() => IsComputed && !IsPersisted;
/// <summary>
/// Determines whether [is self reference field] [the specified field definition].
/// </summary>
/// <param name="fieldDef">The field definition.</param>
/// <returns><c>true</c> if [is self reference field] [the specified field definition]; otherwise, <c>false</c>.</returns>
public bool IsSelfRefField(FieldDefinition fieldDef)
{
return fieldDef.Alias != null && IsSelfRefField(fieldDef.Alias)
|| IsSelfRefField(fieldDef.Name);
}
/// <summary>
/// Determines whether [is self reference field] [the specified name].
/// </summary>
/// <param name="name">The name.</param>
/// <returns><c>true</c> if [is self reference field] [the specified name]; otherwise, <c>false</c>.</returns>
public bool IsSelfRefField(string name)
{
return Alias != null && Alias + "Id" == name
|| Name + "Id" == name;
}
/// <summary>
/// Clones the specified modifier.
/// </summary>
/// <param name="modifier">The modifier.</param>
/// <returns>FieldDefinition.</returns>
public FieldDefinition Clone(Action<FieldDefinition> modifier = null)
{
var fieldDef = new FieldDefinition
{
Name = Name,
Alias = Alias,
FieldType = FieldType,
FieldTypeDefaultValue = FieldTypeDefaultValue,
TreatAsType = TreatAsType,
PropertyInfo = PropertyInfo,
IsPrimaryKey = IsPrimaryKey,
AutoIncrement = AutoIncrement,
AutoId = AutoId,
IsNullable = IsNullable,
IsIndexed = IsIndexed,
IsUniqueIndex = IsUniqueIndex,
IsClustered = IsClustered,
IsNonClustered = IsNonClustered,
IsRowVersion = IsRowVersion,
FieldLength = FieldLength,
Scale = Scale,
DefaultValue = DefaultValue,
CheckConstraint = CheckConstraint,
IsUniqueConstraint = IsUniqueConstraint,
ForeignKey = ForeignKey,
GetValueFn = GetValueFn,
SetValueFn = SetValueFn,
Sequence = Sequence,
IsComputed = IsComputed,
IsPersisted = IsPersisted,
ComputeExpression = ComputeExpression,
CustomSelect = CustomSelect,
BelongToModelName = BelongToModelName,
IsReference = IsReference,
FieldReference = FieldReference,
CustomFieldDefinition = CustomFieldDefinition,
IsRefType = IsRefType,
};
modifier?.Invoke(fieldDef);
return fieldDef;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,871
|
\section{introduction}
In recent years, distinction between extreme and nonextreme black holes
became an object of intensive studies. A nontrivial peculiarity of the
extreme state consists in that its properties depend crucially on the way
the limiting transition is performed. Consider, for instance, the
Reissner-Nordstr\"{o}m black hole as the simplest example. If its mass $m$
is put equal to its charge $e$, the proper distance between the horizon and
any other point outside it diverges, the topology corresponding to the
annulus of an infinite size. As a result, the vicinity of the horizon
responsible for the entropy does not contribute to the Euclidean action.
This property gave grounds to reason that classical extreme black holes
possess unusual thermodynamic properties in the sense that their temperature
is arbitrary and is not connected with the Hawking one (which is zero in the
limit in question), while the entropy $S=0$ \cite{hawking}. On the other
hand, as was shown in \cite{zasl1}, \cite{zasl2}, there exists a limiting
transition $m\rightarrow e$ in the topological sector of nonextreme black
holes such that the proper distance between the horizon and points outside
it remains finite. Although either the surface gravity or the Hawking
temperature in this case are equal to zero, the physical temperature
defining properties of a thermodynamic ensemble is finite, and the entropy $%
S $ has the Bekenstein-Hawking value $A/4$, where $A$ the a horizon area.
Thus, we have two types of limiting states of black holes with essentially
different properties. For the sake of shortness we will further refer to the
first type as 1 and to the second type as 2. Types 1 and 2 possess different
Euler characteristics \cite{bin}. A sharp distinction between them manifests
itself not only in thermodynamics and Euclidean approach but also in
Lorentzian quantum geometrodynamics \cite{claus}. In fact, the states of
type 2 represent the direct product of two-dimensional spaces --- for
example, $AdS_{2}xS_{2}$ (see below in a more detail). In this sense, they
are not real black holes. However, as they arise as a result of the limiting
transition from the true ones whose horizons certainly possess thermodynamic
properties, it is necessary to elucidate what happens to these properties in
the limiting states.
Up to now our treatment concerned classical black holes. Meanwhile, studying
quantum effects in the background of extreme black holes is of special
interest. If a black hole is in state 1 there are good grounds to believe
that the principal conclusion $S=0$ made in \cite{hawking} for classical
black holes loses its validity when effects of backreaction from quantum
fields surrounding a hole are taken into account. These effects force the
temperature to take the Hawking value $T_{H}=0$ since otherwise the
stress-energy tensor of quantum fields diverges on a horizon \cite{anders},
so the possibility of thermodynamic description of extreme black holes
becomes questionable. In this respect, we are faced with the curious
situation when even weak switching on dynamical interaction between a hole
and its quantum environment changes drastically thermal properties of the
system.
Correspondingly, the question arises about the role of quantum effects for
state 2. In the paper \cite{ent} for the particular case of the Bertotti -
Robinson (BR) metric \cite{br}, which is the limiting form of the
Reissner-Nordstr\"{o}m black hole \cite{zasl2}, the quantum correction to
thermodynamic entropy $S_{q}$ was found. It turned out that this correction
does not change the thermodynamic properties of a system dressed by a
quantum field as compared with a bare one and, moreover, $S_{q}=0$. In the
present paper we generalize this result and show that it retains its
validity for a wide set of the limiting state belonging to class 2
independently of the particular form of the metric. Thus, the role of
quantum effects in thermal properties proves to be very different for both
types of states.
We consider also the influence of quantum effects on geometrical properties
of the type 2 states. First of all, even in the absence of quantum effects
the nature of a horizon in state 2 is different from that of a black hole
whose limiting form the state 2 represents. Thus, for the BR metric, the
existence and properties of a horizon is an observer-dependent effect due to
geodesic incompleteness in some accelerated frames \cite{lap}, so the
horizon of a Reissner-Nordstr\"{o}m black hole turns into an acceleration
one. The geometry of quantum-corrected acceleration horizons possesses two
main features. It retains the general form of the direct product of two
two-dimensional spacetimes typical of the classical counterpart and, in this
sense, our approach confirms the recent observation made for the particular
case of the geometry which represents the direct product of anti-de$%
\mathop{\rm Si}%
$tter space with a sphere \cite{ads}. However, we argue that the concrete
type of geometry may be changed by quantum backreaction in an essential way.
For example, it can lead to the appearance of a second, cosmological-like
horizon which is absent for a classical counterpart of the metric.
In turn, the general structure of a metric and, in particular, the
difference between an acceleration horizon as compared to a black hole one,
affects the thermodynamics in what concerns the formulation of the general
first law. In the black hole case, the usual picture implies that the
horizon radius is a free parameter allowed to vary, whereas the metric on a
boundary of a system may be kept fixed. However, for a typical metric with
an acceleration horizon the radius of a sphere entering an angular part of a
metric is constant, so if it is kept fixed the first law turns into an empty
identity. Therefore, we are faced here with a somewhat unusual situation
when the first law acquires a nontrivial meaning only under condition that a
boundary metric itself is varied. Another peculiarity consists in that the
quasilocal energy density of such a system coming from a gravitational
action is identically zero and nonzero contribution stems entirely from a
reference metric (usually chosen as a flat one).
The paper is organized as follows. In Sec. II we derive the general
expression for the entropy of quantum massless radiation valid in spacetimes
with either black hole horizons or acceleration ones and show that in the
latter case, $S_{q}=0$. This derivation relies on the definition of the
stress-energy tensor, its conservation law, the scale properties of massless
radiation, and does not use the first law.
In Sec. III we give an qualitative explanation to the found property $%
S_{q}=0 $ as connected with the Unruh effect in, generally speaking, curved
spacetimes. We discuss relationship between different pairs of spacetimes
which represent Minkowski - Rindler analogues in curved manifolds and show
how in some limit this analogy becomes literal coincidence.
In Sec. IV, we show that if in the formulation of the first law one properly
accounts for spatial gravitational stresses and the pressure of quantum
fields, this law confirms the property $S_{q}=0$.
In Sec. V, we derive the explicit form of the quantum-corrected geometry and
find that there are three qualitatively different types of solutions
depending on whether the curvature of submanifold in time-radial directions
is negative, positive or zero.
In Sec. VI, we summarize briefly the results obtained and mark some possible
problems for future research.
\section{entropy of hawking radiation in terms of stress-energy tensor}
Consider the Euclidean metric of the form
\begin{equation}
ds^{2}=d\tau ^{2}b^{2}+\alpha ^{2}dy^{2}+r^{2}(y)d\omega ^{2}\text{.}
\label{genmet}
\end{equation}
If $r$ is not constant, it can be chosen as a new radial variable and we
arrive at a generic spherically symmetrical spacetime. The special class of
metric arises when $r=const\equiv r_{+}$. It just corresponds to the
limiting transition to the extreme state of nonextreme black holes since in
that limit all points of a manifold take the same value of $r$ \cite{zasl2}.
In the particular case $b=r_{+}\sinh y$, $\alpha =r_{+}$ we obtain the BR
spacetime. While deriving in this section the formula for quantum entropy,
we will not specify further the coefficients $b$ and $\alpha $.
The total entropy of the system is equal to
\begin{equation}
S=S_{0}+S_{q}\text{.} \label{total}
\end{equation}
Here $S_{0}$ is the Bekenstein-Hawking entropy $S_{0}=A/4$ where $A$ is the
area of the event horizon and $S_{q}$ is the entropy of Hawking radiation.
While $S_{0}$ appears as the result of the zero-loop approximation in the
path-integral approach \cite{gib}, $S_{q}$ is due to quantum fields.The
quantity $S_{0}$ is determined solely by one characteristics of the event
horizon and in this sense manifests the universality of laws of black hole
physics. On the contrary, the quantity $S_{q}$ depends strongly on the kind
of fields and concrete details of matter distribution. If $r$ is not
constant, the expression for the entropy of Hawking radiation in terms of
the renormalized stress-energy tensor is obtained for a wide class of metric
\begin{equation}
ds^{2}=d\tau ^{2}f(\frac{r_{+}}{r})+dr^{2}f^{-1}+r^{2}d\omega ^{2}\text{, }%
d\omega ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2} \label{met}
\end{equation}
in \cite{rad}. Here $r_{+}$ is a horizon radius. The Schwarzschild metric
belongs to this class with $f(x)=1-x$. In the state of thermal equilibrium
the entropy of massless quantum field in this background inside a cavity of
a radius $r_{B}$ is equal to \cite{rad}
\begin{equation}
S_{q}=16\pi ^{2}r_{+}\left| f^{\prime }(1)\right|
\int_{+}^{r_{B}}drr^{2}(T_{r}^{r}-T_{0}^{0}-T_{\mu }^{\mu }\ln \frac{r_{B}}{r%
})\text{,} \label{ent1}
\end{equation}
where $T_{\mu }^{\nu }$ are the components of the renormalized stress-energy
tensor in the Hartle-Hawking state. As such a tensor is finite on a horizon
in an orthonormal frame, all components entering (\ref{ent1}) are finite and
the entropy converges. It is worth stressing that this is just thermodynamic
entropy which reveals itself in physical experiments but not a
statistical-mechanical part of it which for black holes has no direct
physical meaning and even diverges \cite{frolov}. This expression contains,
apart from two first terms typical of a classical thermal gas, also a purely
quantum anomaly part which is necessary for the general first law to hold
\cite{rad}.
Meanwhile, we are willing to obtain the entropy of quantum field valid for
the BR spacetime
\begin{equation}
ds^{2}=r_{+}^{2}(d\tau ^{2}\sinh ^{2}x+dx^{2}+d\omega ^{2}) \label{ber}
\end{equation}
and its generalization (\ref{genmet}). As the coefficient at $d\omega ^{2}$
may now be constant, we cannot use the result (\ref{ent1}) of \cite{rad}
directly. Below we suggest derivation suitable for both cases, when $r$ can
be either variable or constant.
It is convenient to normalize the radial coordinate in such a way that $y=0$
at a horizon and $y=1$ on the boundary, the period of the Euclidean time $%
\beta _{0}$ is chosen to be $2\pi $. It is implied that a system is situated
in a cavity of a finite size. Let the surface area of a horizon be equal to $%
\pi r_{+}^{2}$. We choose $y=l/l_{B}$, $l$ is a proper distance from a
horizon $(l=l_{B}$ for a boundary) and assume that metric coefficients take
the form
\begin{equation}
r=r_{+}q(l/r_{+})\equiv r_{+}q(zy)\text{, }b=r_{+}d(l/r_{+})\equiv r_{+}d(zy)%
\text{, }\alpha =l_{B}\text{, }z=l_{B}/r_{+} \label{coef}
\end{equation}
which embraces both cases (\ref{ber}) and (\ref{met}). Now we will show how
the expression for the entropy can be recovered from the components of the
stress-energy tensor.
Let us consider the variation of the metric which preserves the form (\ref
{coef}), so only parameters of the metric are allowed to change. This metric
has two such parameters - for instance, $z$ and $r_{+}$. Then the variation
of the Euclidean action $I$ of quantum field inside a cavity splits to two
parts: $\delta I=\delta _{1}I+\delta _{2}I$. Here the first term has the
standard form:
\begin{equation}
\delta _{1}I=\frac{1}{2}\int d^{4}x\sqrt{g}T_{\mu \nu }\delta g^{\mu \nu }%
\text{,} \label{1}
\end{equation}
$x^{0}=\tau $, $x^{1}=y$, $x^{2}=\theta $, $x^{3}=\phi $. It is worth
bearing in mind, however, that this term does not exhausts the total
variation since the formula (\ref{1}) implies that the metric on a boundary
is fixed. Meanwhile, we consider generic variation which affects the
boundary surface. Below we will see that the term $\delta _{2}I\sim \delta
r_{B}$ can be recovered from the scale properties of the action.
The terms with the variation of the metric can be written as
\begin{equation}
T_{\mu \nu }\delta g^{\mu \nu }=-2T_{0}^{0}\frac{\delta b}{b}%
-(T_{2}^{2}+T_{3}^{3})\frac{\delta r}{r}-T_{1}^{1}\frac{\delta \alpha }{%
\alpha }\text{.} \label{var}
\end{equation}
It follows from (\ref{coef}) that $\delta b/b=\delta r_{+}/r_{+}+\delta
zyd^{\prime }/d$, $\delta r/r=\delta r_{+}/r_{+}+\delta zyq^{\prime }/q$, $%
\delta \alpha /\alpha =\delta z/z+\delta r_{+}/r_{+}$, where the prime
denotes the derivative with respect to argument. Performing integration over
angle variables and Euclidean time we obtain for the first part of
variation: $\delta _{1}I/8\pi ^{2}=-\frac{\delta r_{+}}{r_{+}}\int_{0}^{1}dy%
\sqrt{\tilde{g}}T_{\mu }^{\mu }-\delta z\int_{0}^{1}dyy\sqrt{\tilde{g}}%
[T_{0}^{0}\frac{d^{\prime }}{d}+T_{1}^{1}z^{-1}+(T_{2}^{2}+T_{3}^{3})\frac{%
q^{\prime }}{q}]$ where $\sqrt{\tilde{g}}=b\alpha r^{2}=r_{+}^{4}zq^{2}d$
takes into account the fact that the factor $8\pi ^{2}$ due to integration
over angles and Euclidean time is already singled out. Now let us make use
of the conservation law $T_{1;\nu }^{\nu }\equiv \frac{(T_{1}^{\mu }\sqrt{g})%
}{\sqrt{g}}-\frac{1}{2}\frac{\partial g_{\alpha \beta }}{\partial x_{1}}%
T^{\alpha \beta }=0$. Taking into account the explicit form of the metric (%
\ref{genmet}) and scale properties of it we have: $\frac{(q^{2}dT_{1}^{1})^{%
\prime }}{q^{2}d}-[\frac{d^{\prime }}{d}T_{0}^{0}+(T_{2}^{2}+T_{3}^{3})\frac{%
q^{\prime }}{q}]=0$. By substituting this expression into $\delta _{1}I$ we
obtain after integration by parts:
\begin{equation}
\frac{\delta _{1}I}{8\pi ^{2}}=-\frac{\delta r_{+}}{r_{+}}\int_{0}^{1}dy%
\sqrt{\tilde{g}}T_{\mu }^{\mu }-\delta z\frac{\sqrt{\tilde{g}}}{z}%
T_{1}^{1}(z)\text{.} \label{var1}
\end{equation}
Let us write down also the term $\delta _{2}I/8\pi ^{2}\equiv \gamma \delta
r_{B}$. It follows from (\ref{coef}) that $\delta r_{B}=\delta
r_{+}q+q^{\prime }r_{+}\delta z=\delta r_{+}\frac{r_{B}}{r_{+}}+q^{\prime
}r_{+}\delta z$, whence
\begin{equation}
\delta I/8\pi ^{2}=\delta r_{+}(\gamma q-\frac{1}{r_{+}}\int_{0}^{1}dy\sqrt{%
\tilde{g}}T_{\mu }^{\mu })+\delta z[\gamma q^{\prime }r_{+}-\frac{\sqrt{%
\tilde{g}}}{z}T_{1}^{1}(z)]\text{.} \label{del}
\end{equation}
This general formula describes the response of the action to the change of
two variables $r_{+}$ and $z$.
Now let us take into account scale properties of the action of the massless
quantum field. It is a dimensionless quantity which must depend on
dimensionless arguments. In general, one can compose two such combinations
from the parameters of the problem: $z$ and $l_{0}/r_{+}$, where $l_{0}$ is
the Planck length. Correspondingly, one can write down $I=I(z,l_{0}/r_{+})$.
However, in the semiclassical approximation, when $l_{0}\ll r_{+}$ and all
effects of high-order loops are neglected, only the first parameter is
relevant: $I=I(z,0)\equiv I(z)$ plus negligible corrections. (Let me recall
that in the Scwharzschild case when either the entropy or the action of
quantum fields are calculated explicitly, they depend on one variable $%
r_{B}/r_{+}$ only, where $r_{B}$ is a radius of a system \cite{mat};
meanwhile, now we use $z=l_{B}/r_{+}$ instead of $r_{B}/r_{+}$). Thus, the
semiclassical action of massless fields depends only on one variable $z$: $%
I=I(z)$. I\ stress that this fact, as we will see below, is consistent with
the presence of quantum anomaly terms in the action.
This means that the coefficient at $\delta r_{+}$ should be equal to zero,
whence
\begin{equation}
\gamma =(r_{+}q)^{-1}\int_{0}^{1}dy\sqrt{\tilde{g}}T_{\mu }^{\mu }\text{.}
\label{gamma}
\end{equation}
By substitution into the part proportional to $\delta z$ we obtain
\begin{equation}
\delta I/8\pi ^{2}=\delta z[\frac{q^{\prime }}{q}\int_{0}^{1}dy\sqrt{\tilde{g%
}}T_{\mu }^{\mu }-\frac{\sqrt{\tilde{g}}}{z}T_{1}^{1}]\text{.} \label{2del}
\end{equation}
From dimension grounds it also follows that the stress-energy tensor has the
general form
\begin{equation}
T_{\mu }^{\nu }=r_{+}^{-4}t_{\mu }^{\nu }(r/r_{+})\equiv r_{+}^{-4}f_{\mu
}^{\nu }\text{,} \label{tensor}
\end{equation}
where $t_{\mu }^{\nu }(q(zy))\equiv $ $f_{\mu }^{\nu }(zy)$. In eq. (\ref
{2del}) the value of $T_{1}^{1}$ is to be taken at the boundary where $y=1$.
After simple transformations we have
\begin{equation}
\frac{1}{8\pi ^{2}}\frac{dI}{dz}=\frac{q^{\prime }}{q}%
\int_{0}^{z}dxd(x)q^{2}(x)f_{\mu }^{\mu }(x)-f_{1}^{1}(z)d(z)q^{2}(z)\text{.}
\label{der}
\end{equation}
Now we may find $I$ by direct integration. The constant of integration is
determined by the demand that $I=0$ when $z=0=l_{B}$ (no room for
radiation). Changing the order of integration in the first term we may get
rid of a double integral. Using the thermodynamic formula $I=-\int d^{4}x%
\sqrt{g}T_{0}^{0}-S_{q}$ let us write down the result at once for the
entropy:
\begin{equation}
S_{q}=\int_{0}^{z}dxq^{2}(x)d(x)[f_{1}^{1}-f_{0}^{0}-f_{\mu }^{\mu }\ln
\frac{q(z)}{q(x)}]\text{.} \label{S}
\end{equation}
Returning to dimensional variables we may rewrite this formula as
\begin{equation}
S_{q}=\int dV_{4}(T_{1}^{1}-T_{0}^{0}-T_{\mu }^{\mu }\ln \frac{r_{B}}{r})
\label{2S}
\end{equation}
Here $dV_{4}=\beta _{0}d^{3}x\sqrt{g}$ is the element of Euclidean
four-volume which in our particular coordinate system has the form (after
integration over angles and time variable) $dV_{4}$ $=8\pi
^{2}dxr_{+}^{3}q^{2}(x)d(x)$ where $x=yz$. Let me stress that the
contribution of conformal anomalies, as seen from (\ref{2S}), is taken into
account.
The formula (\ref{2S}) holds for any spacetime whose metric has the form (%
\ref{coef}). There are two typical classes of it. In the first case, the
metric can be rewritten as
\begin{equation}
ds^{2}=d\tau ^{2}U(r)+dr^{2}V^{-1}(r)+r^{2}d\omega ^{2}\text{.}
\label{spher}
\end{equation}
where $U$ and $V$ depend on $r_{+}$ via combination $r_{+}/r$. In
particular, the Schwarzschild metric belongs to this class, in which case $%
\beta _{0}=4\pi r_{+}$, $dV_{4}=16\pi ^{2}r_{+}r^{2}dr$ and we return to the
result (\ref{ent1}) describing a nonextreme black hole. The second class can
be obtained from (\ref{spher}) by a well-defined limiting transition to the
extreme state such that a local Tolman temperature remains finite nonzero
quantity at any point outside a horizon. In so doing, $r\rightarrow r_{+}$
for all points of manifolds. As a result, the metric takes the form (\ref
{genmet}) with $r(y)=r_{+}=const$ and $b$ typically representing a
combination of hyperbolic functions (see \cite{zasl2} for details). In so
doing, the components $T_{\mu }^{\nu }$ of the stress-energy tensors in an
orthonormal frame in any spacetime with a regular horizon pick up their
values from a horizon: $\lim_{r\rightarrow r_{+}}T_{\mu }^{\nu }(r)=T_{\mu
}^{\nu }(r_{+})$. Then the regularity condition on a horizon $%
T_{0}^{0}-T_{1}^{1}=0$ holds for a whole manifold. With the condition $%
r=r_{+}$, this means that both terms in entropy (\ref{S}), (\ref{2S})
(either ''normal'' or ''anomalous'' ones) are equal to zero, so $S_{q}=0$.
In particular, the previous result \cite{ent} for the Bertotti-Robinson
spacetime (for which $q=\sinh zy$ and $T_{\mu }^{\nu }\sim \delta _{\mu
}^{\nu }r_{+}^{-4}$\cite{kof}, \cite{paul}) is reproduced.
It is also worth paying attention to the degenerate case corresponding to
the ''ultraextreme'' case with $U^{\prime \prime }(r_{+})=V^{\prime \prime
}(r_{+})=0$. Then the limiting procedure elaborated in \cite{zasl2} leads to
the metric which is a direct product of two-dimensional Rindler space and a
sphere:
\begin{equation}
ds^{2}=d\tau ^{2}l^{2}+dl^{2}+r_{+}^{2}d\omega ^{2}\text{.} \label{rin2}
\end{equation}
In spite of $T_{\mu }^{\nu }\neq 0$ due to effects of curvature of
spacetime, the entropy of radiation $S_{q}=0$ according to the general
properties $T_{1}^{1}-T_{0}^{0}=0$ and $r=r_{+}$.
One reservation is in order here. The possibility to replace components $%
T_{\mu }^{\nu }$ by their horizon values due to taking the limit under
discussion implies that a horizon itself is regular in the original metric
in the extreme state. This is not the case for dilatonic black holes and, as
a result, the coefficient at the angular part does not turn into a constant
in this limit but retains some dependence on $l$ \cite{dil}.
Correspondingly, some residual dependence on $l$ may survive for $T_{\mu
}^{\nu }$ and there is no reason to expect $S_{q}=0$ in dilatonic
backgrounds. We will not, however, discuss this case here further.
Thus, for a generic metric obtained as a finite-temperature extreme limit of
nonextreme black holes with a regular extreme state the entropy of quantum
massless Hawking radiation $S_{q}=0$. Thus, the total entropy of such
systems dressed by Hawking radiation (which turn into spacetimes with
acceleration horizons in the limit under consideration) is equal to those of
bare ones.
\section{unruh effect and limits of spacetimes}
What physical explanation can be suggested for the property $S_{q}=0$?
Metrics discussed above share the following feature: they are obtained by a
special kind of the limiting transition $r\rightarrow r_{+}$. As a result,
spacetime picks up the sharp strip of near horizon geometry which expands
into a whole manifold with a finite Euclidean four-volume, so this is not
approximation but the example of taking the ''spacetime limit'' procedure
which maps an original manifold onto a new one \cite{ger}. In so doing, a
new spacetime inherits properties of a vicinity of a horizon where a metric
looks like that perceived by an accelerated observer, so as a matter of fact
we deal with the Unruh effect \cite{bir}. This effect, however, has a pure
kinematic nature and does not need the existence of a true black horizon; in
a sense, it is too week to gain nonzero entropy of Hawking radiation as
there are no ''true'' quanta of it. (To avoid possible confusion in
terminology, let me stress that we distinguish here the Unruh and Hawking
effects as connected with the presence of acceleration and true black hole
horizons, correspondingly. Thus, we use these terms in a way different from
the book \cite{wald}, where the thermal properties of the Hartle-Hawking
state are prescribed, by definition, to the Unruh effect in a curved
spacetime independently of the nature of a horizon --- in particular, in the
Schwarzschild background, i.e. in a true black hole metric. On the other
hand, the term ''Hawking effect'' is used therein to describe a dynamical
process of particles creation.)
Such a role of an acceleration horizon could serve as one more manifestation
of the kinematical nature of Hawking radiation which may or may not be
connected with an entropy associated with a horizon \cite{viss}. However, it
is worth noting that in our case, in contrast to what was discussed in \cite
{viss}, the zero-loop entropy connected with the information loss does exist
and kinematical properties of a spacetime reveal themselves only in
cancellation of the one-loop part of entropy.
In general, metric obtained after the limiting transition in question, is
curved; if quantum backreaction is neglected, it is the BR spacetime which
is nothing else than a direct product of anti-de Sitter space and a sphere $%
(AdS_{2}\times S_{2})$ since the curvature of $(\tau ,r)$ submanifold is a
negative constant. Such a spacetime has three independent Killing vectors
\cite{lap} and an observer moving along a Killing orbit may feel horizons
with nonzero or zero Hawking temperature or see the absence of a horizon at
all that justifies purely kinematic nature of the effect in question.
In this sense there is some analogy in relationship between different
sections of BR spacetime, on one hand, and relationship between the
Minkowski and Rindler spaces, on the other one. Now we show that in the
properly adjusted ''large mass limit'' $r_{+}\rightarrow \infty $ this
analogy turns into literal correspondence. Let us write down the Lorentzian
form of the BR metric relevant for description of the extreme limit of
nonextreme black holes (BR1):
\begin{equation}
ds^{2}=r_{+}^{2}(-dt^{2}\sinh ^{2}x+dx^{2}+d\theta ^{2}+d\phi ^{2}\sin
^{2}\theta )\text{.} \label{br}
\end{equation}
This metric possesses a horizon at $x=0$ which, however, is not of black
hole type but rather is an acceleration horizon. One can perform the
transformation into another frame in which an observer moving along orbits
of another Killing vector will see no horizons at all \cite{lap}. Namely,
after the transformation
\begin{equation}
\cosh t\sinh x=\sinh \chi \text{, }\cosh x=\cos \tilde{t}\cosh \chi
\label{tr}
\end{equation}
we arrive at the metric BR2
\begin{equation}
ds^{2}=r_{+}^{2}(-d\tilde{t}^{2}\cosh ^{2}\chi +d\chi ^{2}+d\theta
^{2}+d\phi ^{2}\sin ^{2}\theta )\text{.} \label{2br}
\end{equation}
Let us perform the transformation $\theta =\vartheta +\pi /2$, $x=l/r_{+}$, $%
\tilde{t}=T/r_{+}$, $\chi =Z/r_{+}$, $\vartheta =X/r_{+}$, $\phi =Y/r_{+}$.
Then after the limit $r_{+}\rightarrow \infty $ is taken, the metric (\ref
{br}) turns into the Rindler one
\begin{equation}
ds^{2}=-dt^{2}l^{2}+dl^{2}+dX^{2}+dY^{2}\text{,} \label{4rin}
\end{equation}
while (\ref{2br}) becomes the Minkowski metric
\begin{equation}
ds^{2}=-dT^{2}+dZ^{2}+dX^{2}+dY^{2}\text{.} \label{mink}
\end{equation}
Expanding eq. (\ref{tr}) in powers of $r_{+}^{-1}$ and retaining main
non-vanishing terms, we obtain the formulae
\begin{equation}
Z=l\cosh t\text{, }Z^{2}-T^{2}=l^{2} \label{zt}
\end{equation}
which describe just the connection between the Minkowski and Rindler
metrics. As far as the entropy of thermal gas is concerned, it remains zero
in the process of the limiting transition for both sections of the BR
spacetime (\ref{br}), (\ref{2br}). Thus, the analogy between the
Rindler/Minkowski and BR1/BR2 metrics in what concerns pure kinematic nature
of Hawking radiation and the property $S_{q}=0$ becomes literal in the limit
at hand.
It is worthwhile to note that, strictly speaking, the result $S_{q}=0$ for
the Rindler spacetime (\ref{4rin}) does not follow from the general formula (%
\ref{2S}) directly since the metric (\ref{4rin}) does not belong (in
contrast to (\ref{rin2})) to the limiting class of metrics (\ref{genmet})
for which (\ref{2S}) was derived. However, the above limiting relation makes
this property transparent since $S_{q}($Rindler$)=\lim S_{q}($BR$)=0$.
Nevertheless, it is also instructive to trace another kind of limiting
transition --- directly from the Schwarzschild metric since, as we will see,
it exhibits features similar to those for the limiting transition which
brings a nonextreme Reissner-Nordstr\"{o}m black hole to the extreme limit.
Consider the Schwarzschild metric
\begin{equation}
ds^{2}=-dt^{2}(1-\frac{r_{+}}{r})+dr^{2}(1-\frac{r_{+}}{r}%
)^{-1}+r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\text{,} \label{Sch}
\end{equation}
where -$\pi \leq \phi \leq \pi $, $0\leq \theta \leq \pi $. Here $r_{+}=2M$
is a horizon radius of a black hole with mass $M$. We will assume that a
black hole is situated in a cavity whose boundary ensures thermal
equilibrium between a black hole itself and its Hawking radiation (the
Hartle-Hawking state). Then, as was shown in \cite{rad}, the entropy of
massless Hawking radiation is described by eq. (\ref{ent1}) with $\left|
f^{\prime }(1)\right| =1$. Due to scale properties of massless quantum
radiation (\ref{tensor}) this equation can be rewritten as
\begin{equation}
S_{q}=16\pi ^{2}\int_{w}^{1}duu^{-4}[f_{r}^{r}-f_{0}^{0}-f_{\mu }^{\mu }\ln
(u/w)]\text{,} \label{ent}
\end{equation}
where $w=r_{+}/r$. The formulas (\ref{ent1}), (\ref{ent}) hold for any kind
of massless radiation including either bosons with generic type of coupling
to gravity or fermions \cite{mat}.
In the canonical ensemble the event horizon radius $r_{+}$ is not arbitrary
but is a function of either $r_{B}$ or a local Tolman temperature $\beta
^{-1}$on a boundary according to the equation \cite{york86}
\begin{equation}
\beta =4\pi r_{+}\sqrt{1-\frac{r_{+}}{r_{B}}}\text{.} \label{tol}
\end{equation}
Accounting for a finite size of a system has the crucial consequences for
thermodynamics: it allows one to define the canonical ensemble for black
holes in a self-consistent way, leads to the appearance of the stable branch
of solutions, etc. \cite{york86}. Now we will consider the large mass limit
which takes into account properly that the system has a finite size, so $%
r_{+}\leq r\leq r_{B}$. We will also assume that the limiting transition
preserves the value of $\beta $. These assumptions mean that the process of
limiting transition is performed in such a way that $r_{+}\rightarrow \infty
$, $r_{B}\rightarrow \infty $, while a square root in eq. (\ref{tol}) tends
to zero and $r_{+}/r\rightarrow 1$ for all points of the manifold. Thus, the
coordinate $r$ becomes degenerate and is to be properly rescaled. In a
similar way, as the inverse Hawking temperature $T_{H}^{-1}=4\pi
r_{+}\rightarrow \infty $, the time coordinate needs to be rescaled too.
Before taking such a limit, let us perform the change of variable $\theta
=\vartheta +\pi /2$, so $-\pi /2\leq \vartheta \leq \pi /2$ and introduce
new coordinates according to
\begin{equation}
\vartheta =X/r_{+}\text{, }\phi =Y/r_{+}\text{, }t=\tau /2\pi T_{H}=2\tau
r_{+}\text{, }r=r_{+}+l^{2}/4r_{+}\text{.} \label{new}
\end{equation}
Then after the limiting transition at hand the original metric (\ref{Sch})
turns into the Rindler metric (\ref{4rin}). It follows from (\ref{tensor})
that $T_{\mu }^{\nu }\rightarrow 0$ in accordance with the fact that in the
state of thermal equilibrium the stress-energy tensor in the Hartle -
Hawking state cancels for the Rindler metric \cite{ginz}. It is worth noting
that in the Hartle - Hawking state, the components $T_{\mu }^{\nu }$ of the
stress-energy tensor in an orthonormal frame are finite on the event horizon
of the Schwarzschild black hole, so the quantities $f_{\mu }^{\nu }$ are
also finite at $u\rightarrow 1$. Therefore, it is seen from (\ref{ent}) that
$S_{q}\rightarrow 0$ in the limit at hand: the thermodynamical entropy of
Hawking radiation cancels for a Rindler wedge. In a sense, thermal gas of
Rindler quanta is a rather peculiar object from the thermodynamic point of
view: in spite of its temperature being nonzero, either its energy defined
as $-\int d^{3}x\sqrt{g}T_{0}^{0}$ or the entropy are equal to zero. I
stress that the statement $S_{q}=0$ is related just to the thermodynamic
entropy (i.e. the quantity just having direct physical meaning) and should
not be confused with the properties of statistical-mechanical one \cite
{solod}, \cite{zerb} (in a similar way, the property $S_{q}=0$ for a thermal
gas in the BR background \cite{ent} should not be confused with the behavior
of quantum correction to the entropy of a black hole itself \cite{man}).
It is worth stressing that the limiting procedure for any spacetime is not
unique and depends, for example, on a particular choice of a coordinate
system in which parameters of a system tend to their limiting values \cite
{ger}. In our case the limit $r_{+}\rightarrow \infty $ is distinguished by
the demand that a local temperature on a boundary is fixed, so it has clear
thermodynamic meaning. In so doing, however, the boundary itself drastically
changes: a sphere turns into a plane. As in the limit at hand $%
r_{+}\rightarrow \infty $ in such a way that $r_{+}/r_{B}\rightarrow 1$, the
zero-loop entropy of a black hole, equal to its Bekenstein-Hawking value,
behaves like $S_{0}=\pi r_{+}^{2}\simeq \pi r_{B}^{2}\simeq A/4$ where $%
A=\int dXdY$ is the surface area of a plane $l=const$, so $S_{0}$ diverges
but the entropy per unit area is finite \cite{lf}. It is worthwhile to note
that the derivation of the formula for the entropy $S_{0}$ in \cite{lf}
relies at once on the metric of the Rindler wedge and the term $A/4$ comes
from a boundary, so the connection between this term and the horizon
remained not quite clear (any surface $l=const$ has the infinite area $A$
for the Rindler metric). Meanwhile, the limiting transition performed above
clearly shows that the Rindler wedge inherits the formula $S_{0}=A/4$ from
the Schwarzschild spacetime where it originates from an event horizon.
It is interesting that, although $A\rightarrow \infty $, the proper distance
$L$ between a horizon and any other fixed point outside, including a
boundary, is finite. Indeed, in the Schwarzschild metric we have $L=r_{+}[%
\sqrt{x(x-1)}+\ln (\sqrt{x}+\sqrt{x-1})]$ where $x=r/r_{+}$. In the limit $%
r_{+}\rightarrow \infty $ performed in coordinates (\ref{new}), $%
L\rightarrow l$. In other words, we have a plane situated at the proper
distance $l_{B}$ from the origin of coordinates, where $l_{B}$ is the value
of the $l$-coordinate of the boundary in the original Schwarzschild metric,
this plane having the same temperature as a boundary sphere in the
Scwharzschild metric.
In general, the total entropy of a system possessing a horizons comes,
according to (\ref{total}), from either the horizon (the term $S_{0}$)
itself or quantum fields (the term $S_{q}$). The fact that in our case the
entropy is determined by the area of the horizon only, this horizon having
kinematical nature (the acceleration horizon instead of the black hole one),
manifests the kinematic character of the Unruh effect in the given context.
Recently, an interesting interpretation of the Hawking effect as the Unruh
one in some embedding auxiliary flat space of higher dimensionality has been
suggested \cite{des}. On the contrary, in our approach we trace the passage
from one spacetime to another remaining within a physical four-dimensional
curved manifold. And what we want to stress is that the equivalence between
two effects traced in \cite{des} for temperatures and zero-loop entropies,
breaks down for the entropies of quantum radiation.
\section{general first law for acceleration horizons}
The Euclidean canonical action for a spherically-symmetrical bounded
self-gravitating charged system obeying the Hamiltonian constraint and Gauss
law reads \cite{braden}
\begin{equation}
I=\beta E-S-\beta \phi e\text{,} \label{action}
\end{equation}
where $\beta $ is an inverse Tolman temperature on a boundary, $\phi $ is a
blueshifted potential difference between the horizon and boundary, and $e$
is charge. For a given set of boundary data ($\beta $, $r_{B}$, $\phi $) a
small variation in a horizon radius gives, according to the action principle
$\delta I=0$, the form of the first law under conditions that all the field
equations are satisfied, so terms arising due to equations of motions cancel
\cite{quasi}. The energy $E=4\pi r_{B}^{2}\varepsilon $, where quasilocal
energy density $\varepsilon $ entering a thermodynamic energy $E$ is equal
to ($k-k_{0}$)$/8\pi $ \cite{canon}, \cite{quasi}, $k$ is an extrinsic
curvature of two-dimensional boundary embedded into a three-dimensional
space, $k_{0}$ is that for the same boundary metric embedded into a
reference flat space to have $E=0=I$ for a flat metric. In a
spherically-symmetrical spacetime of the form (\ref{spher}) $E=r_{B}[1-\sqrt{%
V(r_{B})}]$, where the first contribution corresponds to the flat space term
$k_{0}$.
In attempting to apply the first law to metrics under discussion which have $%
r=cons=r_{+}$ one immediately faces the following oddities. The term with $k$
in energy is equal to zero identically. It follows either from definition of
$k$ or from the above formula for $E$. Indeed, the metric in question is
obtained by the limit $r\rightarrow r_{+}$ for all points of manifold
including a boundary. As a result, the coefficient $V$ in the formula for $E$
picks up its value from a horizon where $V=0$. Apart from this, the quantity
$r_{+}$ cannot be any longer considered as a free parameter independent of
boundary data. Now the two-dimensional metric induced on a horizon coincides
with that of a boundary. These two circumstances do not mean, however, that
the first law loses its sense. Rather, it leads us to the necessity to
consider its extended form including from the very beginning the
contribution from the changes of a boundary metrics. According to \cite
{quasi}, \cite{action}, such a contribution can be represented as $-\frac{1}{%
2}\int \delta \sigma _{ab}s^{ab}\sqrt{\sigma }\beta $ where indices $a$,$b$
are related to a two-dimensional boundary with the metric $\sigma _{ab}$, $%
s^{ab}$ are components of spatial stresses. For a spherically symmetrical
spacetime with $\beta =const$ on a boundary this reduces to $\lambda \delta
r_{B}$ where $\lambda =-4\pi r_{B}\beta s_{a}^{a}$. It follows from eq.
(6.9) of \cite{quasi} that, in our notations, $\lambda =2\pi [b-\frac{%
\partial (br)}{\partial l}]_{B}.$ Apart from gravitational contribution, we
must take into account the change of the action of quantum fields $8\pi
^{2}\gamma \delta r_{B}$ with $\gamma $ from eq. (\ref{gamma}). Comparing
with (\ref{action}), we have
\begin{equation}
\beta \delta E-\beta \phi \delta e-\delta S=2\pi \{[b-\frac{\partial (br)}{%
\partial l}]_{B}+4\pi r_{B}^{-1}\int_{0}^{1}dy\sqrt{\tilde{g}}T_{\mu }^{\mu
}\}\delta r_{B}\text{.} \label{first}
\end{equation}
In such a form the first law must hold for any spacetime of the form (\ref
{genmet}) under the presence of an electromagnetic field. Now we apply it to
the spacetimes which are the $r\rightarrow r_{+}$ limits of (\ref{spher}) in
the sense discussed above.
First, consider the classical BR spacetime for which in (\ref{genmet}) $%
b=r_{+}\sinh l/r_{+}$, $T_{\mu }^{\nu }$ is neglected. The energy $%
E=r_{B}=r_{+}$. Integrating the Maxwell equation $F_{;\mu }^{0\mu }\equiv
(F^{01}\sqrt{g})_{^{\prime }}/\sqrt{g}=0$ it is easy to find the value of $%
\phi \equiv b_{B}^{-1}[A_{0}(1)-A_{0}(0)]$, where $A_{0}(1)-A_{0}(0)$ is the
difference of electrostatic potentials $A_{0}$ between a horizon and
boundary: $\phi =\tanh l/2r_{+}$ where we have taken into account that for a
BR spacetime the charge $e=r_{+}$ (see for details below). Substituting $%
\beta =2\pi b_{B}$ and $S=\pi r_{+}^{2}$ into (\ref{first}) we see that the
first law is satisfied.
Let now quantum backreaction be taken into account, the total stress-energy
tensor $T_{\mu }^{\nu (tot)}=T_{\mu }^{\nu (em)}+T_{\mu }^{\nu }$
representing the sum of contributions from an electromagnetic field and
quantum one. For a metric (\ref{genmet}) with $r=r_{+}=const$ the
nonvanishing components of the Einstein tensor are $%
G_{0}^{0}=-1/r_{+}^{2}=G_{1}^{1}$, $G_{2}^{2}=G_{3}^{3}=b^{-1}\frac{\partial
^{2}b}{\partial l^{2}}$. If the Gauss law $\frac{\partial A_{0}}{\partial l}%
=eb/r_{+}^{2}$ is taken into account, the electromagnetic part of the
energy-momentum tensor has the standard form $T_{\mu }^{\nu (em)}=e^{2}/8\pi
r_{+}^{4}diag(-1,-1,1,1)$. For a massless radiation the stress-energy in the
BR background is
\begin{equation}
T_{\mu }^{\nu }=\frac{B}{8\pi }\delta _{\mu }^{\nu }r_{+}^{-4} \label{tenbr}
\end{equation}
\cite{kof}, \cite{paul}, where $B=const$ and the factor $(8\pi )^{-1}$ is
introduced for convenience. Einstein equations give us $e^{2}=r_{+}^{2}+B$, $%
b=\rho \sinh \rho ^{-1}l$ where $\rho ^{-2}=r_{+}^{-2}(1+2B/r_{+}^{2})$. In
the main approximation with respect to $B$ we have $\gamma =\frac{B}{2\pi
r_{+}}(\cosh l_{B}/r_{+}-1)$. From the Gauss law it follows that $\phi
\equiv b^{-1}[A_{0}-A_{0}(0)]=\frac{e}{r_{+}^{2}\rho }\tanh \frac{\rho l}{2}$%
. Substituting these expressions into (\ref{first}), we may check directly
that the general first law is satisfied provided $\delta S=2\pi r_{+}\delta
r_{+}$. Integrating this equality we obtain that $S=\pi r_{+}^{2}+c$ where $%
c $ is some constant. Here the first term represent Bekenstein - Hawking
entropy whereas the second one is responsible for Hawking radiation. From
the demand that the second contribution vanishes when a boundary approaches
the surface of a horizon (no room for radiation) we obtain that $c=0$. Thus,
we arrive at the same conclusion as was made above: entropy of Hawking
radiation in the BR background is equal to zero exactly.
\section{quantum-corrected geometry of spacetimes with acceleration horizons}
Limiting geometries found in \cite{zasl2} relied on the general assumptions
of the limiting transition from nonextreme black hole metrics to the extreme
state with a finite local temperature in any point between a horizon and
physical boundary. No field equations with or without backreaction of
quantum fields on a metric were used in \cite{zasl2}. Meanwhile, account for
such equations restricts strongly the possible type of limiting metrics. As
follows from the formulae of the previous section, the quantum-corrected BR
spacetime has the form
\begin{eqnarray}
ds^{2} &=&d\tau ^{2}\rho ^{2}\sinh ^{2}\frac{l}{\rho }+dl^{2}+r_{+}^{2}d%
\omega ^{2}\text{, }\phi =\frac{e}{r_{+}^{2}\rho }\tanh \frac{\rho l}{2}%
\text{,} \label{cor} \\
\rho ^{-2} &=&r_{+}^{-2}(1+\frac{2B}{r_{+}^{2}})\text{, }e^{2}=r_{+}^{2}+B%
\text{.} \nonumber
\end{eqnarray}
There is also another solution
\begin{equation}
ds^{2}=d\tau ^{2}\rho ^{2}e^{2\rho l}+dl^{2}+r_{+}^{2}d\omega ^{2}
\label{ext}
\end{equation}
with the same $\rho $.
Recently it was argued by Solodukhin that the product spacetime $%
AdS_{2}\times S_{2}$ is an exact solutions of semiclassical field equations
with quantum backreaction taken into account \cite{ads}. In fact, as in $(r$,%
$\tau )$ submanifold the curvature $R_{2}=-2\rho ^{-2}$ is constant and $%
R_{2}<0$, our formula (\ref{cor}) is in conformity with this statement. This
metric is based on the perturbative expression for $T_{\mu }^{\nu }$ valid
in the region $\left| B\right| \ll r_{+}^{2}$. It is instructive, however,
to extend (not quite rigorously) a semiclassical approach to pure quantum
domain for which $\left| B\right| \sim r_{+}^{2}$. Then for $B=-\left|
B\right| $ the $(_{2}^{2})$ equation gives us a qualitatively new type of
solutions if $\left| B\right| \geq r_{+}^{2}/2$:
\begin{eqnarray}
ds^{2} &=&d\tau ^{2}\sigma ^{2}\sin ^{2}\frac{l}{\sigma }+dl^{2}+r_{+}^{2}d%
\omega ^{2}\text{, }\phi =\frac{e\sigma }{r_{+}^{2}}\tan \frac{l}{2\sigma }%
\text{,} \label{sin} \\
\sigma ^{-2} &=&r_{+}^{-2}(\frac{2\left| B\right| }{r_{+}^{2}}-1)\text{, }%
e^{2}=r_{+}^{2}-\left| B\right| \text{.} \nonumber
\end{eqnarray}
Formally, the metric (\ref{sin}) is obtained from (\ref{cor}) by the
substitution $\rho =i\sigma $. It is seen from (\ref{sin}) that the solution
of the form (\ref{sin}) may exist only under the condition
\begin{equation}
r_{+}^{2}/2\leq \left| B\right| \leq r_{+}^{2} \label{ineq}
\end{equation}
, i.e. for strong backreaction, and in this sense is pure quantum, the
curvature of $(r$,$\tau )$ submanifold $R_{2}=2\sigma ^{-2}=const\geq 0$. In
these respects the $(r$,$\tau )$ part of (\ref{sin}) resembles the one found
in \cite{2d} for 2D dilaton gravity. Thus, in addition to the $AdS_{2}\times
S_{2}$ found in \cite{ads}, in our problem there exists also spacetime which
is a direct product of de Sitter space and a sphere $(dS_{2}\times S_{2})$.
As for this solution $T_{\mu }^{\nu (em)}=0$ and $B<0$, the energy density
is positive everywhere including a horizon. In this respect it can be
considered as a BR-like counterpart of black holes which may possess the
extreme state due to positive energy density on a horizon whose existence
was qualitatively conjectured in \cite{balb}.
For the solution under discussion the ratio of squared radii of two pieces
of spacetime $0\leq r_{+}^{2}/\sigma ^{2}\leq 1$. The minimum value of this
ration is achieved at $R_{2}=2\sigma ^{-2}=0$, $\left| B\right| =r_{+}^{2}/2$
when we obtain the spacetime $Rindler_{2}\times S_{2}$ (\ref{rin2}) with the
electrostatic potential $\phi =2^{-3/2}l/r_{+}$. In contrast to (\ref{cor})
- (\ref{sin}), the solution at hand has no 2D counterpart: in the latter
case \cite{2d} a metric can be flat only under condition that either an
electromagnetic field or backreaction cancel whereas now either each
contribution separately or their sum differs from zero: $%
T_{0}^{0(tot)}=T_{1}^{1(tot)}=-1/8\pi r_{+}^{2}$. Such a spacetime can be
regarded as the example of physical realization of the ultraextreme limit of
nonextreme black holes \cite{zasl2} that shows how a Rindler metric may
appear as a nontrivial result of special tuning between electromagnetic
forces and quantum backreaction: $e^{2}=\left| B\right| ^{2}$ and tangential
stresses vanish, $T_{2}^{2(tot)}=T_{3}^{3(tot)}=0$.
The maximum value of $r_{+}^{2}/\sigma ^{2}$ corresponds to $e=0$, $%
B=-r_{+}^{2}$, $\sigma =r_{+}$. The possibility $e=0$ due to quantum effects
was suggested by Solodukhin \cite{ads} for the $AdS_{2}\times S_{2}$
solution. Our formulae, however, do not admit such a possibility for the $%
AdS_{2}\times S_{2}$ case since it is inconsistent with the property $%
R_{2}<0 $ according to (\ref{cor}). This difference in properties of
solutions under discussion can be explained by the fact that we are dealing
with a conformally invariant scalar field, whereas in \cite{ads} this field
has minimal coupling. Meanwhile, for $dS_{2}\times S_{2}$ solutions $e=0$ is
indeed possible, in which case radii of both two-dimensional subspaces
coincide.
It is instructive to suggest qualitative explanation of rather unusual
consequences in the structure of spacetime caused by quantum backreaction.
The key moment consists in the structure of the energy-momentum tensor (\ref
{tenbr}). It is seen from eq. (\ref{tenbr}) that the stress-energy tensor in
the BR metric mimics the effect of the cosmological constant: $T_{\mu }^{\nu
}=\Lambda _{eff}\delta _{\mu }^{\nu }$, where the effective cosmological
constant $\Lambda _{eff}=-Br_{+}^{-4}$. This constant is absent on the
classical level and is caused in our case by quantum effects entirely. If $%
B<0$, $\Lambda _{eff}>0$. If a system possesses either an electric charge or
the positive cosmological constant, we have, in general, the
Reissner-Nordstrom-de Sitter solution (RNdS) with three horizons - the inner
one $r_{i}$, the outer black hole horizon $r_{o}$ and the cosmological one $%
r_{c}$. In the particular case, when the radii $r_{o}$ and $r_{c}$ merge,
one obtains the charged version of the Nariai solution \cite{nar}. It can be
obtained as the so-called cold limit of the RNdS metric \cite{romans}. In
this limit, the volume in the region $r_{o}<r<r_{c}$ remains finite despite $%
r_{o}\rightarrow r_{c}$, and the new metric, arising as a result of the
limiting transition, looks just like (\ref{sin}). The surface gravity of
each horizon in question tends to zero (that motivates the name ''cold'')
but it is essential that the physical Tolman temperature in every point
outside the horizon remains finite nonzero quantity (in Ref. \cite{dist}
such a kind of limiting transitions is considered in a general setting
without specifying the concrete type of the metric). Substituting $\left|
B\right| =\Lambda _{eff}r_{+}^{4}$ into eq. (\ref{ineq}), we obtain
inequalities inherent to the charged Nariai solution, $e^{2}<\Lambda
_{eff}r_{+}^{4}$ and $\Lambda _{eff}r_{+}^{2}<1$ (the modern discussion of
the Nariai solution and its properties as well as a number of references can
be found in \cite{bousso}). The case of the equalities corresponds to the
ultracold one (see below). The physical interest in spacetimes under
discussion is dictated, in particular, by their role in pair creation of
black holes in cosmological backgrounds (see, for instance, Refs. \cite
{mann95}, \cite{cosm} and references therein).
In the case when all three horizons merge ($r_{i}\rightarrow
r_{o}\rightarrow r_{c}$) the so-called ultracold limit of the RNdS
spacetimes arises \cite{romans}. The corresponding Euclidean metric reads (%
\ref{rin2}) \cite{mann95} and, thus, coincides with the quantum-corrected
geometry obtained by us above.
Thus, the structure of the resulting spacetime depends on the sign of $%
\Lambda _{eff}$ and can be thought of as the result of different types of
limiting transitions to the extreme state. If $\Lambda _{eff}<0$, we have $%
AdS_{2}\times S_{2}$ which can be considered as a result of the extreme
limit for nonextreme black holes \cite{zasl1}, \cite{zasl2}. If $\Lambda
_{eff}>0$, the metric has the form $dS_{2}\times S_{2}$ (\ref{sin}) which
appears after the limiting transition to the state with $r_{o}=r_{c}\neq
r_{i}$, or $Rindler_{2}\times S_{2}$ when radii of all three horizons merge.
Let me stress that the qualittaively new point as compared with Refs. \cite
{romans}, \cite{bousso}, \cite{mann95}, \cite{cosm} and references therein
consists in that the $\Lambda $-term in our problem is absent classically,
so the appearance of analogs of the limiting forms of the RNdS solutions is
a pure quantum effect. It is also worth noting that we did not consider the
RNdS with the consequent limiting transition but started at once with the
BR-like spacetimes and showed how account for backreaction of quantum fields
may change the properties of a spacetime.
The constant $B$, responsible for quantum effects, can be written as $%
al_{0}^{2}$, where $l_{0}$ is the Planck length, $a$ is numerical
coefficient. According to (\ref{cor}) - (\ref{sin}), the solution with $e=0$
is possible only for $a=-\left| a\right| <0$. In particular, a massless
conformally invariant field for which $a=(2880\pi ^{2})^{-1}>0$ \cite{paul}
seems to be not a suitable candidate for such type of solutions. If $e=0$,
the radius $r_{+}$ which measures the curvature of a sphere acquires
planckian scale in agreement with Solodukhin's observation: $r_{+}=\left|
a\right| l_{0}$ (In fact, even $r_{+}\ll l_{0}$, but one can attain $%
r_{+}\sim l_{0}$ for a sufficiently large number of field species).
One reservation is in order. From the formal viewpoint, the metric (\ref{cor}%
) was based on a semiclassical expression for the stress-energy tensor $%
T_{\mu }^{\nu }=(8\pi )^{-1}Br_{+}^{-4}\delta _{\mu }^{\nu }$ calculated on
a given BR background, so the extension to the domain $\left| B\right| \sim
r_{+}^{2}$ is nothing else than extrapolation. Nevertheless, the striking
similarity to the 2D case where a semiclassical stress-energy tensor is
known exactly, strongly supports the validity of found solutions. More
rigorous justification of possibilities indicated in \cite{ads} and in the
present paper needs a fully self-consistent quantum treatment. One cannot
exclude in advance the existence of pure quantum solutions with $e=0$ for
any kind of quantum field.
Anyway, the indicated class of solutions (\ref{sin}) indebted entirely to
quantum effects hints that strong backreaction can modify the structure of
spacetime qualitatively and, in particular, change the sign of curvature.
Drastic changes may also happen not only to solutions (\ref{cor})-(\ref{sin}%
) themselves but also with black holes whose near-horizon geometry may be
approximated by these constant curvature solutions. One such possibility
consists in the existence of an extreme quantum Schwarzschild-like black
hole with a zero surface gravity \cite{balb}, \cite{ads}. Here we would like
to draw attention to another possibility: the appearance, according to (\ref
{sin}), of the solution with a cosmological horizon.
Thus, on one hand, quantum effects preserve the general form of the metric
as a direct product of two two-dimensional submanifolds in accordance with
\cite{ads}. On the other hand, the concrete set of possible kinds of such a
structure includes not only spacetimes with $R_{2}<0$ indicated in \cite{ads}
but also those with both other possible types $R_{2}>0$ and $R_{2}=0$.
\section{conclusion}
In general, one may distinguish three areas of application of thermodynamic
approach to systems with horizons: nonextreme black holes, extreme black
holes and acceleration horizons. The present paper is devoted to the third
case and, in this sense, fills a gap in the relationship between
thermodynamics and horizon mechanics. A typical representative of the
spacetimes with an acceleration horizon is the BR metric. The interest to
different aspects of BR-like spacetimes has increased in recent years \cite
{recent}. Here, we considered quantum-corrected BR-like spacetimes and
showed that their acceleration horizons exhibit the following universal
property: the entropy of Hawking massless radiation $S_{q}=0$. The procedure
of taking spacetimes limits, with the help of which the metrics with an
acceleration horizons are obtained from black hole ones, showed that the
result $S_{q}=0$ is intimately connected with the Unruh effect rather than
with the Hawking one. The general first law does have sense for the metrics
in question in spite of some restrictions on the variation procedure which
now implies that the boundary radius is to be varied together with that of
the horizon. If formulated properly, this law confirms that the entropy of
radiation does not contribute to thermodynamics of a system. The result $%
S_{q}=0$ can serve as a test for checking various renormalization schemes in
calculations of quantum entropy of black holes which possess the extreme
state.
While quantum effects do not reveal themselves directly in thermodynamics of
acceleration horizons, they can have crucial consequences for a structure of
spacetime. In particular, the strong backreaction seems to lead to the
possible change of the sign of two-dimensional $R_{2}$ curvature of $(r$,$%
\tau )$ submanifold and the appearance of the quantum version of the Nariai
solution with cosmological horizon but without the cosmological constant or
to the possibility to have a flat $(r$,$\tau )$ submanifold as an exact
solution of {\it semiclassical} field equations.
In the present paper we restricted ourselves by the case of massless fields
since their scale properties simplify the problem at once in two points: (i)
the action depends on one variable and (ii) the thermal stress-energy tensor
has the general structure (\ref{tensor}) (see Sec.II above). Meanwhile, it
is of interest to obtain the formula for the entropy of quantum massive
fields in terms of the stress-energy tensor which would replace eq. (\ref{2S}%
) derived for massless ones, and check the validity of the property $S_{q}=0$%
. The case of massive fields is especially important in connection with the
issue of quantum renormalization of the black hole entropy. It was shown in
\cite{dem} without using the brick-wall model that Pauli-Villars
regularization correctly reproduces the Bekenstein-Hawking entropy, if the
gravitational constant is properly renormalized (this approach was
elaborated further for many-dimensional cases in \cite{kim}). Therefore, the
desired formula for the thermodynamic entropy, being combined with the
results of \cite{dem}, would enable us, in particular, to trace in detail
the behavior of different parts of the black entropy near the extreme state.
We hope to address this issue in a subsequent research.
Apart from extending the approach to another kinds of fields, it is also
worthwhile to consider more general geometries - in particular, spacetimes
which may be obtained by the spacetime limits taken in the background of
distorted black holes \cite{dist}. Of special interest is the problem of
finding either quantum geometries or the stress-energy tensor in a
background with acceleration horizons in a fully self-consistent manner.
In this paper, we restricted ourselves to the limiting states of type 2
which are obtained by certain limiting transitions to the extreme state
within the topological sector of nonextreme black holes. The separate issue
deserving special treatment is the influence of quantum backreaction on
states of type 1 which represent topologically true extreme black holes.
\section{acknowledgments}
I am grateful to Ted Jacobson for comment on my paper \cite{zasl2} with the
interpretation of limiting geometries found therein as describing the Unruh
effect in the $AdS_{2}$ background. I am also grateful to Sergey Solodukhin
for helpful correspondence. This work is supported by the International
Science Education Program, grant \# QSU082068.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,309
|
\section{Introduction}
Tachyon condensation provides an interesting arena in which we can
improve our understanding of string theory in a dynamical set-up.
While the condensation of closed string tachyons, and the associated
decay of spacetimes, is still hampered by conceptual and technical
problems, a lot of progress has recently been made in understanding
the dynamics of open string tachyons. Most of the analysis was
performed directly using boundary conformal field theory in flat
space, initiated by Sen's construction of the boundary states for
decaying D-branes~\cite{Sen:2002nu}, or by using the~$c=1$ matrix
model for the description of the decay of D-branes in 1+1 dimensional
string theory~\cite{McGreevy:2003kb}. In the present work we study the
problem of decaying branes in the set-up of the ``standard'' AdS/CFT
correspondence.
As was argued by Harvey et al.~\cite{Harvey:2000qu}, unstable D-branes
in string-theory are equivalents of ``sphalerons'': they are unstable
solutions located at a saddle point of the potential in string field
theory configuration space, at the top of a non-contractible
loop~\cite{Manton:1983nd}. In the context of the AdS/CFT conjecture,
this correspondence between unstable D-branes and sphalerons in gauge
theory is in fact even more direct. By analysing the \emph{kinematical
properties} of these two systems, it has been argued by Drukker et
al.~\cite{Drukker:2000wx} that the unstable D-particles of string
theory are in precise correspondence with known sphaleron solutions of
the dual gauge theory.
We will study \emph{dynamical properties} of this correspondence. On
the gravity side we start with the results of Lambert et
al.~\cite{Lambert:2003zr} for the spectrum of decaying D-branes in
\emph{flat space}. To compare these results to those which we will
obtain in gauge theory, we ``embed'' the flat-space results in the AdS
space. A priori, there is no reason to expect that the flat space
results of the decay should be valid for branes in an AdS
background. However, since the D-particles in question are fully
localised in the bulk space, one expects that the flat space results
should carry over, at least when the radius of the AdS is large.
There are two properties of the spectrum of the decaying brane that we
want to compare with the dual gauge theory calculation. The first
property of the spectrum is constrained by the symmetries of the
system, and concerns emission amplitudes for the states on the leading
Regge trajectory. By slightly refining the calculation
of~\cite{Lambert:2003zr} we find \cite{Peeters:2004rd} that all
emission amplitudes for these states are zero. The same result is
separately recovered on the gauge theory side by evaluating the number
operator for the corresponding dual composite operators.
More important is a second property of the spectrum, observed
in~\cite{Lambert:2003zr}, which reflects genuine dynamical features of
the decay. There is strong evidence~\cite{Sen:2002nu,Lambert:2003zr}
that the open strings decay \emph{fully} into closed string states,
i.e.~that there is no open string remnant left after the decay. This
conclusion is also supported by the matrix model calculations
of~\cite{McGreevy:2003kb}. As shown in~\cite{Lambert:2003zr}, the
emission amplitudes are exponentially suppressed with the level of the
emitted string, at least for high levels (however, due to the
exponential growth of the available states, most of the energy of the
brane gets transferred into a high-density cloud of very massive
closed string states).
In the remainder of this report we focus on two issues in the dual
gauge theory on the boundary. The first issue is the construction of
the time-dependent gauge theory solution which is the analogue of
Sen's time-dependent boundary state in boundary conformal field
theory. The second issue is how, given this time dependent solution,
one can reproduce the two properties of the spectrum of the decaying
particle mentioned above.
\section{D-particle $\leftrightarrow$ sphaleron correspondence: statics}
Before addressing the dynamical properties of the correspondence, let
us first briefly revise its basic static
properties~\cite{Drukker:2000wx}. Gauge theory
sphalerons~\cite{Manton:1983nd} are static solutions of the equations
of motion, associated to saddle points whose existence is guaranteed
by the existence of a non-contractible loop in (compact) configuration
space. Whereas the sphaleron solution on~$\mathbb{R}^4$ in
Yang-Mills-Higgs theory, found by Klinkhamer and Manton
\cite{Klinkhamer:1984di}, is very complicated and not known
analytically, the situation is much simpler on $S^3\times\mathbb{R}$
in pure Yang-Mills theory. To construct the sphaleronic gauge
configuration in the SU(2) gauge theory, one starts from the
instanton solution on $\mathbb{R}^4$,
\begin{equation}
\label{e:ansatz}
A_{\mu} = f(r) (\partial_{\mu}U) U^{\dagger}, \quad \quad
U=\frac{x^{\mu}\sigma_{\mu}}{r},
\quad \quad r^2 = x_0^2+x_i^2 \,,
\end{equation}
where $f= r^2/(r^2 + a^2)$. This function interpolates between two
pure gauge configurations (i.e.~the two vacua) $f(r=0)=0$ and
$f(r=\infty)=1$. When $f(r)=1/2$, the system is at the top of the
potential barrier, see figure~\ref{f:configspace}. By taking $f=1/2$
everywhere one gets a singular solution to the equation of motion on
$\mathbb{R}^4$, which is the so-called ``meron''. The $f=1/2$ solution
is, however, also a solution on $S^3\times \mathbb{R}$, since this
manifold can be conformally mapped to $\mathbb{R}^4$ and Yang-Mills
theory in four dimensions is conformally invariant. The solution
obtained in this way is the Euclidean version of the ``sphaleron'',
and is non-singular.
\begin{figure}[t]
\hskip 3em\hbox{\vbox{\psfrag{E}{$\!\!\!\!E_{\text{p}}$}
\includegraphics*[width=.25\textwidth]{configspace.eps}}\hskip-.6\textwidth
\vbox{\psfrag{S(r)}{\small $S_{\text{inst}}(r)$} \psfrag{r=a,
f(a)=1/2}{\small \vbox{\hbox{$r=a$}\hbox{$f=\frac{1}{2}$}}}
\psfrag{r=0}{\small $r=0$} \psfrag{s}{\rnode{sphal_t}{\phantom{s}}}
\psfrag{r}{\small $r \rightarrow$}
\includegraphics*[width=.5\textwidth]{instsphal.eps}}}
\caption{The picture on the left shows in a schematic way the
existence of a non-contractible loop in configuration space, as well
as the presence of the sphaleron (red dot) at the saddle point. The
picture on the right shows the action density $S(r)$ of the
instanton in the Euclidean theory, together with the special
configuration at $r=a$ which is used to construct the sphaleronic
particle in the Lorentzian theory.}
\label{f:configspace}
\end{figure}
The Lorentzian version is the same, since the time component of the
potential of the sphaleron is zero. The solution is completely
time-independent and has infinite action, corresponding to a
sphaleronic particle which is sitting at the top of the potential.
As far as a generalisation of the previous construction to SU($N$)
gauge theory is concerned, the general sphaleron configuration is not
known. However, an interesting special configuration has been given
in~\cite{Drukker:2000wx}. It is obtained by replacing the Pauli
matrices in~\eqn{e:ansatz} with Clifford algebra generators according
to
\begin{equation}
\label{e:sunsu2}
\sigma_\mu \rightarrow \gamma_\mu
= \begin{pmatrix}
\sigma_\mu & 0 & \cdots & 0 \\
0 & \sigma_\mu & \cdots & 0 \\
\vdots & \vdots & \ddots & 0 \\
0 & 0 & \cdots & \sigma_\mu
\end{pmatrix}\, .
\end{equation}
The mass of this sphaleronic particle is $k$~times the mass of the
SU(2) particle (where $k$ is the number of sigma matrices
in~\eqn{e:sunsu2} and $2k < N$). It was also shown that the number
unstable modes is increased from one (for SU(2)) to $k^2$.
It has been argued by Drukker et al.~\cite{Drukker:2000wx} that the
(non-supersymmetric) sphaleronic saddle points in the gauge theory are
preserved as the 't~Hooft coupling is increased, despite the fact that
the precise form of the potential receives quantum corrections. The
main reason for this is that these sphaleronic saddle points are
linked to the underlying non-contractible loops in configuration
space. Furthermore, they are linked to the (supersymmetric) instanton
configuration which is present both at strong and weak coupling. Thus
the sphaleronic particle in the Yang-Mills theory on $S^3 \times R$
has, at weak coupling, been conjectured to be dual to the unstable
D-particle in the AdS.
A number of arguments has been given~\cite{Drukker:2000wx} in support
of this correspondence. Firstly, both D-particles and sphaleronic
particles are static with respect to the global AdS time. Secondly,
since the D-particle is located at the origin of the AdS space (in
global coordinates), it is ``projected'' in a homogeneous fashion to
the boundary, in agreement with the fact that the sphaleronic particle
is homogeneously spread over the $S^3$. Thirdly, the D-particle in the
bulk is a source for the gravitational and dilaton field (while it
does not source the RR forms), which is in agreement with the
(non)vanishing expectation values of the dual gauge
operators. Finally, in the case of the more general
sphaleron~\eqn{e:sunsu2}, the number of unstable modes on both sides
agrees.
\section{D-particle $\leftrightarrow$ sphaleron correspondence: dynamics}
To study the dynamics of the decaying D-particles from the gauge
theory perspective let us, as a first step, construct the time
dependent gauge configuration describing the sphaleron decay. We
restrict to the decay modes which preserve spherical symmetry by
making the following ansatz,
\begin{equation}
\label{ansatz}
A = f(t)\, \Sigma^i \sigma_i\,,
\end{equation}
where $\Sigma^i$ are the three left-invariant one-forms. To deduce
what is the unknown function $f(t)$ we plug the ansatz into the action
and derive the action for this function. The value of the action for
our ansatz is
\begin{figure}[t]
\begin{center}
\includegraphics*[width=.5\textwidth]{configplot.eps}
\caption{The functions $f_\pm(t)$ of the decaying sphaleron on $S^3$ as
given in~\eqn{e:fsol}, together with the kinetic and potential
energy (with normalisation as given in~\eqn{e:theenergy} and $R=1$).}
\label{f:foft}
\end{center}
\end{figure
\begin{equation}
\label{e:reducedYM}
S = -\frac{1}{4 g_{\text{YM}}^2} \int\!{\rm d}t{\rm d}\Omega\, F_{\mu\nu}F^{\mu\nu}
= \frac{24\, \vol(S^3)}{4\, g_{\text{YM}}^2} \int\! \frac{{\rm d}t}{R}
\left(\frac{R^2}{2}\dot{f}^2 - 2 f^2 (1-f)^2\right) \, ,
\end{equation}
where $\vol(S^3)\equiv 2\pi^2$ denotes the volume of the unit sphere
and~$R$ is the radius of $S^3$. The equation of motion for the
function~$f$ is
\begin{equation}
\label{single}
R^2\,\ddot{f} + 4f(1-f)(1-2f) = 0 \, .
\end{equation}
When integrated once, this equation yields a conserved quantity,
namely the energy (i.e.~the component $T_{00}=48 \vol(S^3) E$)
\begin{equation}
\label{e:theenergy}
E = R^2\,\dot{f}^2 + 4 f^2 (1-f)^2 \,
\end{equation}
which is simplest to integrate analytically
for~$E=\tfrac{1}{4}$. There are two solutions, corresponding to the
fact that the sphaleron can roll down on either side of the potential,
to the vacua with Chern-Simons number one and zero respectively. The
final result reads (see figure~\ref{f:foft})
\begin{equation}
\label{e:fsol}
f_\pm(t) = \frac{1}{2}\left(\frac{\pm\sqrt{2}}{\cosh\left(\frac{\sqrt{2}}{R} (t-t_0)\right)}
+ 1 \right)\, .
\end{equation}
This solution describes a configuration that starts from the potential
maximum at $t=-\infty$ (with zero velocity and acceleration), rolls
down the hill and up the other side, where it arrives at
$t=t_0$.\footnote{After we had derived this solution, we learned that
it has been obtained before~\cite{Gibbons:1994pq} albeit in a
different context.}
The periodicity of the whole process is natural from the AdS
perspective. Since AdS effectively acts as a box, the cloud of
outgoing radiation is refocused to the origin of the space, where it
arrives as fine-tuned radiation and ``re-builds'' the D-particle. In
this sense the D-particle never decays, since there is no real
dissipation of the energy in the system. However, in the limit of
large AdS radius, our flat-space intuition should (at least
approximately) hold. A natural point in time, which should be
associated to the decayed brane, is the point where the sphaleron has
rolled down to the the bottom of the potential, i.e.~when all
potential energy has been converted to kinetic energy (see
figure~\ref{f:rolling}).
\begin{figure}[t]
\begin{center}
\psfrag{tb}{\raise -1ex\hbox{$t_b$}}
\psfrag{decay}{\!\!\!``decayed D0''}
\includegraphics*[width=.3\textwidth]{rolling.eps}
\end{center}
\caption{The evolution of the sphaleron. As it rolls down, it reaches
a point where all potential energy has been converted to kinetic
energy. This is what we will call the ``decayed D-particle'',
despite the fact that the decay products will eventually come back
as fine-tuned radiation to ``re-build'' the D-particle.}
\label{f:rolling}
\end{figure}
Near the bottom the solution is
\begin{equation}
\label{e:Afull}
A_{\mu} = \tilde{f}(t)\, U^\dagger (\partial_\mu U) \, , \quad
\tilde{f} = f-1 \, .
\end{equation}
with $\tilde{f}\approx 0$, which means that the derivative part of the
field strength, rather than the non-linear (commutator) part, is
dominant. The solution becomes a solution of the free Yang-Mills
equations of motion on~$S^3 \times \mathbb{R}$ (written in the
radiation gauge: $A_0 = \nabla_i A^i=0$),
\begin{equation}
\label{e:linearisedEOM}
\Big( -\partial_t^2 + \frac{1}{R^2} \big(\nabla_{S^3}^2 - 2\big) \Big) A^{\text{lin.}}_i =
0\, .
\end{equation}
Indeed, one can easily see that as~$t \rightarrow t_{\text{bottom}}$
the solution~\eqn{e:Afull} with $f$ given by~\eqn{e:fsol} is
very well approximated by the following solution of the linearised
equation of motion~\eqn{e:linearisedEOM}:
\begin{equation}
\label{e:Anearbottom}
A_i^{\text{lin.}} =
-\frac{1}{4}\sin\left(\frac{2(t-t_{\text{bottom}})}{R}\right) \,
U^\dagger (\partial_i U) \,.
\end{equation}
Hence near the bottom of the valley, one can think about the
Yang-Mills configuration as dual to a coherent state of
non-interacting closed string states which are the product of the
D-particle decay. Our goal will then be to determine the numbers of
various (gravity) ``particles'' in this final coherent state. What we
precisely mean by this will be explained in the next section. Let us
first construct this coherent state.
The fact that our solution abelianises near the bottom of the
potential allows us to apply the standard machinery to write down the
coherent state. By expanding the classical, free Yang-Mills gauge
potential in terms of spherical vector harmonics, one can read off the
amplitudes for different modes, and write a coherent state as
\begin{equation}
\label{coherent}
|c\rangle = {\mathcal C}\, \exp\left( g_{\text{YM}}^{-2} \sum_{J,M,y}
\Tr\big(A_{JMy}\, \hat{a}^{\dagger}_{JMy}\big) \right) |0\rangle\,,
\end{equation}
where $A_{JMy}$ are the coefficients appearing in the Fourier
decomposition of the classical sphal\-eron configuration and the
normalisation factor~${\mathcal C}$ is chosen such that $|c\rangle$ is
of unit norm.
For this to be a legitimate state in the Hilbert space, one has to
make sure that it satisfies all constraints. It is easy to see that
creation operators in~\eqn{coherent} lead to physical excitations in
the free theory. However, once $g_{\text{YM}}$ is turned on, Gauss' law
implies that only singlets can be excited. This means that all
non-singlet states in~\eqn{coherent} have to be projected out. In
practice, however, we will neither write this projector nor construct
the projected state explicitly. This is because our calculations
always involve projections of the coherent state onto states which
themselves are color singlets. Therefore the singlet projection is
imposed implicitly throughout.
\section{Particles in the AdS/CFT correspondence}
In the AdS/CFT correspondence we have a relation between string states
in the bulk and operators in the boundary. These operators are, via
the operator--state mapping, interpreted to create ``particles'' in
the bulk theory at a particular point on the boundary. That is, one
needs to solve for the wave equation of the dual field in the bulk in
the presence of a delta source inserted at the boundary. This means
that the states created in the bulk are not eigen-momentum states, an
attribute which one usually associates to the notion of a particle in
field theories. However, since the AdS/CFT correspondence is
formulated in position space rather than momentum space, these
definitions are natural in this context. On the other hand, our
string calculation in~\cite{Peeters:2004rd} is a flat space
calculation, and for us it will be more natural to use the standard
notion of particles in the bulk as angular momentum
eigenstates. Therefore, we will first have to construct boundary
operators that are dual to bulk angular momentum eigenstates.
The operator--state correspondence is usually discussed in the context
of radial quantisation of conformal field theories (see
e.g.~\cite{Fubini:1973mf} for a discussion in a four-dimensional
context). One first Wick rotates $\mathbb{R}\times \mathbb{R}^3$ to
the Euclidean regime and then performs a conformal transformation such
that the origin of $\mathbb{R}^4$ corresponds to $t=-\infty$ in the
original frame. Operators inserted at the origin are then in
one-to-one correspondence with states in the Hilbert space. The entire
procedure can, however, be formulated without doing the conformal
rescaling, which is more natural in our setup since, as we have
discussed before, the gauge field configuration on $\mathbb{R}\times
S^3$ is non-singular while the one on $\mathbb{R}^4$ is singular.
The state corresponding to an operator with conformal weight~$w$ is
obtained by multiplying with the appropriate exponential of Euclidean
time and taking the limit $\tau\rightarrow-\infty$ (keeping only the
regular part):
\begin{equation}
\label{e:Ostatedef}
|\hat{O}^{(m)}_{\text{weight-}w}\rangle = \lim_{\tau\rightarrow-\infty}
\,\Big\{e^{-w\tau}
\hat{O}^{(m)}_{\text{weight-}w}(\tau)\Big\}
\big|0\big\rangle
\equiv \hat{O}^{\dagger(m)}_{\text{weight-}w} |0\rangle\, .
\end{equation}
The last expression shows the shorthand notation that we will use in
order not to clutter expressions unnecessarily. The hermitian
conjugate of an operator is given by
\begin{equation}
\Big(\hat{O}(\tau)\Big)^\dagger
= \hat{O}^\dagger(-\tau)\, .
\end{equation}
This procedure mimics the operator--state mapping on $\mathbb{R}^4$
but avoids technical problems related to solutions which become
singular after the conformal transformation.
The operators which we use in~\eqn{e:Ostatedef} are independent of the
angular coordinates on the sphere, i.e.~they are obtained from the
position dependent operators as follows
\begin{equation}
\label{e:pola}
\hat{O}_{w}^{(m)}(\tau) = K^{(m)}_{w} \int_{S^3}\! {\rm d}\Omega\,
\, \hat{O}_{w}^{\mu_1...\mu_s}(\tau,\phi_i)\, Y_{\mu_1...\mu_s}^{(m)}(\phi_i) \, .
\end{equation}
Here $Y^{(m)}$ denote the lowest lying tensor spherical harmonics for
a given spin~$s$. The index~$m$ labels the degeneracy of such
harmonics. The normalisation constants~$K^{(m)}_w$ are chosen such
that the states constructed using~\eqn{e:Ostatedef} are of unit norm.
Note that the multiplication with the time dependent exponent
in~\eqn{e:Ostatedef} selects out composite operators of the required
conformal dimension, but when one expresses these operators in terms
of elementary creation and annihilation operators, one explicitly sees
that different operators~$\hat{O}$ are not orthogonal. It is only
after the integration~\eqn{e:pola} that one obtains a set of
orthogonal states.
There are many subtleties related to the fact that operators $\hat{O}$
are composite operators rather than elementary gauge
operator. Firstly, the multi-particle states cannot simply be obtained
by acting repeatedly with the~$\hat{O}^\dagger$ operators on the
vacuum. States generated in this way are \emph{not orthogonal}, not
even in the $N\rightarrow\infty$ limit when the number of operators
becomes large as well. Starting from the naive states
$(\hat{O}^\dagger)^n|0\rangle$ one has to subtract terms in order to
achieve orthogonality. For the same reason, there is no simple number
operator which can be used to count the number of composite
excitations in a given state. It is true that
\begin{equation}
\label{e:OdaggerO}
{} [\hat{O}, \hat{O}^\dagger ] = 1 + {\mathcal O}(N^{-2}) \,,
\end{equation}
and one might expect that this leads to a well-defined number
operator~$\hat{O}^\dagger \hat{O}$. However, the coefficients that
multiply the~$1/N^2$ corrections in~\eqn{e:OdaggerO} are operators,
not c-numbers. As a consequence, the strength of the~$1/N^2$
corrections depends on the state in which the number operator is
evaluated,
\begin{equation}
\label{noco}
\langle n | \hat{O}^\dagger \hat{O} | n\rangle = n + \sum_i \frac{c_i(n)}{N^{2i}}\,.
\end{equation}
The numbers $c_i(n)$ can become arbitrarily large
when~$n\rightarrow\infty$. Since the coherent state contains such
highly excited states, the operator $\hat{O}^\dagger \hat{O}$ cannot
be used as a number operator, not even in the $N\rightarrow\infty$
limit.\footnote{An proper number operator for composite particles,
which produces the exact occupation number rather than an expression
which is only correct up to $N^{-2}$ corrections, has been constructed
by~\cite{Brittin:1980ev}. However, their operator is very
complicated and difficult to handle in practice. We prefer to follow a
different route here.} We will encounter an explicit manifestation of
these problems in the next section, when we start counting particles
in the coherent state, and then on a concrete example we will
illustrate how one can deal with them.
Let us end this section with a comment on alternatives to the coherent
state~\eqn{coherent}. From the point of view of the dual string
theory, it might seem more natural to construct a coherent state using
the composite operators~$\hat{O}^\dagger_J$ in the exponent, rather
than the elementary ones~$\hat{a}^\dagger$. After all, the~$\hat{O}_J$
correspond to elementary string excitations. However, a state of
the form
\begin{equation}
|\tilde c\rangle = \tilde{{\mathcal C}} \exp\Big(
\sum_i O^{\text{class.}}_i\,\hat{O}^\dagger_i
\Big)|0\rangle
\end{equation}
is not a coherent state in the standard sense since the
expectation value of an operator in this coherent state does not equal
the classical value of that operator,
\begin{equation}
\big\langle \tilde c\big|\, \hat{O}_i\, \big|\tilde c\big\rangle \not= O^{\text{class.}}_i\,,
\end{equation}
not even up to $1/N$ corrections. The reason for this is essentially
given in equation~\eqn{noco}, with~$|n \rangle$ now being given
by~$|n\rangle = \big(\hat{O}^\dagger_i\big)^n\,|0 \rangle$. This is
our prime motivation to use~\eqn{coherent} as the sphaleron coherent
state.
\section{Particle counting}
Starting from the coherent state~\eqn{coherent} we now want to extract
information from it about particle numbers in the decay product. By
particle counting, we mean counting of the states constructed in the
previous section.
Due to the problems explained around~\eqn{e:OdaggerO}, one cannot use
the ``standard'' number operator $\hat{O}^\dagger\hat{O}$. Instead we
will simply decompose the coherent state on the basis of
multi-particle states. Subsequently we will, using these
probabilities, calculate the average energies and particle numbers.
The probability of finding a multi-particle state consisting of $p_1$
particles of type $O_{J_1}$, $p_2$ particles of type $O_{J_2}$ etc.,
is given by
\begin{equation}
\label{expectations}
{\mathcal P}(p_1;p_2;\ldots;p_M) :=
\frac{\left|\, \Big\langle
(\hat{O}_{J_1})^{p_1} \ldots (\hat{O}_{J_M})^{p_M}\,
\Big|\,c\Big\rangle \, \right|^2}{\Big\langle \big( \hat{O}_{J_1} \big)^{p_1} \ldots
\big( \hat{O}_{J_M} \big)^{p_M}\, \Big|\,
\big( \hat{O}_{J_1} \big)^{p_1} \ldots
\big( \hat{O}_{J_M} \big)^{p_M} \Big\rangle\,\big\langle
c\big|c\big\rangle}\,.
\end{equation}
For this to work it is of course crucial that the basis of
multi-particle states is constructed to be orthogonal. By definition,
the average number of particles of the type~$\hat{O}_{J_i}$ present in
the coherent state is now given by
\begin{align}
\label{e:numbers}
N(J_i) &:=
\sum_{p_1=0}^\infty \cdots \sum_{p_M=0}^\infty
p_i\, {\mathcal P} (p_1;p_2;\ldots;p_M)\,.
\intertext{The energy stored in these particles, as measured with
respect to the global time in the bulk, is given by the conformal
dimension of the corresponding operators. Therefore, the total energy
is given by the expression}
\label{e:energies}
E(J_i) &:=
\sum_{p_1=0}^\infty \cdots \sum_{p_M=0}^\infty
\Delta_{J_i}\, p_i\, {\mathcal P} (p_1;p_2;\ldots;p_M) \, ,
\end{align}
where $\Delta_{J_i}$ is the conformal dimension of the operator
$\hat{O}_{J_i}$. For a generic operator, the calculation of the
numerators in~\eqn{expectations} reduces to evaluating the classical
expression of the (abelianised) operator using the positive frequency
part of the decayed solution. Hence, by considering only the
numerators in~\eqn{expectations} we can deduce which particles are
\emph{absent} from the decay spectrum. In particular one can easily
deduce that expectation values of the operators dual to the graviton,
NS-NS two form and all twist two operators are zero.\footnote{Note
that the expression which vanishes is the energy momentum tensor
evaluated on the positive frequency part of the solution:~$|\langle 0
|\hat{T}_{\mu\nu}|c \rangle|^2 =
|T_{\mu\nu}(A^+_{\text{coherent}})|^2=0$. On the other hand, the
classical expression for the energy momentum tensor of the full
configuration is non-zero:~$T_{\mu\nu}(A^+ + A^-) \neq 0$.} By
slightly refining the calculation of~\cite{Lambert:2003zr} we have
found that all emission amplitudes for these states are zero in string
theory as well~\cite{Peeters:2004rd}. The absence of the
gravitational radiation is not surprising, since the decay is
spherically symmetric. We also believe that absence of the other
states is dictated by some underlying symmetry arguments.
Thus, to explore the genuine symmetry aspects of the decay we need
to concentrate on the states for which~\eqn{expectations} does not
vanish. The main technical problem arises when evaluating the denominators
of~\eqn{expectations}. To illustrate this, let us consider a
``simplified'' model, based on a non-abelian scalar field. This model
exhibits all of the technical subtleties associated to the
determination of the decay products. The crucial ingredients of the
vector coherent state, namely that it is constructed from the
lowest-lying spherical harmonics and that it depends
non-perturbatively on the coupling constant, are preserved by this toy
model. It, however, avoids the inessential technical complications
associated to the evaluation of tensor spherical harmonics in the
numerators of~\eqn{expectations}.
The coherent state for a given classical configuration in this
non-abelian scalar theory is given by
\begin{equation}
\label{e:simplecoh}
|c\rangle = {\mathcal C} \exp\left( \frac{1}{g_{\text{YM}}^2} \Tr\big( a\,
\hat{a}^\dagger\big) \right) |0\rangle\,, \qquad {\mathcal C} = \exp\left(
-\frac{1}{g_{\text{YM}}^2} \Tr\big( a^\dagger a\big)\right)\,.
\end{equation}
This mimics the construction~\eqn{coherent}. The unit normalised (at
leading order in $1/N$ expansion), single-trace operators which create
particles in the out vacuum are
\begin{equation}
\label{sample}
\hat{O}_J^\dagger = \frac{1}{\sqrt{\strut J (g_{YM}^2 N)^J}} \Tr\big( (\hat{a}^\dagger)^J \big) \, .
\end{equation}
These operators are coordinate independent operators, obtained using a
procedure similar to~\eqn{e:pola}.
With the above normalisation of the operator, the numerators and hence
probabilities in~\eqn{expectations} depend on the Yang-Mills coupling
in a non-perturbative fashion,
\begin{equation}
\label{argu}
\Big|\langle 0 | \big(\hat{O}_J\big)^p |c \rangle\Big|^2 = {\mathcal C}^2
\left| \frac{ \Tr\big((a^+)^J\big)}{\sqrt{\strut J (
g_{YM}^2 N)^J}} \right|^{2p} \, \equiv \frac{{\mathcal C}^2}{
J^p} \, \left({\frac{\eta_J^{2}}{\lambda^{J}}} \right)^p
\, ,
\end{equation}
(where the last equality defines $\eta_J$; note that it is of the
order~$N$ for the configuration~\eqn{e:sunsu2} and generically scales
as the number of D-particles). This reflects the fact that our
original sphaleron configuration is a non-perturbative solution of the
equations of motion. Note also that the only way in which the
coupling~$\lambda$ appears in~\eqn{e:numbers} and~\eqn{e:energies} is
through the combination~$\eta_J^2/ \lambda^J$.
The complicated part of the calculation of the average particle
numbers and energies is the computation of the norms for the states
with an arbitrary number of particles. The norm of the state with
$p$~identical particles can be written as
\begin{equation}
\label{expansion}
\begin{aligned}
\big\langle (\hat{O}_J)^p\, (\hat{O}_J^\dagger)^p \big\rangle &=
p!\, \big\langle (\hat{O}_J)\, (\hat{O}_J^\dagger)\big\rangle^p
+ \binom{p}{2}^2 \big\langle
(\hat{O}_J)^2\,(\hat{O}_J^{\dagger})^2\big \rangle_{\text{connected}}
(p-2)! \big\langle (\hat{O}_J)\, (\hat{O}_J^\dagger) \big\rangle^{(p-2)} \\
& +\binom{p}{3}^2 \langle \hat{O}_J^3 \hat{O}_J^{\dagger 3} \, \rangle_{\text{connected}} (p-3)! \langle
\hat{O}_J \hat{O}_J^\dagger \rangle^{(p-3)} \\[1ex]
&+ \binom{p}{2}^2 \binom{p-2}{2}^2 \big\langle (\hat{O}_J)^2
(\hat{O}_J^{\dagger})^2 \big\rangle_{\text{connected}}^2 \frac{(p-4)!}{2!} \langle
\hat{O}_J \hat{O}_J^\dagger \rangle^{(p-4)} + ...
\end{aligned}
\end{equation}
The first term is at a leading order independent of $1/N$, the second
is suppressed as~$1/N^2$, the last two terms both scale as~$1/N^4$, and
so on. A similar but more complicated expansion can be written for
states involving more than one type of particle.
Naively, one might expect that in the large-$N$ limit, all but the
leading term~$p!$ in this expansion can be omitted. However, this
would produce an exponential dependence on the expectation values for
the operators $\hat{O}_J$ in formula~\eqn{e:numbers}. Since the
arguments of the exponent~\eqn{argu} \emph{increase} with conformal
dimension~$J$, one would conclude that the number of particles
produced during the decay \emph{increases} with the mass of the
particle. It is easy to see that this kind of truncation
of~(\ref{expansion}) does not make sense in the case of the
\emph{non-perturbative} coherent state~\eqn{e:simplecoh}, as it would
actually produce probabilities~\eqn{expectations} which are larger
than one. The point is that since the numerator~\eqn{argu} is very
large, the maximal probabilities are attained for large
values~$p^{\text{max}}$ of~$p$. Moreover,~$p^{\text{max}}$ grows
with~$N$, hence in the large-$N$ limit the sub-leading terms
in~\eqn{expansion} become more and more relevant, and are actually
\emph{comparable} to the leading term.
In trying to estimate how fast the norms (\ref{expansion}) have to
grow with~$p$, one can see that even an exponential growth of the
norms, say as~$p!\, \gamma^p$ ($\gamma=\text{const}.$), does not lead
to reasonable results. Namely, if we consider the
expression~$\sum_{p} {\mathcal P}(J,p)$, which has to be smaller than one,
and assume exponential growth of norms, we would find that this sum
behaves as
\begin{equation}
\label{e:pos-prob}
\sum_{p=0}^\infty {\mathcal P}(J,p) = {\mathcal C}^2 \sum_{p=0}^\infty
\frac{1}{p!} \left(\frac{\eta_J^2}{\lambda^J \gamma}\right)^p =
\exp\left(\frac{\eta_J^2}{\lambda^J \gamma}\right) \exp\left(- \frac{N}{\lambda}
\Tr(a^\dagger a)\right)\,.
\end{equation}
Hence we see that even when $N\rightarrow\infty$ (while keeping
$\lambda$ arbitrary but smaller than one) the result will always be
larger than~1 for some value of~$J$. Since the calculation of the
average number of particles requires a summation over all~$J$, we
conclude that we cannot assume this behavior of the norms.\footnote{Note
that if we would have had a perturbative coherent state instead of a
\emph{non-perturbative} one, the classical expectation values~$a$ in
(\ref{e:simplecoh}) would be of the form $a = g_{YM} \eta$, with
$\eta$ a number independent of the coupling constant. Hence formula
(\ref{e:pos-prob}) would be replaced with
\begin{equation}
\label{e:pert-prob}
\sum_{p=0}^\infty {\mathcal P}(J,p) = {\mathcal C}^2 \sum_{p=0}^\infty
\frac{1}{p!} \left(\frac{\eta_J^2}{N^J \gamma}\right)^p =
\exp\left(\frac{\eta_J^2}{N^J \gamma}\right) \exp\left(- \Tr(a^\dagger a)\right)\,.
\end{equation}
We now see that a truncation to the first term in~\eqn{expansion}
(i.e.~setting $\gamma=1$) produces reasonable results for the
probabilities~\eqn{expectations}.}
The situation which we face here is similar in spirit to the
double-scaling BMN limit. As observed in~\cite{Kristjansen:2002bb}
and~\cite{Constable:2002vq}, in the limit $N \sim J^2
\rightarrow\infty$ correlators in general receive contributions from
non-planar graphs of all genera. In this case, a new expansion
parameter~$J^2/N$ appears. In our case, $N\rightarrow\infty$ as well,
but now the additional parameter which becomes large is the value of
the~$p_i$ for which the sum~\eqn{e:energies} has its maximum term. It
would be interesting to understand whether our system also exhibits a
double-scaling limit in which some ratio of powers of~$p$ and~$N$ is
kept fixed.
\section{Calculation of norms and numerical results}
\label{s:U4results}
In order to determine the correct values of the norms of the states,
it is useful to write the norms of multi-particle states in terms of
correlators of a complex matrix model,
\begin{multline}
\label{e:mcintegral}
\big\langle 0 \big| \Big[ \big( \hat{O}_{J_1} \big)^{p_1} \ldots
\big( \hat{O}_{J_n} \big)^{p_n}\Big] \, \Big[
\big( \hat{O}^\dagger_{J_n} \big)^{p_n} \ldots
\big( \hat{O}^\dagger_{J_1} \big)^{p_1}\Big] \, \big|0
\big\rangle\\[1ex]
= \int\!{\rm d}A{\rm d}\bar A\, \Big[\big( {O}_{J_1} \big)^{p_1} \ldots
\big( {O}_{J_n} \big)^{p_n}\Big] \, \Big[
\big( {O}^\dagger_{J_n} \big)^{p_n} \ldots
\big( {O}^\dagger_{J_1} \big)^{p_1}\Big]
\exp\Big( - \Tr( A^\dagger A ) \Big)\,.
\end{multline}
The measure used here is simply a separate integral over the real and
imaginary parts of the complex matrix $A$, normalised to give unit
result when all~$p_i$ in the expression above are zero,
\begin{equation}
\int\!{\rm d}A{\rm d}\bar A = \pi^{-N}\prod_{a,b=1}^N {\rm d}(\Real
A_{ab})\,\,\, {\rm d}(\Imag A_{ab})\,.
\end{equation}
This approach has been used
by~\cite{Kristjansen:2002bb,Beisert:2002bb} in order to compute
several special cases of~\eqn{e:mcintegral} analytically. It is still
an open problem to extend those exact results to the entire class of
correlators, in particular to general situations for
which~$p_i>2$. Because we will need these very general correlators, we
have decided to use an alternative approach, in which the integral is
evaluated using Monte-Carlo methods. This provides us with a
technically straightforward way to extract the norms for arbitrary
operator insertions, even for very large~$p_i$. Our results will, for
this reason, of course be restricted to a fixed value for~$N$ and
computer resources put a practical limit on the maximum value that can
be handled (we will take $N=4$). Nevertheless, we will see that
interesting results can be obtained this way.
In the U(4) theory there are only two operators which create
physical states (using only the creation operator for the lowest-lying
spherical harmonics). These are~$\Tr\big((a^\dagger)^2\big)$ and
$\Tr\big((a^\dagger)^4\big)$.\footnote{The restriction to the
zero-mode of the scalar field is motivated by the full sphaleron
solution of the earlier sections, which only turns on the lowest
spherical vector harmonics. Naturally, in the full U(4) there are
also operators of the form $\Tr(D_\mu\phi D_\nu\phi)$. However, in the
oscillator picture these are turned on by the oscillators that create
the higher spherical tensorial harmonics.} The proper linear
combinations of these operators are
\begin{equation}
\hat{O}_2^\dagger = \Tr( a^\dagger a^\dagger )\,,\qquad
\hat{O}_4^\dagger = \Tr( a^\dagger a^\dagger a^\dagger a^\dagger ) -
\frac{2N^2 + 1}{N(N^2+2)} \Tr( a^\dagger a^\dagger ) \Tr( a^\dagger
a^\dagger )\,.
\end{equation}
These lead to $\langle \hat{O}_4\,|\, \hat{O}_2
\hat{O}_2\rangle=0$. Multi-particle states will generically not be
orthogonal, but in our case this turns out to be far less important
than the $1/N^2$~corrections to the norms. We will for simplicity also
use a classical configuration for which
\begin{equation}
\frac{\eta_4}{N} = \left(\frac{\eta_2}{N}\right)^2 = \frac{\eta}{N}\,,
\end{equation}
where the $\eta_J$ are defined in~\eqn{argu}. Closer inspection of the
coherent state of the sphaleron given in~\eqn{coherent} shows that the
expectation values of e.g.~the $\Tr(F_{mn} F^{mn})$ and~$\Tr(F_{m
n}F^{m n} F_{rs} F^{rs})$ states are similarly related.
The energy radiated into $O_{J=2}$ and $O_{J=4}$ particles can be computed
using formula~\eqn{e:energies}, summed over a suitably large range of
values for $p_2$ and $p_4$. In our particular case, this formula
reduces to
\begin{equation}
\label{e:cutoffsum}
E(J,p^{\text{cutoff}}_2, p^{\text{cutoff}}_4) =
\sum_{p_2=0}^{p_2^{\text{cutoff}}} \sum_{p_4=0}^{p_4^\text{cutoff}}
\left|\frac{\eta_2^2}{\lambda^2}\right|^{p_2}
\left|\frac{\eta_4^2}{\lambda^4}\right|^{p_4}
\frac{J p_J}{2^{p_2} 4^{p_4}}
\frac{{\mathcal C}^2}{\langle 0|\, (\hat{O}_2)^{p_2} (\hat{O}_4)^{p_4}\,
(\hat{O}_4^\dagger)^{p_4} (\hat{O}_2^\dagger)^{p_2}\,
|0\rangle\,\langle c|c\rangle}\,.
\end{equation}
and the maximum values of $p_2$ and $p_4$ which are included in the
sum should be taken sufficiently large as to include at least the
maximum term in the sum. This requirement is indeed met in our
numerical approach. We have computed the ratio of energies in the
$J=2$ and $J=4$ particles using successive approximations
of~\eqn{e:cutoffsum}, for larger and larger $p^{\text{cutoff}}_2$ and
$p_4^{\text{cutoff}}$, for a range of couplings.\footnote{From the
gravity point of view, in case of large $N$, the $p^{\text{cutoff}}$
should always be such that the total energy (i.e.~conformal dimension)
carried by this multi particle state is smaller than~$N^2$ in order to
neglect back reaction. In the case of small~$N$, such as discussed
here, constants of order one become relevant, and this rough estimate
is no longer sufficient. For example, it turns
out~\cite{Peeters:2004rd} that the maximal probability for the number
of particles of type ${\mathcal O}_2$ is larger than $N^2$, but the
total energy carried by these particles is still smaller than the
energy of the brane (once constants of order one have been taken into
account).} A typical example is plotted in
figure~\ref{f:energies}. One clearly sees that the asymptotic value of
the ratio $E(4)/E(2)$, given by the exponent of the asymptotic height
difference between the two surfaces, is smaller than one. We therefore
conclude that our calculation predicts that higher-energy states in
the decay product are suppressed with respect to the lower-energy
ones. This is in qualitative agreement with alternative calculations
of this decay process~\cite{Lambert:2003zr}.
It would be very interesting to extend our analysis to higher-rank
gauge groups, perhaps by obtaining an analytic expression for the
norms of the states. For~$N>4$, there are more than two gauge singlet
states, and it becomes possible to determine the suppression factor as
a function of the energy in more detail. We leave this for future
investigations.
\psfrag{10}{$\scriptstyle 10$}
\psfrag{20}{$\scriptstyle 20$}
\psfrag{40}{$\scriptstyle 40$}
\psfrag{60}{$\scriptstyle 60$}
\psfrag{-20}{$\scriptstyle ~-20$}
\psfrag{-40}{$\scriptstyle ~-40$}
\psfrag{-60}{$\scriptstyle ~-60$}
\psfrag{-80}{$\scriptstyle ~-80$}
\psfrag{0}{$\scriptstyle 0$}
\psfrag{p2}{$\scriptstyle p_2^\text{cutoff}$}
\psfrag{p4}{$\scriptstyle p_4^\text{cutoff}$}
\begin{figure}[t]
\begin{center}
\includegraphics*[width=.6\textwidth]{energies_math.eps}
\makebox[0pt]{\hspace{-1.3cm}\raisebox{2.2cm}{\includegraphics*[height=5cm]{energies_overlay.eps}}}
\makebox[0pt]{\hspace{-19cm}\raisebox{5.4cm}{\includegraphics*[width=2cm]{enlarge.eps}}}
\caption{Successive approximations to the logarithm of the total
energy radiated in the $J=2$ particles (light, blue surface) and $J=4$
particles (dark, red surface). The $x$ and $y$ axes label the maximum
value of $p_2$ and $p_4$ in the sum~\eqn{e:cutoffsum}. The values
asymptote to the full result in the upper left corner of the
graph. While the present plot shows energies, qualitatively similar
plots are obtained for the particle numbers.}
\label{f:energies}
\end{center}
\end{figure}%
\section{Summary and outlook}
We have presented the formalism to analyse the decay of unstable
D-branes in the~$\text{AdS}_5 \times S^5$ background by considering
the dual gauge theory.\footnote{Such dynamical features of the
correspondence have meanwhile also been studied in the context of
large spinning strings~\cite{Peeters:2004pt}.} Our results show
qualitative agreement with previous work on D-particle decay, and our
work provides a basis for further study of non-perturbative dynamical
features of the correspondence.
A relevant way of improving on our results would be to determine
analytical expressions for the norms required in
section~\ref{s:U4results} (using the construction of states in terms
of group
characters~\cite{Balasubramanian:2001nh,Corley:2001zk,Kristjansen:2002bb}). This
would allow one to extend the results obtained there to large values
of~$N$. Also, as we have explained, due to the non-perturbative
nature of the initial sphaleron configuration, the computation of the
decay product requires information from a regime in which
both~\mbox{$N\rightarrow\infty$} as well as the number of particles
~$p\rightarrow\infty$. Knowing the norms of states analytically should
allow us to understand this double limit. This may perhaps circumvent
the need to calculate the norms of states exactly when calculating the
energy distribution in the final state. Finally, it would be
interesting to understand how quantum corrections can be incorporated
into our formalism, in order to see how much they influence the
qualitative characteristics of the decay product.
\section*{Acknowledgements}
We would like to thank Gleb Arutyunov, Rajesh Gopakumar, Justin
David, Stefano Kovacs, Charlotte Kristjansen, Shiraz Minwalla, Jan
Plefka and Ashoke Sen for discussions.
\vspace{-10ex}
\begingroup\raggedright
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,459
|
package org.jivesoftware.openfire.filetransfer;
/**
* An event listener for File Transfer related events.
*
* @author Guus der Kinderen, guus.der.kinderen@gmail.com
*/
public interface FileTransferEventListener
{
/**
* Invoked when a file transfer is about to start.. The interceptor can either modify the file transfer or
* throw a FileTransferRejectedException. The file transfer went sent to the interceptor can be in two states, ready
* and not ready. The not ready state indicates that this event was fired when the file transfer request was sent by
* the initiator. The ready state indicates that the file transfer is ready to begin, and the channels can be
* manipulated by the interceptor.
* <p>
* It is recommended for the the sake of user experience that when in the not ready state, any processing done on
* the file transfer should be quick.
*
* @param transfer the transfer being intercepted (never null).
* @param isReady true if the transfer is ready to commence or false if this is related to the
* initial file transfer request. An exception at this point will cause the transfer to
* not go through.
* @throws FileTransferRejectedException if the request was rejected
*/
void fileTransferStart( FileTransfer transfer, boolean isReady ) throws FileTransferRejectedException;
/**
* Invoked when a file transfer was completed. Events are generated for events that succeeded, but also for those
* that failed.
*
* @param transfer the transfer being intercepted (never null).
* @param wasSuccessful false when an exception was thrown during file transfer, otherwise true.
*/
void fileTransferComplete( FileTransfer transfer, boolean wasSuccessful );
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,884
|
Clean Transportation
Water & Conservation
Are you a CleanTechie?
Better Place Refuels Electric Car Commuters In Israel
by Ceylan Thomson August 7, 2009 1 comment
Refueling newly developed electric cars in Israel may be one step closer to being commonplace with an agreement reached between Israel Railways and the Better Place electric car and energy terminal company.
Better Place, which we've covered in depth currently in the process of developing practical electric powered cars, as well as recharging stations for them, has agreed to install up to 220 charging terminals in railway parking lots in a number of stations. They are Bat Galim, Central Haifa, Acre, Beit Yehoshua, Herzliya, Hod Hasharon, Rosh Ha'ayin, Petah Tikva Segula, Kiryat Arie Petah Tikva, Bnei Brak, and Pe'atei Modi'in.
Get your electric engines roaring? The agreement was reached by Yitzhak Harel, director -general of Israel Railways, and Moshe Kaplinsky, CEO of Better Place. The terms of the agreement will mean that Better Place will construct the "charging posts" in designated sections of the train station parking areas, so railway commuters can charge up their electric powered vehicles while they are traveling to and from various locations by train.
Special signs will be erected to instruct car owners how to use the charging stands, which will be available for the hundreds – if not thousands of all electric vehicles that are hoped to be available to people within the next two years.
Israel Railroad director Harel was quoted as saying that the train works as an "economic growth stimulator," and he added
"it contributes to improving the quality of life by charging thousands of cars that drive along Israel's roads and eliminating airborne vehicle pollutants."
Better Place began operations in 2007 and has been featured in previous Green Prophet articles, including one on May 14, 2009 as Better Place Tests Battery Replacement Technology in Japan. In this article, an experimental electric battery replacement station was being tested in Japan.
Another article also notes that electric cars are expected to be driven in large quantities world by the year 2030.
The actual use of these recharging stations won't be a reality for a few years, since virtually no total electric cars are presently available in Israel. Several thousand electric-gasoline hybrids (with dynamos) are being driven though; but they still utilize gasoline for more than 80% of their total distance usage, with electric motors only being used for in-town driving or on clogged motorways (electric motors are designed to power cars up to a maximum speed of 40 km per hour).
But any start is good start. And we like how Better Place targeted the commuter audience. And companies like Teva. Smart marketing move.
This article originally appeared on Green Prophet.
Better Placeelectric carIsrael Railwaysrecharging station
Ceylan Thomson
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A toast to the enzyme cocktail
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Ethanol: Remixed
The new Tesla Model S is… very sexy
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Tesla's new competitor – and they are bankrupt!
The Future of Electric Vehicles May Be Here Sooner Than We Think - CleanTechies April 13, 2015 - 12:10 pm
[…] France, the Netherlands, Belgium and Norway have already begun piloting their own projects. Beyond Europe there have also been significant developments in EV implementation. As reported on the CleanTechies Blog earlier this month, Better Place and Israel Railways have agreed on installing 220 charging stations in Israel. […]
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,520
|
is the 31st single by Japanese singer and voice actress Nana Mizuki, released on January 14, 2015 by King Records.
The full version MV wasn't released on YouTube; instead, it was included on Nana Clips 7.
Track listing
"Eden" (エデン)
Lyrics: Nana Mizuki
Composition: Shinichi Fujimori (Aobozu)
Arrangement: Hitoshi Fujima (Elements Garden)
Theme song (January Edition) for NTV Show Freshen Up! (スッキリ!!)
Theme song for animelo mix TV commercial
"No Limit"
Lyrics: Nana Mizuki
Composition: Jun Suyama
Arrangement: Jun Suyama
Opening Theme for anime television series [[Dog Days (Japanese TV series)|Dog Days{{}}]]
"Shūmatsu no Love Song" (終末のラブソング, Love Song of the End)
Lyrics: Nana Mizuki
Composition: Eriko Yoshiki
Arrangement: Hitoshi Fujima (Elements Garden)
2nd Ending Theme for anime television series Cross Ange: Rondo of Angels and Dragons "Necessary"
Lyrics: Goro Matsui
Composition: Zetta
Arrangement: EFFY
Insert song for anime television series Cross Ange: Rondo of Angels and Dragons''
Charts
Oricon Sales Chart (Japan)
References
2015 singles
Nana Mizuki songs
Songs written by Nana Mizuki
2015 songs
King Records (Japan) singles
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,611
|
Q: Request header for uploading the image using twitpic in blackberry? I want to upload an image to the user's twitter account. For this, I need a URL which I will get in from of a json response.
I know the URL where the request should be sent i.e. 'http://api.twitpic.com/2/upload.json', but what should be the parameters for the request that I don't know.
Any Help?
A: which language you are using?
For now, check the examples here http://twittown.com/forums/showthread.php?t=768 and using TwitPic + OAuth to upload a photo + tweet to Twitter (.NET C#) - why no tweet?
They detailed every thing how to Twitpic including the OAuth
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,818
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{"url":"https:\/\/answers.ros.org\/question\/296898\/trying-to-apply-a-rotation-of-one-frame-to-another-multiple-callback-transform\/","text":"# Trying to apply a rotation of one frame to another. Multiple callback transform?\n\nI have two frames set up: prism_frame and imu_frame. The IMU frame is translated in XYZ for (0, 0, 0.3) and get's the rotations via a callback function using the standard tutorial approach:\n\nnode.subscribe(\"\/imu_data\", 10, &imuCallback) where the callback function set's up the frame with prism_frame as the parent frame and imu_frame as the child\n\nThe prism_frame gets tracked and has also a subscription to local xyz data using a callback functions which also defines a frame. My Problem is that the prism_frame is in a rigid relation to the imu_frame and therefore should also get the rotations.\n\nI tried to solve it with two different callback function on the same frame setup with prism_frame as parent and imu_frame as child, but it didn't worked out since im overwriting them constantly... i really don't know how to solve this issue although it actually should be a simple one.. I was wondering if there is a possibility to process more than one message callbacks to define a frame?\n\nCurrently i'm trying to set um a lookupTransform in order to perform a tf2::doTransform on the frames, but i have the feeling that i don't really got the idea of how to use this correctly...\n\n## here is the code:\n\n#include <ros\/ros.h>\n#include <tf2\/LinearMath\/Quaternion.h>\n#include <tf2_ros\/transform_listener.h>\n#include <geometry_msgs\/PointStamped.h>\n#include <geometry_msgs\/TransformStamped.h>\n#include <sensor_msgs\/Imu.h>\n\nvoid ts2prismCallback(const geometry_msgs::PointStampedConstPtr& msg){\ngeometry_msgs::TransformStamped transformStamped;\n\n\/\/ ROS_INFO(\"msg x: %f\", msg->point.x);\ntransformStamped.child_frame_id = \"prism_frame\";\n\/\/ transform from map to tachy_frame with (1,1,1)\ntransformStamped.transform.translation.x = msg->point.x;\ntransformStamped.transform.translation.y = msg->point.y;\ntransformStamped.transform.translation.z = msg->point.z;\ntf2::Quaternion q;\nq.setRPY(0,0,0);\ntransformStamped.transform.rotation.x = q.x();\ntransformStamped.transform.rotation.y = q.y();\ntransformStamped.transform.rotation.z = q.z();\ntransformStamped.transform.rotation.w = q.w();\n\n}\n\nvoid prismRotationCallback(const sensor_msgs::ImuConstPtr& msg){\ngeometry_msgs::TransformStamped transformStamped;\n\ntransformStamped.child_frame_id = \"imu_frame\";\ntransformStamped.transform.translation.x = 0;\ntransformStamped.transform.translation.y = 0;\ntransformStamped.transform.translation.z = 0.3;\ntf2::Quaternion q;\ntransformStamped.transform.rotation.x = msg->orientation.x;\ntransformStamped.transform.rotation.y = msg->orientation.y;\ntransformStamped.transform.rotation.z = msg->orientation.z;\ntransformStamped.transform.rotation.w = msg->orientation.w;\n\n}\n\nint main(int argc, char** argv){\nros::init(argc, argv, \"tf_ts2prism\");\nros::NodeHandle node;\n\nros::Subscriber sub_ts = node.subscribe(\"\/ts_points\", 10, &ts2prismCallback);\nros::Subscriber sub_imu = node.subscribe(\"\/imu\/data_raw\", 10, &prismRotationCallback);\n\nros::spin();\nreturn 0;\n}\n\n\nand now the listener code\n\n#include <ros\/ros.h>\n#include \"std_msgs\/String.h\"\n#include \"sensor_msgs\/Imu.h\"\n#include \"geometry_msgs\/PointStamped.h\"\n#include <tf2\/LinearMath\/Quaternion.h>\n#include <tf2_ros\/buffer.h>\n#include <tf2_ros\/transform_listener.h>\n#include <tf2_geometry_msgs\/tf2_geometry_msgs.h ...","date":"2021-09-24 16:45:15","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\": 0.31633713841438293, \"perplexity\": 7457.578204840843}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"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-2021-39\/segments\/1631780057558.23\/warc\/CC-MAIN-20210924140738-20210924170738-00491.warc.gz\"}"}
| null | null |
/* Generated code */
import CompanyAnsweringRuleWeeklyScheduleInfoRequest from './CompanyAnsweringRuleWeeklyScheduleInfoRequest';
import RangesInfo from './RangesInfo';
interface CompanyAnsweringRuleScheduleInfo {
/**
* Weekly schedule. If specified, ranges cannot be specified
*/
weeklyRanges?: CompanyAnsweringRuleWeeklyScheduleInfoRequest;
/**
* Specific data ranges. If specified, weeklyRanges cannot be specified
*/
ranges?: RangesInfo;
/**
* Reference to Business Hours or After Hours schedule = ['BusinessHours', 'AfterHours']
*/
ref?: 'BusinessHours' | 'AfterHours';
}
export default CompanyAnsweringRuleScheduleInfo;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,822
|
\section{Introduction}
\label{sec:Intro}
Given a bivariate random vector $(X,Y)$ with joint probability distribution function $F_{X,Y}(x,y)=\mathbb{P}(X\leq x,Y\leq y)$ it is possible to assess uncertainty about one of the random variables conditioning on certain values of the other, for example through the univariate conditional probability distribution of $Y$ given $X=x,$ that is $F_{Y|X}(y\,|\,x)=\mathbb{P}(Y\leq y\,|\,X=x).$ As a point estimate for a future value of $Y$ given $X=x$ we may calculate central tendency measures with $F_{Y|X}$ such as the mean (whenever it exists) or the median (which always exists in the continuous case) which will depend on the conditioning value $x$ and therefore such point estimates depending on $x$ may be denoted by $\mu(x)$ and are called \textit{regression function} for $Y$ given $X=x.$
\medskip
As a consequence of \textit{Sklar's Theorem} \cite{Skl59} for continuous random variables there exists a unique copula $C$ such that the joint probability distribution function $F_{X,Y}(x,y)=C(F_X(x),F_Y(y))$ where $F_X(x)=\mathbb{P}(X\leq x)$ and $F_Y(y)=\mathbb{P}(Y\leq y)$ are the marginal probability distribution functions of $X$ and $Y,$ respectively. As explained in \cite{Nel06}, the conditional distribution of $Y$ given $X=x$ can be obtained by
\begin{equation}
F_{Y|X}(y\,|\,x)\,=\,\frac{\partial C(u,v)}{\partial u}\Big|_{u\,=\,F_X(x)\,,\,v\,=\,F_Y(y)}
\label{eq:condcdf}
\end{equation}
and therefore to find the median regression function for $Y$ given $X=x$ whenever $F_{Y|X}$ is a continuous distribution function, we proceed as follows:
\medskip
\noindent\textbf{Algorithm 1}
\begin{enumerate}
\item Set $\partial C(u,v)/\partial u\,=\,1/2\,;$
\item solve for the regression function of $V=F_Y(Y)$ given $U=F_X(X)=u,$ and obtain $v=\psi(u)\,;$
\item replace $u$ by $F_X(x)$ and $v$ for $F_Y(y)\,;$
\item solve for the regression function of $Y$ given $X=x:$
\begin{equation}
y\,=\,\mu(x)\,=\,F^{(-1)}_Y(\psi(F_X(x))).
\label{eq:regcurve}
\end{equation}
\end{enumerate}
It is worth to notice that since $F_X$ and $F_Y$ only explain the individual (marginal) probabilistic behavior of the continuous random variables $X$ and $Y,$ respectively, then the information about their dependence for regression purposes is contained in $\psi.$ A survey of copula-based regression models may be found in \cite{KolPai09} and estimation/inference procedures for such purpose in \cite{NohGhoBou13}.
\section{Piecewise monotone regression}
\label{sec:Piecewise}
In \cite{DetHecVol14} it is argued that when the regression function is non-monotone, copula-based regression estimates do not reproduce the qualitative features of the regression function under commonly used parametric copula families. This occurs because very often such parametric copulas lead to monotone regression functions, but in case there is evidence that the underlying regression function is non-monotone a \textit{piecewise regression} approach may be applied in order to break up a non-monotone relationship into a piecewise monotonic one, and then seek for the best copula fit for each piece.
\medskip
Piecewise (or segmented) monotone regression for $Y$ given $X=x$ is defined by partitioning the support of $X$ into a finite number of intervals such that restricted to each one it is possible to obtain a monotone regression function. For example, instead of (\ref{eq:condcdf}) we may obtain something like
\begin{equation}
F_{Y|X}(y\,|\,x)\,=\,\begin{cases}
\,\frac{\partial}{\partial u}C_1(u,v)\Big|_{u\,=\,F_{X|X\leq\,b}(x)\,,\,v\,=\,F_Y(y)}, & \text{ if } x\leq b, \\
\,\frac{\partial}{\partial u}C_2(u,v)\Big|_{u\,=\,F_{X|X>\,b}(x)\,,\,v\,=\,F_Y(y)},& \text{ if } x > b,
\end{cases}
\label{eq:condcdf2}
\end{equation}
with $C_1$ and $C_2$ two different copulas, $b$ is called a \textit{break-point} for explanatory variable $X,$ and where $F_{X|X\leq\,b}$ and $F_{X|X>\,b}$ are the conditional distribution functions of $X$ given $X\leq b$ and $X>b,$ respectively. This may be justified in terms of the \textit{gluing copula} technique \cite{SibSto08} as explained in \cite{ErdDia10} for the particular case of vertical section gluing and bivariate copulas. Specifically, given two bivariate copulas $C_1$ and $C_2,$ and a fixed value $0<\theta<1$ (gluing point), we may scale $C_1$ to $[0,\theta]\times[0,1]$ and $C_2$ to $[\theta,1]\times[0,1]$ and \textit{glue} them into a single copula
\begin{equation}
C_{1,2,\theta}(u,v)\,=\,\begin{cases}
\,\theta C_1(\frac{u}{\theta},v), & 0\leq u\leq\theta,\\
\,(1-\theta)C_2(\frac{u-\theta}{1-\theta},v)+\theta v, & \theta\leq u\leq 1.
\end{cases}
\label{eq:gluing}
\end{equation}
Then
\begin{equation}
\frac{\partial}{\partial u}C_{1,2,\theta}(u,v)\,=\,\begin{cases}
\,\frac{\partial}{\partial u}C_1(\frac{u}{\theta},v), & 0\leq u\leq\theta,\\
\,\frac{\partial}{\partial u}C_2(\frac{u-\theta}{1-\theta},v), & \theta\leq u\leq 1,
\end{cases}
\label{eq:gluingdu}
\end{equation}
and by (\ref{eq:condcdf})
\begin{eqnarray}
F_{Y|X}(y\,|\,x) &=& \frac{\partial C_{1,2,\theta}(u,v)}{\partial u}\bigg|_{u\,=\,F_X(x)\,,\,v\,=\,F_Y(y)} \nonumber \\
&=& \begin{cases}
\,\frac{\partial}{\partial u}C_1(\frac{F_X(x)}{\theta},F_Y(y)), & 0\leq F_X(x)\leq\theta,\\
\,\frac{\partial}{\partial u}C_2(\frac{F_X(x)-\theta}{1-\theta},F_Y(y)), & \theta\leq F_X(x)\leq 1,
\end{cases} \nonumber \\
&=& \begin{cases}
\,\frac{\partial}{\partial u}C_1(u,v)\Big|_{u\,=\,F_{X|X\leq\,b}(x)\,,\,v\,=\,F_Y(y)}, & x\leq F^{(-1)}_X(\theta)=b, \\
\,\frac{\partial}{\partial u}C_2(u,v)\Big|_{u\,=\,F_{X|X>\,b}(x)\,,\,v\,=\,F_Y(y)}, & x > F^{(-1)}_X(\theta)=b,
\end{cases}
\label{eq:condcdf3}
\end{eqnarray}
since $F_{X|X\leq\,b}(x)=\mathbb{P}(X\leq x\,|\,X\leq b) = \mathbb{P}(X\leq x)/\mathbb{P}(X\leq b) = F_X(x)/\theta$ and\linebreak $F_{X|X>\,b}(x) = \mathbb{P}(b<X\leq x)/\mathbb{P}(X>b) = (F_X(x)-\theta)/(1-\theta).$ The result obtained in (\ref{eq:condcdf3}) leads to a regression function of the form
\begin{equation}
\mu(x)\,=\,\begin{cases}
\,\mu_1(x), & \text{ if } x\leq b, \\
\,\mu_2(x), & \text{ if } x > b,
\end{cases}
\label{eq:regcurve2}
\end{equation}
where, for example, if $\mu_1(x)$ is an increasing function and $\mu_2(x)$ a decreasing one, then $\mu(x)$ is non-monotone.
\begin{example}\label{ex:NelsenEj33}
From example 3.3 in \cite{Nel06} if a probability mass $0<\theta<1$ is uniformly distributed on the line segment joining $(0,0)$ to $(\theta,1),$ and a probability mass $1-\theta$ is uniformly distributed on the line segment joining $(\theta,1)$ to $(1,0),$ see Fig. \ref{fig:Example1}, the underlying copula for a random vector $(X,Y)$ of continuous Uniform$(0,1)$ random variables with such non-monotone dependence is given by
\begin{equation}
C_{\theta}(u,v)\,=\,\begin{cases}
\,u, & 0\leq u\leq\theta v\leq \theta, \\
\,\theta v, & 0\leq\theta v<u<1-(1-\theta)v, \\
\,u+v-1, & \theta\leq 1-(1-\theta)v\leq u\leq 1.
\end{cases}
\label{eq:NelsenEj33A}
\end{equation}
\begin{figure}[t]
\sidecaption
\includegraphics[scale=.30]{Example1.pdf}
\caption{Example \ref{ex:NelsenEj33}. Left: $(X,Y)$ dependence. Right: underlying copula (\ref{eq:NelsenEj33A}).}
\label{fig:Example1}
\end{figure}
By construction we have that $\mathbb{P}(Y=\frac{x}{\theta}\,|\,X=x)=1$ whenever $0\leq x\leq\theta$ and $\mathbb{P}(Y=\frac{1-x}{1-\theta}\,|\,X=x)=1$ whenever $\theta\leq x\leq 1,$ which implies that the regression function of $Y$ given $X=x$ is
\begin{equation}
\mu(x)\,=\,\begin{cases}
\,\,\,\,\frac{x}{\theta}, & 0\leq x\leq\theta, \\
\,\frac{1-x}{1-\theta}, & \theta\leq x\leq 1,
\end{cases}
\label{eq:NelsenEj33B}
\end{equation}
clearly a non-monotone function: linearly increasing for $0\leq x\leq\theta$ and linearly decreasing for $\theta\leq x\leq 1,$ which suggests in this case that the underlying dependence might be split by means of the gluing copula technique in terms of two copulas, with $\theta$ as gluing point. Indeed, let $C_1(u,v)=\min\{u,v\}$ (the Fr\'echet-Hoeffding upper bound that represents the case when one variable is almost surely an increasing function of the other) and $C_2(u,v)=\max\{u+v-1,0\}$ (the Fr\'echet-Hoeffding lower bound that represents the case when one variable is almost surely a decreasing function of the other), then applying (\ref{eq:gluing}) it is straightforward to verify that the resulting gluing copula $C_{1,2,\theta}$ is equal to (\ref{eq:NelsenEj33A}).
\medskip
Therefore, the same regression function obtained in (\ref{eq:NelsenEj33B}) could be obtained in two pieces: the first one in terms of the random vector $(X_1,Y)$ with underlying copula $C_1$ and where the distribution of $X_1$ is the conditional distribution of $X$ given $X\leq\theta,$ which turns to be uniform$(0,\theta),$ and the second one in terms of the random vector $(X_2,Y)$ with underlying copula $C_2$ and where the distribution of $X_2$ is the conditional distribution of $X$ given $X>\theta,$ which turns to be uniform$(\theta, 1).$ Applying (\ref{eq:condcdf}) to the first piece we obtain the following:
\begin{eqnarray}
F_{Y|X_1}(y\,|\,x) &=& \frac{\partial}{\partial u}C_1(u,v)\Big|_{u\,=\,\frac{x}{\theta}\,,\,v\,=\,y} \nonumber \\
&=& \begin{cases}
\,1, & \text{ if } y\geq\frac{x}{\theta}\,, \\
\,0, & \text{ if } y < \frac{x}{\theta}
\end{cases}
\label{eq:NelsenEj33C}
\end{eqnarray}
from which we get $\mu_1(x)=\frac{x}{\theta}$ whenever $0\leq x\leq\theta,$ and similarly from $F_{Y|X_2}(y\,|\,x)$ we obtain $\mu_2(x)=\frac{1-x}{1-\theta}$ whenever $\theta\leq x\leq 1,$ as expected.$\qquad_{\blacksquare}$
\end{example}
For simplicity's sake, the case for a single break-point has been analyzed, but the analogous idea may be applied for finitely many break-points. For each interval $I$ induced in the support of the explanatory variable, the conditional distribution of $Y$ given $X=x$ is obtained by
\begin{equation}
F_{Y|X}(y\,|\,x)\,=\,\frac{\partial}{\partial u}C_I(u,v)\Big|_{u\,=\,F_{X|X\in\,I}(x)\,,\,v\,=\,F_Y(y)}
\label{eq:condcdf4}
\end{equation}
and with it the regression function $\mu(x)$ for $x\in I$ may be calculated.
\section{Dependence and regression}
\label{sec:DepReg}
In this section the concepts of quadrant and regression dependence by \cite{Leh66} are recalled.
\begin{definition}\label{def:PQD}
A bivariate random vector $(X,Y)$ or its joint distribution function $F_{X,Y}$ is \textit{positively quadrant dependent} and abbreviated as $\text{PQD}(X,Y)$ if
\begin{equation}
\mathbb{P}(X\leq x, Y\leq y)\,\geq\,\mathbb{P}(X\leq x)\mathbb{P}(Y\leq y)\,,\qquad \text{for all }x \text{ and } y,
\label{eq:PQD}
\end{equation}
and \textit{negatively quadrant dependent} $\text{NQD}(X,Y)$ if (\ref{eq:PQD}) holds with the inequality sign reversed.
\end{definition}
In the particular case where both $X$ and $Y$ are continuous random variables with underlying copula $C,$ as an immediate consequence of \textit{Sklar's Theorem} \cite{Skl59} we have that $\text{PQD}(X,Y)$ is equivalent to $C(u,v)\geq uv$ for all $u,v$ in $[0,1],$ and
$\text{NQD}(X,Y)$ with this last inequality sign reversed. From \cite{Nel06} we have the following:
\begin{definition}\label{def:order}
If $C_1$ and $C_2$ are copulas, we say that $C_1$ is \textit{smaller than} $C_2$ (or $C_2$ \textit{is larger than} $C_1$), and write $C_1\prec C_2$ (or $C_2\succ C_1$) if $C_1(u,v)\leq C_2(u,v)$ for all $u,v$ in $[0,1].$
\end{definition}
This point-wise partial ordering of the set of copulas is called \textit{concordance ordering.} It is a partial order rather than a total order because not every pair of copulas is comparable. However, there are families of copulas that are totally ordered. We will call a totally ordered parametric family $\{C_{\theta}\}$ of copulas \textit{positively ordered} if $C_{\alpha}\prec C_{\beta}$ whenever $\alpha\leq\beta;$ and \textit{negatively ordered} if $C_{\alpha}\succ C_{\beta}$ whenever $\alpha\leq\beta.$ Many of well known one-parameter families of copulas are totally ordered and include $\Pi(u,v)=uv,$ and hence have subfamilies of PQD and NQD copulas.
\medskip
As mentioned in \cite{Nel06} one form to calculate \textit{Spearman's concordance measure} is
\begin{equation}
\rho_{\,C}\,=\,12\int\!\!\!\!\int_{[0,1]^2}\big[\,C(u,v)-uv\,\big]\,dudv\,=\,12\int\!\!\!\!\int_{[0,1]^2}C(u,v)\,dudv\,-\,3\,,
\label{eq:Spearman}
\end{equation}
and hence $\rho_{\,C}/12$ can be interpreted as a measure of ``average'' quadrant dependence (both positive and negative) for continuous random variables whose copula is $C.$ Closely related to (\ref{eq:Spearman}) is the $L_1$ distance between $C$ and the (sometimes called) independence copula $\Pi(u,v)=uv$ known as \textit{Schweizer-Wolff's dependence measure} \cite{SchWol81} defined as
\begin{equation}
\sigma_{\,C}\,=\,12\int\!\!\!\!\int_{[0,1]^2}\big|C(u,v)-uv\big|\,dudv\,.
\label{eq:Schweizer}
\end{equation}
Two main differences (among others) are that $-1\leq\rho_{\,C}\leq 1$ in contrast to $0\leq\sigma_{\,C}\leq 1,$ and that $\sigma_{\,C}=0$ if and only $X$ and $Y$ are independent (that is $C=\Pi$) while $\rho_{\,C}=0$ does not necessarily imply independence. Moreover, as explained in \cite{Nel06}:
\begin{quote}\label{qu:Nelsen}
Of course, it is immediate that if $X$ and $Y$ are PQD, then $\sigma_{\,X,Y}=\rho_{\,X,Y}\,;$ and that if $X$ and $Y$ are NQD, then $\sigma_{\,X,Y}=-\rho_{\,X,Y}\,.$ Hence for many of the totally ordered families of copulas presented in earlier chapters (e.g., Plackett, Farlie-Gumbel-Morgenstern, and many families of Archimedean copulas), $\sigma_{\,X,Y}=|\rho_{\,X,Y}|.$ But for random variables $X$ and $Y$ that are neither PQD nor NQD, i.e., random variables whose copulas are neither larger nor smaller than $\Pi,$ $\sigma$ is often a better measure than $\rho$ [\ldots]
\end{quote}
\begin{definition}\label{def:PRD}
A random variable $Y$ is \textit{positively regression dependent} on a random variable $X$ and abbreviated as $\text{PRD}(Y|X)$ if
\begin{equation}
F_{Y|X}(y\,|\,x)\,=\,\mathbb{P}(Y\leq y\,|\,X=x)\quad\text{is non-increasing in } x,
\label{eq:PRD}
\end{equation}
and \textit{negatively regression dependent} $\text{NRD}(Y|X)$ if (\ref{eq:PRD}) is non-decreasing in $x.$
\end{definition}
From theorems 5.2.4 and 5.2.12 in \cite{Nel06} or from Lemma 4 in \cite{Leh66} we have the following:
\begin{corollary}
Given $(X,Y)$ a bivariate random vector:
\begin{itemize}
\item[a)] If $\,\text{PRD}(Y|X)\,$ then $\,\text{PQD}(X,Y).$
\item[b)] If $\,\text{NRD}(Y|X)\,$ then $\,\text{NQD}(X,Y).$
\end{itemize}
\label{cor:QDimpliesRD}
\end{corollary}
By arguments explained in \cite{Nel06} the reverse implications in Corollary \ref{cor:QDimpliesRD} do not necessarily hold.
\begin{corollary}
If $(X,Y)$ are continuous random variables with underlying copula $C$ then:
\begin{itemize}
\item[a)] $\text{PRD}(Y|X)$ if and only if for any $v$ in $[0,1]$ and for almost all $u,$ $\partial C(u,v)/\partial u$ is non-increasing in $u;$
\item[b)] $\text{NRD}(Y|X)$ if and only if for any $v$ in $[0,1]$ and for almost all $u,$ $\partial C(u,v)/\partial u$ is non-decreasing in $u.$
\end{itemize}
\label{cor:NelsenThm5210}
\end{corollary}
In case the conditional expectation exists it is possible to obtain a \textit{mean regression function}
\begin{equation}
\mu(x) \,=\, \mathbb{E}(Y\,|\,X=x) \,=\, \int_0^{\,\infty}[1-F_{Y|X}(y\,|\,x)]\,dy\,-\,\int_{-\infty}^{\,0} F_{Y|X}(y\,|\,x)\,dy\,,
\label{eq:regmedia}
\end{equation}
and in case $F_{Y|X}(y\,|\,x)$ is a continuous function of $y$ then it is possible to obtain a \textit{median regression function}
\begin{equation}
\mu(x) \,=\,\text{median}(Y\,|\,X=x) \,=\, F_{Y|X}^{(-1)}(0.5\,|\,x)\,.
\label{eq:regmediana}
\end{equation}
\begin{proposition}
Let $\mu(x)$ be a mean or median regression function:
\begin{itemize}
\item[a)] If $\text{PRD}(Y|X)$ then $\mu(x)$ is non-decreasing.
\item[b)] If $\text{NRD}(Y|X)$ then $\mu(x)$ is non-increasing.
\end{itemize}
\label{prop:regresion}
\end{proposition}
\begin{proof}
If $\text{PRD}(Y|X)$ then for all $x_1<x_2$
\begin{eqnarray}
-F_{Y|X}(y\,|\,x_1) &\leq& -F_{Y|X}(y\,|\,x_2)\,, \label{eq:PRD2} \\
1 - F_{Y|X}(y\,|\,x_1) &\leq& 1 - F_{Y|X}(y\,|\,x_2)\,. \label{eq:PRD3}
\end{eqnarray}
Integration of (\ref{eq:PRD2}) on $\,]-\infty,0]$ and of (\ref{eq:PRD3}) on $[0,\infty[\,,$ and adding the results according to the inequalities it is obtained $\mu(x_1)=\mathbb{E}(Y\,|\,X=x_1)\leq\mathbb{E}(Y\,|\,X=x_2)=\mu(x_2),$ as required. Now from (\ref{eq:PRD2}) we have $F_{Y|X}(y\,|\,x_1)\geq F_{Y|X}(y\,|\,x_2),$ and since $F_{Y|X}(y\,|\,x)$ is non-decreasing in $y$ for any $x$ so is $F_{Y|X}^{(-1)}(u\,|\,x)$ as a function of $u$ and therefore $F_{Y|X}^{(-1)}(u\,|\,x_1)\leq F_{Y|X}^{(-1)}(u\,|\,x_2),$ hence $\mu(x_1)=\text{median}(Y\,|\,X=x_1)=\,$$F_{Y|X}^{(-1)}(0.5\,|\,x_1)\leq\,$$F_{Y|X}^{(-1)}(0.5\,|\,x_2)=\,$$\text{median}(Y\,|\,X=x_2)=\mu(x_2),$ as required.$\qquad_{\blacksquare}$
\end{proof}
But the reverse implications in this last proposition do not necessarily hold, as it can be easily verified by similar arguments.
\begin{example}\label{ex:NelsenEj33bis}
Continuing with Example \ref{ex:NelsenEj33}, applying formulas (\ref{eq:Spearman}) and (\ref{eq:Schweizer}) it is straightforward to verify that Spearman's $\rho_{\theta}=2\theta-1$ and Schweizer-Wolff's $\sigma_{\theta}=\theta^2+(\theta-1)^2,$ and since $0<\theta<1$ then $|\rho_{\theta}|<\sigma_{\theta}$ and therefore neither we have PQD nor NQD, and neither PRD nor NRD. Moreover, if $\theta=\frac{1}{2}$ then $\rho_{1/2}=0$ but this does not imply independence since $\sigma_{1/2}=\frac{1}{2}$ (its minimum possible value, by the way). See Fig. \ref{fig:Examples23} (left).$\qquad_{\blacksquare}$
\end{example}
\section{Change-point detection}
\label{sec:Chngpt}
The ideas explained in the previous sections may be useful in tackling the concerns raised by \cite{DetHecVol14} when the dependence relationship between random variables implies a non-monotone regression function, considering that the most common families of parametric copulas lead to monotone regression functions, and a possible solution might be to break up such dependence into pieces such that within each one the dependence implies a piecewise monotone regression function, and possibly one of the common families of parametric copulas may have an acceptable fit for each piece. In pursuing this objective, when dealing with data from which the dependence has to be estimated, a methodology to find break-point candidates, that is \textit{change-point detection,} becomes necessary.
\medskip
\begin{definition}\label{def:diagonal}
The \textit{diagonal section} of a copula $C$ is a function $\delta_{\,C}:[0,1]\rightarrow[0,1]$ given by $\delta_{\,C}(t)=C(t,t).$
\end{definition}
Since every copula $C$ is bounded by the Fr\'echet-Hoeffding bounds $\max\{u+v-1,0\}\leq C(u,v)\leq\min\{u,v\}$ then $\max\{2t-1,0\}\leq\delta_{\,C}(t)\leq t.$ If $C=\Pi$ (independence copula) then $\delta_{\,\Pi}(t)=t^2.$ If $(X,Y)$ is a random vector of continuous random variables with underlying copula $C$ and $\text{PQD}(X,Y)$ or $\text{NQD}(X,Y)$ then $C(u,v)\geq uv$ or $C(u,v)\leq uv,$ respectively, for all $(u,v)$ in $[0,1]^2,$ and therefore $\delta_{\,C}(t)\geq t^2$ or $\delta_{\,C}(t)\leq t^2,$ respectively, for all $t$ in $[0,1].$ Hence, if there exist $t_1,t_2$ in $[0,1]$ such that $\delta_{\,C}(t_1)<t_1^2$ and $\delta_{\,C}(t_2)>t_2^2$ then neither $\text{PQD}(X,Y)$ nor $\text{NQD}(X,Y),$ and consequently this would imply that neither $\text{PRD}(Y|X)$ nor $\text{NRD}(Y|X).$ In case of this last scenario, this would not necessarily imply that a mean or median regression function $\mu(x)$ is non-monotone since Proposition \ref{prop:regresion} is a one-way implication, but at least raises the question and leads to propose and analyze break-point candidates. The following result is straightforward:
\begin{proposition}
Let $C_1$ and $C_2$ be two copulas such that $C_1(u,v)\geq uv$ and $C_2(u,v)\leq uv$ for all $(u,v)\in[0,1]^2,$ and let $0<\theta<1.$ Then the diagonal section of the resulting gluing copula $C_{1,2,\theta}$ as in (\ref{eq:gluing}) satisfies
\begin{equation}
\delta_{1,2,\theta}(t) \begin{cases}
\,\geq t^2, & \text{ if } \,0\leq t\leq\theta, \\
\,=\theta^2, & \text{ if } \,t = \theta, \\
\,\leq t^2, & \text{ if } \,\theta\leq t\leq 1.
\end{cases}
\label{eq:gluingDiag}
\end{equation}
\label{prop:gluingDiag}
\end{proposition}
Since the diagonal section $\delta_{\,C}$ of any copula $C$ is a continuous function, see \cite{Nel06}, we may choose and analyze as possible break-point candidates those where crossings between $\delta_{\,C}$ and $\delta_{\,\Pi}$ take place.
\begin{example}\label{ex:NelsenEj33bis2}
Continuing with Example \ref{ex:NelsenEj33}, from formula (\ref{eq:NelsenEj33A}) the corresponding diagonal section is:
\begin{equation}\label{diagEjemplo}
\delta_{\theta}(t)\,=\,C_{\theta}(t,t)\,=\,\begin{cases}
\,\theta t\,, & t\leq\frac{1}{2-\theta}\,, \\
\,2t-1\,, & t\geq\frac{1}{2-\theta}\,.
\end{cases}
\end{equation}
If $0<t\leq\frac{1}{2-\theta}$ then $\delta_{\theta}(t)\geq t^2$ if and only if $t\leq\theta.$ If $\frac{1}{2-\theta}\leq t<1$ then $\delta_{\theta}(t)\leq t^2.$ Since $0<\theta<1$ then $\theta<\frac{1}{2-\theta}$ and therefore we conclude that $\delta_{\theta}(t)\geq t^2$ if and only if $t\leq\theta,$ and $\delta_{\theta}(t)\leq t^2$ if and only if $t\geq\theta.$ Hence, we would propose $t=\theta$ as break-point candidate, as expected. See Fig. \ref{fig:Examples23} (right).$\qquad_{\blacksquare}$
\end{example}
\begin{figure}[t]
\sidecaption
\includegraphics[scale=.30]{Examples23.pdf}
\caption{Left: $|\rho_{\theta}| $ (dashed line) and $\sigma_{\theta}$ (continuous line) in Example \ref{ex:NelsenEj33bis}. Right: $\delta_{\theta}$ (thick line) and $\delta_{\Pi}$ (thin line) in Example \ref{ex:NelsenEj33bis2}.}
\label{fig:Examples23}
\end{figure}
\begin{example}\label{Dette}
This is one of the examples used in \cite{DetHecVol14} to raise concerns about the use of copulas when the dependence relationship between random variables implies a non-monotone regression function. Let $\varepsilon$ be a Normal$(0,1)$ random variable, a constant $k^2 = 0.01,$ and $X$ a Uniform$(0,1)$ random variable independent from $\varepsilon.$ Now define the random variable:
\begin{equation}\label{eq:Yreg}
Y\,=\,(X\,-\,0.5)^2\,+\,k\varepsilon\,.
\end{equation}
Then the conditional distribution of $Y$ given $X=x$ is Normal$\big((x-0.5)^2,k^2\big)$ and therefore the corresponding mean regression function is given by:
\begin{equation}\label{eq:YregFunc}
\mu(x)\,=\,\mathbb{E}(Y\,|\,X=x)\,=\,(x\,-\,0.5)^2\,,\quad 0\leq x\leq 1,
\end{equation}
clearly a non-monotone regression function (decreasing when $x\leq 0.5,$ increasing when $x\geq 0.5).$ Since the joint probability density of $(X,Y)$ is given by $f_{X,Y}(x,y)=f_X(x)f_{Y|X}(y\,|\,x)$ then:
\begin{eqnarray}\label{eq:FXY}
F_{X,Y}(x,y) &=& \int_{-\infty}^{\,x}f_X(r)\int_{-\infty}^{\,y}f_{Y|X}(s\,|\,x)\,dsdr = \int_{-\infty}^{\,x}f_X(r)F_{Y|X}(y\,|\,r)\,dr \nonumber \\
&=& \begin{cases}
\,0\,, & \text{ if } x\leq 0, \\
\displaystyle{\int_0^{\,x}\Phi\Big(\frac{y-(r-0.5)^2}{k}\Big)\,dr\,,} & \text{ if } 0<x<1, \\
\displaystyle{\int_0^1\Phi\Big(\frac{y-(r-0.5)^2}{k}\Big)\,dr\,,} & \text{ if } x\geq 1,
\end{cases}
\end{eqnarray}
\noindent where $\Phi$ is the distribution function for a Normal$(0,1)$ random variable. From (\ref{eq:FXY}) it is possible obtain the following expression for the marginal distribution function of $Y:$
\begin{equation}\label{eq:FY}
F_Y(y) \,=\, F_{X,Y}(+\infty,y) \,=\, \int_0^1\Phi\Big(\frac{y-(r-0.5)^2}{k}\Big)\,dr.
\end{equation}
Hence, by Sklar's Corollary 2.3.7 in \cite{Nel06} it is possible to obtain the following expression for the underlying copula of $(X,Y):$
\begin{equation}\label{eq:copula}
C(u,v) \,=\, F_{X,Y}\big(F_X^{(-1)}(u), F_Y^{(-1)}(v)\big) \,=\, \int_0^{\,u}\Phi\Big(\frac{F_Y^{-1}(v)-(r-0.5)^2}{k}\Big)\,dr\,,
\end{equation}
and consequently the diagonal section of such copula is given by:
\begin{equation}\label{eq:diagonal}
\delta_C(t) \,=\, C(t,t) \,=\, \int_0^{\,t}\Phi\Big(\frac{F_Y^{-1}(t)-(r-0.5)^2}{k}\Big)\,dr.
\end{equation}
\begin{figure}[t]
\sidecaption
\includegraphics[scale=.30]{Example4A.pdf}
\caption{Example \ref{Dette}. Left: Level curves (thick style) of copula (\ref{eq:copula}) versus level curves (thin style) of product (or independence) copula. Right: Diagonal section (thick style) of copula (\ref{eq:copula}) versus diagonal section (thin style) of product (or independence) copula.}
\label{fig:Example4A}
\end{figure}
\noindent In Fig. \ref{fig:Example4A} (left) we may notice crossings between copula (\ref{eq:copula}) level curves (thick style) and the product (or independence) copula $\Pi(u,v)=uv$ level curves (thin style), with the following interpretation: thick curve below thin curve implies $C(u,v)\geq\Pi(u,v)$ and thick curve above thin curve implies $C(u,v)\leq\Pi(u,v).$ In Fig. \ref{fig:Example4A} (right) the graph of the diagonal section (\ref{eq:diagonal}) is compared to the graph of the diagonal section of $\Pi$ from where we get as gluing point candidate $u=\theta=1/2.$
\medskip
\noindent Then we proceed to a \textit{gluing copula decomposition} by means of (\ref{eq:gluing}) where $C_{1,2,\theta}=C.$ For $0\leq u\leq\theta$ we get $\theta C_1(\frac{u}{\theta},v)=C(u,v),$ and if we let $u_*=\frac{u}{\theta}\in[0,1]$ then:
\begin{equation}\label{eq:copula1}
C_1(u_*,v) \,=\, \frac{1}{\theta}C(\theta u_*,v) \,=\, 2\int_0^{\,u_*/2}\Phi\Big(\frac{F_Y^{-1}(v)-(r-0.5)^2}{k}\Big)\,dr\,,
\end{equation}
and therefore:
\begin{equation}\label{eq:parcialcopula1}
\frac{\partial}{\partial u_*}C_1(u_*,v) \,=\, \Phi\Big(\frac{F_Y^{-1}(v)-0.25(1-u_*)^2}{k}\Big)\,,
\end{equation}
where clearly (\ref{eq:parcialcopula1}) is a non-decreasing function of $u_*$ which by Corollary \ref{cor:NelsenThm5210} implies NRD for copula $C_1,$ and consequently NQD by Corollary \ref{cor:QDimpliesRD}. Also, by Proposition \ref{prop:regresion} we get that a regression function $\mu_1(x)$ based on $C_1$ will lead to a non-increasing function of $x.$ See Fig. \ref{fig:Example4B} (left) for the level curves of $C_1$ (thick style) versus the level curves (thin lines) of $\Pi(u,v)=uv,$ where all the level curves of $C_1$ are above the corresponding ones to $\Pi$ implying that $C_1(u,v)\leq\Pi(u,v),$ as expected.
\begin{figure}[t]
\sidecaption
\includegraphics[scale=.30]{Example4B.pdf}
\caption{Example \ref{Dette}. Left: Level curves (thick style) of copula (\ref{eq:copula1}) versus level curves (thin style) of product (or independence) copula. Right: Level curves (thick style) of copula (\ref{eq:copula2}) versus level curves (thin style) of product (or independence) copula.}
\label{fig:Example4B}
\end{figure}
\medskip
\noindent Similarly, for $\theta\leq u\leq 1$ we get $(1-\theta)C_2(\frac{u-\theta}{1-\theta},v) + \theta v=C(u,v)$ and if we let $u_*=\frac{u-\theta}{1-\theta}\in[0,1]$ then:
\begin{eqnarray}
C_2(u_*,v) &=& \frac{C((1-\theta)u_*+\theta,v)-\theta v}{1-\theta} \nonumber \\
&=& 2\int_0^{(u_*+1)/2}\Phi\Big(\frac{F_Y^{-1}(v)-(r-0.5)^2}{k}\Big)\,dr\,-\,v\,,
\end{eqnarray}\label{eq:copula2}
and therefore:
\begin{equation}\label{eq:parcialcopula2}
\frac{\partial}{\partial u_*}C_2(u_*,v) \,=\, \Phi\Big(\frac{F_Y^{-1}(v)-0.25u_*^2)}{k}\Big)\,,
\end{equation}
where clearly (\ref{eq:parcialcopula2}) is a non-increasing function of $u_*$ which by Corollary \ref{cor:NelsenThm5210} implies PRD for copula $C_2,$ and consequently PQD by Corollary \ref{cor:QDimpliesRD}. Also, by Proposition \ref{prop:regresion} we get that a regression function $\mu_2(x)$ based on $C_2$ will lead to a non-decreasing function of $x.$ See Fig. \ref{fig:Example4B} (right) for the level curves of $C_2$ (thick style) versus the level curves (thin lines) of $\Pi(u,v)=uv,$ where all the level curves of $C_2$ are below the corresponding ones to $\Pi$ implying that $C_2(u,v)\geq\Pi(u,v),$ as expected.
\medskip
\noindent In summary, the dependence between $X$ and $Y$ induced by (\ref{eq:Yreg}), which by construction has a regression function $\mu(x)$ that is non-monotone, has an underlying copula $C$ given by (\ref{eq:copula}) with a diagonal section $\delta_C$ given by (\ref{eq:diagonal}) that gives as gluing point candidate $\theta=1/2,$ leading to a gluing copula decomposition as in (\ref{eq:gluing}) where $C_1$ is NQD and NRD and therefore leads to a non-increasing regression function $\mu_1(x),$ and where $C_2$ is PQD and PRD and therefore leads to a non-decreasing regression function $\mu_2(x),$ that is:
\begin{equation}\label{eq:nomonotreg}
\mu(x) \,=\, \begin{cases}
\,\mu_1(x)\,\downarrow\,, & u=F_X(x)=x\leq\theta=1/2\,, \\
\,\mu_2(x)\,\uparrow\,, & u=F_X(x)=x\geq\theta=1/2.
\end{cases}
\end{equation}
In this example it was possible to obtain a gluing copula decomposition as in (\ref{eq:gluing}) of the underlying copula $C$ into $C_1$ and $C_2$ being these last two copulas NQD and PQD, respectively, and therefore candidates to be approximated by well known totally ordered families of copulas.$\qquad_{\blacksquare}$
\end{example}
\section{Final remarks}
\label{sec:Finrmks}
If $(X,Y)$ is a bivariate random vector of continuous random variables with an underlying copula $C$ such that $|\rho_C|<\sigma_C$ then $C$ is neither PQD nor NQD and therefore neither PRD nor NRD. Many of well known parametric families of copulas are totally ordered (that is, PQD and/or NQD) and in such case they have to be discarded as admissible copulas for $(X,Y).$ To face this challenge, in the present work it has been proposed a \textit{gluing copula decomposition} of $C$ into totally ordered copulas that combined may lead to a non-monotone regression function.
\begin{acknowledgement}
The present work was partially supported by project IN115817 from Programa de Apoyo a Proyectos de Investigaci\'on e Innovaci\'on Tecnol\'ogica (PAPIIT) at Universidad Nacional Aut\'onoma de M\'exico.
\end{acknowledgement}
\bibliographystyle{spmpsci}
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\section{Introduction}
The binary search tree (BST) is the canonical comparison-based implementation of the dictionary data type for maintaining ordered sets. Dynamic BSTs can be re-arranged after every access via rotations and pointer moves starting from the root. Various ingenious techniques have been developed for dynamically maintaining balanced BSTs, supporting search, insert, delete, and other operations in time $O(\log{n})$, where $n$ is the size of the dictionary (see e.g.~\cite[\S\,6.2.2]{Knuth3}, \cite[\S\,5]{Mehlhorn84}).
In several applications where the access sequence has strong \emph{locality of reference}, the worst-case bound is too pessimistic (e.g.\ in list merging, adaptive sorting, or in various geometric problems). A classical technique for exploiting locality is \emph{finger search}. In finger search trees, the cost of an access is typically $O(\log{d})$,\footnote{To simplify notation, we let $\log{(x)}$ denote $\log_2{(\max\{2,x\})}$.} where $d$ is the difference in rank between the accessed item and a \emph{finger} ($d$ may be much smaller than $n$). The finger indicates the starting point of the search, and is either given by the user, or (more typically) it points to the previously accessed item. Several special purpose tree-like data structures have been designed to support finger search.\footnote{The initial 1977 design of Guibas et al.~\cite{guibas} was refined and simplified by Brown and Tarjan~\cite{BrownTarjan} and by Huddleston and Mehlhorn~\cite{Huddleston}. Further solutions include~\cite{Tsakalidis, TarjanWyk, Kosaraju, KaplanTarjan}, see also the survey~\cite{BrodalSurvey}. Randomized treaps~\cite{SeidelA96} and skip lists~\cite{SkipList} can also support finger search.}
In 1983, Sleator and Tarjan~\cite{ST85} introduced Splay trees, a particularly simple and elegant ``self-adjusting'' BST algorithm. In 2000, Cole et al.~\cite{finger1,finger2} showed that Splay matches (asymptotically) the efficiency of finger search, called in this context the \emph{dynamic finger} property. This is remarkable, since Splay uses no explicit fingers; every search starts from the root. The result shows the versatility of the BST model, and has been seen as a major (and highly nontrivial) step towards ``dynamic optimality'', the conjecture of Sleator and Tarjan that Splay trees are constant-competitive.
BSTs can also adapt to other kinds of locality. The \emph{working set} property~\cite{ST85} requires the amortized cost of accessing $x$ to be $O(\log{t})$, where $t$ is the number of distinct items accessed since the last access of $x$. Whereas dynamic finger captures proximity in keyspace, the working set property captures proximity \emph{in time}. In 2001, Iacono~\cite{IaconoUnified} proposed a \emph{unified} property that generalizes both kinds of proximity. Informally, a data structure with the unified property is efficient when accessing an item that is \emph{close to some recently accessed item}.
{It is not known whether any BST data structure has the unified property.}
Recently, Iacono and Langerman~\cite{LI16} studied the \emph{lazy finger} property (Bose et al.~\cite{BoseDIL14}), and showed that an online algorithm called Greedy BST\footnote{Greedy BST was discovered by Lucas in 1988~\cite{Luc88} and later independently by Munro~\cite{Mun00}. Demaine et al.~\cite{DHIKP09} transformed it into an online algorithm.} satisfies it. The lazy finger property requires the amortized cost of accessing $x$ to be $O(d)$, where $d$ is the distance (number of edges) from the previously accessed item to $x$ in the best \emph{static reference tree}.
This property is stronger than the dynamic finger property~\cite{BoseDIL14}, and it is not known to hold for Splay.
In this paper we study a generalization of the lazy finger property; instead of a single finger stationed at the previously accessed item, we allow $k$ fingers to be moved around arbitrarily. An access is performed by moving any of the fingers to the requested item. Cost is proportional to the \emph{total} distance traveled by the fingers. We assume that the fingers move according to an optimal strategy, in an optimally chosen static tree, with a priori knowledge of the entire access sequence.
The cost of this optimal \emph{offline} execution with $k$ fingers is an intrinsic measure of complexity of a query sequence, and at the same time a benchmark that algorithms in the classical model can attempt to match.
Parameter $k$ describes the strength of the bound: the case $k=1$ is the lazy finger, at the other extreme, at $k=n$, each item may have its own finger, and all accesses are essentially free.
Our main result is a family of new \emph{online}\footnote{An \emph{online} BST algorithm can base its decisions only on the current and past accesses. An \emph{offline} algorithm knowns the entire access sequence in advance.} dynamic BST algorithms (in the standard model, where every access starts at the root), matching the $k$-finger optimum on sufficiently long sequences, up to an overhead factor with moderate dependence on $k$ and no dependence on the dictionary size or on the number of accesses in the sequence.
Our online BST combines three distinct techniques: (1) an offline, one-finger BST simulation of a multi-finger execution (the technique is a refinement of an earlier construction~\cite{CombineBST}), (2) online $k$-server algorithms that can simulate the offline optimal multi-finger strategy, and (3) a multiplicative-weights scheme for learning a tree metric in an online fashion.
The fact that ``vanilla'' BSTs can, with a low overhead, simulate a much more powerful computational model further indicates the strength and versatility of the BST model.
As an application,
we show that our online BST algorithms satisfy a restricted form of the \emph{unified property}; previously no (online or offline) BST was known to satisfy such a property.
If there is a constant-competitive BST algorithm, then it must match our $k$-finger bounds.
The two most promising candidates, Splay and Greedy BST (see e.g.\ \cite{in_pursuit}) were only shown (with considerable difficulty) to satisfy variants of the one-finger, i.e.\ lazy finger property. To obtain our online BSTs competitive for other values of $k$, we combine sophisticated tools developed for other online problems, as well as our refinement of a previous (highly nontrivial) construction for simulating multiple fingers. These facts together may hint at the formidable difficulty (more pessimistically: the low likelihood) of attaining dynamic optimality by simple and natural BST algorithms such as Splay or Greedy.
\subparagraph*{BST and finger models. Main results.}
Now, we introduce the formal statements of our results.
In the dynamic BST model a sequence of keys
{is} accessed in a binary search tree (BST), and after each access, the tree can be reconfigured via a sequence of rotations and pointer moves starting from the root. (There exist several alternative but essentially equivalent models, see~\cite{Wilber,DHIKP09}.) Denote the space of keys (or elements) by $[n]$. For a sequence $X = (x_1,\ldots, x_m) \in [n]^m$, denote by ${\sf OPT}(X)$ the cost of the optimal offline BST for accessing $X$.\footnote{To avoid technicalities, we only consider \emph{access} (i.e.\ successful search) operations and assume $m \geq n$.}
Arguably the most important question in the BST model is the dynamic optimality conjecture, i.e.\ the existence of an online BST whose cost is $O(\mbox{\sf OPT}(X))$ for every $X$.
A BST {\em optimality property} is an inequality between ${\sf OPT}(X)$ and some function $f(X)$, that holds in the BST model. (If ${\sf OPT}(X) \leq f(X)$ for all $X$ is a BST optimality property, then every $O(1)$-competitive algorithm must cost at most $O(f(X))$.)
Several natural BST properties have been suggested over the last few decades. For instance, the \emph{static finger} property~\cite{ST85} states $\mbox{\sf OPT}(X) =O(\mbox{\sf SF}(X))$, for $\mbox{\sf SF}(X) = \sum_t \log |x_t - j|$, where
$j \in [n]$ is a fixed element (finger). The \emph{static optimality} property~\cite{ST85} is $\mbox{\sf OPT}(X) =O(\mbox{\sf SO}(X))$, where $\mbox{\sf SO}(X) = \min_R \sum_{i} d_R(x_i)$. Here $R$ is a \emph{static} BST, and $d_R(x)$ is the depth of $x$ in $R$.
For the \emph{dynamic finger} property~\cite{ST85}, $\mbox{\sf DF}(X) =\sum_t \log |x_t - x_{t+1}|$, and for \emph{working set}~\cite{ST85}, $\mbox{\sf WS}(X) = \sum_t \log \rho_t(x_t)$, where $\rho_t(a)$ is the number of distinct keys accessed between time $t$ and the last time at which $a$ was accessed (all keys assumed accessed at time zero).
In 2001, Iacono~\cite{IaconoUnified} initiated the study of a property that would ``unify'' the latter two notions of efficiency and exhibited a data structure (not a BST) achieving this property. This \emph{unified bound} is defined as $\mbox{\sf UB}(X) = \sum_{t} \min_{t' < t} \log (|x_t - x_{t'}| + \rho_{t}(x_{t'}))$. Dynamic finger and working set are in general, not comparable. On the other hand, $\mbox{\sf UB}(X) \leq \mbox{\sf DF}(X)$, and $\mbox{\sf UB}(X) \leq \mbox{\sf WS}(X)$ clearly hold, justifying the name of the unified bound.
Despite several attempts, the question whether the unified bound is a valid BST property remains unclear; it was shown in~\cite{DerryberryS09} that $\mbox{\sf OPT}(X) = O(\mbox{\sf UB}(X) +m\log \log n)$, and in~\cite{unified,IaconoUnified} that the unified bound is valid in some other (non-BST)
models\footnote{Another attempt to study the bounds related to the unified bound was done in~\cite{fresh-finger}.}.
We show that a unified bound with ``bounded time-window'' holds in the BST model:
\begin{theorem}
\label{thm: weak UB}
For every integer $\ell \geq 1$, every sequence $X$ and some fixed function $\beta(\cdot)$,
\[\mbox{\sf OPT}(X) \leq \beta(\ell) \cdot \mbox{\sf UB}^{\ell}, \mbox{~~where~~~~} \mbox{\sf UB}^{\ell} = \sum_t \min_{t'\in [t-\ell,t)} \log \big(|x_t - x_{t'}| + \rho_t(x_{t'})\big) . \]
\end{theorem}
Observe that $\mbox{\sf UB}(X) = \mbox{\sf UB}^{m}(X) \leq \cdots \leq \mbox{\sf UB}^1(X) = \mbox{\sf DF}(X)$.
Prior to our work it was not known whether the theorem holds when $\ell=2$, i.e.\ no known BST property subsumes this property even when $\ell =2$.
Thus, Theorem~\ref{thm: weak UB} establishes the first BST property that combines the efficiencies of time- and keyspace-proximity without an additive term.\footnote{The proof of Theorem~\ref{thm: weak UB} implies in fact a stronger, \emph{weighted} form, which we omit for ease of presentation.}
Recently Bose et al.\ \cite{BoseDIL14} introduced the \emph{lazy finger} property, $\mbox{\sf LF}(X) = \min_R \sum_{i} d_R(x_i,x_{i+1})$. Here distance is measured in a static reference BST $R$, optimally chosen for the entire sequence. The lazy finger bound can be visualized as follows: accesses are performed in the reference tree by moving a unique finger from the previously accessed item to the requested item. The lazy finger property is rather strong: Bose et al.\ show that it implies the dynamic finger and static optimality properties, which in turn imply static finger.
Our main tool in proving Theorem~\ref{thm: weak UB} is a generalization of the lazy finger property allowing multiple fingers.
The model is motivated by the famous $k$-server problem.
For an input sequence $X \in [n]^m$ and a static BST $R$ with nodes associated with the keys in $[n]$, we have $k$ servers located initially at arbitrary nodes in $R$.
At time $t=1,\ldots, m$, the request $x_t$ arrives, and we move a server of our choice to the node of $R$ that stores $x_t$.
The cost for serving a sequence $X$ is equal to the total movement in $R$ to serve the sequence $X$.
Denote by $\mbox{\sf F}^{k}_R(X)$ the cost of the optimal (offline) strategy that serves sequence $X$ in $R$ with $k$ servers, minimized over all possible initial server locations.
Let $\mbox{\sf F}^{k}(X) = \min_R \mbox{\sf F}^{k}_R(X)$.
We call $\mbox{\sf F}^{k}(X)$ the {\em $k$-finger cost} of $X$.
We remark that the value of $\mbox{\sf F}^{k}_R(X)$ is polynomial-time computable for each $R$, $k\in {\mathbb N}$, and $X \in [n]^m$ by dynamic programming. Clearly, $\mbox{\sf F}^{1}(X) \geq \mbox{\sf F}^{2}(X) \geq \cdots \geq \mbox{\sf F}^{n}(X)$ holds for all $X$.
We first show that one can simulate any $k$-finger strategy in the BST model, in a near-optimal manner. In particular, we prove the following tight result.
\begin{theorem}
\label{thm: LF simulation}
$\mbox{\sf OPT}(X) \leq O(\log k) \cdot \mbox{\sf F}^{k}(X)$.
\end{theorem}
The proof of Theorem~\ref{thm: LF simulation} is a refinement of an earlier argument~\cite{CombineBST}, improving the overhead factor from $O(k)$ to $O(\log{k})$.
The logarithmic dependence on $k$ is, in general, the best possible. To see this, consider a sequence $S$ of length $m$, over $k$ distinct items with average cost $\Omega{(\log{k})}$ (e.g.\ a random sequence from $[k]^m$ does the job). While $\mbox{\sf OPT}{(S)} = \Theta(m\log{k})$, clearly $\mbox{\sf F}^{k}(X)=O(m)$, as each of the $k$ items can be served with its own private finger.
In the definition of $\mbox{\sf F}^k(X)$ we assume a \emph{static} reference tree $R$ for the $k$-finger execution.
The offline BST simulation in the proof of Theorem~\ref{thm: LF simulation} works in fact (with the same overhead) even if $R$ is \emph{dynamic}, i.e.\ if the multi-finger adversary can perform rotations at any of the fingers. In this case, however, the $k$-finger bound is too strong to be useful; already the $k=1$ case captures the dynamic BST optimum.
Our next result is the online counterpart of Theorem~\ref{thm: LF simulation}. In this case, the restriction that $R$ is static is essential.
\begin{theorem}
\label{thm: online simulation}
There exists an online randomized BST algorithm whose cost for serving $X \in [n]^m$, is
$O\bigl((\log k)^7\bigr) \cdot \mbox{\sf F}^{k}(X) + \rho(n)$, for some fixed function $\rho(\cdot)$.
\end{theorem}
The result can be interpreted as follows. On sufficiently long access sequences, there is an online BST algorithm (in fact, a family of them) competitive with the $k$-finger bound, up to an overhead factor with moderate dependence on $k$.
The randomized algorithm (as is standard in the online setting) assumes an oblivious adversary that does not know in advance the outcomes of the algorithm's random coin-flips.
The use of randomness seems essential to our approach. We propose as intriguing open questions to find a deterministic online BST with comparable guarantees and to narrow the gap between the online and offline results.
Due to its substantial amount of computation (outside the BST model), our online algorithm is of theoretical interest only. Nonetheless, the connection with the $k$-server problem allows us to ``import'' several techniques to the BST problem; some of these, such as the \emph{double coverage} heuristic for $k$-server~\cite{chrobak} are remarkably simple and may find their way to practical BST algorithms.
The strength of the $k$-finger model lies in the $k$-server abstraction. In order to establish a BST property of the form $\mbox{\sf OPT}(X) \leq \beta(\ell) \cdot O(g(X))$, it is now sufficient to prove $\mbox{\sf F}^{\ell}(X) \leq \left(\beta(\ell)/\log{\ell}\right) \cdot O(g(X))$.
In other words, our technique reduces the task of bounding the cost in the BST model to designing $k$-server strategies, which typically admits much cleaner combinatorial arguments. We illustrate this approach by showing that the unified property with a fixed time-window holds in the BST model.
\begin{theorem}
\label{thm: strategy for min-dist}
For some fixed functions $\alpha(\cdot),\gamma(\cdot)$, we have:~
$\mbox{\sf F}^{\alpha(\ell)}(X) \leq \gamma(\ell) \cdot \mbox{\sf UB}^{\ell} $.
\end{theorem}
Theorems~\ref{thm: strategy for min-dist} and~\ref{thm: LF simulation} together imply Theorem~\ref{thm: weak UB}. Moreover, Theorem~\ref{thm: online simulation} implies that the property holds for \emph{online} BST algorithms (we later specify the involved functions).
The $k$-finger approach can be used to show further BST properties. For example, we connect \emph{decomposability} (refer to \textsection\,4 for definitions) and finger properties by showing that
even one finger is enough to obtain the \emph{traversal} property in significantly generalized form.
\begin{theorem} \label{thm:decomp}
Let $X$ be a $d$-decomposable sequence. Then $\mbox{\sf F}^{1}(X) = O(\log d) \cdot |X| $.
\end{theorem}
As a corollary, using the recent result by Iacono and Langermann~\cite{LI16}, we resolve an open problem in~\cite{FOCS15}, showing that Greedy costs at most $O(\log d)\cdot |X|$ on every $d$-decomposable sequence, matching the lower bound in~\cite{FOCS15}.\footnote{Independently of our work, Goyal and Gupta~\cite{goyalgupta} showed the same result using a charging argument.}
In another direction, we connect multiple fingers and generalized monotone sequences.
In~\cite{FOCS15}, we showed that $\mbox{\sf OPT}(X) \leq |X|\cdot 2^{O(d^2)}$ on every $d$-monotone sequence $X$; a sequence is $d$-monotone if it can be decomposed into $d$ increasing or $d$ decreasing sequences.
Using the $k$-finger technique, we show the stronger BST property $\mbox{\sf OPT}(X) \leq O(d \log d) \cdot |X|$.
Concerning simple and natural BST algorithms (Splay and Greedy), we give evidence that the strongest results in the literature may still be far from settling the dynamic optimality conjecture. To this end, we describe a class of sequences for which increasing the number of fingers by one can create an $\Omega(\log n)$ gap.
More precisely, we show the following:
\begin{theorem} \label{thm:hierarchyx}
For every integer $k$, there is a sequence $S_k$ such that $\mbox{\sf F}^{k-1}(S_k) = \Omega(\frac{n}{k} \log (n/k))$ but $\mbox{\sf F}^{k}(S_k) = O(n)$.
\end{theorem}
Theorem~\ref{thm:hierarchyx} shows that the multi-finger bounds form a fine-grained hierarchy. For small $k$, our online algorithm (Theorem~\ref{thm: online simulation}) can match these bounds (up to a constant factor). However, any online BST (such as Splay or Greedy) must also match
the dependence of $O(\log k)$ in the upper bound of $O(\log k) \cdot F^{k}(X)$, in order to be constant-competitive.
\subparagraph*{Techniques. The $k$-server problem.}
The $k$-server problem, introduced by Manasse, McGeoch, and Sleator~\cite{Manasse} in 1988 is a central problem in online algorithms: Is there an online deterministic strategy for serving a sequence of requests by moving $k$ servers around, with a total movement cost at most $k$ times the optimal offline strategy? The question in its original form, for arbitrary metric spaces, remains open. Nonetheless, the problem has inspired a wealth of results and a rich set of techniques, many of which have found applications outside the $k$-server problem. A full survey is out of our scope, we refer instead to some prominent results~\cite{Fiat, Koutsoupias, Seiden,Raghavan, Bartal, Bansal}, and the surveys~\cite[\S\,10, \S\,11]{Borodin},~\cite{kserver_survey}. Most relevantly for us, Chrobak and Larmore~\cite{chrobak} gave in 1991, an intuitive, deterministic, $k$-competitive algorithm for \emph{tree metrics}, and the very recently announced breakthrough of Lee~\cite{Lee}, building on Bubeck et al.\ \cite{Bubeck}, gives an $O\bigl((\log{k})^6\bigr)$-competitive randomized algorithm for arbitrary metrics.
Our online BST algorithm relies on an online $k$-server in an almost black box fashion (the metric space underlying the $k$-server instance is induced by a static reference BST). Thus, improvements for $k$-server would directly yield improvements in our bounds. Despite the depth and generality of $k$-server (e.g.\ it also models the caching/paging problem), to our knowledge it has previously not been related to the BST problem.\footnote{In his work on a generalized $k$-server problem, Sitters~\cite{Sitters} asks whether the work-function (WF) technique~\cite{Koutsoupias} for $k$-server may have relevance for BSTs. Indeed, we can use WF as an $O(k)$-competitive component of our online BSTs, but for our special case of tree-metrics, the technique of~\cite{chrobak} is much simpler. Whether WF may be used (in different ways) to obtain competitive BSTs remains open.}
It is known that in an arbitrary metric space with at least $k+1$ points, no deterministic online algorithm may have a competitive ratio better than $k$. In the randomized case the lower bound $\Omega(\log{k}/\log{\log{k}})$ holds, see e.g.~\cite{kserver_survey}. (The lower bounds thus apply for a metric induced by a BST, for all $k<n$.) These results imply a remarkable separation between the $k$-server and BST problems. Dynamic optimality would require, by Theorem~\ref{thm: LF simulation}, a BST cost of $O(\log{k})\cdot \mbox{\sf F}^k$. To match this, an online BST may not implicitly perform a deterministic $k$-server execution, since, in that case its overhead would have to be $\Omega{(k)}$. This indicates that improving Theorem~\ref{thm: online simulation} will likely require tools significantly different from $k$-server, which is surprising, given the similarity of the two formulations.
Our online BST learns the metric induced by the optimal reference tree using a multiplicative weights update (MWU) scheme. The technique has a rich history, and a recent emergence as a powerful algorithmic tool (we refer to the survey of Arora, Hazan, and Kale~\cite{AroraSurvey}). MWU or closely related techniques have been used previously in data structures (including for BST-related questions), see e.g.\ \cite{Blum, BlumBurch, in_pursuit, Kalai}. Specifically, Iacono~\cite{in_pursuit} obtains, using MWU, an online BST that is constant-competitive on sufficiently long sequences, \emph{if any online BST is constant-competitive}. As we relate online BSTs with an offline strategy, the results are not directly comparable.
\subparagraph*{Further open questions and structure of the paper.}
The main open question raised by our work is whether natural algorithms such as Splay or Greedy match the properties of our new BST algorithms. (This must be the case, if Splay and Greedy are, as conjectured, $O(1)$-competitive). We suggest the following easier questions.
Do Splay or Greedy satisfy the unified bound with a time-window of $2$ steps? Does Splay satisfy the lazy finger or the $2$-monotone bounds?
Does Greedy satisfy the $2$-finger bound?
Except for Theorems~\ref{thm: LF simulation} and \ref{thm:decomp}, the factors in our results are not known to be tight. Improving them may reveal new insight about the power and limitations of the BST model.
In \S\,\ref{sec2} we describe our offline BST simulation. In \S\,\ref{sec3} we describe our new family of online algorithms. In \S\,\ref{sec4} we prove the main applications and further observations.
\section{Offline simulation of multi-finger BSTs (Theorem~\ref{thm: LF simulation})} \label{sec2}
Let $k \in {\mathbb N}$ , let $T$ be a BST on $[n]$, and let $X = (x_1,\dots,x_m) \in [n]^m$ be an access sequence.
A $k$-\emph{finger strategy} consists of a sequence $\vec{f} \in [k]^m$ where $f_t \in [k]$ specifies the finger that serves access $x_t$. Let $\vec{\ell} \in [n]^k$ be the {\em initial vector}, where $\ell_i \in [n]$ gives the initial location of finger $i$.
The cost of strategy $(\vec{f},\vec{\ell})$ is $\mbox{\sf F}^{k}_{T,\vec{f}, \vec{\ell}}(X) = \sum_{t=1}^m (1+d_T(x_t, x_{\sigma(f_t,t)}))$ where $\sigma(i,t) = \max\{j < t \mid f_j = i\}$ is the location of finger $i$ before time $t$, and $\sigma(i,1) = \ell_i$.
Let $\mbox{\sf F}^{k}_T(X) = \min_{\vec{f}, \vec{\ell}} \mbox{\sf F}^{k}_{T,\vec{f}, \vec{\ell}} (X)$.
In other words, for a fixed BST $T$ on keyset $[n]$, $\mbox{\sf F}^{k}_T(X)$ is the \emph{$k$-server} optimum for serving $X$ in the metric space of the tree $T$. (Note that the tree is unweighted, and the distance $d_T(\cdot,\cdot)$ counts the number of edges between two nodes in $T$.)
We define $\mbox{\sf F}^{k}(X) = \min_T \mbox{\sf F}^{k}_{T}(X)$.
It is clear form the definition that $\mbox{\sf F}^{1}(X) \geq \mbox{\sf F}^{2}(X) \geq \cdots \geq \mbox{\sf F}^{n}(X) = m$ for all $X$.
Observe that we implicitly assume that during every access at most one server moves. In addition, we may assume that if some server is already placed at the requested node, then no movement happens. Algorithms with these two restrictions are called \emph{lazy}. As argued in the $k$-server literature (see e.g.\ \cite{kserver_survey}), non-lazy server movements can always be postponed to a later time, keeping track of the ``virtual'' locations of servers. In other words, every $k$-server algorithm can be simulated by a lazy algorithm, without additional cost. We therefore assume throughout the paper that $k$-server/$k$-finger executions are lazy.
Consider some (lazy) $k$-finger execution $(\vec{f},\vec{\ell})$ in tree $T$, for access sequence $X$. We can view $\vec{f}$ as an explicit sequence of elementary steps $\ensuremath{\mathcal S} = \ensuremath{\mathcal S}^k_{T, \vec{f}, \vec{\ell}}$, where in each step we move one of the fingers to its parent or to one of its children in $T$. We further allow $\ensuremath{\mathcal S}$ to contain rotations at a finger in $T$ (although $k$-finger strategies as described above do not generate rotations). The position of a finger is maintained during a rotation.
We show how $\ensuremath{\mathcal S}$ can be simulated in a standard dynamic BST. If in $\ensuremath{\mathcal S}$ a finger visits a node, then the (single) pointer in the BST also visits the corresponding node, therefore all accesses are correctly served in the BST. Every elementary step in $\ensuremath{\mathcal S}$ is mapped to (amortized) $O(\log{k})$ elementary steps (pointer moves and rotations) in the BST. This immediately implies Theorem~\ref{thm: LF simulation}, since, if we can simulate an arbitrary $k$-finger execution, then indeed we can simulate the optimal $k$-finger execution on the best static tree. Assuming that the intial conditions $T$ and $\vec{\ell}$ are known, the steps of $\ensuremath{\mathcal S}$ are simulated one-by-one, without any lookahead. Thus, insofar as the $k$-finger execution is \emph{online}, the BST execution is also online (this fact is used in \S\,\ref{sec3}).
Let us describe simulation by a standard BST $T'$ of a $k$-finger execution $\ensuremath{\mathcal S}$ in a BST $T$. The construction is a refinement of the one given by Demaine et al.\ \cite{CombineBST}, see also~\cite{persistent_tries}. (We improve the overhead factor from $O(k)$ to $O(\log{k})$.)
The main ingredients are: (1) Making sure that each item with a finger
on it in $T$ has depth at most $O(\log k)$ in $T'$. (In~\cite{CombineBST}, each finger may have depth up to $O(k)$ in $T'$.) (2) Implementing a deque data structure within $T'$
so that each finger in $T$ can move to any of its neighbors, or perform a rotation, with cost
$O(\log k)$ amortized. (In~\cite{CombineBST}, this cost is $O(1)$ amortized.)
Given these ingredients, to move a finger $f$ to its neighbor $x$ in $T$, we
can simply access $f$ from the root of $T'$ in $O(\log k)$ steps,
and then move $f$ to $x$ in $T'$ in $O(\log{k})$ amortized steps, with a similar approach for a rotation at $f$. Hence, the overhead factor is $O(\log k)$. We sketch the main technical ideas, postponing the details to Appendix~\ref{sec:simulation lazy fingers}.
Consider the tree $S$ induced by the current fingers and the paths connecting them in $T$. The tree $S$ consists of finger-nodes and non-finger nodes of degree 3 (both types of nodes are called \emph{pseudo-fingers}), and paths of non-finger nodes of degree 2 connecting pseudo-fingers with each other, called \emph{tendons}. Tendons can be compressed into a BST structure that allows their traversal between the two endpoints in $O(1)$ steps.
We maintain $S$ as a root-containing subtree of our BST $T'$, called the \emph{hand}. Due to the compression of the tendons, the relevant part of $S$ has size $O(k)$. The description so far, including the terminology, is identical to the one in~\cite[\S\,2]{CombineBST}. Our construction differs in the fact that it maintains the hand, i.e.\ the compressed representation of $S$ as a \emph{balanced} BST. This guarantees the reachability of fingers in $O(\log{k})$ instead of $O(k)$ steps, i.e.\ property (1).
When a finger in $T$ moves or performs a rotation, the designation of some (pseudo)finger, or tendon nodes may change. Such changes can be viewed as the insertion or deletion of items in the tendons. As these operations happen only at certain places within the tendons, they can be implemented efficiently. We implement tendons with the same BST-based \emph{deque} as~\cite{CombineBST}. The construction appears to be folklore, we describe it in Appendix~\ref{deque_proof} for completeness.
We depart again from~\cite{CombineBST}, as the operation affecting the (pseudo)finger and tendon nodes can trigger a re-balancing of the hand, which may again require $O(\log{k})$ operations to fix, i.e.\ property (2). Any efficient balancing strategy (e.g.\ red-black tree) may be used.
\section{Online simulation of multi-finger BSTs (Theorem~\ref{thm: online simulation})} \label{sec3}
Consider the optimal (offline) $k$-finger execution $\vec{f}$ for access sequence $X \in [n]^m$, with static reference tree $T$ and initial finger-placement $\vec{\ell}$. We wish to simulate it by a dynamic \emph{online} BST.
The construction proceeds in two stages: (1) A simulation of $\vec{f}$ by a sequence $\ensuremath{\mathcal S}$ of steps that describe finger-movements and rotations-at-fingers, starting from an arbitrary BST $T_0$ and arbitrary finger locations $\vec{\ell}_0$. The sequence $\ensuremath{\mathcal S}$ is \emph{online}, i.e.\ it is constructed without knowledge of the optimal initial state $T$,$\vec{\ell}$, and it correctly serves the sequence $X$, as its elements are revealed one-by-one. (2) A step-by-step simulation of $\ensuremath{\mathcal S}$ by a standard BST algorithm using the result of \S\,\ref{sec2}. Since $\ensuremath{\mathcal S}$ is online, the BST algorithm is also online.
As before, we denote by $\mbox{\sf F}^k(X) = \mbox{\sf F}^{k}_{T,\vec{f}, \vec{\ell}}(X)$ the cost of the optimal offline execution. Observe that this is exactly the $k$-server optimum with the tree metric defined by $T$ and initial configuration of servers $\vec{\ell}$. If $T$ and $\vec{\ell}$ were known, we could conclude part (1) by running an arbitrary \emph{online} $k$-server algorithm defined on tree metrics.
To this end, we mention two online $k$-server algorithms, the deterministic ``double coverage'' algorithm of Chrobak and Larmore~\cite{chrobak} (Algorithm~A) and the very recently announced randomized algorithm of Lee~\cite{Lee, Bubeck} (Algorithm~B). It is known that the cost of Algorithms~A, resp.\ B is at most $k$-times, resp.\ $O((\log{k})^{6})$ times $\mbox{\sf F}^k$. We only describe Algorithm~A, as it is particularly intuitive. To obtain the claimed result, we need the much more complex Algorithm~B. (By using Algorithm~A we get an overall factor $O(k \log{k})$.)
During the execution of Algorithm~A, given a current access request $x_t$, call those servers (fingers) \emph{active}, whose path to $x_t$ in $T$ does not contain another server. If several servers are in the same location, one of them is chosen arbitrarily to be active. Algorithm~A serves $x_t$ as follows: as long as there is no server on $x_t$, move all active servers one step closer to $x_t$. Observe that as servers move, some of them may become inactive. Algorithm~A (as described) may need to move multiple servers during one access. It can, however, easily be transformed into a lazy algorithm, as discussed in \S\,\ref{sec2}.
Remains the issue that the optimal initial $T$ and $\vec{\ell}$ are not known. Let $B_1, \dots, B_N$ be instances of an online $k$-server algorithm (in our case Algorithm~B), one for each combination of initial tree $T$ and initial server-placement $\vec{\ell}$. Note that $N = O(4^n \cdot {n^k})$. Let ${\mathcal{M}}$ be a ``meta-algorithm'' that simulates all $B_j$'s for $j=1,\dots,N$, competitive on sufficiently long input with the best $B_j$. Algorithm~${\mathcal{M}}$ processes $X$ in epochs of length $M = n \log n$, executing in the $i$-th epoch, for $i=1,\dots,\lceil m/M \rceil$, some $B_{\tau(i)}$ according to a (randomized) choice $\tau(i)$.
Suppose that $\vec{\ell}^*$ and $T^*$ describe the state of $B_{\tau(i)}$ chosen by ${\mathcal{M}}$ at the beginning of the $i$-th epoch. To switch to the state $\vec{\ell}^*$, $T^*$, ${\mathcal{M}}$ takes $O(n \log{n})$ elementary steps: (1) rotate the current tree to a \emph{balanced} tree using any of the fingers ($O(n)$ steps), (2) move all fingers to their location in $\vec{\ell}^*$ ($k$ times $O(\log{n})$ steps), (3) use an arbitrary finger $f$ to rotate the tree to $T^*$ ($O(n)$ steps), (4) move $f$ back to its location in $\vec{\ell}^*$ ($O(n)$ steps). {Since $M = n \log n$}, the cost of switching can be amortized over the epoch.
The choice of $B_{\tau(i)}$ for epoch $i$ is done according to the multiplicative-weights (MW) technique~\cite{AroraSurvey}, based on the past performance of the various algorithms. Our \emph{experts} are the online executions $B_1, \dots, B_N$, our $i$-th \emph{event} is the portion of $X$ revealed in the $i$-th epoch, the \emph{loss} of the $j$-th expert for the $i$-th event is the \emph{cost} of $B_j$ in the $i$-th epoch. Let $C_{max}$ denote the maximum possible loss of an expert for an event (we may assume $C_{max} \leq n \cdot M$).
It follows from the standard MW-bounds~\cite[Thm.\ 2.1]{AroraSurvey}, that for an arbitrary $\varepsilon \in (0,1)$, the cost of ${\mathcal{M}}$ on $X$ is at most
{$\min_j (1+\varepsilon)\mathcal{C}_j + \displaystyle\frac{C_{max} \cdot \ln{N}}{\varepsilon}$, where $\mathcal{C}_j$ is the cost of expert $B_j$ for the entire $X$; in particular, $B_j$ may correspond to the optimal offline choice $\vec{\ell}$, $T$, in which case $\mathcal{C}_j=O((\log{k})^6) \cdot \mbox{\sf F}^k(X)$.}
Thus, for e.g.\ $\varepsilon = 1/2$, we obtain that the cost of ${\mathcal{M}}$ on $X$ is at most $O((\log{k})^6) \cdot \mbox{\sf F}^k(X) + O(n^3 \log^2{n} )$. The output of ${\mathcal{M}}$ is an \emph{online} sequence $\ensuremath{\mathcal S}_\mathcal{M}$ of rotations and finger moves, starting from an arbitrary initial state $T_0$ and $\vec{\ell}_0$. Note that while ${\mathcal{M}}$ needs to evaluate the costs and current states for all experts in all epochs (an extraordinary amount of computation), only one of the experts interacts with the tree at any time. Thus, $\ensuremath{\mathcal S}_\mathcal{M}$ is a standard sequence of steps which can be simulated by a standard BST algorithm according to Theorem~\ref{thm: LF simulation}, at the cost of a further $O(\log{k})$ factor. This concludes the proof of Theorem~\ref{thm: online simulation}.
\section{Applications of the multi-finger property}
\label{sec4}
In this section we show that every BST algorithm that satisfies the $k$-finger property also satisfies the unified bound with fixed time-window (Application 1), is efficient on decomposable sequences (Application 2), and on generalized monotone sequences (Application 3).
\subparagraph*{Application 1. Combined space-time sensitivity (Theorem~\ref{thm: strategy for min-dist}).}
Recall the definition of $\mbox{\sf UB}^{\ell}$ in Theorem~\ref{thm: weak UB} for a sequence $X=(x_{1},\dots, x_{m}) \in [n]^m$. We connect this quantity with the $k$-finger cost, from which Theorem~\ref{thm: strategy for min-dist} immediately follows.
\begin{theorem}
For every $\ell$, $F^{(\ell!)}(X)=O(\ell!)\cdot{\sf UB}^{ {\ell}}(X)$.\label{thm:LF less than kmin}
\end{theorem}
Since we are only concerned with the case when $\ell$ is constant, we may drop the term $\rho_t(x_{t'})$ in the definition of $\mbox{\sf UB}^{\ell}$ (whose value is always between $1$ and $\ell$).
We prove Theorem~\ref{thm:LF less than kmin} via
another bound in which distances are measured in a static reference BST:\quad
$
\displaystyle\ell\text{-DistTree}_{T}(X)=\sum_{i=1}^{m}\min_{i-\ell\le j<i}\left\{d_{T}(x_{i},x_{j}) + 1\right\}$. \footnote{We let $x_0$ denote the root of $T$, and distances involving negative indices are defined to be $+\infty$.}
\begin{lemma}
$\min_{T}\ell\text{-DistTree}_{T}(X)=O({\sf UB}^{ {\ell}}(X))$.\label{thm:distree to dist}\end{lemma}
\begin{proof}
By \cite[Thm.\ 4.7]{SeidelA96}, there is a randomized BST $\tilde{T}$ such that the expected distance between
elements $i$ and $j$ is $E[d_{\tilde{T}}(i,j)]=\Theta(\log|i-j|)$. Therefore,
\begin{align*}
\min_{T}\ell\text{-DistTree}_{T}(X) \le E[\sum_{i=1}^{m}\min_{i-\ell\le j<i}\{d_{\tilde{T}}(x_{i},x_{j})+1\}]
= \sum_{i=1}^{m}E[\min_{i-\ell\le j<i}\{d_{\tilde{T}}(x_{i},x_{j})+1\}]\\
\quad \le \sum_{i=1}^{m}\min_{i-\ell\le j<i}\{E[d_{\tilde{T}}(x_{i},x_{j})+1]\}
= \sum_{i=1}^{m}\min_{i-\ell\le j<i}\{O(\log|x_{i}-x_{j}|)\}
= O({\sf UB}^{ {\ell}}(X)). \quad \quad \qedhere
\end{align*}
\end{proof}
It is now sufficient to show that $\mbox{\sf F}^{(\ell!)}_{T}(X)=O(\ell!)\cdot\ell\text{-DistTree}_{T}(X)$, for all $X$ and $T$, i.e.\ to describe an $(\ell !)$-finger strategy in $T$ for serving $X$ with the given cost.
At a high level, our strategy is the following: (1) Define a \emph{virtual tree} ${\mathcal T}(X)$ whose nodes are the \emph{requests} $x_i$ for $i=1,\dots,m$. The virtual tree captures the \emph{proximities} between the requests, with each $x_i$ having as parent the \emph{nearest} request $x_j$ within a fixed time-window before time $i$. Edges in ${\mathcal T}(X)$ are given as weights the distances between requests in $T$. Note that the virtual tree is not necessarily binary. (2) Define a recursive structural decomposition of the tree ${\mathcal T}(X)$, with the property that certain blocks of this decomposition contain requests in non-overlapping time-intervals. (3) Describe a multi-finger strategy on ${\mathcal T}(X)$ for serving the requests, which induces a multi-finger strategy on $T$ with the required cost. (The strategy takes advantage of the decomposition in (2).)
We describe the steps more precisely, deferring some details to Appendix~\ref{weakuniapp}.
\newcommand{\mathsf{start}}{\mathsf{start}}
\newcommand{\mathsf{end}}{\mathsf{end}}
\newcommand{\mathsf{nf}}{\mathsf{nf}}
\subparagraph*{The virtual tree.}
Given a number $\ell$, $X \in [n]^m$, and a BST $T$ over $[n]$ with root $r$, the virtual tree ${\mathcal T} = {\mathcal T}{(\ell,T,X)}$ is a rooted tree with vertex-set $\{(i,x_{i}) \mid i\in[m]\}\cup\{(0,x_{0})\}$, where $x_0 = r$ is the root of $T$ and $(0,x_{0})$ is the root of ${\mathcal T}$.
The \emph{parent} of a non-root vertex $(i,x_{i})$ in ${\mathcal T}$ is
$(j,x_{j})=\arg\min_{j \in [i-\ell,i)}\{d_{T}(x_{i},x_{j})\}$. In words, $(j,x_j)$ is the request at most $\ell$ steps before $(i,x_i)$, closest to $x_i$ (in $T$).
For each edge $e=((j,x_{j}),(i,x_{i}))$, we define the weight $w_{{\mathcal T}}(e)=d_{T}(x_{i},x_{j})+1$.
For each subtree $H$ of ${\mathcal T}$, let $w_{{\mathcal T}}(H)$
be the total weight of its edges. Observe that $w_{{\mathcal T}}({\mathcal T})=\ell\text{-DistTree}_{T}(X)$.
\subparagraph*{Structure and decomposition of the virtual tree.} We say that a vertex $(i,x_{i})$
is \emph{before} (or \emph{earlier than}) $(j,x_{j})$ if $i<j$, otherwise it is \emph{after} (or \emph{later than}).
For every subtree $H$ of ${\mathcal T}$ we denote the earliest vertex in $H$ as $\mathsf{start}(H)$ and the latest vertex in $H$ as $\mathsf{end}(H)$. The \emph{time-span} of $H$, denoted $\mbox{\sf span}(H)$, is $(t_{1},t_{2}]$ where $(t_{1},x_{t_{1}})=\mathsf{start}(H)$ and $(t_{2},x_{t_{2}})=\mathsf{end}(H)$, and $H$ is \emph{active} at time $t$ if $t \in \mbox{\sf span}(H)$.
We describe a procedure to decompose ${\mathcal T}{(\ell,T,X)}$ into directed paths (for the purpose of analysis), defining the key notions of \emph{$i$-body }and \emph{$i$-core}. The procedure is called on a subtree $H$ of ${\mathcal T}$, and the top-level call is $\textsf{decompose}({\mathcal T},\ell)$.
\vspace{-0.05in}
\noindent\rule{\textwidth}{0.4pt}
\vspace{-0.06in}
\noindent\textbf{procedure} $\textsf{decompose}(H,i)$:
\vspace{-0.02in}
\begin{enumerate}
\item If $H$ has no edges, return.
\item Let $C(H)$ be the path from $\mathsf{start}(H)$ to $\mathsf{end}(H)$.
\item Call $C(H)$ an $i$-core of $H$, and call $H$ the $i$-body of $C(H)$.
\item For each connected component $H'$ in $H\setminus C(H)$ invoke $\textsf{decompose}(H',i-1)$.
\end{enumerate}
\vspace{-0.05in}
\noindent\rule{\textwidth}{0.4pt}
Observe that ${\mathcal T}$ itself is an $\ell$-body.
Each $i$-body $H$ consists of its $i$-core
$C(H)$ and a set of $(i-1)$-bodies
that are connected
components in $H\setminus C(H)$. For each of those $(i-1)$-bodies
$H'$, we say that $H$ is a \emph{parent} of $H'$, defining a tree-structure over bodies. Observe that the number of ancestor bodies of an $i$-body (excluding itself) is $\ell-i$. We make a sequence of further structural observations about the virtual tree and its decomposition.
\begin{lemma}[\ref{app:struc_decomp}]
\label{lem:struc_decomp}
\begin{enumerate}[(i)]
\item At every time $t$, there are at most
$\ell$ active edges in ${\mathcal T}{(\ell,T,X)}$.
\item The $i$-cores of the decomposition, for $1\le i\le \ell$, partition the vertices of ${\mathcal T}$.
\item Let $H$ be an $i$-body. At any time during the time-span of $H$, among the $(i-1)$-bodies with parent $H$ at most $i-1$ are active.\label{lem:i-1 active children}
\item Let $H$ be an $i$-body. The $(i-1)$-bodies with parent $H$ can be partitioned into $(i-1)$ groups ${\cal H}_{1},\dots,{\cal H}_{i-1}$ such that, for $1\le j\le i-1$
and $H',H''\in{\cal H}_{j}$, the time-spans of $H'$ and $H''$ are disjoint.\label{thm:separate group}
\end{enumerate}
\end{lemma}
\subparagraph*{The strategy for moving fingers.}
For
two vertices $(i,x_{i})$ and $(j,x_{j})$ in the virtual tree ${\mathcal T} = {\mathcal T}{(\ell,T,S)}$, \emph{moving a finger} $f$ from $(i,x_{i})$ to $(j,x_{j})$
means the following: let $P=((i_{1},x_{i_{1}}),\dots,(i_{k},x_{i_{k}}))$
be the unique path from $(i,x_{i}) = (i_{1},x_{i_{1}})$ to $(j,x_{j})=(i_{\ell},x_{i_{\ell}})$ in ${\mathcal T}$.
For $j=1,\dots,k-1$, we iteratively move a finger $f$ from $x_{i_{j}}$
to $x_{i_{j+1}}$ using $d_{T}(x_{i_{j}},x_{i_{j+1}})$ steps. Hence,
the total number of steps is at most $w_{{\mathcal T}}(P)$.
By \emph{serving an access in an $i$-body $H$}, we mean that, for each $(j,x_{j})\in V(H)$, at time $j$ there is a finger move
to $x_{j}$ in $T$. For each $i \le \ell$, let $\mathsf{nf}(i)$ be the number
of fingers used for serving accesses in an $i$-body. We define $\mathsf{nf}(1)=1$
and $\mathsf{nf}(i)=1+(i-1)\cdot\mathsf{nf}(i-1)$, thus, by induction, $\mathsf{nf}(i)\le i!$ for all $i \leq \ell$.
We now describe the strategy for moving fingers. Let $F$ be a set
of fingers where $|F|=\mathsf{nf}(\ell)$. At the beginning all fingers are at $(0,x_0)$. (In the reference tree $T$, all fingers are initially at the root $x_0$.)
For $1\le j\le m$, we call $\mbox{\sf access}({\mathcal T},F,(j,x_{j}))$, defined below for an $i$-body $H$, set of fingers $F$, and $u\in V(H)$.
\noindent\rule{\textwidth}{0.4pt}
\vspace{-0.06in}
\noindent\textbf{procedure} $\textsf{access}(H,F,u)$:
\vspace{0.05in}
Let $C = C(H)$ be the $i$-core of $H$, with $C=\{u_{1},\dots,u_{k'}\}$, where $u_{k}$ is before $u_{k+1}$ for each $k$.
For $1\le j\le i-1$, let
${\cal H}_{j}$ be the $j$-th group of the $(i-1)$-bodies with parent $H$ (${\cal H}_{j}$ defined in Lemma~\ref{lem:struc_decomp}(iv)).
The $i$-bodies in ${\cal H}_{j}$ are ordered by their time-span.
That is, suppose ${\cal H}_{j}=\{H'_{1},\dots,H'_{\ell'}\}$. For
each $\ell$, if $\mbox{\sf span}(H'_{\ell})=(a_{1},a_{2}]$ and $\mbox{\sf span}(H'_{\ell+1})=(b_{1},b_{2}]$,
then $a_{2}\le b_{1}$. Fingers in $F$ are divided into $i$ groups
$F_{1},\dots,F_{i-1},\{f_{i}\}$, where $|F_{j}|=\mathsf{nf}(i-1)$, for $j\le i-1$,
and $f_{i}$ is a single finger.\smallskip
\vspace{-0.06in}
\begin{enumerate}
\item If $u\in C$, then move $f_{i}$ to $u$ from the predecessor node of $u$ in $C$. If $u=\mathsf{end}(H)$, then move $F$ from $\mathsf{end}(H)$ to $\mathsf{start}(H)$.
\item
{Else let $u\in V(H')\setminus V(C)$ where $H'\in{\cal H}_{j}$. If $u = \mathsf{start}(H')$ and $H'$ is the first $(i-1)$-body in ${\cal H}_j$, move $F_j$ from $\mathsf{start}(H)$ to $\mathsf{start}(H')$. Perform $\mbox{\sf access}(H',F_{j},u)$. If $u=\mathsf{end}(H')$ and if $H'$ is the last in ${\cal H}_{j}$ then move $F_{j}$
from $\mathsf{start}(H')$ to $\mathsf{end}(H)$.
Otherwise, if $u=\mathsf{end}(H')$ and there is a next $(i-1)$-body $H''$ in ${\cal H}_{j}$, then
move $F_{j}$ from $\mathsf{start}(H')$ to $\mathsf{start}(H'')$.}
\end{enumerate}
\vspace{-0.05in}
\noindent\rule{\textwidth}{0.4pt}
In order to give the reader more intution, we give an alternative description. A $1$-body $H$ consists only of its $1$-core $C(H)$. We use one finger and move it through $C(H)$. For $i > 1$, an $i$-body $H$ decomposes in its $i$-core $C(H)$ and $i-1$ groups ${\cal H}_1$ to ${\cal H}_{i-1}$ of $(i-1)$-bodies. Initially, we have $\mathsf{nf}(i)$ fingers on $\mathsf{start}(H)$. We use one finger to move down the $i$-core. We use a group $F_j$ of $\mathsf{nf}(i-1)$ fingers for the $j$-group ${\cal H}_j$. Let $H_1$, \ldots $H_p$ be the $(i-1)$-cores in ${\cal H}_j$. We first move $F_j$ to $\mathsf{start}(H_1)$. Then we use the strategy recusively to move $F_j$ through $H_1$. Once the group of fingers has reached $\mathsf{end}(H_1)$, we move them to $\mathsf{start}(H_2)$, and so on. Once the fingers have reached $\mathsf{end}(H_p)$, we move them back to $\mathsf{start}(H)$. We coordinate (this is not really necessary) the movement of the fingers by the order of the accesses in the access sequence $X$.
From the description of $\mbox{\sf access}$ it is clear that all
accesses in ${\mathcal T}$ are served and that $\mathsf{nf}(\ell)$ fingers are sufficient.
It remains to bound the total number of steps all fingers move. For
an $i$-body $H$, let $\mbox{\sf cost}(H)$ be the total cost of calling
$\mbox{\sf access}(H,F,u)$ for all $u\in H$.
Let ${\cal H}$ denote the set of $(i-1)$-bodies with parent $H$. Let $C^{+}(H)$ denote the $i$-core $C(H)$ augmented with the edges connecting $C(H)$ to the $(i-1)$-bodies in ${\cal H}$.
Then:
\begin{lemma}[\ref{app:finger}]
$\mbox{\sf cost}(H)\le2\cdot\mathsf{nf}(i)\cdot w_{{\mathcal T}}(C^{+}(H))+\sum_{H'\in{\cal H}}\mbox{\sf cost}(H')$.\label{thm:cost_H}\end{lemma}
By induction, we obtain $\mbox{\sf cost}(H)\le2\cdot i!\cdot w_{{\mathcal T}}(H)$. (For $i=1$ we have $H = C(H)$.)
\noindent Since $\mathsf{nf}(\ell)\le \ell!$, we
have that $\mbox{\sf F}^{(\ell!)}_{T}(X)\le \mbox{\sf F}^{\mathsf{nf}(\ell)}_{T}(X) \leq \mbox{\sf cost}({\mathcal T})$.
By the previous claim we have $\mbox{\sf cost}({\mathcal T})\le 2\cdot (\ell!)\cdot w_{{\mathcal T}}({\mathcal T})=2 \cdot (\ell!)\cdot\ell\text{-DistTree}_{T}(X)$, concluding the proof.
\subparagraph*{Application 2. Decomposable sequences (Theorem~\ref{thm:decomp}).}
Let $\sigma = (\sigma(1),\ldots, \sigma(n))$ be a permutation.
For $a,b: 1 \leq a< b \leq n$, we say that $[a,b]$ is a {\em block} of $\sigma$ if $\set{\sigma(a),\ldots, \sigma(b)} = \set{c,\ldots, d}$ for some integer $c,d \in [n]$.
A {\em block partition} of $\sigma$ is a partition of $[n]$ into $k$ blocks $[a_i, b_i]$ such that $(\bigcup_i [a_i, b_i]) \cap \ensuremath{\mathbb N} = [n]$.
For such a partition, for each $i=1,\ldots, k$, consider a permutation $\sigma_i\in S_{b_i -a_i+1}$ obtained as an order-isomorphic permutation when restricting $\sigma$ on $[a_i, b_i]$.
For each $i$, let $q_i \in [a_i, b_i]$ be a representative element of $i$.
The permutation $\tilde \sigma \in [k]^k$ that is order-isomorphic to $\set{\sigma(q_1),\ldots, \sigma(q_k)}$ is called a {\em skeleton} of the block partition.
We may view $\sigma$ as a {\em deflation} $\tilde \sigma[\sigma_1, \ldots, \sigma_k]$.
A permutation $\sigma$ is $d$-decomposable if
$\sigma = (1)$, or
$\sigma = \tilde \sigma[\sigma_1,\ldots, \sigma_{d'}]$ for some $d' \leq d$ and each permutation $\sigma_i$ is $d$-decomposable (we refer to~\cite{FOCS15} for alternative definitions). Permutations that are $2$-decomposable are called \emph{separable}~\cite{separable}, and this class includes preorder traversal sequences~\cite{ST85} as a special case.
To show Theorem~\ref{thm:decomp}, it is sufficient to define a reference tree $T$ and a one-finger strategy for serving a $d$-decomposable sequence $X$ in $T$ with cost $O(\log{d}) \cdot |X|$. (Appendix~\ref{app:decomp}.)
Combined with the Iacono-Langerman result~\cite{LI16} that Greedy BST has the lazy finger property, we conclude that
the cost of Greedy on any $d$-decomposable sequence $X$ is at most $O(\log d)\cdot |X|$.
The result is tight and strengthens our earlier bound~\cite{FOCS15} of $|X| \cdot 2^{O(d^{2})}$.
\subparagraph*{Application 3. Generalized monotone sequences.}
A sequence $X \in [n]^m$ is \emph{$k$-monotone}, if it can be partitioned into $k$ subsequences (not necessarily contiguous), all increasing or all decreasing. This property has been studied in the context of adaptive sorting, and special-purpose structures have been designed to exploit the $k$-monotonicity of input sequences (see e.g.\ \cite{Moffat, Levcopoulos}). Our results show that BSTs can also adapt to such structure.
\begin{theorem}
Let $X$ be a $k$-monotone sequence. Then
$\mbox{\sf F}^k(X)=O(k) \cdot |X|$.
\end{theorem}
It follows that $\mbox{\sf OPT}(X) \leq O(k \log {k}) \cdot |X|$ for $k$-monotone sequences.\footnote{The result holds, in fact, for the more general case, when each $X_i$ is either increasing or decreasing.} The simulation is straightforward. Let $\{X_1, \dots, X_k\}$ be a partitioning of $X$ into increasing sequences (such a partition can be found online).
Let $T$ be an arbitrary static BST over $[n]$.
Consider $k$ fingers $f_{1},\dots,f_{k}$, initially all on $1$.
For accessing $x_j \in X_i$, move finger $f_i$ to $x_j$. Observe that over the entire sequence $X$, each finger does only an in-order traversal of $T$, taking $O(n)$ steps. Thus, $\mbox{\sf F}^k_{T}(X)=O(nk)$.
A lower bound of $\Omega(n \log{k})$ follows from enumerative results: for sufficiently large $n$, the number of $k$-monotone permutations $X \in [n]^n$ is at least $k^{\Omega(n)}$ (implied by e.g.~\cite{regev1981}). Therefore, by a standard information-theoretic argument (see e.g.\ \cite[Thm.\ 4.1]{Blum}), there exists a $k$-monotone permutation $X \in [n]^n$ with $\mbox{\sf OPT}(X) = \Omega(n \log k)$.
\subparagraph*{Further results.}
\label{sec5}
\label{sec:separation}
We state our hierarchy result (Theorem~\ref{thm:hierarchyx}), also implying a weak separation between $k$-finger bounds and ``monotone'' bounds.
\begin{theorem}[Appendix~\ref{sec:proof-hierarchy}]
\label{thm:hierarchy}
For all $k$ and infinitely many $n$, there is a $k$-monotone sequence $S_k$ of length $n$, such that:
\begin{itemize}
\item $F^{k-1}(S_k) = \Omega(\frac{n}{k} \log (n/k))$
\item $F^{k}(S_k) = O(n)$ (independent of $k$).
\end{itemize}
\end{theorem}
In addition, we show a separation between the $k$-finger property and the working set property, showing that for all $k$ and infinitely many $n$, there are sequences $S$ and $S'$ of length $n$, such that $\mbox{\sf WS}(S) = o(\mbox{\sf F}^{k}(S))$, and $\mbox{\sf F}^{k}(S') = o(\mbox{\sf WS}(S))$. (Appendix~\ref{sec:WS-LF}.)
\newpage
\paragraph*{Acknowledgements}
Parinya Chalermsook is supported by European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No.\ 759557) and by Academy of Finland Research Fellows, under grant No.\ 310415. L\'{a}szl\'{o} Kozma is supperted through ERC consolidator grant No.\ 617951. Thatchaphol Saranurak is supported by European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 715672, and by the Swedish Research Council (Reg.\ No.\ 2015-04659).
We thank Nikhil Bansal and Greg Koumoutsos for insightful discussions.
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Given a random matrix $A$, the question of fundamental interest is: \emph{how likely is $A$ to be invertible}, and, more quantitatively, {\em well conditioned}?
These questions can be expressed in terms of the singular values
$\sigma_1(A)\geq\dots\geq \sigma_n(A) \ge 0$, which are defined as the square roots of the
eigenvalues of $A^\mathsf{T} A$.
The extreme singular values are especially interesting. They can be expressed as
\begin{equation} \label{eq: s1 sn}
\sigma_1(A)=\max_{x\in{{\mathbb S}^{n-1}}} |Ax|
\quad \text{and} \quad
\sigma_n(A)=\min_{x\in{{\mathbb S}^{n-1}}} |Ax|,
\end{equation}
where ${{\mathbb S}^{n-1}}$ is the unit Euclidean sphere in $\mathbb R^n$.
In this paper, we will be concerned with the smallest singular value $\sigma_n(A)$.
Its value is nonzero if and only if $A$ is invertible,
and the magnitude of $\sigma_n(A)$ provides us with a quantitative measure of invertibility.
The behavior of the smallest singular values of random matrices have been extensively studied \cite{BaiYin, BVW, cook, KKS, LPRT, LitRiv,
Liv, MenPao, RebTikh, Rud-square, RudVer-square, RudVer-general, tatarko, taovu-1, taovu, taovu-annals, taovu-tall,
Tikh, Tikh-nomoments, Tikh1, TikhErd, Versh-tall}. For Gaussian random matrices with i.i.d. $N(0,1)$ entries,
the magnitude of $\sigma_n(A)$ is of order $1/\sqrt{n}$ with high probability. This observation
goes back to von Neumann and Goldstine \cite{Neuman}, and it was rigorously verified, with precise
tail bounds, by Edelman \cite{edelman} and Szarek \cite{szarek}.
Extending this result beyond the Gaussian distribution is non-trivial due to the absence of rotation invariance. After the initial progress by Tao and Vu \cite{taovu-annals} and Rudelson \cite{Rud-square}, the following lower bound on $\sigma_n(A)$ was proved by Rudelson and Vershynin \cite{RudVer-square} for matrices with sub-gaussian, mean zero, unit variance, i.i.d. entries:
\begin{equation} \label{eq: RV}
{\mathbb P}\left\{ \sigma_n(A) \le \frac{\varepsilon}{\sqrt{n}} \right\}
\le C \varepsilon + 2 e^{-cn},
\quad \varepsilon \ge 0.
\end{equation}
This result is optimal up to positive constants $C$ and $c$ (depending only on the subgaussian moment).
It has been further extended and sharpened in various ways \cite{Liv, RebTikh, RudVer-general, tatarko, Versh-tall}.
In particular, Rebrova and Tikhomirov \cite{RebTikh} relaxed
the sub-gaussian assumption on the distribution of the entries to just having unit variance.
It has remained unclear, however, if one can completely drop the assumption of the identical distribution of the entries of $A$. The identical distribution seemed to be crucial in the existing versions of the Littlewood--Offord theory \cite{LitOf}, which allowed to handle arithmetic structures that arise in the invertibility problem for random matrices.
A partial result was obtained recently by Livshyts \cite{Liv} who proved \eqref{eq: RV} under the assumption that the rows of $A$ are identically distributed (the entries must be still independent but not necessarily i.i.d). In the present paper we remove the latter requirement as well, and thus prove \eqref{eq: RV} without any identical distribution assumptions whatsoever.
We only assume the following about the entries of $A$: (a) they are independent; (b) the sum of their second moments is $O(n^2)$, which is weaker than assuming that each entry has unit second moment; (c) their distributions are uniformly anti-concentrated, i.e. not concentrated around any single value. The latter assumption is convenient to state in terms of
the L\'evy concentration function, which for a random variable $Z$ is defined as
$$
\mathcal{L}(Z,t):=\sup_{u\in\mathbb R}{\mathbb P}\{|Z-u|<t\}, \quad t \ge 0.
$$
The following is our main result.
\begin{theorem}[Main] \label{mainthm1}
Let $A$ be an $n\times n$ random matrix whose entries $A_{ij}$ are independent and satisfy
$\sum_{i,j=1}^n \mathbb E A_{ij}^2 \leq Kn^2$ for some $K>0$ and
$\max_{i,j} \mathcal{L}(A_{ij},1)\leq b$ for some $b\in (0,1)$.
Then
$$
{\mathbb P}\left\{\sigma_n(A)\leq \frac{\varepsilon}{\sqrt{n}}\right\}\leq C\varepsilon+2e^{-cn},
\quad \varepsilon \ge 0.
$$
Here $C,c>0$ depend only on $K$ and $b$.
\end{theorem}
We would like to emphasize that prior to this paper even the problem of {\it singularity} of inhomogeneous random
matrices was not resolved in the literature. In particular, it was not known if for an $n\times n$ random matrix $B$
with independent discrete entries
(say, uniformly bounded and with variances separated from zero), the singularity probability
is {\it exponentially small} in dimension. (Theorem 1 of \cite{Liv} only implied a polynomial bound on the singularity probability, without the assumption of i.i.d. rows.)
The following theorem is the primary tool in proving the main result of the paper.
\begin{theorem}[Distances] \label{mainthm2}
For any $K>0$ and $b\in(0,1)$ there are $r, C, c>0$ depending only on $K$ and $b$ with the following property.
Let $A$ be a random $n\times n$ matrix as in Theorem~\ref{mainthm1}.
Denote the columns of $A$ by $A_1,\ldots, A_n$, and define
$$
H_j=\Span \left\{ A_i:\,i\neq j, \; i=1,\dots,n \right\},\quad j\leq n.
$$
Take any $j\leq n$ such that $\mathbb E|A_j|^2\leq r n^2$, and let $v_j$ be a random unit vector orthogonal to $H_j$
and measurable with respect to the $sigma$--field generated by $H_j$. Then
$$
\mathcal{L} \left(\langle v_j, A_j\rangle,\varepsilon \right)
\leq C\varepsilon+2e^{-cn},\quad \varepsilon \ge 0.
$$
In particular, for every such $j$ we have
$$
\Pr{\dist(A_j,H_j)\leq \varepsilon}\leq C\varepsilon+2e^{-cn},\quad \varepsilon \ge 0.
$$
\end{theorem}
Let us
outline how Theorem~\ref{mainthm1} can be deduced from Theorem~\ref{mainthm2}.
The first step follows the argument in \cite{RudVer-square}, which is to
decompose the sphere into compressible and incompressible vectors.
Fix some parameters $\rho,\delta\in (0,1)$, which for simplicity can be thought of as small constants. The set of compressible vectors $\Comp(\delta,\rho)$ consists of all vectors
on the unit sphere ${{\mathbb S}^{n-1}}$ that are within Euclidean distance $\rho$ to $\delta n$-sparse vectors
(those that have at most $\delta n$ nonzero coordinates).
The remaining unit vectors are called incompressible,
and we have the decomposition of the sphere:
$$
{{\mathbb S}^{n-1}} = \Comp(\delta,\rho) \cup \Incomp(\delta,\rho).
$$
By the characterization \eqref{eq: s1 sn} of the smallest singular value,
the invertibility problem reduces to finding a uniform lower bound
over the sets of compressible and incompressible vectors:
\begin{equation} \label{eq: comp incomp}
\Pr{ \sigma_n(A)\leq \frac{\varepsilon}{\sqrt{n}} }
\leq \Pr{ \inf_{x\in \Comp(\delta,\rho)}|Ax| \leq \frac{\varepsilon}{\sqrt{n}} }
+ \Pr{ \inf_{x\in \Incomp(\delta,\rho)}|Ax| \leq \frac{\varepsilon}{\sqrt{n}} }.
\end{equation}
For the compressible vectors, Lemma 5.3 from \cite{Liv} gives the upper bound $2e^{-cn}$
on the corresponding probability in \eqref{eq: comp incomp}.
For the incompressible vectors, we use a version of the ``invertibility via distance'' bound
from \cite{RudVer-square}, which holds for any $n \times n$ random matrix $A$ (regardless of the distribution):
\begin{equation} \label{eq: inv via dist}
\Pr{ \inf_{x\in \Incomp(\delta, \rho)} |Ax |\leq \frac{ \varepsilon \rho}{\sqrt{n}} }
\leq \frac{4}{\delta n} \inf_{J} \sum_{j\in J} \Pr{ \dist(A_j, H_j)\leq \varepsilon },
\end{equation}
where the infimum is over all subsets $J \subset[n]$ of cardinality at least $n-\delta n/2$.
To handle the distances, we apply Theorem~\ref{mainthm2}.
Due to our assumption $\sum_{i,j=1}^n \mathbb E A_{ij}^2 = \sum_{j=1}^n \mathbb E |A_j|^2 \leq Kn^2$,
all except at most $K/r$ terms satisfy $\mathbb E |A_j|^2 \le rn^2$.
Denoting the set of these terms by $J$ and applying Theorem~\ref{mainthm2}, we get
$$
\Pr{ \dist(A_j, H_j)\leq \varepsilon } \le C\varepsilon+2e^{-cn}
\quad \text{for all } j \in J.
$$
Since the cardinality of $J$ is at least $n-K/r \ge n-\delta n/2$ for large $n$,
we can substitute this bound into \eqref{eq: inv via dist} and conclude that
the last term in \eqref{eq: comp incomp} is bounded by
$\lesssim \varepsilon + e^{-cn}$ (recall that $\delta$ is a constant and we suppress it here).
Putting all together, the probability in \eqref{eq: comp incomp} gets bounded
by $\lesssim \varepsilon + e^{-cn}$, as claimed in Theorem~\ref{mainthm1}.
\begin{remark}
Given Theorem~\ref{mainthm1}, the second assertion of Theorem~\ref{mainthm2} can be formally strengthened as follows.
Since the matrix $A$ is shown to be singular with probability at most $2e^{-cn}$,
we have that for any $j\leq n$ and any random unit vector $v_j$ orthogonal to $H_j$,
$|\langle v_j,A_j\rangle|=\dist(A_j,H_j)$ with probability at least $1-2e^{-cn}$.
Hence, the assertion of Theorem~\ref{mainthm2} can be replaced with
$$
\mathcal{L} \left(\dist(A_j,H_j),\varepsilon \right)
\leq C\varepsilon+2e^{-cn},\;\; \varepsilon \ge 0,\quad\mbox{ whenever }\quad \mathbb E|A_j|^2\leq r n^2,
$$
for some $r,c,C>0$ depending only on $K,b$.
\end{remark}
\medskip
An earlier version of Theorem~\ref{mainthm2}, under the assumption that the coordinates of $A_i$ are i.i.d., was obtained by Rudelson and Vershynin \cite{RudVer-square}. They discovered an arithmetic-combinatorial invariant of a vector (in this case, a normal vector of $H_i$), which they called an essential Least Common Denominator (LCD).
The authors of \cite{RudVer-square} proved a strong Littewood--Offord--type inequality
for linear combinations of i.i.d.\ random variables in terms of the LCD of the coefficient vector,
and thus were able to estimate $\mathcal{L} \left( \dist(A_i, H_i),\varepsilon \right)$.
However, in the case when $A_i$ do not have i.i.d.\ coordinates, the essential LCD is no longer applicable.
Moreover, none of the existing Littlewood--Offord--type results
could be used even to show that the distance $\dist(A_i,H_i)$ is zero with an exponentially small probability
(which would allow to conclude that the singularity probability for the inhomogeneous random matrix is exponentially small in dimension).
In the present paper, we develop a {\em randomized} version of the least common denominator
and show how it can handle the non-i.i.d.\ coordinates.
Given a random vector $X$ in $\mathbb R^n$,
and a (deterministic) vector $v$ in $\mathbb R^n$, as well as parameters $L>0$, $u \in (0,1)$,
the Randomized Least Common Denominator of $v=(v_1,\dots,v_n)$ (with respect to the distribution of $X=(X_1,\dots,X_n)$) is
$$
\RLCD^X_{L,u}(v)=\inf \left\{ \theta>0:\,\mathbb E \dist^2(\theta (v_1 \bar X_1,\dots,v_n \bar X_n),\mathbb{Z}^n)<\min(u|\theta v|^2, L^2) \right\},
$$
where $\bar X_i$ denotes a symmetrization of $X_i$ defined as $\bar X_i:=X_i-X_i'$, with $X_i'$ being an independent copy of $X_i$, $i=1,2,\dots,n$
(for the sake of comparison, let us recall that the essential Least Common Denominator for random vectors with i.i.d.\ components
was defined in \cite{RudVer-general} as ${\rm LCD}(v):=\inf
\{ \theta>0:\, \dist(\theta (v_1 ,\dots,v_n ),\mathbb{Z}^n)<\min(u|\theta v|, L) \}$).
In this paper, we establish a few key properties of the RLCD,
in particular, its relation to anti-concentration as well as stability under perturbations of a vector.
Other essential elements of the proof of Theorem~\ref{mainthm2} are
a discretization argument based on the concept of random rounding and
a double counting procedure for
estimating cardinalities of $\varepsilon$--nets.
Those were, in a rather different form, used in \cite{Liv} and \cite{TikhErd}.
In Section~\ref{s: prelims} we discuss some preliminaries and introduce our main tool, the RLCD.
In Section~\ref{s: discretization} we outline the discretization procedure, based on the idea of random rounding.
In Section~\ref{s: double counting} we outline the key result, which informally states that ``lattice vectors are usually nice'', and is based on the idea of double counting.
In Section~\ref{s: proof dist} we combine the results of Sections~\ref{s: discretization} and~\ref{s: double counting}, and prove Theorem~\ref{mainthm2}.
In Section~\ref{s: proof main} we conclude by formally deriving Theorem \ref{mainthm1} from Theorem \ref{mainthm2}.
\begin{remark}
The main results of this paper are stated here for real random matrices, and can be extended to random matrices with complex entries. This was recently done in the preprint \cite{Jain-Silwal} following the approach we presented in the present paper.
\end{remark}
\subsection*{Acknowledgement} The first author is grateful to the mathematics department of UC Irvine for hospitality. The first two authors are grateful to Mark~Rudelson for suggesting this problem. All authors thank the referees for their many useful comments.
\section{Preliminaries} \label{s: prelims}
The inner product in $\mathbb R^n$ is denoted $\langle \cdot,\cdot\rangle$, the Euclidean norm is denoted $|\cdot|$, and the sup-norm is denoted $\|x\|_{\infty}=\max_i |x_i|$. The Euclidean unit ball and sphere in $\mathbb R^n$ are denoted $B_2^n$ and ${{\mathbb S}^{n-1}}$, respectively. The unit cube and the cross-polytope in $\mathbb R^n$ are denoted
$$
B_{\infty}^n = \big\{ x\in\mathbb R^n:\,\|x\|_{\infty}\leq 1 \big\},
\quad
B_1^n = \big\{x\in\mathbb R^n:\,\sum_{i=1}^n |x_i|\leq 1 \big\}.
$$
The integer part of a real number $a$ (i.e., the largest integer which is smaller or equal to $a$) is denoted by $\lfloor a\rfloor$,
and the fractional part by $\{a\}=a-\lfloor a\rfloor$.
The cardinality of a finite set $I$ is denoted by $\sharp I$.
Columns of an $N\times n$ matrix $M$ will be denoted by $M_j,$ for $j=1,\dots,n,$ and the rows will be denoted $M^i,$ with $i=1,\dots,N.$
For a random variable $X$, we denote by $\overline{X}$ the symmetrization of $X$
defined as $\overline{X}=X-X'$, where $X'$ is an independent copy of $X$.
Note that
\begin{equation} \label{eq: symmetrization var}
\mathbb E |\overline{X}|^2 = 2\Var(X),
\end{equation}
where we defined the variance of a random vector $X$ as the covariance
of $X$ with itself, i.e. $\Var(X) = \Cov(X,X) = \mathbb E |X - \mathbb E X|^2$.
\subsection{Decomposition of the sphere}
We shall follow the scheme developed by Rudelson and Vershynin in \cite{RudVer-square},
the first step of which is to decompose the sphere to the set of compressible and incompressible vectors. Such decomposition in some form goes back to earlier works, in particular that of Litvak, Pajor, Rudelson and Tomczak-Jaegermann \cite{LPRT}, and it was used in many papers since then
\cite{RudVer-general, tatarko, Tikh, RebTikh}.
Fix some parameters $\delta,\rho\in(0,1)$ whose values will be chosen later, and define
the sets of sparse, compressible, and incompressible vectors as follows:
\begin{gather*}
\Sparse(\delta) := \left\{ u\in{{\mathbb S}^{n-1}}: \#\supp(u) \le \delta n \right\},\\
\Comp(\delta,\rho) := \left\{ u\in{{\mathbb S}^{n-1}}:\,\dist(u,\Sparse(\delta))\leq \rho \right\}, \\
\Incomp(\delta,\rho) := {{\mathbb S}^{n-1}}\setminus \Comp(\delta,\rho).
\end{gather*}
We will use a result of \cite{Liv},
which gives a good uniform lower bound for $|Ax|$
on the set of compressible vectors:
\begin{lemma}[Lemma 5.3, \cite{Liv}]\label{comp-final}
Let $A$ be an $N\times n$ random matrix with $N \ge n$,
whose entries $A_{ij}$ are independent and satisfy
$\sum_{i=1}^N \sum_{j=1}^n \mathbb E A_{ij}^2 \leq KNn$ for some $K>0$ and
$\max_{i,j} \mathcal{L}(A_{ij},1)\leq b$ for some $b\in (0,1)$.
Then
$$
\Pr{ \inf_{x\in \Comp(\delta,\rho)} |Ax|\leq c\sqrt{N} }
\leq 2e^{-cN}.
$$
Here $\rho, \delta\in(0,1)$ and $c>0$ depend only on $K$ and $b$.
\end{lemma}
The rest of our argument will be about incompressible vectors.
\subsection{Randomized Least Common Denominator}
We will need the following lemma due to Esseen (see Esseen \cite{Ess}, or, e.g., Rudelson--Vershynin \cite{RudVer-square}):
\begin{lemma}[Esseen]\label{essen}
Given a variable $\xi$ with the characteristic function $\varphi(\cdot)=\mathbb E\exp(2\pi {\bf i}\xi\cdot)$,
$$\mathcal{L}(\xi,t)\leq C\int_{-1}^{1} \bigg|\varphi\left(\frac{s}{t}\right)\bigg|\,ds,\quad t>0,$$
where $C>0$ is an absolute constant.
\end{lemma}
Rudelson and Vershynin \cite{RudVer-square, RudVer-general} specialized Esseen's lemma for
weighted sums of independent random variables
$\ip{X}{v} = \sum_{i=1}^n v_i X_i$:
\begin{lemma}\label{concfunc}
Let $X=(X_1,\dots,X_n)$ be a random vector with independent coordinates.
Then for every vector $v\in\mathbb R^n,$ and any $t>0$, we have\footnote{Recall that $\overline{X_i}$ denotes the symmetrization of $X_i$, which we defined in the beginning of Section~\ref{s: prelims}.}
$$\mathcal{L}\left(\langle X, v\rangle,t\right)
\leq C_{\text{\tiny\ref{concfunc}}}\int_{-1}^1 \exp\bigg(-c_{\text{\tiny\ref{concfunc}}}\mathbb E\Big(\sum_{i=1}^n \Big[1-\cos\Big(\frac{2\pi s\overline{X}_i v_i}{t}\Big)\Big]
\,\Big)\bigg)ds.$$
The constants $C_{\text{\tiny\ref{concfunc}}},c_{\text{\tiny\ref{concfunc}}}>0$ are absolute.
\end{lemma}
For completeness, we outline the argument here.
\begin{proof}
Let $\varphi$ be the characteristic function of $\langle X, v\rangle$, and
$\varphi_i$ be the characteristic function of $X_i$. By independence, we have
$$\varphi(s)=\prod_{i=1}^n \varphi_i(sv_i),\quad s\in\mathbb R.$$
By definition of $\overline{X}_i$, we have for each $i\leq n$:
$$
|\varphi_i(sv_i)|=\sqrt{\mathbb E \cos(2\pi sv_i\overline{X}_i)}
\leq \exp \Big(-\frac{1}{2}\left(1-\mathbb E\cos( 2\pi sv_i \overline{X}_i)\right) \Big),\quad s\in\mathbb R,
$$
where the last step uses the inequality $|a|\leq \exp \big( -\frac{1}{2}(1-a^2) \big)$ valid for all $a\in\mathbb R$.
To finish the proof it remains to use Lemma \ref{essen}.
\end{proof}
In analogy with the notion of the essential least common denominator (LCD) developed by Rudelson and Vershynin \cite{RudVer-square, RudVer-general, RudVer-delocalization}, we define a
randomized version of LCD, which will be instrumental in controlling the sums non-identically
distributed random variables.
\begin{definition}
For a random vector $X$ in $\mathbb R^n$,
a (deterministic) vector $v$ in $\mathbb R^n$, and parameters $L>0$, $u \in (0,1)$,
define
$$
\RLCD^X_{L,u}(v)
:=\inf \left\{ \theta>0:\,\mathbb E \dist^2(\theta v\star \overline{X},\mathbb{Z}^n)
<\min(u|\theta v|^2, L^2) \right\}.
$$
Here by $\star$ we denote the Schur product
$$v\star X:=(v_1 X_1,\dots,v_nX_n).$$
\end{definition}
The usefulness of RLCD is demonstrated in the following lemma, which shows how
RLCD controls the concentration function of a sum of independent random variables.
\begin{lemma}\label{smallball-1}
Let $X=(X_1,\dots,X_n)$ be a random vector with independent coordinates.
Let $c_0 > 0$, $L>0$ and $u \in (0,1)$.
Then for any vector $v\in\mathbb R^n$ with $|v|\geq c_0$
and any $\varepsilon \ge 0$, we have
$$
\mathcal{L}(\langle X,v\rangle, \varepsilon)
\leq C\varepsilon
+C\exp(-\widetilde c L^2)
+ \frac{C}{\RLCD^X_{L,u}(v)}.
$$
Here $C>0,\widetilde c>0$ may only depend on $c_0,u$.
\end{lemma}
\begin{proof}
Take any $\varepsilon\geq 1/\RLCD^X_{L,u}(v)$.
By Lemma~\ref{concfunc}, we have
$$\mathcal{L}\left(\langle X, v\rangle,\varepsilon\right)
\leq C_{\text{\tiny\ref{concfunc}}}\int_{-1}^1 \exp\bigg(-c_{\text{\tiny\ref{concfunc}}}\mathbb E\Big(\sum_{i=1}^n \Big[1-\cos\Big(\frac{2\pi s\overline{X}_i v_i}{\varepsilon}\Big)\Big]
\,\Big)\bigg)ds.$$
For each $s\in[-1,1]$ and $i\leq n$ we have
$$
\mathbb E\Big[1-\cos\Big(\frac{2\pi s\overline{X}_i v_i}{\varepsilon}\Big)\Big]
\geq \widetilde c\,\mathbb E \,\dist^2(s\overline{X}_i v_i/\varepsilon,\mathbb Z)
$$
for some universal constant $\widetilde c>0$. Hence,
\begin{align*}
\mathcal{L}\left(\langle X, v\rangle,\varepsilon\right)
&\leq C_{\text{\tiny\ref{concfunc}}}\int_{-1}^1 \exp\bigg(-c_{\text{\tiny\ref{concfunc}}}\widetilde c\,
\mathbb E \,\dist^2(s\overline{X} \star v/\varepsilon,\mathbb Z^n)
\,\bigg)ds\\
&=
C_{\text{\tiny\ref{concfunc}}}\varepsilon\int_{-1/\varepsilon}^{1/\varepsilon} \exp\bigg(-c_{\text{\tiny\ref{concfunc}}}\widetilde c\,
\mathbb E \,\dist^2(s\overline{X} \star v,\mathbb Z^n)
\,\bigg)ds\\
&\leq
C_{\text{\tiny\ref{concfunc}}}\varepsilon\int_{-1/\varepsilon}^{1/\varepsilon} \exp\bigg(-c_{\text{\tiny\ref{concfunc}}}\widetilde c\,
\,\min(u|s v|^2,L^2)
\,\bigg)ds,
\end{align*}
where at the last step we used the definition of RLCD and the assumption on $\varepsilon$.
A simple computation finishes the proof.
\end{proof}
We shall also need the notion of the randomized LCD for matrices.
\begin{definition} \label{def: RLCD matrix}
For an $m\times n$ matrix $M$ with rows $M^{1},\dots, M^m$, and a vector $v\in\mathbb R^n$, define
$$
\RLCD^M_{L,u}(v):=\min_{i=1,\dots,m} \RLCD_{L,u}^{M^i}(v).$$
\end{definition}
Recall the following ``tensorization'' lemma of Rudelson and Vershynin \cite{RudVer-square}:
\begin{lemma}[Tensorization lemma, Rudelson--Vershynin \cite{RudVer-square}]\label{tensorization}
Suppose that $\varepsilon_0\in(0,1)$, $K\geq 1$, and let $Y_1,\dots,Y_m$ be independent random variables such that each $Y_i$ satisfies
$${\mathbb P}\{|Y_i|\leq \varepsilon\}\leq K\varepsilon\quad\mbox{for all }\varepsilon\geq\varepsilon_0.$$
Then
$${\mathbb P}\Big\{\sum_{i=1}^m Y_i^2\leq \varepsilon^2 m \Big\}\leq (CK\varepsilon)^m,\quad \varepsilon\geq\varepsilon_0,$$
where $C>0$ is a universal constant.
\end{lemma}
The tensorization lemma is useful when one wants to control the anti-concentration
of $|Mx|$ where $M$ is an $m \times n$ random matrix with independent rows $M^i$
and $x$ is a fixed vector. Indeed, in this case $|Mx|^2 = \sum_{i=1}^m \ip{M^i}{x}^2$,
and one can use Lemma~\ref{tensorization} for $Y_i := \ip{M^i}{x}$.
Furthermore, one can use Lemma \ref{smallball-1}
to control the concentration function of each $Y_i$. This gives:
\begin{lemma}\label{smallball}
Let $M$ be an $m\times n$ random matrix with independent entries $M_{ij}$.
Let $L>0$, $c_0>0$ and $u\in(0,1)$.
Then for any $x\in\mathbb R^n$ with $|x|\geq c_0$ and any
$\varepsilon \geq C_{\ref{smallball}}\exp(-\widetilde c_{\ref{smallball}} L^2) + C_{\ref{smallball}}/\RLCD^M_{L,u}(x)$,
we have
$
{\mathbb P}\big\{|Mx|\leq \varepsilon\sqrt{m}\big\}
\leq (C_{\ref{smallball}}\varepsilon)^m.$$
Here $C_{\ref{smallball}},\widetilde c_{\ref{smallball}}>0$ may only depend on
$c_0$ and $u$.
\end{lemma}
A crucial property of the RLCD which will enable us to discretize the range of possible realizations
of random unit normals, is {\it stability of RLCD with respect to small perturbations}:
\begin{lemma}[Stability of RLCD]\label{stable-rlcd}
Consider a random vector $X$ in $\mathbb R^n$ with uncorrelated coordinates,
a (deterministic) vector $x$ in $\mathbb R^n$, and parameters $L,u>0$.
Fix any tolerance level $r>0$ that satisfies
\begin{equation} \label{eq: r range}
r^2 \Var(X) \le \frac{1}{8} \min \Big( u|x|^2, \, \frac{L^2}{D^2} \Big)
\end{equation}
where $D=\RLCD^X_{L,u}(x)$.
Then for any $y\in\mathbb R^n$ with $\|x-y\|_{\infty}<r$, we have
$$
\RLCD^X_{2L,4u}(y)
\le \RLCD^X_{L,u}(x)
\le \RLCD^X_{L/2,u/4}(y).
$$
\end{lemma}
\begin{proof}
Note that
\begin{equation*}
\mathbb E |x\star \overline{X} - y\star \overline{X}|^2
= \mathbb E\sum_{i=1}^n \overline{X}_i^2(x_i-y_i)^2
< r^2\mathbb E|\overline{X}|^2
= 2 r^2 \Var(X),
\end{equation*}
where the last identity is \eqref{eq: symmetrization var}.
Since $\RLCD^X_{L,u}(x)=D,$ the definition of RLCD yields
\begin{equation*}
\mathbb E \dist^2(D x\star \overline{X},\mathbb Z^n)= \min (uD^2|x|^2, L^2).
\end{equation*}
By the inequality $(a+b)^2\leq 2a^2+2b^2$, we get
\begin{equation*}
\begin{split}
\mathbb E \dist^2(D y\star \overline{X},\mathbb Z^n)
&\leq 2\mathbb E \dist^2(D x\star \overline{X},\mathbb Z^n)+2\mathbb E|Dx\star \overline{X}-Dy\star \overline{X}|^2\\
&< 2\min (uD^2|x|^2, L^2) + 4D^2 r^2 \Var(X)
\leq 4\min (uD^2|x|^2, L^2),
\end{split}
\end{equation*}
where the last step follows from our assumptions \eqref{eq: r range} on $r$.
By definition of RLCD, this immediately gives
$$
\RLCD^X_{2L,4u}(y) \le D,
$$
which proves the first conclusion of the lemma.
The second conclusion can be derived similarly. For any $\theta<D$,
the definition of RLCD yields
$$
\mathbb E\dist^2(\theta x\star \overline{X},\mathbb Z^n)
\ge \min (u\theta^2|x|^2, L^2).
$$
By the inequality $(a+b)^2\geq a^2/2-b^2$, we get
\begin{align*}
\mathbb E\dist^2(\theta y\star \overline{X},\mathbb Z^n)
&\geq \frac{1}{2} \mathbb E\dist^2(\theta x\star \overline{X},\mathbb Z^n)-\mathbb E|\theta x\star \overline{X} - \theta y\star \overline{X}|^2\\
&\geq
\frac{1}{2} \min (u \theta^2|x|^2, L^2)
- 2 \theta^2 r^2 \Var(X)
\ge \frac{1}{4} \min (u\theta^2|x|^2, L^2),
\end{align*}
where in the last step we used the bound $\theta < D$
and our assumptions \eqref{eq: r range} on $r$. Thus,
$$
\mathbb E\dist^2(\theta y\star \overline{X},\mathbb Z^n)\geq \min (u\theta^2|x|^2/4, L^2/4)\quad\mbox{ for all }\theta\in(0,D),
$$
and, by the definition of RLCD, this immediately gives
$$
\RLCD^X_{L/2,u/4}(y) \ge D,
$$
which proves the second conclusion of the lemma.
\end{proof}
\medskip
The following result is a version of \cite[Lemma~3.6]{RudVer-general}.
\begin{lemma}[Incompressible vectors have large RLCD]\label{l: aux incomp rlcd}
For any $b,\delta,\rho\in(0,1)$ there are $n_0=n_0(b,\delta,\rho)$,
$h_{\ref{l: aux incomp rlcd}}=h_{\ref{l: aux incomp rlcd}}(b,\delta,\rho)\in(0,1)$
and $u_{\ref{l: aux incomp rlcd}}=u_{\ref{l: aux incomp rlcd}}(b,\delta,\rho)\in(0,1/4)$ with the following property.
Let $n\geq n_0$, let $x\in Incomp_n(\delta,\rho)$, and assume that a random vector $X=(X_1,\dots,X_n)$ with independent components
satisfies $\mathcal{L}(X_i,1)\leq b$, $i\leq n$, and $\Var |X|\leq T$, for some fixed parameter $T\geq n$. Then for any $L>0$ we have
$\RLCD^X_{L,u_{\ref{l: aux incomp rlcd}}}(x)\geq h_{\ref{l: aux incomp rlcd}}\cdot\frac{n}{\sqrt{T}}$.
\end{lemma}
\begin{proof}
For clarity of the argument, we shall often hide the parameters $b$, $\delta$, $\rho$, $h_{\ref{l: aux incomp rlcd}}$,
and $u_{\ref{l: aux incomp rlcd}}$ in the notation such as $\lesssim, \gtrsim$;
the reader will find it easy to fill in the details.
By definition of RLCD and since $x$ is a unit vector, it suffices to show that
$$
\mathbb E \dist^2(\theta x\star \overline{X},\mathbb{Z}^n)
\gtrsim \theta^2
\quad \forall \; \theta \in \left(0, h_{\ref{l: aux incomp rlcd}}\cdot\frac{n}{\sqrt{T}}\right).
$$
Suppose that
$$
\mathbb E \dist^2(\theta x\star \overline{X},\mathbb{Z}^n) \ll \theta^2
$$
for some $\theta>0$; we want to show that in this case $\theta \gtrsim \frac{n}{\sqrt{T}}$.
Let $p \in \mathbb Z^n$ denote a closest integer vector to $\theta x\star \overline{X}$;
note that $p$ is a random vector.
Then $\mathbb E \abs[0]{\theta x\star \overline{X}-p}^2 \ll \theta^2$, and Markov's inequality yields that
$\abs[0]{\theta x\star \overline{X}-p} \ll \theta$
with high probability. Dividing both sides by $\theta$ gives
$\abs[0]{x \star \overline{X} - p/\theta} \ll 1$, so another application of Markov's inequality
shows that
$$
\abs[2]{x_i \overline{X}_i - \frac{p_i}{\theta}} \ll \frac{1}{\sqrt{n}}
\quad \text{for $n-o(n)$ coordinates $i$}.
$$
Moreover, $\mathbb E \abs[1]{\overline{X}}^2 = 2 \Var \abs{X} \le 2T$ by \eqref{eq: symmetrization var}.
So a similar double application of Markov's inequality shows that, with high probability,
$$
\abs[1]{\overline{X}_i} \lesssim \sqrt{\frac{T}{n}}
\quad \text{for $n-o(n)$ coordinates $i$}.
$$
Furthermore, incompressible vectors are ``spread'' in the sense that
$$
I \coloneqq \Big\{ i: \; \abs{x_i} \asymp \frac{1}{\sqrt{n}} \Big\}
\quad \text{satisfies} \quad
\abs{I} \gtrsim n.
$$
This fact is easy to check; a formal proof can be found in \cite[Lemma~3.4]{RudVer-square}.
Finally, the assumption on the concentration function shows that ${\mathbb P} \big\{\abs[1]{\overline{X}_i} \ge 1\big\} \ge b$.
By the independence of $\overline{X}_i$'s
this implies that, with high probability,
$$
\abs[1]{\overline{X}_i} \ge 1
\quad \text{for $b\abs{I}/2 \gtrsim n$ coordinates $i \in I$}
$$
(this conclusion follows by considering the sum of independent indicator variables ${\bf 1}_{\{|\overline{X}_i|\geq 1\}}$, $i\in I$).
Taking the intersection of these events and sets of coordinates, we see that
with high probability there must exist a coordinate $i$ for which we have simultaneously
the following three bounds:
$$
\abs[2]{x_i \overline{X}_i - \frac{p_i}{\theta}} \ll \frac{1}{\sqrt{n}}, \quad
1 \le \abs[1]{\overline{X}_i} \lesssim \sqrt{\frac{T}{n}}, \quad
\abs{x_i} \asymp \frac{1}{\sqrt{n}}.
$$
Then, using the triangle inequality, we get
$$
\abs[2]{\frac{p_i}{\theta}}
\ge \abs[1]{x_i \overline{X}_i} - o \Big( \frac{1}{\sqrt{n}} \Big)
\ge \frac{c}{\sqrt{n}} \cdot 1 - o \Big( \frac{1}{\sqrt{n}} \Big)
> 0.
$$
Thus $p_i \ne 0$, and since $p_i$ is an integer, we necessarily have $\abs{p_i} \ge 1$.
On the other hand, a similar application of the triangle inequality gives
$$
\abs[2]{\frac{p_i}{\theta}}
\le \abs[1]{x_i \overline{X}_i} + o \Big( \frac{1}{\sqrt{n}} \Big)
\lesssim \frac{1}{\sqrt{n}} \cdot \sqrt{\frac{T}{n}} + o \Big( \frac{1}{\sqrt{n}} \Big)
\lesssim \frac{\sqrt{T}}{n}.
$$
This yields that $\theta \gtrsim \abs{p_i}\cdot\frac{n}{\sqrt{T}} \ge \frac{n}{\sqrt{T}}$, as claimed.
\end{proof}
\section{Discretization} \label{s: discretization}
In this section we outline the required discretization results. They essentially follow from the results in Section 3 of \cite{Liv}, however they are not stated there in the form we need, and thus we repeat certain arguments here.
\begin{definition}[Discretization, part 1]
Given a vector of weights $\alpha \in \mathbb R^n$ and a resolution parameter $\varepsilon>0$,
we consider the set of approximately unit vectors
whose coordinates are quantized at scales $\alpha_i \varepsilon/\sqrt{n}$.
Precisely, we define
$$
\Lambda_{\alpha}(\varepsilon)
:= \Big(\frac{3}{2}B_2^n\setminus\frac{1}{2}B_2^n \Big)
\cap \left(\frac{\alpha_1 \varepsilon}{\sqrt{n}} \mathbb Z \times \dots \times \frac{\alpha_n \varepsilon}{\sqrt{n}} \mathbb Z \right).
$$
\end{definition}
\begin{lemma}[Rounding] \label{keylemmarounding}
Fix any accuracy $\varepsilon\in (0,1/2)$,
a weight vector $\alpha \in [0,1]^n$, and any (deterministic) $N\times n$ matrix $A$
whose columns we denote $A_i$.
Then for any $x\in{{\mathbb S}^{n-1}}$ one can find $y\in \Lambda_{\alpha}(\varepsilon)$ such that
$$
\norm{x-y}_{\infty}\leq \frac{\varepsilon}{\sqrt{n}}
\quad \text{and} \quad
\abs{A(x-y)} \le \frac{\varepsilon}{\sqrt{n}} \Big( \sum_{j=1}^n \alpha^2_j \abs{A_j}^2 \Big)^{1/2}.
$$
\end{lemma}
\begin{proof}
Our construction of $y$ is probabilistic and amounts to \emph{random rounding} of $x$.
The technique of random rounding has been used in computer science
(see the survey by Srinivasan \cite{Srin}, papers \cite{KA}, \cite{KV}),
asymptotic convex geometry \cite{KlLiv}
and random matrix theory \cite{Liv, Tikh}.
A random rounding of $x \in {{\mathbb S}^{n-1}}$ is a random vector $y$ with independent coordinates that takes values in
the $\Lambda_{\alpha}(\varepsilon)$ and satisfies
$\mathbb E y = x$
and
\begin{equation} \label{eq: random rounding}
\abs{x_j-y_j} \le \frac{\alpha_j \varepsilon}{\sqrt{n}}, \quad j=1,\ldots,n,
\quad \text{for any realization of $y$}.
\end{equation}
One can construct such a distribution of $y$ by rounding each coordinate of $x$
up or down, at random, to a neighboring point in the lattice $(\alpha_j \varepsilon/\sqrt{n}) \mathbb Z$.
The identity $\mathbb E y = x$ can be enforced by choosing the probabilities
of rounding up and down accordingly.\footnote{Precisely, if
$x_j= (\alpha_j \varepsilon/\sqrt{n})(k_j+ p_j)$ for some $k_j \in \mathbb Z$ and $p_j \in [0,1)$,
we let $y_j$ take value $(\alpha_j \varepsilon/\sqrt{n})k_j$ with probability $1-p_j$
and value $(\alpha_j \varepsilon/\sqrt{n})(k_j + 1)$ with probability $p_j$. Clearly,
this yields $\mathbb E y = x$.}
To check that $y$ indeed takes values in $\Lambda_{\alpha}(\varepsilon)$,
note that the bound in \eqref{eq: random rounding} and the assumption that $\alpha_i \in [0,1]$
imply
\begin{equation} \label{eq: norminf x-y}
\norm{x-y}_{\infty}\leq \frac{\varepsilon}{\sqrt{n}}
\quad \text{for any realization of $y$}.
\end{equation}
It follows that $\norm{x-y}_2 \le \varepsilon < 1/2$, and since $\norm{x}_2=1$, this implies
by triangle inequality that $1/2 < \norm{y}_2 < 3/2$. This verifies that the random vector
$y$ takes values in $\Lambda_{\alpha}(\varepsilon)$ as we claimed.
Finally, we have
\begin{align*}
\mathbb E \abs{A(x-y)}^2
&= \mathbb E \abs[3]{\sum_{j=1}^n (x_j-y_j) A_j}^2
= \sum_{i=1}^n \mathbb E (x_j-y_j)^2 \cdot \abs{A_j}^2
\quad \text{(since $\mathbb E(x_j-y_j) = 0$)} \\
&\le \frac{\varepsilon^2}{n} \sum_{j=1}^n \alpha_j^2 \abs{A_j}^2
\quad \text{(using the bound in \eqref{eq: random rounding})}. \\
\end{align*}
Combining this with $\eqref{eq: norminf x-y}$, we conclude that
there exists a realization of the random vector $y$
that satisfies the conclusion of the lemma.
\end{proof}
\begin{lemma} \label{lem: ell1 net}
Let $M \ge 1$. There exists a subset $\Xi \subset \mathbb R^n_+$ of cardinality
at most $(CM)^n$ and such that the following holds.
For every vector $x \in \mathbb R^n_+$ with $\norm{x}_1 \le Mn$ there exists
$y \in \Xi$ such that $\norm{y}_1 \le (M+1)n$ and $y \ge x$ coordinate-wise.
\end{lemma}
\begin{proof}
Define $y \coloneqq \lceil x \rceil$ where the ceiling function is applied coordinate-wise.
Then $\norm{y}_1 \le \norm{x}_1 + n \le (M+1)n$ as claimed.
In particular, there are as many vectors $y$ as there are integer points in the $\ell_1$-ball
$\{z \in \mathbb R^n \;:\; \norm{z}_1 \le (M+1) n\}$.
According to classical results (see \cite[Exercise 29]{PS}, \cite{Schutt}),
the number of integer points in this ball is bounded by $(CM)^n$ (see also \cite{KlLiv} for a similar covering argument).
The lemma is proved.
\end{proof}
Fix $\kappa>e$ and consider the set
\begin{equation}\label{Omega}
\Omega_{\kappa}:=\Big\{\alpha \in [0,1]^n: \; \prod_{j=1}^n \alpha_j\geq \kappa^{-n}\Big\}.
\end{equation}
The following result is a corollary of \cite[Lemma~3.11]{Liv}.
\begin{lemma} \label{netsonnets}
For any $\kappa>e$ there exists a subset $\mathcal{F}\subset \Omega_{e\kappa}$ of cardinality
at most $({C}{\log\kappa})^n$
and such that the following holds.
For every vector $\beta\in\Omega_{\kappa}$ there exists $\alpha\in \mathcal{F}$
such that $\alpha \le \beta$ coordinate-wise.
\end{lemma}
\begin{proof}
Apply Lemma~\ref{lem: ell1 net} for $x = -\log \beta$, $y = -\log \alpha$ (defined coordinate-wise)
and $M = \log \kappa$.
\end{proof}
\begin{definition}[Discretization -- part 2]\label{def: disc part 2} Assuming the dimension $n$ fixed, for the parameters $\kappa>e$ and $\varepsilon>0$, we shall use notation
\begin{equation}\label{mainLambda}
\Lambda^{\kappa}(\varepsilon):=\bigcup_{\alpha\in\mathcal{F}} \Lambda_{\alpha} (\varepsilon),
\end{equation}
with $\mathcal{F}$ being the set whose existence is guaranteed by Lemma \ref{netsonnets}.
\end{definition}
\begin{remark}\label{rem: card of Lambda}
It is immediate from the above definition that for any $\kappa>e$ there is $C_\kappa>0$ depending
only on $\kappa$ such that $\sharp \Lambda^{\kappa}(\varepsilon)\leq \sum\limits_{\alpha\in\mathcal F}\sharp \Lambda_{\alpha} (\varepsilon)\leq (C_\kappa /\varepsilon)^n$
for every $\varepsilon\in(0,1]$.
\end{remark}
\medskip
The following notion from \cite{Liv} will help us to control the norms of the columns $A_j$ of
an $N\times n$ matrix $A$ in the absence of any distributional assumptions on $A_j$:
$$
\mathcal{B}_{\kappa}(A)
\coloneqq \min \Big\{ \sum_{j=1}^n \alpha_j^2 |A_j|^2 \;:\; \alpha \in \Omega_\kappa \Big\}.
$$
\begin{theorem} \label{determin}
Fix $\varepsilon \in (0,1/2)$, $\kappa>e$, and
any (deterministic) $N\times n$ matrix $A$.
Then for every $x\in {{\mathbb S}^{n-1}}$ one can find $y\in \Lambda^{\kappa}(\varepsilon)$ so that
$$
\norm{x-y}_{\infty}\leq \frac{\varepsilon}{\sqrt{n}}
\quad \text{and} \quad
\abs{A(x-y)} \le \frac{\varepsilon}{\sqrt{n}} \sqrt{\mathcal{B}_{\kappa}(A)}.
$$
\end{theorem}
\begin{proof}
By Lemma \ref{keylemmarounding}, for any $x\in{{\mathbb S}^{n-1}}$
we can find $y\in \Lambda^{\kappa}(\varepsilon)$ that approximates $x$ in the $\ell_\infty$ norm
as required, and such that
\begin{align*}
\abs{A(x-y)}
&\le \frac{\varepsilon}{\sqrt{n}}
\Big( \min_{\alpha\in \mathcal{F}} \sum_{j=1}^n \alpha_j^2 \abs{A_j}^2 \Big)^{1/2}
\le \frac{\varepsilon}{\sqrt{n}}
\Big( \min_{\beta\in \Omega_{\kappa}} \sum_{j=1}^n \beta_j^2 \abs{A_j}^2 \Big)^{1/2}
\quad \text{(by Lemma~\ref{netsonnets})}\\
&= \frac{\varepsilon}{\sqrt{n}} \sqrt{\mathcal{B}_{\kappa}(A)}.
\end{align*}
The proof is complete.
\end{proof}
Lastly, we recall the important property concerning the large deviation behavior of $\mathcal{B}_{\kappa}$; here Lemma 3.11 from \cite{Liv} is quoted with a specific choice of parameters.
\begin{lemma}[Lemma 3.11 from \cite{Liv}]\label{ldpB}
Let $A$ be a random matrix with independent columns.
Then for any $\kappa>e$, we have
$$
\Pr{ \mathcal{B}_{\kappa}(A)\geq 2\mathbb E\|A\|_\mathrm{HS}^2 }
\leq \left(\frac{\kappa}{\sqrt{2}}\right)^{-2n}.
$$
\end{lemma}
\medskip
Finally, we are ready to state the main result of this section, which will follow as a corollary of Lemma \ref{stable-rlcd}, Theorem \ref{determin} and Lemma \ref{ldpB}. Given $\gamma>0, \omega\in (0,1), D>0$, and a distribution of a random matrix $M,$ we shall use notation
\begin{equation*}\label{levelsets}
\begin{split}
S^M_{\omega,\gamma}(D)&:=\left\{x\in\frac{3}{2}B_2^n\setminus\frac{1}{2}B_2^n:\, \RLCD^M_{\gamma\sqrt{n},\omega}(x)\in [D,2D]\right\},\\
\tilde{S}^M_{\omega,\gamma}(D)&:=\left\{x\in\frac{3}{2}B_2^n\setminus\frac{1}{2}B_2^n:\,\RLCD^M_{2\gamma\sqrt{n},4\omega}(x)\leq 2D,\,\, \RLCD^M_{0.5\gamma\sqrt{n},0.25\omega}(x)\geq D\right\}
\end{split}
\end{equation*}
for the level sets of the RLCD.
\begin{theorem}[Approximation]\label{main-nets}
Fix any $\varepsilon\in(0,0.1)$, $\kappa>e,$ $\gamma>0,$ $\omega\in (0,1),$ $K>0$. Let $M$ be an $m\times n$ random matrix with independent columns, and whose rows $M^i$ satisfy
\begin{equation}\label{cond}
\varepsilon^2 \Var(M^i) \le \frac{1}{8} \min \Big( \omega n, \, \frac{\gamma^2n^2}{D^2} \Big),
\quad i=1,\ldots,m.
\end{equation}
Then, with probability at least $1-(\kappa/\sqrt{2})^{-2n}$, for every $x\in{{\mathbb S}^{n-1}}\cap S^M_{\omega,\gamma}(D)$ there exists $y\in \Lambda^{\kappa}(\varepsilon) \cap \tilde{S}^M_{\omega,\gamma}(D)$ such that
\begin{equation}\label{concl}
\|x-y\|_\infty\leq \frac{\varepsilon}{\sqrt{n}},\quad \abs{M(x-y)} \leq \frac{\sqrt{2}\varepsilon}{\sqrt{n}} \Big( \mathbb E \norm{M}^2_\mathrm{HS} \Big)^{1/2}.
\end{equation}
\end{theorem}
\begin{proof}
Lemma \ref{ldpB} says that the event
$$
\mathcal{E}:=\{\mathcal{B}_{\kappa}(M)\leq 2\mathbb E\norm{M}^2_\mathrm{HS}\}
$$
occurs with probability at least $1-(\kappa/\sqrt{2})^{-2n}$.
Fix any realization of the random matrix $M$ for which this event happens.
Let $y$ be the approximation of $x$ given by Theorem \ref{determin}. Then (\ref{concl}) follows from the conclusion of Theorem \ref{determin} and the definition of our event. The fact that $y\in \tilde{S}^M_{\omega,L}(D)$ follows from Lemma \ref{stable-rlcd} (applied with $r=\epsilon/\sqrt{n}$) together with the assertion of Theorem \ref{determin} (applied with $A=M$): indeed, the assumption (\ref{cond}) allows us to appeal to Lemma \ref{stable-rlcd}.
\end{proof}
\section{Anti-concentration on lattice points} \label{s: double counting}
The goal of this section is to study anti-concentration properties of random sums with
coefficients taken from sets of the form
\begin{equation} \label{eq: Lambda}
\Lambda:=\left(\frac{3}{2}B_{2}^n\cap\big\{x\in\mathbb R^n:\;
\sharp\{i:\;|x_i|\geq \frac{\rho}{\sqrt{n}}\}\geq \delta n\big\}\right)
\cap \left( \frac{{\lambda}_1}{\sqrt{n}} \mathbb Z \times \cdots \times \frac{{\lambda}_n}{\sqrt{n}} \mathbb Z \right).
\end{equation}
The main result of this section is the following
\begin{theorem}[Most lattice points are unstructured] \label{mainprop}
For any $U\geq 1$, $b\in(0,1)$ and $\delta,\rho\in(0,1/2]$ there exist $n_0=n_0(U,b,\delta,\rho)$,
$\gamma=\gamma(U,b,\delta,\rho)\in(0,1)$
and $u=u(b,\delta,\rho)\in(0,1/4)$ such that the following holds.
Let $n\geq n_0$.
Consider a random vector $X$ in $\mathbb R^n$ with independent components $X_i$ that satisfies
$$
\Var(X)\leq \frac{1}{8}(1-b)\delta \gamma^2 n^2
\quad \text{and} \quad
\max_i \mathcal{L}(X_i,1)\leq b.
$$
Fix numbers ${\lambda}_1,\ldots,{\lambda}_n$ satisfying $6^{-n} \le {\lambda}_i \le 0.01$
and let $W$ be a vector uniformly distributed on the set $\Lambda$ defined in \eqref{eq: Lambda}.
Then
$$
{\mathbb P}_W \Big\{ \RLCD^X_{\gamma\sqrt{n},u}(W)< \min_i 1/{\lambda}_i \Big\} \leq U^{-n}.
$$
\end{theorem}
The above theorem will be used to control the cardinality of $\varepsilon$-nets
on the set of ``typical'' realizations of unit normal vectors to the spans of columns of our random matrix,
and forms a crucial step in the proof of Theorem~\ref{mainthm2}.
The idea of using double counting to verify structural properties of random normals
was applied earlier in \cite{TikhErd}.
We start with an observation that will allow us to reduce the Euclidean ball $\frac{3}{2}B_{2}^n$
by a parallelotope in the definition of $\Lambda$.
\begin{lemma}\label{l: ball to rect}
There is a universal constant $C_0>0$ with the following property.
For any $n\geq 1$, there is a collection of parallelotopes $\mathcal P=\{P_i\}$ in $\mathbb R^n$
of cardinality at most $2^{C_0n}$, such that
\begin{itemize}
\item Each $P_i$ is centered at the origin, with the edges parallel to the coordinate axes;
\item Each edge of $P_i$ is of length at least $2/\sqrt{n}$;
\item $\frac{3}{2}B_2^n\subset\bigcup\limits_i P_i\subset 3B_2^n$.
\end{itemize}
\end{lemma}
\begin{proof}
First, standard volumetric estimates imply that there is a covering of $\frac{3}{2}B_2^n$ by parallel translates of the cube $\frac{1}{2\sqrt{n}}B_\infty^n$,
of cardinality at most $2^{C_0n}$ for a universal constant $C_0>0$.
Let $\{x_i\}_{i\in I}$ be a collection of at most $2^{C_0n}$ points in $\frac{3}{2}B_2^n$ such that each of the cubes from the covering
contains at least one point $x_i$ from the collection.
Now, define $\mathcal P=\{P_i\}_{i\in I}$ by taking, for each $i\in I$, $P_i:=\widetilde P_i+\frac{1}{\sqrt{n}}B_\infty^n$,
where $\widetilde P_i$ is the unique parallelotope centered at the origin, and with $x_i$ being one of its vertices.
It is elementary to check that the collection satisfies the required properties.
\end{proof}
\begin{lemma}\label{l: aux1}
For any $b\in(0,1)$ and $\delta,\rho\in(0,1/2]$, there exists $n_0=n_0(b,\delta,\rho)$
such that the following holds.
Let $n\geq n_0$ and $\gamma \in (0,1)$.
Fix any subset $J \subset [n]$
and consider a fixed (deterministic) vector $x \in \mathbb R^n$ satisfying
\begin{equation} \label{eq: x conditions}
|x|^2\leq \frac{1}{4}(1-b)\delta \gamma^2 n^2
\quad \text{and} \quad
\sharp\{i\in J:\;|x_i|\geq 1\} \geq \frac{1}{2}(1-b)\delta n.
\end{equation}
Furthermore, fix numbers ${\lambda}_1,\ldots,{\lambda}_n$ satisfying $6^{-n} \le {\lambda}_i \le 0.01$ and
a vector $a = (a_1,\dots,a_n)$ satisfying
$\abs{a} \le 3$ and $\min a_i \ge 1/\sqrt{n}$.
Consider the parallelotope $P:=\prod_{i=1}^n [-a_i,a_i]$,
and define
$$
\Lambda':=\left\{ w \in P:\; |w_i| \geq \frac{\rho}{\sqrt{n}} \; \forall i\in J \right\}
\cap \left( \frac{{\lambda}_1}{\sqrt{n}} \mathbb Z \times \cdots \times \frac{{\lambda}_n}{\sqrt{n}} \mathbb Z \right).
$$
Let $W$ be a random vector uniformly distributed on $\Lambda'$.
Then, for $D \coloneqq \min_i 1/{\lambda}_i$, we have
\begin{equation}\label{eq: outcome}
{\mathbb P}\Big\{ \min_{\theta \in (0,D)} \dist(\theta W\star x,\mathbb{Z}^n)^2 < \min \big( c|\theta W|^2/2,16\gamma^2n \big) \Big\}
\leq (C\gamma)^{cn},
\end{equation}
where $C,c>0$ depending only on $b,\delta,\rho$.
\end{lemma}
\begin{proof}
{\bf Step 1. Halving the set $I$.}
The assumptions on $x$ imply that the set
$$
I:=\big\{i\in J:\;1\leq |x_i|\leq \gamma\sqrt{n}\big\}
\quad \text{satisfies} \quad
\sharp I \ge \frac{1}{4} (1-b)\delta n.
$$
Next, let $\mu=\mu(x)$ be a median of the set $\{ a_i\abs{x_i} :\; i \in I \}$.
Thus, each of the subsets
$$
I' \coloneqq \{i\in I :\; \;a_i\abs{x_i}\leq \mu\}
\quad \text{and} \quad
I'' \coloneqq \{i\in I :\; a_i\abs{x_i}\geq \mu\}
$$
contains at least a half of the elements of $I$:
\begin{equation} \label{eq: halves sizes}
\min( \sharp I', \sharp I'')
\ge \frac{1}{2} \sharp I
\ge \frac{1}{8} (1-b)\delta n
\ge cn,
\end{equation}
where $c>0$ depends only on $b$ and $\delta$.
Take $\theta\in(0,D)$ and consider two cases.
\medskip
{\bf Step 2. Ruling out small multipliers $\theta$.}
We claim that the range for $\theta$ in \eqref{eq: outcome} can automatically
be narrowed to $(\frac{1}{2\mu}, D)$.
To check this, it suffices to show that for any $\theta \in (0, \frac{1}{2\mu}]$,
the bound
\begin{equation} \label{eq: small multipliers}
\dist(\theta W\star x,\mathbb{Z}^n)^2 \ge c |\theta W|^2/2
\end{equation}
holds deterministically, i.e. for any realization of the random vector $W$.
By construction, the coordinates $W_i$ of $W$ for $i \in I$ are uniformly distributed
in lattice intervals, namely
\begin{equation} \label{eq: Wi range}
W_i \sim \Unif \Big( \Big[ \frac{\rho}{\sqrt{n}}, a_i \Big] \cap \frac{{\lambda}_i}{\sqrt{n}} \mathbb Z \Big),
\quad i \in I.
\end{equation}
This means in particular that the coordinates of $\theta W\star x$ for $i \in I'$ satisfy
$$
\theta \abs{W_i x_i}
\le \theta a_i \abs{x_i}
\le \theta \mu \le \frac{1}{2},
$$
where we used the definition of $I'$ and the smallness of $\theta$.
This bound in turn yields
$$
\dist(\theta \abs{W_i x_i}, \mathbb Z)
= \theta \abs{W_i x_i}
\ge \theta \cdot \frac{\rho}{\sqrt{n}} \cdot 1
$$
where in the last step we used the range of $W_i$ from \eqref{eq: Wi range}
and the definition of $I$.
Square both sides of this bound and sum over $i \in I'$ to get
$$
\dist(\theta W\star x, \mathbb Z^n)^2
\ge \frac{\theta^2 \rho^2}{n} \sharp I'
\ge c \theta^2 \rho^2
\ge c_0 \theta^2 \abs{W}^2/2,
$$
where we used \eqref{eq: halves sizes},
suppressed $\rho$ into $c_0$, and
noted that $\abs{W}^2 \le \abs{a}^2 \le 9$
by definition of $W$ and assumption on $a$.
We have proved \eqref{eq: small multipliers}.
\medskip
{\bf Step 3. Handling a fixed multiplier $\theta$.}
Due to the previous step, our remaining task is to show that
$$
{\mathbb P}\Big\{ \min_{\theta \in (1/2\mu,D)} \dist(\theta W\star x,\mathbb{Z}^n)^2
16\gamma^2n \Big\}
\leq (C\gamma)^{cn}.
$$
To do this, let us first estimate the probability
that $\dist(\theta W\star x,\mathbb{Z}^n)^2 <49\gamma^2n$
for a {\em fixed} multiplier\footnote{Extending the range by $1$ will be help us in the next step to unfix $\theta$;
increasing the constant factor $16$ to $49$ will help us run a net approximation argument in Step 4.} $\theta \in (1/2\mu,D+1)$.
Let $i \in I''$. Recall from \eqref{eq: Wi range} that the random variable $\abs{W_i}$
is uniformly distributed in a lattice interval whose diameter is at least
$$
a_i - \frac{\rho}{\sqrt{n}} - \frac{2{\lambda}_i}{\sqrt{n}}
\ge \frac{a_i}{3};
$$
here we used the assumptions $a_i \ge 1/\sqrt{n}$, $\rho \le 1/2$ and ${\lambda}_i \le 0.01$.
Thus, the random variable $\theta \abs{W_i x_i}$,
i.e. the absolute value of a coordinate of $\theta W\star x$,
is distributed in a lattice interval of diameter at least
$$
\frac{a_i}{3} \theta \abs{x_i}
\ge \frac{\theta \mu}{3}
\ge \frac{1}{6};
$$
here we used the definition of $I''$ and the largeness of $\theta$.
Moreover, the step of that lattice interval (the distance between any adjacent points) is
$$
\frac{{\lambda}_i}{\sqrt{n}} \theta \abs{x_i}
\le {\lambda}_i \theta \gamma
\le {\lambda}_i (D+1) \gamma
\le 2\gamma;
$$
here we used the definition of $I$, the range of $\theta$, the definition of $D$,
and the assumption that ${\lambda}_i \le 0.01$.
The random variable $\theta \abs{W_i x_i}$
that is uniformly distributed on a lattice interval of diameter at least $1/6$
and with step at most $2\gamma$ satisfies
$$
\Pr{\dist(\theta \abs{W_i x_i}, \mathbb Z) < \varepsilon} \le C\varepsilon
\quad \text{for any } \varepsilon \ge 4 \gamma,
$$
where $C$ is an absolute constant.
Squaring the distances, summing them over $i \in I''$
and using Tensorization Lemma~\ref{tensorization},
we conclude that
$$
\Pr{ \dist(\theta W\star x,\mathbb{Z}^n)^2 < \varepsilon^2 \sharp I'' }
\le (C'\varepsilon)^{\sharp I''}
\quad \text{for any } \varepsilon \ge 4 \gamma.
$$
Recall from \eqref{eq: halves sizes} that $\sharp I'' \ge cn$.
Hence, substituting $\varepsilon = C_0 \gamma$ with sufficiently large $C_0$
(depending on $c$ and thus ultimately on $b$ and $\delta$), we get
$$
\Pr{ \dist(\theta W\star x,\mathbb{Z}^n)^2 < 49 \gamma^2 n }
\le (C''\gamma)^{cn}.
$$
\medskip
{\bf Step 4. Unfixing the multiplier $\theta$.}
It remains to make the distance bound hold simultaneously for all $\theta$ in the
range $(1/2\mu,D)$. To this end, we use a union bound combined with a
discretization argument. To discretize the range of $\theta$, consider the
lattice interval
$$
\Theta \coloneqq \Big( \frac{1}{2\mu},D \Big) \cap \frac{1}{\sqrt{n}} \mathbb Z.
$$
For sufficiently large $n$, its cardinality can be bounded as follows:
$$
\sharp \Theta \le (D+1) \sqrt{n}+1
\le (6^n + 1)\sqrt{n} + 1 \le 7^n;
$$
here we used that $D = \min_i(1/{\lambda}_i)$ by definition, and ${\lambda}_i \ge 6^{-n}$ by assumption.
The construction of $\Theta$ shows that any $\theta \in (1/2\mu,D)$ can be approximated
by some $\theta_0 \in \Theta$ in the sense that
$$
\theta \le \theta_0 \le \theta + \frac{1}{\sqrt{n}}.
$$
Note in particular that $\theta_0$ falls in the range $(1/2\mu, D+1)$, which
we handled in the previous step of the proof.
Recall that we need to bound the probability of the event
$$
\mathcal{E} \coloneqq
\Big\{ \min_{\theta \in (1/2\mu,D)} \dist(\theta W\star x,\mathbb{Z}^n) < 4\gamma \sqrt{n} \Big\}.
$$
Suppose this event occurs. Let $\theta$ be the multiplier that realizes the minimum
and consider an approximation $\theta_0 \in \Theta$ as above.
By triangle inequality, it satisfies
$$
\dist(\theta_0 W\star x,\mathbb{Z}^n)
< 4\gamma \sqrt{n} + \abs{\theta_0 - \theta} \abs{W \star x}.
$$
By construction, we have $\abs{\theta_0 - \theta} \le 1/\sqrt{n}$
and
$$
\abs{W \star x}
\le \norm{W}_\infty \abs{x}
\le 3 \gamma n;
$$
here we used that $\norm{W}_\infty \le \norm{a}_\infty \le |a| \le 3$ by definition
of $W$ and assumptions on $a$,
as well as $\abs{x} \le \gamma n$ by assumption on $x$.
Thus,
$$
\dist(\theta_0 W\star x,\mathbb{Z}^n)
\le 7 \gamma n.
$$
For each fixed $\theta_0$, the result of the previous step of the proof
shows that the probability of this event is at most $(C''\gamma)^{cn}$.
As we know, the number of possible choices of $\theta$ is at most $\sharp \Theta \le 7^n$.
Thus, the union bound gives
$$
{\mathbb P}(\mathcal{E}) \le 7^n (C''\gamma)^{cn} \le (C\gamma)^{cn}.
$$
This completes the proof of the lemma.
\end{proof}
\begin{remark}
Note that with our choice of parameters, $\Lambda'$ is non-empty, and therefore $W$ is well-defined in the Lemma above.
\end{remark}
From Lemma \ref{l: aux1} we deduce
\begin{lemma}\label{l: aux 03982}
For any $U\geq 1$, $b\in(0,1)$ and $\delta,\rho\in(0,1/2]$,
there exist $n_0=n_0(U,b,\delta,\rho)$,
$\gamma=\gamma(U,b,\delta,\rho)\in(0,1)$
and $u=u(b,\delta,\rho)\in(0,1/4)$ such that the following holds.
Let $n\geq n_0$, and let $J$ be a fixed subset of $[n]$ of cardinality at least $\delta n$.
Further, consider a random vector $X$ in $\mathbb R^n$
with independent components $X_i$ that satisfies
$$
\mathbb E|X|^2\leq \frac{1}{8}(1-b)\delta \gamma^2 n^2
\quad \text{and} \quad
\max_i \mathcal{L}(X_i,1) \leq b.
$$
Consider a set $\Lambda'$ described in Lemma~\ref{l: aux1}
and a random vector $W$ uniformly distributed on $\Lambda'$.
Then
$$
{\mathbb P}_W\big\{\RLCD^X_{\gamma\sqrt{n},u}(W)< \min_i 1/{\lambda}_i \big\}\leq U^{-n}.
$$
\end{lemma}
\begin{proof}
We apply a simple argument based on change of integration order, or a ``double-coun\-ting'' trick.
Without any loss of generality, we can assume that the random vector $X$ is uniformly distributed
on a finite set $\mathcal X:=\mathcal X_1\times\dots\times \mathcal X_n$, so that for any $x\in \mathcal X$, we have
$$
{\mathbb P}\{X=x\}=\frac{1}{\sharp \mathcal X}.
$$
Indeed, this follows from a simple fact that any multidimensional distribution $\zeta=(\zeta_1,\dots,\zeta_n)$
with independent components
can be approximated by a discrete distribution $\tau=(\tau_1,\dots,\tau_n)$
of the above form, so that
$$
\sup\limits_{\theta\in[0,6^n]}
\sup\limits_{v\in S^{n-1}}\big|\mathbb E \dist^2(\theta (v_1 \bar \zeta_1,\dots,v_n \bar \zeta_n),\mathbb{Z}^n)-
\mathbb E \dist^2(\theta (v_1 \bar \tau_1,\dots,v_n \bar \tau_n),\mathbb{Z}^n)
\big|
$$
is arbitrarily small. Then the definition of RLCD would imply that proving the required assertion for $\tau$
implies corresponding assertion for $\zeta$, perhaps with a different choice of $\gamma,u,n_0$.
Set $\mathcal X':=\{x\in\mathcal X:\;\mbox{$x$ satisfies \eqref{eq: x conditions}}\}$.
In view of our assumptions on $X$ (and assuming that $n$ is sufficiently large), we have
$$
{\mathbb P}\{X\in\mathcal X'\}\geq 1/4,
$$
while, in view of the assertion of Lemma~\ref{l: aux1} and summing over $x \in \mathcal{X}'$, we get
\begin{align}\label{eq: 3948724098217}
\sharp\big\{(x,w)\in\mathcal X'\times\Lambda':
&\min_{\theta \in (0,D)} \dist(\theta w\star x,\mathbb{Z}^n)^2 \ge \min(c|\theta w|^2/2,16\gamma^2n)\big\}\\
&\geq \big(1-(C\gamma)^{cn}\big)\,\sharp\mathcal X'\,\sharp\Lambda',\nonumber
\end{align}
where $D = \min_i 1/{\lambda}_i$.
This implies
\begin{multline*}
\sharp\big\{w\in \Lambda':
\sharp\{x\in\mathcal X': \min_{\theta \in (0,D)} \dist(\theta w\star x,\mathbb{Z}^n)^2 \ge \min(c|\theta w|^2/2,16\gamma^2n)\}
\geq \sharp\mathcal X'/4\big\}\\
\geq \big(1-2(C\gamma)^{cn}\big)\,\sharp\Lambda'
\end{multline*}
(indeed, if the last assertion were not true, we would get that the cardinality of the set in \eqref{eq: 3948724098217}
was bounded above by $(1-2(C\gamma)^{cn})\,\sharp\Lambda'\,\cdot\,\sharp\mathcal X'
+2(C\gamma)^{cn}\,\sharp\Lambda'\,\cdot\,\sharp\mathcal X'/4\leq (1-3(C\gamma)^{cn}/2)\sharp\mathcal X'\,\sharp\Lambda'$).
Back from counting to probabilities, we get from the last bound and the estimate $\sharp\mathcal X'/4\geq \sharp X/16$:
$$
\sharp\big\{w\in \Lambda':\, \min_{\theta \in (0,D)} \mathbb E_X\,\dist(\theta w\star X,\mathbb{Z}^n)^2
\ge \min(c|\theta w|^2/32,\gamma^2n) \big\}
\geq
\big(1-2(C\gamma)^{cn}\big)\,\sharp\Lambda'.
$$
This can be equivalently rewritten with $u \coloneqq c/32$ as
$$
\sharp\big\{w\in \Lambda':\;\RLCD^X_{\gamma\sqrt{n},u}(w) > D\big\}\geq \big(1-2(C\gamma)^{cn}\big)\,\sharp\Lambda',
$$
and the result follows by taking any $\gamma \in (0,1)$ satisfying $2(C\gamma)^{cn}\leq U^{-n}$.
\end{proof}
\medskip
\begin{proof}[Proof of Theorem~\ref{mainprop}]
Without loss of generality, $\mathbb E X=0$, so that $\Var(X)=\mathbb E|X|^2$.
We obtain the results as a combination of Lemmas~\ref{l: ball to rect} and \ref{l: aux 03982}.
To do so, note that $\Lambda$ can be covered by $2^{C_1n}$ sets of the type $\Lambda'$
(one for each paralellotope and a support set $J$). Then the probability measures on
$\Lambda$ and a given $\Lambda'$ are within $2^{C_1n}$ from each other. Thus the probability in the conclusion of Theorem~\ref{mainprop} is bounded by $2^{C_1n} U^{-n} \le (cU)^{-n}$.
It remains to re-define $U\to cU$ to get the result.
\end{proof}
\section{Proof of Theorem \ref{mainthm2}} \label{s: proof dist}
In this section, we split the Euclidean unit sphere $S^{n-1}$ into
{\it level sets} collecting (incompressible) unit vectors having comparable RLCD.
To show that with a high probability the normal vector does not belong to a level set with a small RLCD,
we consider a discrete approximating set whose cardinality is well controlled from above,
by using a combination of Theorem \ref{main-nets} and Theorem~\ref{mainprop}.
In view of the stability property of RLCD, the event that the normal vector has a small RLCD
is contained within the event that one of the vectors in the approximating set has a small RLCD.
We then apply the small ball probability estimates for individual vectors, combined with the union bound,
to show that the latter event has probability close to zero.
\medskip
For any $D\geq 1$, $\gamma,u\in(0,1)$, and an $m\times n$ random matrix $M$, define, as before,
$$S_D(M,\gamma,u):=\{v\in {{\mathbb S}^{n-1}}:\, \RLCD^M_{\gamma\sqrt{n},u}\in [D,2D]\}.$$
As the first step, we combine the approximation Theorem~\ref{main-nets} with Theorem~\ref{mainprop} to obtain
\begin{proposition}\label{p: discrete complete}
For arbitrary $b,\rho,\delta\in(0,1)$, $U\geq 1$ and $K\geq 1$ there exist
$n_{\ref{p: discrete complete}}=n_{\ref{p: discrete complete}}(b,\delta,\rho,U,K)$,
$u_{\ref{p: discrete complete}}=u_{\ref{p: discrete complete}}(b,\delta,\rho)
\in(0,u_{\ref{l: aux incomp rlcd}}(b,\delta,\rho))$, $\gamma_{\ref{p: discrete complete}}
=\gamma_{\ref{p: discrete complete}}(b,\delta,\rho,U,K)\in(0,1/2)$
with the following property.
Let $D\geq 1$ and $0<\varepsilon\leq 1/D$.
Let $n\geq n_{\ref{p: discrete complete}}$,
$m\geq 1$,
and let $M$ be an $m\times n$ matrix with independent entries $M_{ij}$ such that
$\mathcal{L}(M_{ij},1)\leq b$ for all $i,j$;
$$
\Var(M^\top e_i)\leq \frac{1}{8}\min\Big((1-b)\delta\gamma_{\ref{p: discrete complete}}^2 n^2,
\varepsilon^{-2} u_{\ref{p: discrete complete}}n\Big)
$$
for every $i\leq m$, and
$$
\mathbb E\|M\|_\mathrm{HS}^2\leq K n^2.
$$
Then there is a non-random set $\Lambda\subset\mathbb R^n$ of cardinality at most $(\varepsilon U)^{-n}$
having the following properties:
\begin{itemize}
\item For any $y\in \Lambda$, we have $3/2\geq |y|\geq 1/2$;
\item For any $y\in \Lambda$, $\RLCD^M_{\gamma_{\ref{p: discrete complete}} \sqrt{n}/2,u_{\ref{p: discrete complete}}/4}(y)\geq D$ and
$\RLCD^M_{2\gamma_{\ref{p: discrete complete}} \sqrt{n},4u_{\ref{p: discrete complete}}}(y)\leq 2D$;
\item With probability at least $1-e^{-n}$, for any $x\in S_D(M,\gamma_{\ref{p: discrete complete}},
u_{\ref{p: discrete complete}})\cap \Incomp(\delta,\rho)$ there is
$y\in \Lambda$ with $\|x-y\|_\infty\leq \varepsilon/\sqrt{n}$ and $|M (x-y)|\leq \varepsilon\sqrt{n}$.
\end{itemize}
\end{proposition}
\begin{proof}
Set $\kappa:=5$, and let $C_\kappa>0$ be the constant from Remark~\ref{rem: card of Lambda}.
Let $U\geq 1$, $U':=100\sqrt{2K}U C_\kappa/\rho$, and set
$$n_{\ref{p: discrete complete}}:=n_0(U',b,\delta,\rho/2),\;
\gamma=\gamma_{\ref{p: discrete complete}}:=\gamma(U',b,\delta,\rho/2),\;
u=u_{\ref{p: discrete complete}}:=u(b,\delta,\rho/2)\in(0,\frac{1}{4}),$$
where the functions $n_0(\cdot),\gamma(\cdot),u(\cdot)$
are taken from Theorem~\ref{mainprop}.
Finally, set
$$\varepsilon':=\frac{\rho\varepsilon}{100\sqrt{2\max(K,1)}}\in(0,0.01),$$
and let $\Lambda^\kappa(\varepsilon')$ be as in Definition~\ref{def: disc part 2}.
Let $\Lambda$ be a subset of all vectors $y\in \Lambda^\kappa(\varepsilon')$
such that
$$\RLCD^M_{\gamma \sqrt{n}/2,u/4}(y)\geq D\quad\mbox{ and }\quad
\RLCD^M_{2\gamma \sqrt{n},4u}(y)\leq 2D,$$
and, such that the $\ell_\infty$--distance
of $y$ to $\Incomp(\delta,\rho)$ is at most $\varepsilon'/\sqrt{n}$. Note that the last condition implies that for any $y\in\Lambda$,
$\sharp\{i\leq n:\;|y_i|\geq \rho/(2\sqrt{n})\}\geq \delta n$, see the argument in Lemma 3.4 from \cite{RudVer-square}.
By our choice of $\varepsilon'$ and the condition on the matrix, we have
$$
(\varepsilon')^2 \Var(M^\top e_i)\leq \frac{1}{8}\frac{\gamma^2 n^2}{D^2};\quad
(\varepsilon')^2 \Var(M^\top e_i)\leq \frac{1}{8} un.
$$
Then, according to Theorem~\ref{main-nets}, with probability at least $1-(5/\sqrt{2})^{-2n}$ for any incompressible vector $x\in S_D(M,\gamma,u)$
there is a vector $y\in \Lambda$ such that $\|x-y\|_\infty \leq \varepsilon'/\sqrt{n}$ and
$|M(x-y)|\leq \sqrt{2}\varepsilon'\sqrt{K}\sqrt{n}\leq \varepsilon\sqrt{n}$.
It remains to estimate the cardinality of $\Lambda$.
We recall that
$$
\Lambda^{\kappa}(\varepsilon')=\bigcup_{\alpha\in\mathcal{F}} \Lambda_{\alpha} (\varepsilon'),
$$
where the collection $\mathcal F$ of parameters $(\alpha_1,\dots,\alpha_n)\in(0,1]^n$ is given by Lemma~\ref{netsonnets}.
Fix for a moment any $(\alpha_1,\dots,\alpha_n)\in\mathcal F$, and set
$\lambda_i:=\alpha_i\varepsilon'\in(0,0.01]$, $i\leq n$.
Observe that $1/\lambda_i\geq 1/\varepsilon'> 2/\varepsilon\geq 2D$, $i\leq n$.
Hence, we can apply Theorem~\ref{mainprop} to obtain
$$
\sharp(\Lambda\cap \Lambda_{\alpha} (\varepsilon'))\leq \sharp\Lambda_{\alpha} (\varepsilon')\,(U')^{-n}.
$$
Taking the union over all $(\alpha_1,\dots,\alpha_n)\in\mathcal F$, we then get
$$
\sharp\Lambda\leq (U')^{-n}\sum\limits_{\alpha\in\mathcal F}\sharp\Lambda_{\alpha} (\varepsilon')
\leq (\varepsilon U)^{-n},
$$
where at the last step we used our definition of $U'$.
\end{proof}
Next, we combine the discrete approximation set introduced above, with the small ball probability of Lemma~\ref{smallball}:
\begin{proposition}\label{pevelsets}
For any $b,\rho,\delta\in(0,1)$ and $K\geq 1$ there are $n_{\ref{pevelsets}}=n_{\ref{pevelsets}}(b,\delta,\rho,K)$,
$u_{\ref{pevelsets}}=u_{\ref{pevelsets}}(b,\delta,\rho)\in(0,u_{\ref{l: aux incomp rlcd}}(b,\delta,\rho))$,
$\gamma_{\ref{pevelsets}}=\gamma_{\ref{pevelsets}}(b,\delta,\rho,K)\in(0,1/2)$
and $\gamma'_{\ref{pevelsets}}=\gamma'_{\ref{pevelsets}}(b,\delta,\rho,K)$ with the following property.
Let $n\geq n_{\ref{pevelsets}}$, $e^2\leq D\leq D_0\leq e^{\gamma'_{\ref{pevelsets}} n}$,
$0\leq k\leq n/\ln D_0$, $m:=n-k$,
and let $M$ be an $m\times n$ random matrix with independent entries $M_{ij}$ such that
$\mathcal{L}(M_{ij},1)\leq b$ for all $i,j$;
\begin{equation}\label{eq: aux 498750983275}
\Var(M^i)\leq \frac{1}{64}\min\Big((1-b)\delta\gamma_{\ref{pevelsets}}^2 n^2,D_0^2 u_{\ref{pevelsets}}n\Big)
\end{equation}
for every $i \le m$,
and
$$
\mathbb E\|M\|_\mathrm{HS}^2\leq K n^2.
$$
Let $M^{(1)}$ be the matrix obtained from $M$ by removing the first row. Then
$$
{\mathbb P}\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)\cap S_D(M,\gamma_{\ref{pevelsets}},u_{\ref{pevelsets}})$
s.t.\ $\RLCD^{M^{(1)}}_{\gamma_{\ref{pevelsets}}\sqrt{n},u_{\ref{pevelsets}}}(x)\geq D_0$, $M^{(1)}x=0$}\big\}
\leq 2e^{-n}.
$$
\end{proposition}
\begin{proof}
First, we should carefully define the parameters.
We choose $u:=u_{\ref{p: discrete complete}}(b,\delta,\rho)$.
Next, set $U:=2e^3 C_{\ref{smallball}}^2$,
where $C_{\ref{smallball}}$ is taken from Lemma~\ref{smallball} with parameters
$c_0:=1/2$ and $u/4$, and we assume without loss of generality that $C_{\ref{smallball}}\geq 1$.
Finally, take $\gamma:=\gamma_{\ref{p: discrete complete}}(b,\delta,\rho,U,K)$,
$\gamma':=\widetilde c_{\ref{smallball}} \gamma^2/4\leq 1$.
Let $e^2\leq D\leq D_0\leq e^{\gamma' n}$, and let random matrix $M$ satisfy the assumptions of the proposition.
Let $\Lambda$ be the set defined in Proposition~\ref{p: discrete complete} with $\varepsilon:=1/D_0$.
Set
$${\mathcal E}_D:=\big\{
\mbox{$\exists$ $x\in \Incomp(\delta,\rho)\cap S_D(M,\gamma,u)$
s.t.\ $\RLCD^{M^{(1)}}_{\gamma\sqrt{n},u}(x)\geq D_0$, $M^{(1)}x=0$}\big\}.$$
Note that whenever $x$ and $y$ are two vectors in $\mathbb R^n$ with
$\RLCD^{M^{(1)}}_{\gamma\sqrt{n},u}(x)\geq D_0$ and $\|x-y\|_\infty\leq \frac{1}{D_0\sqrt{n}}$,
then necessarily $\RLCD^{M^{(1)}}_{\gamma\sqrt{n}/2,u/4}(y)\geq D_0$
(as follows from Lemma~\ref{stable-rlcd}).
Hence, applying Proposition~\ref{p: discrete complete}, we get
\begin{align*}
{\mathbb P}({\mathcal E}_D)
&\leq e^{-n}+{\mathbb P}\big\{\mbox{There is $y\in \Lambda$ with $|M^{(1)} y|\leq \sqrt{n}/D_0$ and
$\RLCD^{M^{(1)}}_{\gamma\sqrt{n}/2,u/4}(y)\geq D_0$}\big\}\\
&\leq e^{-n}+\sharp\Lambda\,\,\sup\limits_{y}{\mathbb P}\big\{|M^{(1)} y|\leq \sqrt{n}/D_0\big\}\\
&\leq e^{-n}+(D_0/U)^n\,\,\sup\limits_{y}{\mathbb P}\big\{|M^{(1)} y|\leq \sqrt{n}/D_0\big\},
\end{align*}
where the supremum is taken over all vectors $y\in\frac{3}{2}B_2^n\setminus \frac{1}{2}B_2^n$
with $\RLCD^{M^{(1)}}_{\gamma\sqrt{n}/2,u/4}(y)\geq D_0$.
Fix any $y$ satisfying the above conditions.
Set $\widetilde\varepsilon:=2C_{\ref{smallball}}/D_0$ and observe that, by our conditions on $D_0$,
$$
\widetilde\varepsilon\geq C_{\ref{smallball}}\exp(-\widetilde c_{\ref{smallball}} \gamma^2 n/4)
+ C_{\ref{smallball}}/\RLCD^{M^{(1)}}_{\gamma\sqrt{n}/2,u/4}(y).
$$
Applying Lemma~\ref{smallball}, we then obtain
\begin{align*}
{\mathbb P}\{|M^{(1)} y|\leq \sqrt{n}/D_0\}&\leq{\mathbb P}\{|M^{(1)} y|\leq 2\sqrt{m-1}/D_0\}\\
&\leq{\mathbb P}\{|M^{(1)} y|\leq \widetilde\varepsilon\sqrt{m-1}\}\leq
(C_{\ref{smallball}}\widetilde\varepsilon)^{m-1}.
\end{align*}
Taking the supremum over all admissible $y$, we then get
$$
{\mathbb P}({\mathcal E}_D)\leq e^{-n}+(D_0/U)^n\,(C_{\ref{smallball}}\widetilde\varepsilon)^{m-1}\leq e^{-n}+
D_0^{n-m+1}U^{-n}\big(2C_{\ref{smallball}}^2\big)^n.
$$
The result follows by the choice of $U$ and the condition on $m$.
\end{proof}
Our proof of Theorem~\ref{mainthm2}, in the case $\Var(A_j)=\Theta(n)$, $j=1,2,\dots,n$,
is a straightforward application of Proposition~\ref{pevelsets} (taking a dyadic sequence of level sets),
together with results of \cite{Liv} on invertibility over compressible vectors.
The fact that in our model some columns may have variances much greater than $n$
adds some complexity to the proof because the relation \eqref{eq: aux 498750983275}
for such columns may hold true only for ``large enough'' $D_0$ leaving a gap in the treatment
of small values of the parameter.
We deal with this issue in the statement below by carefully splitting the event in question
into subevents and invoking Lemma~\ref{l: aux incomp rlcd}
that allows to deterministically bound RLCD in terms of the variance.
\begin{proposition}\label{p: many cases}
Let $b,\delta,\rho\in(0,1)$ and $K\geq 1$ be parameters,
and let $u_{\ref{pevelsets}}$, $\gamma_{\ref{pevelsets}}$ be taken
from Proposition~\ref{pevelsets}.
Then there are $n_{\ref{p: many cases}}(b,\delta,\rho,K)$ and $\gamma'_{\ref{p: many cases}}(b,\delta,\rho,K)$ with the following property.
Let $n\geq n_{\ref{p: many cases}}$, let $n\times n$ matrix $A$ be as in the statement of Theorem~\ref{mainthm2},
and let $j\leq n$ be such that
$$
\Var(A_j)\leq \min\Big(h_{\ref{l: aux incomp rlcd}}^2 e^{-4} n^2,\frac{1}{64}(1-b)\delta\gamma_{\ref{pevelsets}}^2 n^2\Big),
$$
where $h_{\ref{l: aux incomp rlcd}}$ is taken from Lemma~\ref{l: aux incomp rlcd}.
Then
$$
{\mathbb P}\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)$ orth.\ to $A_i$, $i\neq j$, with
$\RLCD^{A_j}_{\gamma_{\ref{pevelsets}}\sqrt{n},u_{\ref{pevelsets}}}(x)\leq e^{\gamma'_{\ref{p: many cases}}n}$}\big\}
\leq 2^{-n/2}.
$$
\end{proposition}
\begin{proof}
We will assume that $n$ is large, and that $\gamma'>0$ is a small parameter whose
value can be recovered from the proof below.
Without loss of generality, $j=1$.
Let $A'$ be the submatrix of $A$ composed of all columns $A_i$
satisfying
$$
\Var(A_i)\leq \min\Big(h_{\ref{l: aux incomp rlcd}}^2 e^{-4} n^2,
\frac{1}{64}(1-b)\delta\gamma_{\ref{pevelsets}}^2 n^2\Big).
$$
We note that the number of columns of $A'$ is at least $n-K/\min\big(h_{\ref{l: aux incomp rlcd}}^2 e^{-4},
\frac{1}{64}(1-b)\delta\gamma_{\ref{pevelsets}}^2\big)$.
Further, let $M$ be the transpose of $A'$,
and denote by $W$ the submatrix of $M^{(1)}$ formed by removing rows with variances at least $n^{9/8}$.
The proof of the statement is reduced to estimating probability of the event
$$
{\mathcal E}':=\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)$ with $M^{(1)}x=0$ and
$\RLCD^{A_1}_{\gamma_{\ref{pevelsets}}\sqrt{n},u_{\ref{pevelsets}}}(x)\leq e^{\gamma'n}$}\big\}.
$$
We can write
\begin{align*}
{\mathbb P}({\mathcal E}')
\leq
&\sum\limits_{\log_2 n-1\leq \ell\leq \gamma'n\log_2 e}
{\mathbb P}\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)\cap S_{2^\ell}(M,\gamma_{\ref{pevelsets}},u_{\ref{pevelsets}})$ with $M^{(1)}x=0$}\big\}
\\
&+
{\mathbb P}\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)$ with $M^{(1)}x=0$ and $\RLCD^M_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)<n$}\big\}.
\end{align*}
The first sum can be estimated directly by applying Proposition~\ref{pevelsets}
with $D_0:=D:=2^\ell$, $\log_2 n-1\leq \ell\leq \gamma' n\log_2 e$
(note that the relation \eqref{eq: aux 498750983275} is fulfilled for such $D$ for all rows of $M$,
and that the proposition can be applied as long as $K/\min\big
(h_{\ref{l: aux incomp rlcd}}^2 e^{-4},\frac{1}{64}(1-b)\delta\gamma_{\ref{pevelsets}}^2\big)\leq
1/\gamma'$).
Further, the condition that $\RLCD^M_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)<n$ implies that either $\RLCD^W_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)<n$ or
$\RLCD^W_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)\geq n$ and $\RLCD^{M^q}_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)< n$ for some row $M^q$ of $M$.
Hence, we get
\begin{align*}
{\mathbb P}({\mathcal E}')
\leq 2n\cdot 2e^{-n}
&+\sum_{q}{\mathbb P}\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)$ with $Wx=0$ and $\RLCD^W_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)\geq n$}\\
&\hspace{2cm}\mbox{and $\RLCD^{M^q}_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)< n$}\big\}\\
&+
{\mathbb P}\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)$ with $Wx=0$ and $\RLCD^W_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)<n$}\big\}.
\end{align*}
To estimate the sum, we apply Lemma~\ref{l: aux incomp rlcd} which, together with our restrictions on the variances,
allows to deterministically bound the RLCD with respect to $M^q$ by $e^2$.
Thus, we get
\begin{align*}
&{\mathbb P}\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)$ with $Wx=0$ and $\RLCD^W_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)\geq n$}\\
&\hspace{2cm}\mbox{and $\RLCD^{M^q}_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)< n$}\big\}\\
&={\mathbb P}\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)$ with $Wx=0$ and $\RLCD^W_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)\geq n$}\\
&\hspace{2cm}\mbox{and $e^2\leq \RLCD^{M^q}_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)< n$}\big\}.
\end{align*}
Splitting the interval $[e^2,n]$ into dyadic subintervals and applying Proposition~\ref{pevelsets} with $D_0:=n$
and for the matrix formed by concatenating $W$ and $M^q$,
we get an upper bound $2e^{-n}\log_2 n$ for the probability.
In order to estimate probability of the event
$$
\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)$ with $Wx=0$ and $\RLCD^W_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(x)<n$}\big\},
$$
we apply Lemma~\ref{l: aux incomp rlcd}; this time the definition of $W$ implies that
$\RLCD$ with respect to each row is deterministically bounded from below by $n^{3/8}$, for a sufficiently large $n$.
Again, splitting of the interval $[n^{3/8},n]$ into dyadic subintervals reduces the question to estimating events
of the form
$$
\big\{\mbox{$\exists$ $x\in \Incomp(\delta,\rho)\cap S_D(W,\gamma_{\ref{pevelsets}},u_{\ref{pevelsets}})$ with $Wx=0$}\big\}
$$
for some $D\in [n^{3/8},n]$. Taking $D_0:=D$, one can see that the condition \eqref{eq: aux 498750983275}
is fulfilled for all rows of $W$, and that the difference between the number of columns and rows of $W$
is clearly less than $n/\ln D_0$. Thus, Proposition~\ref{pevelsets} is applicable.
Summarizing, we get ${\mathbb P}({\mathcal E}')\leq C' ne^{-n}\ln n$
for a universal constant $C'>0$. The result follows for all sufficiently large $n$.
\end{proof}
Now, we are in position to prove Theorem~\ref{mainthm2}.
\begin{proof}[Proof of Theorem~\ref{mainthm2}]
We will assume that $n$ is large.
We start by recording a property of $A$ which follows immediately from Lemma 2.1 (that is, \cite[Lemma~5.3]{Liv}):
For any $j\leq n$, with probability at least $1-e^{-c_1 n}$ any unit vector
orthogonal to $\{A_i,\; i\neq j\}$,
is $(\delta,\rho)$--incompressible
for some $\delta,\rho\in(0,1)$ depending only on $b,K$ (here, $c_1\in(0,1)$ depends only on $b,K$).
Indeed, let $j\leq n$, let $B$ be the $n\times (n-1)$ matrix formed from $A$
by removing $A_j$, and define $M:=B^\mathsf{T}.$ Then
$$
\Pr{\exists x\in\Comp(\delta,\rho)\mbox{ orthogonal to }H_j}
\leq \Pr{\inf_{x\in \Comp(\delta,\rho)} |Mx|=0}
\leq e^{-c_1 n},
$$
where in the last passage Lemma 2.1 (that is, \cite[Lemma~5.3]{Liv}) was used.
Set
$$
r:=\min\Big(h_{\ref{l: aux incomp rlcd}}^2 e^{-4},
\frac{1}{64}(1-b)\delta\gamma_{\ref{pevelsets}}^2\Big),
$$
where $h_{\ref{l: aux incomp rlcd}}$ and $\gamma_{\ref{pevelsets}}$ are defined in respective lemmas with the parameters $b,K,\delta,\rho$.
Pick any index $j\leq n$ such that $\Var(A_j)\leq rn^2$, and let $v$
be a random unit vector orthogonal to $H_j$ and measurable with respect to the sigma-field generated by $H_j$.
Applying Proposition~\ref{p: many cases} together with the above observation, we get
$$
\mbox{$v$ is $(\delta,\rho)$--incompressible and
$\RLCD^{A_j}_{\gamma_{\ref{pevelsets}}\sqrt{n},
u_{\ref{pevelsets}}}(v) \geq e^{\gamma'_{\ref{p: many cases}} n}$}
$$
with probability at least $1-e^{c_1 n}-2^{-n/2}$.
Application of Lemma~\ref{smallball-1} finishes the proof.
\end{proof}
\begin{remark}
In our proof, the Randomized Least Common Denominator acts like a mediator
in the relationship between anticoncentration properties of matrix-vector products and
cardinalities of corresponding discretizations (nets),
following the ideas developed in \cite{RudVer-square}.
A crucial element of our argument is the fact that RLCD is stable with respect to small perturbations of the vector, which we quantify in Lemma~\ref{stable-rlcd}.
An alternative approach recently considered in \cite{TikhErd}
is based on
directly estimating the concentration function for ``typical'' points on a multidimensional lattice.
The argument of \cite{TikhErd} uses as an important step certain stability properties
of the L\'evy concentration function and of small ball probability estimates
for linear combinations of Bernoulli random variables.
However, in the general (non-Bernoulli) setting, and with different distributions of entries
of the matrix, obtaining satisfactory
stability properties similar to those in \cite{TikhErd} seems to be a very non-trivial problem, in the situation when the approximation is done by a random vector. We note here that in our net construction the approximating vector is, indeed, random, and depends on the realization of the matrix.
On a technical level, since RLCD is a structural (geometric) property, its stability follows from relatively
simple computations, while the L\'evy concentration function is much more difficult to control;
in particular, the Esseen lemma provides only an upper bound for the concentration function,
hence cannot be relied on when studying its stability.
\end{remark}
\section{Proof of the Theorem \ref{mainthm1}} \label{s: proof main}
In this section we formally derive Theorem \ref{mainthm1} from Theorem \ref{mainthm2},
using a modification of the ``invertibility via distance'' lemma from \cite{RudVer-square}.
\begin{lemma}[Invertibility via distance]\label{invdist}
Let $A$ be any $n\times n$ random matrix. Fix a pair of parameters $\delta, \rho\in (0,\frac{1}{2})$, and assume that $n\geq 4/\delta$. Then, for any $\varepsilon>0,$
$${\mathbb P}\left\{\inf_{x\in \Incomp(\delta, \rho)} |Ax|\leq \varepsilon \frac{\rho}{\sqrt{n}}\right\}
\leq \frac{4}{\delta n} \inf\limits_{\substack{I\subset[n],\\ \sharp I=n-\lfloor \delta n/2\rfloor}}\sum_{j\in I}
{\mathbb P}\{\dist(A_j, H_j)\leq \varepsilon\},$$
where $H_j$ denotes the subspace spanned by all the columns of $A$ except for $A_j.$
\end{lemma}
\begin{proof}
Fix any $I\subset[n]$ with $\sharp I=n-\lfloor \delta n/2\rfloor$,
and consider event
$$
{\mathcal E}:=\left\{\inf_{x\in \Incomp(\delta, \rho)} |Ax|\leq \varepsilon \frac{\rho}{\sqrt{n}}\right\}.
$$
Fix any realization of the matrix $A$ such that the event holds, i.e.\ there exists a vector
$x\in \Incomp(\delta, \rho)$ with $|Ax|\leq \varepsilon \frac{\rho}{\sqrt{n}}$.
In view of the definition of the set $\Incomp(\delta, \rho)$,
there is a subset $J_x\subset[n]$ of cardinality $\lfloor \delta n\rfloor$ such that
$|x_i|\geq \rho/\sqrt{n}$ for all $i\in J_x$, whence
$$
\dist(A_i,H_i)\leq |x_i|^{-1}\,|Ax|\leq \varepsilon,\quad i\in J_x.
$$
Note that $J_x\cap I$ has cardinality at least $\lfloor \delta n\rfloor-\lfloor \delta n/2\rfloor\geq \delta n/4$.
Thus,
$$
{\mathcal E}\subset\big\{\sharp\{i\in I:\,\dist(A_i,H_i)\leq \varepsilon\}\geq \delta n/4\big\}
$$
It remains to note that
$$
{\mathbb P}\big\{\sharp\{i\in I:\,\dist(A_i,H_i)\leq \varepsilon\}\geq \delta n/4\big\}
\leq \frac{4}{\delta n}\mathbb E\,\sharp\{i\in I:\,\dist(A_i,H_i)\leq \varepsilon\}.
$$
\end{proof}
\textbf{Proof of Theorem \ref{mainthm1}.} The theorem follows from Lemma \ref{comp-final} (that is, Lemma 5.3 from \cite{Liv}), Lemma \ref{invdist} and Theorem \ref{mainthm2},
by taking $I_0:=\{i\in[n]:\;\mathbb E|A_i|^2\leq rn^2\}$ and noting that, in view of the assumption $\mathbb E\|A\|_\mathrm{HS}^2\leq Kn^2$,
we have $\sharp I_0=n-K/r\geq n-\lfloor \delta n/2\rfloor$ for all sufficiently large $n$, so that for all large enough $n$
$$
{\mathbb P}\left\{\inf_{x\in \Incomp(\delta, \rho)} |Ax|\leq \varepsilon \frac{\rho}{\sqrt{n}}\right\}
\leq \frac{4}{\delta n} \sum_{j\in I_0}
{\mathbb P}\{\dist(A_j, H_j)\leq \varepsilon\}.
$$
$\square
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Amatepec es una congregación del municipio de Ilamatlán ubicado en la región de la Huasteca Baja del estado mexicano de Veracruz.
Geografía
La localidad de Amatepec se sitúa en las coordenadas geográficas , a una elevación de 751 metros sobre el nivel del mar.
Demografía
Según el Conteo de Población y Vivienda 2020, efectuado por el Instituto Nacional de Estadística y Geografía, la localidad de Amatepec tiene 734 habitantes, de los cuales 348 son del sexo masculino y 386 del sexo femenino. La tasa de fecundidad es de 2.72 hijos por mujer y tiene 226 viviendas particulares habitadas.
Véase también
Huasteca Baja
Referencias
Localidades del municipio de Ilamatlán
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Q: How to add to array from list and return array I am at an internship and have a problem I have tried to solve for three days, please help.
My problem: I have a list of data and it has to me converted into an array.
TemplateSummary[] TransformTemplates(IEnumerable<Template> templates)
{
foreach (Template template in templates)
{
template.ToArray();
}
return ();
}
This does not work and I have tried one milion different things, I just can't figure it out.
Template:
In Template:
public class Template:IEntity
{
public virtual int Id { get; set; }
public virtual string TemplateName { get; set; }
public virtual string Content { get; set; }
}
In ExempelAccess:
public IEnumerable<Template> ListAllTemplates()
{
this.session.Query<Template>().ToList();
return ListAllTemplates();
}
public int Create(string templateName, string content)
{
using (var tx = session.BeginTransaction())
{
var template = new Template { Content = content, TemplateName = templateName };
session.Save(template);
tx.Commit();
return template.Id;
}
}
In ExempelEngine:
TemplateSummary[] TransformTemplates(IEnumerable<Template> templates)
{
foreach (Template template in templates)
{
template.ToArray();
}
return ();
}
Templatesummary:
In IExempelManager:
public interface IExempelManager
{
[OperationContract]
TemplateSummary[] ListTemplates();
[OperationContract]
void Create(string templateName, string content);
}
[DataContract]
public class TemplateSummary
{
[DataMember]
public int Id { get; set; }
[DataMember]
public string TemplateName { get; set; }
[DataMember]
public string Content { get; set; }
}
In ExempelManager:
public TemplateSummary[] ListTemplates()
{
return ListTemplates();
}
In ClientFactory:
public TemplateSummary[] ListTemplates()
{
return Channel.ListTemplates();
}
In ExempelViewModel
public ObservableCollection<TemplateSummary> TemplateSummaryResult { get; set; }
In ExempelEngine:
TemplateSummary[] TransformTemplates(IEnumerable<Template> templates)
{
foreach (Template template in templates)
{
template.ToArray();
}
return ();
}
public TemplateSummary[] ListTemplates()
{
return ();
}
These are all the references that are in the solution.
A: According to this comment you want to get this method
TemplateSummary[] TransformTemplates(IEnumerable<Template> templates)
{
return templates.Select(x => new TemplateSummary
{
Id = x.Id,
TemplateName = x.TemplateName,
Content = x.Content
}).ToArray()
}
Do not forget to include using System.Linq; at the file head.
A: If I can assume that there is a method that summarizes a template - TemplateSummary SummarizeTemplate(Template template) - then this is all you need:
TemplateSummary[] TransformTemplates(IEnumerable<Template> templates)
{
return templates.Select(t => SummarizeTemplate(t)).ToArray();
}
For a more detailed answer you need to explain how SummarizeTemplate would work.
This is what you need:
TemplateSummary[] TransformTemplates(IEnumerable<Template> templates)
{
return templates.Select(t => new TemplateSummary()
{
Id = t.Id,
TemplateName = t.TemplateName,
Content = t.Content,
}).ToArray();
}
A: Assuming that the ToArray on Template returns TemplateSummary[] the problem is resolvable using SelectMany
TemplateSummary[] TransformTemplates(IEnumerable<Template> templates)
{
return templates.SelectMany(i => i.ToArray()).ToArray();
}
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 7,864
|
Q: unordered map behavior in cpp I have this 2 codes,
Code 1
#include<bits/stdc++.h>
using namespace std;
int main()
{
unordered_map<int,int>m;
int nums[]={1,1,3,4,5};
for(auto x:nums)
{
m[x]++;
}
for(auto x:m)
{
cout<<x.first<<" "<<m[x.first]<<endl;
}
return 0;
}
Code 2
#include<bits/stdc++.h>
using namespace std;
int main()
{
unordered_map<int,int>m;
int nums[]={1,1,3,4,5}; int k=2;
for(auto x:nums)
{
m[x]++;
}
for(auto x:m)
{
cout<<x.first<<" "<<m[x.first+k]<<endl;
}
return 0;
}
The output of the first code is
5 1
4 1
1 2
3 1
But the output of the second code is
5 0
4 0
5 0
7 0
I am not getting why the output of the 2nd code is like this, shouldn't it be like
5 0 //5 is the index and m[5+2]=0 hence 0
4 0
1 1 //since m[1+2]=1
3 1
......................................................................................................................................................................................................................
A: When you do m[x.first + k], you're modifying m when x.first + k is not already in m. According to en.cppreference.com:
If an insertion occurs and results in a rehashing of the container, all iterators are invalidated. [...]
If this occurs, you're invalidating iterators while iterating the map since for (auto x: m) is just a hidden way of iterating using iterators. Thus, your code has undefined behavior (you're using invalidated iterators), hence what you are seeing (and also why other people might see something different).
A: operator[], on a map or unordered_map, has the potential to change the container, because (as you're relying on) when you call it with a key which isn't already present, it immediately adds it with a default-constructed value. As such, it's not safe to evaluate m[foo] while looping over m unless you're certain that foo is already present.
I'm not sure what this code is trying to do, but assuming it's just trying to print 0 when the value for the shifted key isn't present, you could do unordered_map::find directly on that shifted key and check whether the returned iterator is valid, and if not, print 0 instead.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,595
|
Q: Im using gaesessions with flex in a gae web application. but the cookie delete itself between ajax request. why? Im using gaesessions with flex in a gae web application. but the cookie delete itself between ajax request. why?
A: just fixed it. it was a problem with the appengine_config.py file where the cookie key was changing at every request. xD
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,610
|
Susan Collins' decision to give Maine's support to the border wall is disheartening and seems yet another sign of democracies escalating erosion under the autocratic bullying of the Trump administration. I spent years working along the vastly beautiful Texas/Mexico border and there is a reason Representatives from this region are against an environmentally devastating divide that will not serve its own purpose. This includes Will Hurd, a Republican and former CIA operations officer in Afghanistan who represents 820 miles of border communities from El Paso to San Antonio. Hurd has referred to walls as the most expensive and least effective way to secure the border. We need 5.7 billion dollars for infrastructure that will actually keep us safer; new roads, new bridges, updated water systems, and better schools, not a massive dysfunctional symbol of presidential ego fueled by racism.
I agree about punishing businesses looking only at the bottom line. However, it's the responsibility of Congress to establish the rules through legislation. Good luck convincing Pingree and Golden to go against their leaders - using the term very loosely - who are counting every vote they can get in the horde crossing the border.
Scott, I didn't mention " emergency ", I questioned you saying " there is no danger "
Scott Erb, it is beyond absurd for you to say that there is "no danger". Maybe you could go tell that to the wife and 6 month old boy of officer Ronil Singh who was shot and killed the day after Christmas by criminal illegal alien Gustavo Perez Arriaga after Singh pulled him over for drunk driving. So to break that down for you, he comes here ILLEGALLY, is driving drunk, and shoots/kills a police officer. His wife and little boy are now permanently separated from their husband and father but somehow you way too many others think there is no danger. Sadly, this is just one story of thousands that happen every year in this country but we usually never hear about it because it doesn't fit the PC narrative. Unbelievable.
RRJ, life has danger everywhere - in large cities, in desolate areas. But there is no emergency or danger due to the lack fo a wall. It is weak, feeble thinking to look at any crime committed by someone who entered the country and go crazy. The crime rate is LOWER for illegal aliens than American citizens, so you have just as much danger from your fellow countryfolk as from the southern border. Your approach reminds me of how Nazis would tell stories about Jewish criminals in order to prove that Germans should fear Jews. Yours has the same morality.
Yes, Jews in Germany often committed crimes. So did Germans. When the Nazis would point to Jewish crimes they'd say "this proves Jews are a danger!"
Scott, sometimes I just have to shake my head at your comments.
RRJ gives an example of an illegal immigrant taking a life, that could have possibly been prevented by a security wall. Maybe not, but we don't know for certain.
You turn around and call this commenter " weak and feeble thinking " and compare building a security wall to rounding up millions of Jews and murdering them.
Scott, you're right, we've already got enough criminals in our country. Even more reason to have stronger border security (which should include physical barriers where it makes sense) to help prevent even more criminals from entering illegally. I'm all for immigration, it just needs to be done legally and we need to make sure we are properly vetting the people coming in.
BTW, your Nazi talk is ridiculous and you should know better. They were putting Jews into ovens. All we're asking is for people to follow the legal process of entering our country and not break our laws. That's not too much to ask and a far cry from a concentration camp and being burned to death.
Funny how lots of the same people who say illegal immigrants killing American citizens is a small problem not worthy of laws targeting it will also support restrictions on gun owners because" even if they only save one life it's worth it"
Obama built 134 miles of wall. Was that also a massive dysfunctional symbol of presidential ego fueled by racism?
Pelosi now says a wall is immoral, although she was sort-of OK with it just a couple months agp. If she really believes that, where is the House bill to tear down Obama's immoral wall?
That's a great question Frostproof. I would also like to know the answer to that and while at it maybe tell us what happened to the rest of the billions of dollars they had to use. That kind of money should have done the entire border not just 134 miles.
BY the way Eddie you also make a GREAT point. It's called hypocrisy.
if you all want a wall so desperately, get Mexico to pay for it. Trump promised this hundreds of times as well.
Yes I agree…Lets go back to the Letter!
The President, El Trumpster, wants $5.7 Billion (haven't checked on the latest figures, etc., I could be off by a few mil), for the "Wall" Hmm? The Patriots are scheduled to play the Super Bowl in a freshly built state of the art, with an amazing sky view SUUper-SUUper Stadium for…? Wait for it! —-a cost to build, a Kool 1.7 to 2.5 Billion, (depending on who you ask with all the add ins and contributions/ funding from others, not just the out-of-pocket money from Mr. Home Depot). So, others have argued already, to put in perspective what the cost for the Wall relative to the National budget is, and realized cost/benefits from having a first line of defense barrier/the "WALL", so I wont dwell on that tiny fraction/cost. Again, wait for it! Lets see...Obby "The One" Cannobee's, Grand plan to save the Country was to spend $832 billion in the infamous "esssstimulus package" which after many trials and tribulations, "Fun-da-mentally" did not amount to much, in other words all cost/spending and no reciepts. Ok, some roads and bridges were built and a few jobs were saved, (BTW, thanks to his own parties' failures and "Job-Lock", the record high-unemployment needing saving). OH…and his other famous savior moment, that of the, resurrection of GM! So in the end, as I stated not much of a cost/benefit was achieved. Continuing with rising unemployment, increased SNAP/Welfare costs and lost homes from failed government programs , etc. just to name a few of the losses which when you add all these factoids in a real-life balance sheet, yielded miner gains, in fact losses. However, this great lack of achievement is rarely opined or referred to on any mainstream source as historical facts.
You seem to have ignored a few facts in your slam of Obama. He inherited the largest recession since the Great Depression, caused once again, by a Republican. Once he took over the economy slowly, and steadily, recovered. He took criticism for this slow, steady recovery, but if you look at the graphs it's apparent the present economy is a linear, or nearly linear, graph which trended upward after Bush "Crashed the ambulance" to borrow a Mark Knopfler line. Tax cuts for the rich, the battle cry of Republicans, didn't work for Reagan, or either Bush, and they won't work for Trump, who took an economy cruising along at a positive rate, and blew up the deficit, which happened in every tax cut for the rich in the past. Your kids and grandchildren will love trying to pay for his record deficit, and don't spend that promised tax cut until you get your refund! It's all going to the rich, while our taxes won't go down, and may even go up.
Yes, when someone says they are going to make something happen, I fully expect them to do it. Trump said Mexico would pay for the wall, so why did he increase his demand from 5.2 million in December, to 5.7 million and shut down the government? It was all to try to fulfill a campaign boast. I call it that and not a promise, as a promise is only given by an honest person, which nobody can claim Trump is. He's lied like he always does: unfortunately you seem to have drunk his cool aid and ignore his blatant lies. A laser, really? Why do you think Trump says and does all these crazy things? He just keeps pointing your laser at something even more ridiculous, and we tend to forget about the other lies while we explore his latest claim. Tariffs are paid by foreign companies? No way, they're a tax on us, not foreigners. ISIS is dead, we won? I remember GW saying that on the deck of an aircraft carrier many years ago while we are still fighting over there now. I also find it ironic, but completely in character, that he claims a bunch of people seeking refuge and sanctuary are a threat to our democracy, while ever single expert testified that they are not a threat to our country. I tend to believe the experts who have been on the front lines, and behind enemy lines, when it comes to security, rather than someone who claimed his personal Vietnam was chasing skirts. What an insult to our veterans who fought over there, while he, like GW, dodged the draft.
Obama wasn't perfect, and I disagreed with many of his policies, but at least he had policies instead of tweets designed to distract people from focusing on the past lies he's fed us. He also turned the economy around and got many back to work. I do want to point out that the GM deal was actually started by GW, so he gets a point for that, but fixing something you broke isn't really that great an accomplishment.
A wall should be built to keep ME out. Have you seen or heard me lately. Scary!
Sorry "Me" the arrogance of saying "I saw the border so only I can comment" is an example of fallacious and dishonest reasoning. No one sees the whole issue just getting glimpses of the border. Indeed, it is more likely you get a warped view only seeing and interpreting what you see through your own beliefs. The way to understand it is to examine evidence, and look at information that goes far beyond what any one person can see. Sure, you can use your experience to explain your position, but if you're so arrogant as to think that gives you the answer and everyone else has to shut up - well, I'll laugh in your face at that!
Also, "Me" even if your reasoning was right (it's not, as I just demonstrated), trying to claim personal expertise when using an alias is lame. Anyone can post that they've done anything under an alias.
I stand by comparison. People who point out that immigrants, like any other population, have some criminals are making a point that is meaningless. It's an irrational effort at demonization and fear mongering, and when people use it they need to be called out on the carpet for such dishonest tactics. Unless you want to stop ALL immigration, you'll have some criminals enter, most enter legally. But most criminals are home grown.
As Mr. Erb has pointed out it is important when discussing topics like this that there be some factual information to work with. So, I've attached a couple of links that have some very serious arguments and reasoning to prove that "The Wall" is a mistake and a distraction. The first link is to an extremely well researched article (Immigration, Building a Wall, and Hispanic Crime) by Ron Unz, whose website publishes many articles that are on the very Right side of the picture. The difference between him and Donald Trump is he knows something and can prove it! The information to understand this topic cannot be disseminated in a few tweets.
It's long, a good read to help churn the discussion. And it's not written by one of those evil libs!
The second link is to an article about what two sheriffs who have been protecting the border for many years are seeing and their thoughts on the situation.
No-one in these articles is anonymous, authors or subjects.
In the military you learn about tactical advantage. If your enemy has to go over an obstacle, it slows them down. If they have to go around an obstacle it funnels them to where it's easier to confront and neutralize them. Either way an obstacle gives you a tactical advantage. Allowing you to focus resources into smaller areas and maximizing the effectiveness of a smaller force. How can any rational person dispute the effectiveness of a wall, fence or natural barrier? Use your heads people, it's not that hard to understand.
The only ones I listen to concerning wall or no wall are the men and women who serve our country as Border Patrol Agents and they all say we need walls in certain areas. BUILD THE WALL!!
@Michael Breault "As Mr. Erb has pointed out it is important when discussing topics like this that there be some factual information to work with. " Yes, and the only link Mr. Erb has posted to bolster his opinions was to his own blog. To me that seems a bit self-promoting.
Thank you "Perspective" for wondering and searching for links that I mentioned. Please don't blame Mr. Erb for the missing links. They were missing from my post. I am including them again.
Well "Perspective", it would seem we should ask our esteemed Editor/Moderator why the links are missing.
I've updated the last comment to add some words to the links so they'll appear.
You need to highlight existing text before hitting the link button, otherwise it won't appear in the comment. Alternately, you can link to something, then add text in between the (/a) tags.
@Michael Breault - in my previous comment I was not referring to your post with missing links. I figured it would get straightened out; glad to see it did. I specifically meant Mr. Erb's many posts without a single link, other than to his own blog, to back up his opinions.
Thank you for your time and effort to find links to useful information that we can all ponder.
A number of posters have mentioned their knowledge of the US-Mexican border. It's been more than 20 years since I was there, just in parts of Texas and California, so I wondered how to see what's going there without being able to invest the time and money for a trip like that.
USA Today recently created an interactive map so we could do just that. A simple way to get an idea of the magnitude of a Build the Wall project.
Would the proposed border wall stop this?
The question is moot. The GOP does not want another shut down, and they've told the President they will not support declaring a national emergency. Democrats and Republicans will agree to a bill that will have fencing and technological improvements, and both sides will declare victory. The wall is dead. But the good news is that both parties agree on enhanced border security.
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{"url":"https:\/\/es.b-ok.org\/book\/2363626\/73caca","text":"Principal Using Excel For Principles of Econometrics\n\n# Using Excel For Principles of Econometrics\n\n,\nCategories: Economy\\\\Econometrics\nA\u00f1o: 2011\nEdici\u00f3n: 4th\nEditorial: Wiley\nIdioma: english\nP\u00e1ginas: 484\nISBN 13: 9781118032107\nFile: PDF, 78.18 MB\n\n## Most frequently terms\n\nludo\nThank you so much for this book .\nfinally I found it here .\n24 March 2016 (16:19)\nYou can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.\n1\n\n### America: Imagine a World Without Her\n\nA\u00f1o: 2014\nFile: EPUB, 3.17 MB\n2\n\n### La vita inaspettata. Il fascino di un'evoluzione che non ci aveva previsto.\n\nA\u00f1o: 2011\nIdioma: italian\nFile: PDF, 7.17 MB\n\f\fUsing Excel\nFor Principles of Econometrics, Fourth Edition\n\nUsing Excel\nFor Principles of Econometrics, Fourth Edition\n\nGENEVIEVE BRIAND\nWashington State University\n\nR. CARTER HILL\nLouisiana State University\n\nJOHN WILEY & SONS, INC\nNew York I Chichester I Weinheim I Brisbane I Singapore I Toronto\n\nGenevieve Briand dedicates this work to Tom Trulove\nCarter Hill dedicates this work to Todd and Peter\n\nThis book was set by the authors.\nTo order books or for customer service call 1-800-CALL-WILEY (225-5945)\n\npublication rnay be reproduced, stored in a retrieval system or transmitted in any form or\nby any means, electronic, mechanical, photocopying, recording, scanning or otherwise,\nexcept as permitted under Sections 107 or 108 of the 1976 United States Copyright Act,\nwithout either the prior written permission of the Publisher, or authorization through\npayment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222\nRosewood Drive, Danvers, MA 01923, website www.copyright.corn. Requests to the\nPublisher for permission should be addressed to the Permissions Department, John Wiley\n\n& Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)7486008, website http:\/\/www.wiley.corn\/go\/permissions.\n\nISBN-13\n\n978-111-803210-7\n\nPrinted in the United States of America\n10 9 8 7 6 5 4 3 2 1\n\nPreface\nThis book is a supplement to Principles\n\nof Econometrics, 4th Edition by R. Carter Hill, William E.\n\nGriffiths and Guay C. Lim (Wiley, 2011). This book is not a substitute for the textbook, nor is it a\nstand alone computer manual. It is a companion to the textbook, showing how to perform the\nexamples in the textbook using Excel 2007. This book will be useful to students taking\neconometrics, as well as their instructors, and others who wish to use Excel for econometric\nanalysis.\nIn addition to this computer manual for Excel, there are similar manuals and support for the\nsoftware packages EViews, Gretl, Shazam, and Stata. In addition, all the data for\n\nEconometrics,\n\nlh\n\nin\n\nvarious\n\nformats,\n\nincluding\n\nExcel,\n\nare\n\nPrinciples of\n\navailable\n\nat\n\nhttp:\/\/www.wiley.com\/college\/hill. Individual data files, as well as errata for this manual and the\n\ntextbook, can also be found at http:\/\/principlesofeconometrics.com.\nThe chapters in this book parallel the chapters in\n\nPrinciples of Econometrics, lh. Thus, if you\n\nseek help for the examples in Chapter 11 of the textbook, check Chapter 11 in this book.\nHowever within a Chapter the sections numbers in\n\nPrinciples of Econometrics, lh do not\n\nnecessarily correspond to the Excel manual sections.\nThis work is a revision of\n\nUsing Excel 2007 for Principles of Econometrics, 3rd Edition by\n\nGenevieve Briand and R. Carter Hill (Wiley, 2010). Genevieve Briand is the corresponding\nauthor.\nWe welcome comments on this book, and suggestions for improvement.\n\n*\n\nGenevieve Briand\nSchool of Economic Sciences\nWashington State University\nPullman, WA 99164\n\ngbriand@wsu.edu\nR. Carter Hill\nEconomics Department\nLouisiana State University\nBaton Rouge, LA 70803\n\neohill@lsu.edu\n\n\u00b7\n\nMicrosoft product screen shot(s) reprinted with permission from Microsoft Corporation.\n\niv\n\nOur\n\nuse does not directly or indirectly imply\n\nBRIEF CONTENTS\n\n1.\n\nIntroduction to Excel\n\n2.\n\nThe Simple Linear Regression Model\n\n3.\n\nInterval Estimation and Hypothesis Testing\n\n4.\n\nPrediction, Goodness-of-Fit and Modeling Issues\n\n5.\n\nThe Multiple Linear Regression\n\n6.\n\nFurther Inference in the Multiple Regression Model\n\n7.\n\nUsing Indicator Variables\n\n8.\n\nHeteroskedasticity\n\n9.\n\nRegression with Time Series Data: Stationary Variables\n\n1\n19\n67\n95\n\n143\n154\n\n180\n\n204\n228\n\n10.\n\nRandom Regressors and Moment-Based Estimation\n\n11.\n\nSimultaneous Equations Models\n\n12.\n\nNonstationary Time-Series Data and Cointegration\n\n13.\n\nVector Error Correction and Vector Autoregressive Models\n\n14.\n\nTime-Varying Volatility and ARCH Models\n\n15.\n\nPanel Data Models\n\n16.\n\nQualitative and Limited Dependent Variable Models\n\n278\n294\n310\n\n328\n\n355\n\nA.\n\nMathematical Tools\n\nB.\n\nReview of Probability Concepts\n\nC.\n\nReview of Statistical Inference\n\nIndex\n\n262\n\n391\n\n402\n416\n431\n\n466\n\nv\n\nCONTENTS\n\n1.1\n\nStarting Excel\n\n1\n\n1.2\n\nEntering Data\n\n3\n\n1.3\n\nUsing Excel for Calculations\n\n2.5\n\n3\n\n1.3.1\n\nArithmetic Operations\n\n1.3.2\n\nMathematical Functions\n\n1.4\n\n1.5\n\n1.6\n\nImporting Data into Excel\n1.6.1\n\nRandom Number Generation\n\n2.4.3\n\nThe LINEST Function\n\n2.4.4\n\nRepeated Sampling\n\n4\n\n2.6\n\nNonlinear Relationships\n2.6.1\n\n8\n\n53\n\n53\n\n53\n2.6.lb ScatterPlot ofData\n\n10\n\nRelationship\n\nData Files forPrinciples of\n\n2.6.2\n\n13\n\nA Log-Linear Model\n\n55\n57\n\n2.6.2a Histograms ofPRICE\n\n1.6.2a John Wiley & Sons\n\nand ln(PRJCE)\n\n13\n\n57\n\n2.6.2b Estimating the Model\n\n1.6.2bPrinciples of\n\n61\n\nEconometrics Website\n\n2.6.2c ScatterPlot ofData\n\n14\n1.6.3\n\n50\n\n2.6.la Estimating the Model\n\n10\n\nWebsite\n\n49\n\nVariance and Covariance ofb1 and b2\n\nResources for Economists\n\nEconometrics\n\n45\n\n52\n\n3\n\n6\n\non the Internet\n1.6.2\n\nModel Assumptions\n\n2.4.2\n\n47\n\n1\n\nCHAPTER 1 Introduction to Excel\n\n2.4.1\n\nImporting ASCII Files\n\nwith Fitted Log\u00ad\n\n14\n\nLinear Relationship\n62\n\nCHAPTER 2 The Simple Linear Regression\nModel\n\n2.1\n\n2.7\n\n19\nPlotting the Food Expenditure Data\n2.1.1\n2.1.2\n\nUsing Chart Tools\nEditing the Graph\n\n2.7.2\n\nEstimating the Model\n\n63\n\n21\n\nCHAPTER 3\n\nInterval Estimation and\n\nHypothesis Testing\n\n24\n\n2.1.2c Gridlines and Markers\n25\n\n3.1\n\n2.2.1\n2.2.2\n\nversus Normal\n\n2.3\n\nPlotting a Simple Regression\n\nUsing Excel Built-in Feature\nUsing a Regression Option\n38\n\n2.4\n\nExpected Values of b1 and b2\n\n69\n3.1.1d TINY Function\n\n34\n\n38\n\nEditing the Chart\n\n3.1.1c Percentile Values\n\n34\n\n2.3.2\n\n2.3.4\n\nInterval Estimates\n69\n\n31\n\nUsing TwoPoints\n\n40\n\n68\n\n3.1.1b t-Critical Values and\n\n27\n\n2.3.1\n\n2.3.3\n\nDistribution\n\n27\n\nUsing Excel Regression\nAnalysis Routine\n\n68\n\n3.1.1a The t-Distribution\n\nUsing Least Squares\nEstimators' Formulas\n\n68\n\nThe t-Distribution\n\n26\nEstimating a Simple Regression\n\n67\n\nInterval Estimation\n3.1.1\n\n2.1.2d Moving the Chart\n2.2\n\n65\n\n23\n\n23\n\n2.1.2b Axis Titles\n\n63\n\nHistograms ofHousePrices\n\n19\n\n2.1.2a Editing the Vertical\nAxis\n\nRegression with Indicator Variables\n2.7.1\n\n69\n\n3.1.le Appendix E: Table 2\ninPOE\n3.1.2\n\n71\n\nObtaining Interval Estimates\n71\n\n3.1.3\n\nAn Illustration\n\n71\n\n44\n\nvi\n\n3.4.1\n\n3.1.3a Using the Interval\n\nThep-Value Rule\n88\n\n71\n\n3.4.1b Justification for thep\u00ad\n\n3.1.3b Excel Regression\nDefault Output\n\nValue Rule\n\n73\n3.4.2\n\n3.1.3c Excel Regression\n\n3.4.3\n\nConfidence Level\nOption\n3.1.4\n\nThe TDIST Function\n\nSection 3.3.2\nSection 3.3.3a\n\nSampling\nRevisited\n\nSection 3.3.3b\n\nTemplate\n\nCHAPTER 4\n\n78\n\n3.1.4e The IF Function\n\n79\n\n3.1.4f The OR Function\n\n79\n\n3.1.4g The COUNTIF\nFunction\n\nLeast Squares Prediction\n\n4.2\n\nMeasuring Goodness-of-Fit\n\n81\n\n4.2.2\n\nCorrelation Analysis and R2\n\n82\n\nFunction\n4.3\n\nTwo-Tail Tests with\nAlternative \"Not Equal To\"(:1:)\n82\n82\n\n83\n\n3.3.lb One-Tail Test of an\n84\nTwo-Tail Tests\n\n4.5\n\n4.3.2\n\nChanging the Scale ofy\n\n101\n\n4.3.3\n\nChanging the Scale of x andy\n\nA Linear-Log Food Expenditure Model\n4.4.l\n\nEstimating the Model\n\n4.4.2\n\nScatter Plot of Data with Fitted\n\nUsing Diagnostic Residual Plots\n\n104\n\n105\n108\n\n4.5.1\n\nRandom Residual Pattern\n\n4.5.2\n\nHeteroskedastic Residual\n\n4.5.3\n\nDetecting Model Specification\n\nPattern\n\nEconomic Hypothesis\n87\n3.3.3b Two-Tail Test of\n88\n\n100\n\n108\n\n86\n\n3.3.3a Two-Tail Test of an\n\nThep-Value\n\n100\n\nChanging the Scale of x\n\nLinear-Log Relationship\n\n84\n\nSignificance\n\nThe Effects of Scaling the Data\n4.3.1\n\n104\n\n84\n\nEconomic Hypothesis\n\n99\n\n102\n4.4\n\n3.3.la One-Tail Test of\n\nLeft-Tail Tests\n\nThe Food Expenditure\nExample and the CORREL\n\nAlternative \"Less Than\"(<)\n\n3.4\n\nCoefficient of Determination\nor R2 98\n\nOne-Tail Tests with\n\n3.3.3\n\n98\n\n4.2.1\n\n4.2.3\n\n81\n\n3.3.2\n\n96\n\n98\n\nAlternative \"Greater Than\" (>)\n\nSignificance\n\n95\n\n4.1\n\n80\n\nExamples of Hypothesis Tests\n\nPrediction, Goodness-of-Fit\n\nand Modeling Issues\n\nOne-Tail Tests with\n\nRight-Tail Tests\n\n93\n\n77\n\n3.1.4d The Simulation\n\n3.3.l\n\n93\n\n3.4.3d Two-Tail Test from\n\n75\n\n3.1.4c The LINEST Function\n\nHypothesis Tests\n\n92\n\n3.4.3c Two-Tail Test from\n\n75\n\n3.3\n\n92\n\n3.4.3b Left-Tail Test from\n\n3.1.4b Repeated Random\n\n3.2.3\n\n92\n\nSection 3.3.1b\n\n3.1.4a Model Assumptions\n\n3.2.2\n\n91\n\n3.4.3a Right-Tail Test from\n\nThe Repeated Sampling\n75\n\n3.2.1\n\n89\n\nExamples of Hypothesis Tests\nRevisited\n\n74\n\n3.2\n\n88\n\n3.4.1a Definition ofp-value\n\nEstimator Formula\n\n87\n\nErrors\n4.6\n\n111\n112\n\nAre the Regression Errors Normally\nDistributed?\n\n115\n\nvii\n\n4.6.1\n\n5.3.2a Left-Tail Test of\n\nHistogram of the Residuals\n\nElastic Demand\n\n115\n4.6.2\n\nThe Jarque-Bera Test for\n\n146\n\nNormality using the CHINV\nand CHIDIST Functions\n4.6.3\n\n118\n\nEffectiveness\n\nFood ExpenditureModel\n\n4.7.1\n\n5.5\n\nScatter Plot of Wheat Yield\n5.6\n\n123\n\n4.7.2a Estimating theModel\n125\n4.7.2b Residuals Plot\n\n126\n\n6.1\n\nThe Cubic EquationModel\n\n4.7.3b Residuals Plot\nLog-LinearModels\n\n128\n\n129\n\nA Growth Model\n\n4.8.2\n\nA Wage Equation\n\n4.8.3\n\nPrediction\n\n4.8.4\n\nA Generalized R2Measure\n\n6.2\n\n130\n\nPrediction Intervals\n\n136\n\n139\n\n6.3\n\nTesting the Effect of Advertising: the F\u00ad\n154\n\n6.1.1\n\nThe Logic of the Test\n\n6.1.2\n\nThe Unrestricted and\n\n6.1.3\n\nTest Template\n\n154\n\n155\n\n158\n\nTesting the Significance of theModel\nNull and Alternative\n\n6.2.2\n\nTest Template\n\n6.2.3\n\nExcel Regression Output\n\n159\n\n163\n\n6.4.1\n\nThe Optimal Level of\n\n6.4.2\n\nThe Optimal Level of\n\n163\n\n164\n\n6.5\n\nThe Use of Nonsample Information\n\n6.6\n\nModel Specification\n\n166\n\nCHAPTER 5 The Multiple Linear Regression\n143\n5.1\n\nLeast Squares Estimates Using the\nHamburger Chain Data\n\n143\n6.7\n\n6.6.1\n\nOmitted Variables\nIrrelevant Variables\n\n6.6.3\n\nThe RESET Test\n\n5.3\n\nHypothesis Tests for a Single Coefficient\n\nInsignificance\n\n145\n\n6.7.1\n\n5.3.2\n\nviii\n\nTests of Significance\nOne-Tail Tests\n\n146\n\n145\n\n167\n169\n\n172\n\nPoor Data, Collinearity and\n\nInterval Estimation\n\n5.3.1\n\n167\n\n6.6.2\n\n5.2\n\n145\n\n160\n\nTesting Some Economic\n\nScatter Plot of Data with Fitted\n140\n\n159\n\nThe Relationship between t- and F-Tests\n\nHypotheses\n\nEstimating theModel 139\nA Generalized R2Measure\n\nLog-Log Relationship\n\n6.2.1\n\n161\n6.4\n\n140\n4.9.3\n\n154\n\nHypotheses\n\n132\n\nA Log-LogModel: Poultry Demand\n\n4.9.2\n\n151\n153\n\nRestrictedModels\n\n129\n\n135\n\n4.9.1\n\nLog-LinearModels\n\n159\n\n4.8.1\n\nEquation\n\n5.5.2\n\n149\n\nMeasuring Goodness-of-Fit\n\ntest\n\n126\n\n4.9\n\nLinearModels\n\nCHAPTER 6 Further Inferenee in the\n\n4.7.3a Estimating theModel\n\n4.6.5\n\n149\n\n5.5.1\n\nMultiple Regression Model\n\n126\n\n4.8\n\nInteraction Variables\n\n148\n\nThe Linear EquationModel\n125\n\n4.7.3\n\n147\n\nPolynomial Equations: Extending the\nModel for Burger Barn Sales\n\n122\nover Time\n\n4.7.2\n\n5.4\n\n121\n\nPolynomialModels: An Empirical\nExample\n\nThe Jarque-Bera Test for\nNormality for the Linear-Log\n\n4.7\n\n5.3.2b Right-Tail Test of\n\n6.7.2\n\n176\n\nCorrelationMatrix\n\n176\n\nThe CarMileageModel\nExample\n\n177\n\n180\n\nCHAPTER 7 Using Indicator Variables\n7.1\n\nIndicator Variables: The University\nEffect on House Prices Example\n\n7.2\n\nApplying Indicator Variables\n7.2.1\n7.2.2\n\nEquations for\n\n182\n\nRural Areas\n\nMetropolitan and\n\n223\n8.5\n\n187\n\nGeneralized Least Squares: Unknown\nForm of Variance\n\nData: Stationary Variables\n\n192\n\nThe Difference Estimator: The Project\n\n9.1\n\n193\n\nChange Example\n\n198\n\nFinite Distributed Lags\n\nUS Economic Time Series\n\n9.1.2\n\nAn Example: The Okun's Law\n\nSerial Correlation\n9.2.1\n\n204\n\n8.1\n\nThe Nature ofHeteroskedasticity\n\n8.2\n\nDetecting Heteroskedasticity\n\nand Gt-1\n\nLagrange Multiplier Tests\n\n206\n\n233\n\n206\n\n9.2.2\n\n8.2.2a Using the Lagrange\nPagan Test\n\nSerially Correlated Errors\n237\n\nMultiplier or Breusch\u00ad\n\n9.2.2a Australian Economic\n\n206\n\nTime Series\n\n8.2.2b Using the White Test\n\n237\n\n9.2.2b A Phillips Curve\n\n209\n\n239\n9.2.2c Correlogram for\n\nThe Goldfeld-Quandt\n\nResiduals\n\n210\n\n8.2.3a The Logic of the Test\n\n9.3\n\n211\n\n8.2.3c Wage Equation\n\n212\n\n8.2.3d Food Expenditure\n216\n\n9.4\n\n240\n\nLagrange Multiplier Tests for Serially\nCorrelated Errrors\n\n210\n8.2.3b Test Template\n\nExample\n\n232\n\n9.2.lb Correlogram for G\n\nResidual Plots\n\nExample\n\n232\n\n9.2.la Scatter Diagram for Gt\n\n204\n\n206\n\n8.2.2\n\nTest\n\n232\n\nSerial Correlation in Ouput\nGrowth\n\n8.2.1\n\n8.2.3\n\n228\n\n9.1.1\n\n230\n9.2\n\nCHAPTER 8 Heteroskedasticity\n\n228\n\n228\n\nThe Differences-in-Differences\nEstimator: The Effect of Minimum Wage\n\n241\n\n9.3.1\n\n!-Test Version\n\n9.3.2\n\nT\n\nx\n\nR2 Version\n\n241\n243\n\nEstimation with Serially Correlated\nErrors\n9.4.1\n\n245\nGeneralized Least Squares\n\nHeteroskedasticity-Consistent Standard\n\nEstimation of an AR(1) Error\n\nErrors or the White Standard Errors\n\nModel\n\n219\n8.4\n\n224\n\nCHAPTER 9 Regressions with Time Series\n\nThe Linear Probability Model: A\n\nSTAR Example\n\n8.3\n\n222\n\n8.4.2b GLS Wage Equation\n\n185\n\n191\n\nMarketing Example\n\n7.6\n\n8.4.2a Separate Wage\n\nLog-Linear Models: a Wage Equation\nExample\n\n7.5\n\n222\n\n182\n\nTesting the Equivalence of\nTwo Regressions\n\n7.4\n\nExample\n\n180\n\nQualitative Factors with\nSeveral Categories\n\n7.2.3\n\nGrouped Data: Wage Equation\n\nInteractions Between\nQualitative Factors\n\n7.3\n\n8.4.2\n\n9.4.la The Prais-Winsten\n\nGeneralized Least Squares: Known Form\nof Variance\n8.4.1\n\n245\nEstimator\n\n221\n\n9.4.lb The Cochrane-Orcutt\n\nVariance Proportional to x:\nFood Expenditure Example\n221\n\n245\n\nEstimator\n9.4.2\n\n248\n\nAutoregressive Distributed\nLag (ARDL) Model\n\n252\n\nix\n\n9.5\n\nForecasting\n\n11.1.2a 2SLS Estimates for\n\n254\n\n9.5.1\n\nUsing an Autoregressive (AR)\n\n9.5.2\n\nUsing an Exponential\n\nModel\n\n281\n\n254\n\nSmoothing Model\n9.6\n\nTruffle Demand\n\nMultiplier Analysis\n\n11.1.2b 2SLS Estimates for\nTruffle Supply\n\n257\n\n283\n\n258\n11.2\n\nCHAPTER 10 Random Regressors and\nMoment-Based Estimation\n\n10.1\n\nSupply and Demand Model for the\nFulton Fish Market\n11.2.1\n\n262\n\nThe Reduced Form Equations\n286\n\nOLS Estimation of a Wage Equation\n\n11.2.la Reduced Form\n\n262\n10.2\n\n286\n\nEquation for lnQ\n\nInstrumental Variables Estimation of the\nWage Equation\n10.2.1\n\n286\n\n264\n\nWith a Single Instrument\n\n11.2.1b Reduced Form\n\n264\n\nEquation for lnP\n\n10.2.la First Stage Equation\nfor EDUC\n\n287\n\n264\n\n11.2.2\n\n10.2.lb Stage 2 Least\nSquares Estimates\n\nEstimates\n\n265\n10.2.2\n\nThe Structural Equations or\nStage 2 Least Squares\n290\n\n11.2.2a 2SLS Estimates for\n\nWith a Surplus Instrument\n\nFulton Fish Demand\n\n268\n\n290\n\n10.2.2a First Stage Equation\nfor EDUC\n\n268\n\n10.2.2b Stage 2 Least\n\n10.3\n\nCHAPTER 12 Nonstationary Time-Series\n\nSquares Estimates\n\nData and Cointegration\n\n270\n\n12.1\n\n294\n\nStationary and Nonstationary\n\nSpecification Tests for the Wage\n\nVariables\n\nEquation\n\n12.1.1\n\nUS Economic Time Series\n\n12.1.2\n\nSimulated Data\n\n273\n\n10.3.1\n\nThe Hausman Test\n\n10.3.2\n\nTesting Surplus Moment\nConditions\n\n294\n\n273\n\n274\n\n294\n\n296\n\n12.2\n\nSpurious Regressions\n\n12.3\n\nUnit Root Tests for Stationarity\n\n299\n\n12.4\n\nCointegration\n\n301\n\n306\n\nCHAPTER 11 Simultaneous Equations\nModels\n\n11.1\n\n278\n\nCHAPTER 13 Vector Error Correction and\n\nSupply and Demand Model for Truffles\n\nVector Autoregressive Models\n\n278\n\n13.1\n\n11.1.1\n\n310\n\nThe Reduced Farm Equations\n\n13.1.1\n\nTest for Cointegration\n\n13.1.2\n\nThe VEC Model\n\n13.2\n\nEquation for Q\n279\n11.1.1b Reduced Farm\n\n317\n\n13.2.1\n\nTest for Cointegration\n\n13.2.2\n\nThe VAR Model\n\n13.3.1\n\nThe Univariate Case\n\n13.3.2\n\nThe Bivariate Case\n\n281\n\n318\n\n321\n\nImpulse Responses Functions\n\n280\nThe Structural Equations or\nEstimates\n\n13.3\n\nEstimating a VAR Model\n\n312\n\n315\n\nEquation for P\n\nStage 2 Least Squares\n\nx\n\nEstimating a VEC Model\n\n279\n11.1.1a Reduced Farm\n\n11.1.2\n\n310\n\n323\n323\n325\n\nCHAPTER 14 Time-Varying Volatility and\nARCH Models\n14.1\n\n14.2\n\nEstimation: Different\nCoefficients, Different Error\n\nTime-Varying Volatility\n\n328\n\n14.1.1\n\nReturns Data\n\n14.1.2\n\nSimulated Data\n\nVariances\n\n328\n\nTesting and Forecasting\n14.2.1\n\n15.4.3\n\n328\n\n15.4.4\n\n334\n\n384\n\nSeemingly Unrelated\nRegressions: Testing for\n\n341\n\nContemporaneous Correlation\n\nTesting for ARCH Effects\n\n388\n\n341\n14.2.la Time Series and\nHistogram\n\n342\n\n14.2.lb Lagrange Multiplier\nTest\n14.2.2\n14.3\n\n344\n\nForecasting Volatility\n\nExtensions\n\n347\n\n349\n\n14.3.1\n\nThe GARCH Model\n\n14.3.2\n\nThe T-GARCH Model\n\n14.3.3\n\nThe GARCH-In-Mean Model\n\nCHAPTER 16 Qualitative and Limited\nDependent Variable Models\n16.1\n\nModel\n16.2\n\n349\n350\n\nA. I\n\nCHAPTER 15 Panel Data Models\n\n15.2\n\n355\n\nPooled Least Squares Estimates of Wage\nEquation\n\n16.2.1\n\nCensored Data\n\n16.2.2\n\nSimulated Data\n\n357\n\nA.1.1\n\nExponents\n\nA.1.2\n\nScientific Notation\n\nA.2\n\nConcepts\n\nEstimator for Small\n\nB.1\n\n416\n\nProbabilities Directly\nB.1.2\n\nBINOMDIST\n\n361\n\nFixed Effects Estimates of\n\nThe STANDARDIZE\n\nB.2.2\n\nThe NORMSDIST\n\nFunction\n\nTesting for Random Effects\n\n373\n\nEstimation: Equal Coefficients,\n\nB.2.4\n\nThe NORMDIST\n\nB.2.5\n\nThe NORMINV\n\nFunction\nFunction\nFunction\n\n381\n\nEstimation: Different\n\n423\n\nThe NORMSINV\n\n381\n\nEqual Error Variances\n\n422\n\nB.2.3\n\nRandom Effects Estimation of\nthe Wage Equation\n\n422\n\nB.2.1\n\nFunction\n371\n\n419\n\nThe Normal Distributions\n\n365\n\nSets of Regression Equations\n\n15.4.2\n\nB.2\n\nB.2.6\n\n423\n424\n424\n\nA Template for Normal\n\nCoefficients, Equal Error\n\nDistribution Probability\n\nVariances\n\nCalculations\n\n383\n\n417\n\nComputing Binomial\nProbabilities Using\n\nThe Random Effects Model\n\n15.4.1\n\nComputing Binomial\n\nforN=lO\n\n371\n\n15.4\n\n416\n\nof Wage Equation\n\nWage Equation from Complete\n\n15.3.2\n\ne\n\n413\n\nBinomial Probabilities\nB.1.1\n\n357\n\nEstimator: Estimates\n\n15.3.1\n\n409\n\nAPPENDIX B Review of Probability\n\n15.2.lb The Fixed Effects\n\nPanel\n\n408\n\nLogarithm and the Number\n\nPercentages\n\nDummy Variable\nN\n\n402\n\n402\n\n357\n\n15.2.la The Least Squares\n\n15.3\n\n395\n\n410\n\nEstimates of Wage Equation\nfor SmallN\n\n15.2.2\n\n393\n\n393\n\nMathematical Operations\n\nA.1.3\n\n355\n\nThe Fixed Effects Model\n15.2.1\n\n391\n\nLimited Dependent Variables\n\nAPPENDIX A Mathematical Tools\n\n352\n\n15.1\n\n391\n\nLeast Squares Fitted Linear Probability\n\n424\n\nxi\n\nB.3\n\nDistributions Related to the Normal\n426\nB.3.1\n\nThe Chi-Square Distribution\n426\n\nB.3.2\n\nThe t-Distribution\n\nB.3.3\n\nThe F-Distribution\n\n428\n429\n\nAPPENDIX C Review of Statistical Inference\n431\nC.1\n\nExamining a Sample of Data\n\nC.2\n\nEstimating Population Parameters\nC.2.1\n\n431\n436\n\nCreating Random Samples\n436\n\nC.2.2\n\nEstimating a Population Mean\n438\n\nC.2.3\n\nEstimating a Population\nVariance\n\nC.2.4\n\n438\n\nStandard Error of the Sample\nMean\n\n439\n\nC.3\n\nThe Central Limit Theorem\n\nC.4\n\nInterval Estimation\nC.4.1\n\nInterval Estimation with\nunkown\n\nC.4.2\n\nInterval Estimation with the\n447\n\nMean\nC.5.1\n\n449\nAn Example\n\n450\n\nC.5.2\n\nThe p-value\n\nC.5.3\n\nA Template for Hypothesis\nTests\n\nC.6\n\nu2\n\n446\n\nHip Data\nC.5\n\n439\n\n444\n\n450\n\n451\n\nOther Useful Tests\n\n454\n\nC.6.1\n\nSimulating Data\n\nC.6.2\n\nTesting a Population Variance\n\n454\n\n456\nC.6.3\n\nTesting Two Population Means\n459\n\nC.6.4\n\nTesting Two Population\nVariances\n\nC.7\n\nC.7.1\n\nA Histogram\n\nC.7.2\n\nThe Jacque-Bera Test\n\nIndex 467\n\nxii\n\n461\n\nTesting Population Normality\n\n463\n\n463\n465\n\nCHAPTER\n\n1\n\nIntroduction to Excel\n\nCHAPTER OUTLINE\n1.1 Starting Excel\n\n1.6 Importing Data into Excel\n\n1.2 Entering Data\n\n1.6.1 Resources for Economists on the Internet\n\n1.3 Using Excel for Calculations\n\n1.6.2 Data Files for Principles of Econometrics\n1.6.2a John Wiley & Sons Website\n\n1.3.1 Arithmetic Operations\n1.3.2 Mathematical Functions\n\n1.6.2b Principles of Econometrics Website\n\n1.6.3 Importing ASCII Files\n\n1.5 Saving and Printing your Data\n\n1.1 STARTING EXCEL\nFind the Excel shortcut on your desktop. Double click on it to start Excel (left clicks).\n\nAlternatively, left-click the Start menu at the bottom left comer of your computer screen.\n\ni1\/,; Sta rt\n... \"\n\n'\n\n.:,!o., \"\"\n\nSlide your mouse over All programs, Microsoft Office, and finally Microsoft Office Excel\n2007. Left-click on this last one to start Excel-or better yet, if you would like to create a\n\nshortcut, right-click on it; slide your mouse over Send to, and then select (i.e. drag your mouse\nover and left-click on) Desktop (create shortcut). An Excel 2007 short-cut is created on your\ndesktop. If you right-click on your shortcut and select Rename, you can also type in a shorter\nname like Excel.\n\n1\n\n2\n\nChapter 1\n\nExcel opens to a new file, titled Book I. You can find the name of the open file on the very top of\nthe Excel window, on the Title bar. An Excel file like Bookl contains several sheets. By default,\nExcel opens to Sheet I of Book I. You can figure out which sheet is open by looking at the Sheet\n\ntabs found in the lower left comer of your Excel window.\n-\n\ntitle bar\n\nhelp button\n\nty\/es \ufffd 1-0 fcrmula bar \" cell reference II group of c1>mmand.s v ll_ 11 There are lots of little bits that you will become more familiar with as we go along. The Active cell is surrounded by a border and is in Column A and Row I; its Cell reference is Al. Below the title bar is a Tab list. The Home tab is the one Excel opens to. Under each tab you will find groups of commands. Under the home tab, the first one is the Clipboard group of commands, named after the tasks it relates to. The wide bar including the tab list and the groups of commands is referred to as the Ribbon. The content of the Active cell shows up in the Formula bar (right now, there is nothing in it). Perhaps the most important of all of this is to locate the Help button on the upper right comer of the Excel window. Finally, you can use the Scroll bars and the arrows around them to navigate up-down and right-left in your worksheet. And you have a long way to go: each worksheet in Microsoft Excel 2007 contains 1,048,576 rows and 16,384 columns!!!! Note that your Ribbon might look slightly different than the one shown above. If your screen is bigger, Excel will automatically display more of its available options. For example, in the Styles group of command, instead of the Cell styles button, you might have a colorful display of cell styles. Introduction to Excel 3 1.2 ENTERING DATA We will use Excel to analyze data. To enter labels and data into an Excel worksheet move the cursor to a cell and type. First type X in cell Al. Press the Enter key on your keyboard to get to cell A2 or navigate by moving the cursor with the mouse, or use the Arrow keys (to move right, left, up or down). Fill in the rest as shown below: 1 2 3 4 s 1.3 USING EXCEL FOR CALCULATIONS What is Excel good for? Its primary usefulness is to carry out repeated calculations. We can add, subtract, multiply and divide; and we can apply mathematical and statistical functions to the data in our worksheet. To illustrate, we are going to compute the squares of the numbers we just entered and then add them up. There are two main ways to perform calculations in Excel. One is to write formulas using arithmetic operators; the other is to write formulas using mathematical functions. 1.3.1 Arithmetic Operations Select the Excel Help button in the upper right comer of your screen. In the window of the Excel Help dialog box that pops up, type arithmetic operators and select Search. In the list of results, select Calculation operators and precedence. \ufffdExcel He.Ip - l!ll x (\ufffd ... R.esults 1-25 \ufffdf l'J \ufffd) \ufffd) \ufffd \ufffd Ai '_formulas arithmetic-0perators Standard arithmetic operators are defined as shown below. To close the Excel help dialog box, select the X button found on its upper right comer. Anthmetic operator Mear\\ln'!ll ... (i>lus sign) Addition :J..f.J - (minus sign) Subtract10J11 3-1 Negation -1 Example \"(asterisk) MuJtlplicalicm '3\"3 I (forward slash) DlVISilln JIJ % (percent sign) Percent 2.0% .. (caret) ExponentiaUo-n 3\"2 - r:l \ufffd 4 Chapter 1 Place your cursor in cell Bl, and type X-squared. In cells B2 through B6 below (henceforth referred to as B2:B6), we are going to compute the squares of the corresponding values from cells Let us emphasize that the trick to using Excel efficiently is NOT to re-type values already stored in the worksheet, but instead to use references of cells where the values are stored. So, to compute the square of 1, which is the value stored in cell Al, instead of using the formula =l*l, A2:A6. you should use the formula =A2*A2 or =A2\"2. Place your cursor in cell B2 and type the formula. SUM I A 1 )( ( .. j B f;o x \"\" I c I =A2\"2 I D 1] ill 2 \u2022 (1) a formula always starts with an equal sign; this is how Excel and (2) formulas are not case sensitive, so you could also have typed Then press Enter. Note that: recognizes it is a formula, =a2\"2 instead. Now, we want to copy this formula to cells B3:B6. To do that, place your cursor back into cell B2, and move it to the south-east comer of the cell, until the fat cross turns into a skinny one, as shown below: I A 1 x 3 c 11 11 2. ,_ \ufffd B X-s91.1\u2022nea f .2 Left-click, hold it, drag it down to the next four cells below, and release! Excel has copied the formula you typed in cell B2 into the cells below. The way Excel understands the instructions you gave in cell B2 is \"square the value found at the address A2\". Now, it is important to understand how Excel interprets \"address A2\". To Excel \"address A2\" means \"from where you are at, go left by one cell\"-because this is where A2 is located vis-a-vis B2. In other words, an address gives directions: left-right, up-down, and distances: number of cells away-all in reference to the cell where the formula is entered. So, when we copied the formula we entered in cell B2, which instructed Excel to collect the value stored one-cell away from its left, and then square it-those exact same instructions were given in cells B3:B6. If you place your cursor back into B3, and look at the Formula bar, you can see that, in this cell, these same instructions translate into \"=A3\"2\". B.3 I 1 - I A x ... j B c I =A311.2 D I X::s9':_ared 2 1 1 l 21 4! - \/.I (. \ufffd 1.3.2 Mathematical Functions There are a large number of mathematical functions. Again, the list of functions available in Excel can be found by calling upon our good friend Help button and type Mathematical functions. If you try it, you will be able to see that the list is long. We will not copy it here. Introduction to Excel 5 We did compute the squares of the numbers we had. Now we will add them up-the numbers, and the squares of the numbers, separately. For that, we will be using the SUM function. We first need to select or highlight all the numbers from our table. There are several ways to highlight cells. For this small area the easiest way is to place your cursor in A2, hold down the left mouse button and drag it across the area you wish to highlight-i.e. all the way to cell B6. Here is how your worksheet should look like: A 1 I B x X-sauared 2 1 1 a 2 4 4 3 '9 5 4 16 6 5 025 \u2022 Editing group of command, which is found in the extreme right of the Home tab, and select :r. AutoSum. i%Aut\ufffd\ufffd \ufffd Next, go to the !ii f!IC:!:\"Cl;ear \u2022 Z1f' Sort & Find & Hitt r \u2022 Selt:d \u2022 Editing Excel sums the numbers from each column and places the sum in the bottom cell of each column. The result is: - .A 1 I El x X-squared 2 1 1 3 2 4 4 3 9 5 4 16 5 5 2.5 7 15 55 .. \u2022 Notice that if you select the arrow found to the right of :r. AutoSum you can find a list of additional calculations that Excel can automatically perform for you. Alternatively, you could have placed your cursor in cell the Enter key (and then copied this formula to cell 7 Note that: (1) A7, typed =SUM(A2:A6), and pressed B7). A I l=SUM(A2:::\" 6) B as soon as you type the first letter of your function, a list of all the other available functions that start with the same letter pops up. This can be very useful: if you left click on any of them, Excel gives you its definition; if you double left-click on any of them, it automatically finishes typing the function name for you, and (2) once the function name and the opening parenthesis are typed, Excel reminds you of what the needed need to specify in your function to use it properly. Arguments are, i.e. what else you 6 Chapter 1 Now, you could also have used the Insert function button, which you can find on the left side of the Formula bar . Once your cursor is placed in A 7, select the Insert function button. An Insert function dialog box pops up. You can Select a function you need (highlight it, and select OK), or Search for a function first (follow the instructions given in that window). - --- -- - __ \ufffdl'.EJ Ins-ert Function s_e,,,rch 'fur a function: Tyl?e a.brief' desaiption raf what you Gg [ Or select a 93tegcry : J Mo\u2022t Re<0en tly \"-\u2022mt to do and ther> dick u..,d \ufffd-------\ufffd Select a funttiC!JQ_: \"I In the Function Arguments dialog box that pops up, you need to specify the cell references of the values you want to add. If they are not already properly specified, you can type A2:A6 in the Number 1 window, or place your cursor in the window, delete whatever is in it, and then select A2:A6. Select . OK. Now that you have the formula in A7, copy it into B7 . - Functforn Arguments SUM Number1 - CTJ\ufffd jA2::A6 1.4 EDITING YOUR DATA Before wrapping-up, you want to polish the presentation of your data. It actually has less to do with appearance than with organization and communication. You want to make sure that anyone can easily make sense of your table (like your instructor for example, or yourself for that matter-when you come back to it after you let it sit for a while). We are going to add labels and color\/shade to our table. Hold your cursor over cell A until it turns into an arrow-down; left-click to select the whole column; and select Insert in the Cells group of commands, found left to the Editing group of commands. JS.:i.1 x \u00b7n 2 l 2. 3 z 4 3 _3 4 \ufffd 1 2 3 \ufffd l g iH [ns_ert De\ufffd.e1e li'o\ufffdat C:\ufffdll\u2022 Excel adds a new column to the left of the one you selected. That's where we are going to write our labels. In the new Al cell, type Variables; in cell A2, type Values; in cell A7 type Sum . Introduction to Excel A A B x 1 2 1 v.a\ufffdiables - - 3 7 1 2 - - Values 3 2 4 4 3 5 4 5 5 5 5 L 15 7 - Sum Select column A again, make it Bold (Font group of commands, right to the Clipboard one), and align it Left (Alignment group of commands, right to the Font one). caribri \ufffdI \ufffdl I A \ufffd Ir T1[03 Tl[&\ufffd ,A \ufffd\/ [= \u2022 Ii([\u00a7 fii Font = =lJ\ufffd\u00b7 \/ \ufffdJ I \ufffd\ufffd \ufffd\ufffdl \ufffd wrapT\ufffdxt - \u00b7 Al1gnme-nt Select cells Bl and Cl, and make them Bold. Repeat with cells B7 and C7. Better, but not there yet. Select row 7, make it Italic (next to Bold). Select column B, hold your left-click and drag your mouse over cell C to select column C too; select Center alignment (next to Left). Next, select A2:A6; left-click the arrow next to Merge & Center (on the Alignment group of commands), and select Merge cells. Immediately after, select Middle Align, which is found right above the Center alignment button. AllJJnm\ufffdnt Select Al:C7, left-click the arrow next to the Bottom Border button and select All Borders. 61),r.ilers BJ t::i::! jca;lrnri f B1 I 1! \u00b7\ufffd\u00b7 IK .A1 a Tl T1u::n hrT fnnt r. E':: EJ i .. : .\u2022 ; I EB EB llQtl.Om Bo\u00b7rder Top_ B.order !\u2022ft Bcrd\\'r Ri!<hl Be rder No l\u00b1lorder \ufffdII Bordie\ufffd\ufffd Ocrt1iok Borden t\u00a7 Select A7:C7 (A7:C7, not Al:C7 this time), left-click the arrow next to the Fill Color button, and select a grey color to fill in the cell with. Choose a different color for Al:Cl. 8 Chapter 1 Theme Colms [caJilbri le I rA ATJ \u00b7j I \ufffd \ufffd1 \ufffd. A 1 Fant \ufffd T JI 111 T \ufffd Ii Finally, put your cursor between cells C and D until it turns to a left and right arrow as shown here: + C D Hold it there and double left-click so that the width of column C gets resized to better accommodate the length of the label \"X-squared\". The result is: rtl. 1 A - \ufffd\"''\"\ufffd 7 fsum B c x X-squared - 1-- variables 1 1 2 4 3 9 4 16 5 25 15 55 -- Next, drag your cursor over the Sheetl tab, right-click, select Rename and type in a descriptive name for your worksheet like Excel for POE 1.2-1.4, for Econometrics, 4e-sections Using Excel for Principles of 1.2 through 1.4. Press the Enter key on your keyboard or left-click anywhere on your worksheet. n Excel for 00\u00a3 1.2-:1\ufffd\"1- \/ 5heet2 I \u2022 \ufffd 1 1.5 SAVING AND PRINTING YOUR DATA All you need to do now is to save your Excel file. Select the Save button on the upper left comer of the Excel window. A Save As dialog box pops up. Locate the folder you want to save your file in by using the arrow-down located at the extreme right of the Save in window or browsing through the list of folders displayed below it. Introduction to Excel 9 In the File name window, at the bottom of the Save As dialog box, the generic name Bookl should be outlined. Type the descriptive name you would like to give to your Excel file, like POE Chapter 1. Finally, select Save. File name: Save as !.Ype; lsaOl!J \ufffd==== I I I Excel WorkbMk F.de: name: Save as :[)lpe: I chapter I Excel Workbook POE 1 If you need to create a new folder, use the Create New Folder button found to the right of the Save in window. A New Folder dialog box pops up; it is prompting you for the name you want to give to your new folder, Excel for POE for example. Type it in the Name window and select OK. Finally, select Save. \ufffd c \ufffd\ufffd\ufffdfolder f::!ame: jExcel for POE - = \ufffdCgJ If you would like to print your table, select the Office Button, next to the Save button; go to Print, and select one of the print options. Preview :and \ufffdlliint tl\\le llO<Ument f:rint Se\u2022lect.a p\ufffdinter, nrumb\ufffdr of rnpies,\u00b7and oth .. r pri111tin.g optiorn< before prri\u00b7ntfng. Qukl<Print s\ufffdnd th\u2022 woukbo.olcdi'r\ufffdctly ti\u00a9 tm.e default printer with.a\"! makin9 changes, 1 Eri nt \ufffd\u00b7 17\\ \ufffd Print Prev'iew Preview and rmake <h.anges t<J pages before 'Hinting. \u2022 For more print options, you might want to check out the Page Layout tab, on the upper left of your screen, as well as the Page Layout button on the bottom right of your screen. Hom,; rnsert: P\ufffd.g\u2022 \ufffdaNout To close your file, select the X button on the upper right comer of your screen. ,\ufffd, - \ufffdIx! - !'- . \ufffd 1-' \ufffd 10 Chapter 1 In the next section, we show you how to import data into an Excel spreadsheet. Getting data for economic research is much easier today than it was years ago. Before the Internet, hours would be spent in libraries, looking for and copying data by hand. Now we have access to rich data sources which are a few clicks away. First we will illustrate how convenient sites that make data available in Excel format can be. Then we illustrate how to import ASCII or, text files, into Excel. 1.6 IMPORTING DATA INTO EXCEL 1.6.1 Resources for Economists on the Internet Suppose you are interested in analyzing the GDP of the United States. The website Resources for Economists contains a wide variety of data, and in particular the macro data we seek. Websites are continually updated and improved. We guide you through an example, but be prepared for differences from what we show here. First, open up the website http:\/\/rfe.org\/. RFE: Resources for Economists on the I n te rn et RFE l;\/<>!n@ ISSN 1081-\u00b74248. vol. 1J., No. s May, 2010 RFE Seaoch Editor: .B ill Goffe Dept. of E.oonomics, SUNY Oswego Editori'al As;sistant; Rich Freeh \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 Int m d u ctio n D ta \"onarii=:s; G l o=a rles & Enc do edias E omi>ts. Dep.artments, & UniY c r s itii:-.s. Fore casti ng & Con:.ulting Jobs. Grants. Grad School. & Advice Select the Data link and then select U.S. Macro and Regional Data. Introduction to Excel 11 RFE: Resourcas for Econo mists on the Internet !RFE Ho_, Title Paqe I Oata Tabre of Contentis: Abridged Se.arch Economic Web Sites I I Comolete Contents Search RFE .Data \u2022 \u2022 \u2022 \u2022 \u2022 U.S. M<icro and Re<:J1c0MI Other U.S. Data W0>rld .:ind Non-U.S. Data Finance- and Fina11dal Markets Journal Data and Pmqram A.rcchi11e.s Cla\ufffd This will open up a range of sub-data categories. For the example discussed here, select the Bureau of Economic Analysis (BEA). RFE. \ufffd Resources for Econ omists on the Internet Title Page.\/ Ct.ata I U.S. Macru and Reqion<il Data Table of Contents:: Abn dgi:d I Complete Contents Seard1 Economic \\11\/eb Sites I Search fl.FE RFE S<nrC)-1 U.S. Macro and Regional Data \"Pn\ufffdmary\" maa-o and regiona'f sites that generate data (mau,Y Jong series) \u2022 \u2022 \u2022 Bureau of Ea:ino 11c Anal sis BEA - National Income and ,atiornll and regior1al d.ata P\ufffd\u00b7c;duce Accoun't:s (GDP, etc;), in cl et.ails . \u2022 . Feder.al Reserve Bur:eau of Labor St.ab;tics (6LS) - more th<1_n 25.Q,\ufffdOa Jan>i 12 Chapter 1 Finally, select Gross Domestic Product (GDP). dmw Cl.:lrr.ent Re-leases Latest Information: Federal Recovery Programs amd BEA Slatislics U.S. Economfc Acc.o-unts N'E!U't'S R@leas\u00b7@! Sche-dule CongrE!::!3sion.a1 Quick Data Coni@rences and Meetings. N..ai.-.sroo.m ---RSS llirformation National lnterm1tiona1 Access National Economic Accounts Data Access International Ec\u00b7o\u00b7nomic Accounts D.ata Dig ital tib\u00b7r,,.ry Gross. Domi?S.ti.c Product (GDP) Per;o\u2022n;;;il rm:ome\u00b7 and Ou-tlay5 \u2022 Su.rv.ts-y iJif Olrr\ufffd.t.n\u00b7 _B.11\ufffd.11\u00b7\ufffd\ufffd\ufffd Imteradive Data Tables .. \u2022 (.lii' It- \ufffd Cons:umer Sp:ending t- Comorate Profi ts. t- Fixed \ufffd-sets. t t 11 International lr.i\\l'estment- Position Opcsri3tiorn; of M u1tf n\ufffdtion i3I Como'3'11ie:!:i Satellite Accnunt l'apers. and Working l'all\"'rs Bi3li3rict! of Pd?Jments \ufffd Trade i111 G:oods al'ild Servi't:e\"S \u2022 )ntemcrtioflal Servi,ce.g Survev Forms .aPld Related Materials Rssie\u2022arch arid De,u-elopment. View all lnte.rnati\u00b7onal Accounts Information \u2022\u2022\u2022 \u2022 View all N1ational Actounts Infarm.:atio1T1 \u2022\u2022. Metho-dology P\"f>\"\"' Electromic Reporting wtith The result shows the point we are making. Many government and other web sites make data available in Excel format. Select Current-dollar and \"real\" GDP. Gros.s D:omestic Pmduct (-GDP) News Release; Gmss Domestic Product I PDF verniofl o.fthe Grn-ss Domestic Product release. t inclu-des associated \u2022 \u2022 i,li \ufffd highlights, tables technical note, Note and Beginning with, th,e 2010 Q2 adval'loee\u00b7GDP re\u00b7lease (July 3\u00b70, 20.W), the advam;ed download fili!s (xi,;, -csv, and zip) for th.e NIPA Interactive D.at.a Table.swill be split into two s:ep<arate t im e peri9dsc 1969 to \u00b5resent , .a nd data throu9h t9\u00b759. This is b-eing d()ne in order \\.C) acwmmodate the \u00b7 Current-dollar and \"r\u00b7eal\" GDP 'Exc\ufffdI Percent change from preceding perio \u00b7 ,- el ap\u00b5roa\u00b7ching column limit in Ex-eel 2.0o:i\u00b7 for tallies showing q u.ar\u00b7t e rly series. lnteracti11e T.ables: GDP and the National Income and Pr.oduct Account (NlPAl H1stoncal \\\u00b7abl= Selected Nll?A Vie\u2022\"' Tallies: \u2022 \u2022 \u2022 Te.xt fa.rm at tne ch.ange..s to download P\"ae- ITe:\">tt: the layout for the advancoo Co.mma-delimited format ,cs Port-able document format (PDF' You have the option of saving the resulting Excel file to your computer or storage device, or opening it right away-which we proceed to do next. Do YQU wen: tu open Ill\" saYe Name this file? gdplev. xls. Type: Microsoft Of1fke Excel 97-2003Worksheet, From: 11\\JWW . b ea . g<D'll Ii -'Op en \ufffd \ufffd[ ] 1 _Sa _ v\u00b7_ e_\ufffd 25.CJKB Cancel What opens is a workbook with headers explaining the variables it contained. We see that there is a series of annual data and a quarterly series. Introduction to Excel B F \ufffd I c J _Q___j__ E I I JCurrent-Dollar and \"RealA Gr\u00b7OSS Domestic Product A ,., 1 2 -\ufffd 3 I Quart\ufffdJy Annual - G 13 (S\ufffdasonally adjusted a n n ua l rat.es) _4_ 5 \ufffd GDP\u00b7in 'GDP in\u00b7 6- GDP in hillions of \u00b7 GDP in billions of chained billions of d1ai11ed billions of curr9'nt 2005 current 2005 dollars dollars dollars dollars 7 - 8 - 9, 1929 103.6 977.0 '1.\ufffd47q1 23'7.2 1\/\ufffd2\u00b7.2 10 19'30 '91.2 s92 1 .a 1947q2 240.4 1, 7169.5 \u00b711 1931 76-5 1!34_9 19471q] 244_5 1,7@.0 12 - 1932 SS.:7 725_S. 1_19471q4 254_3 1,7'94,,B 1933 56.4 716.4 1.948i;j1 2-60.3 1,823'.4 - '13 The opened file is \"Read Only\" so you must save it under another name to work with it, graph, run regressions and so on. 1.6.2 Data Files for Principles of Econometrics The book Principles of Econometrics, 4e, uses many examples with data. These data files have been saved as workbooks and are available for you to download to your computer. There are about 150 such files. The data files and other supplementary materials can be downloaded from two web locations: the publisher website or the book website maintained by the authors. 1.6.2a John Wiley and Sons Website Using your web browser, enter the address www.wiley.com\/college\/hill. Find, among the authors named \"Hill\", the book Principles ofEconometrics, 4e. t*- TEXTBOOK P1rfm:::i.p,1'es of 6c:Ooonu\ufffdtrics., 4ttll EdJ1Jirn111 R Carter H ill CLouislan.a State Uni.versity), William E. Griffiths Univers.ity Ctf'Melbourne\u00b7, Australia), Gua: C. Um (University of Melb\u00b7ourne ustra.l ia) January 2011, \u00a92012 Follow the link to Resources for Students, and then Student Companion Site. There, you will find links to supplement materials, including a link to Data Files that will allow you to download all the data definition files and data files at once. 14 Chapter I 1. 6.2b Principles ofEconometrics Website The address for the book website is www.principlesofeconometrics.com. There, you will find links to the Data definitions files, Excel spreadsheets, as well as an Errata list. You can download the data definition files and the Excel files all at once or select individual files. The data definition files contain variable names, variable definitions, and summary statistics. The Excel spreadsheets contain data only; those files were created using Excel 2003. 1.6.3 Importing ASCII Files Sometimes data that you want to use may be provided but in ASCII or text format. To illustrate go to http:\/\/principlesofeconometrics.com. There you will find that one of the formats in which we provide data is ASCII or text files. These are used because they contain no formatting and can be used by almost every software once imported. Favorites. d' Fa11orites _I \ufffdPrinciples I Tools \ufffd \ufffd ::iuggested Sites Help .. lol\/e.b Slice Galler:t .. SJ of Ernnometrics .. ml g iii T Page .,. Safety .. Tools \u2022 lnstriuctor Resourrce s from John Wiley & Sons Data files, PowefPoirit Slides, Tustructo:r's.Mairnal Student, Resources. frnm John Wiley & Sons Datafiles. .and Using Excelfor Principk\ufffd Data files: or POE includes 148 data files in various formats_ Usiri,g the links 'below you can download all files in d01.Vn'load i'ndhiidual fi'le\u00b7s_ The data dennifio.n fil\u00b7es should he downloaded by all Data d'e-finitfon files ASCII riles oiEconometri.c (\u2022.dat) (\u2022_def) are text users_ file\u00b7s conta:ining variable- \u00b7n ames., definitions .and summary statistics_ are text files contai.nin\u00b7g only data. Variable .names are in \ufffd.def files. Select ASCII files and then go to the food data. a \".ZIP format, Introduction to Excel ASCII data files (* .dat) Dnwriload all ilie * .. dat files in are (a) 15 text files containing only data. ZIP format m\u00b7 (b) a s.e1 f- exib'adin!? EXE file (download and double-dick) Select i'ndividual *. dat files from the table below. a irli ne cola ale.oho I c ola2 gQjQ gQ]f meat profits fax medical W!h tax2 andy c o m m w t si growth metrics pube-xp term asp-aras comouter grunfeld mex1co\u00b7 .Q.!..O.l texas banqla1 consumption grunfeld2'. mininqi quizzes the-ories beer grunfeldJ money returns tobit bond \u00a3m cps sm a ll hhSUF\\18V ri_Q!l_ tobitmc 12! ffil.-1 hill lilQ!'.!Jill mroz robberv toodyay br2 cps2 house starts music salary tran Sf!CJrl bro i l e r crime housing ne ls sales truffle-s brum111 csi hwage nels small savirms tun a w demand indpro newbroiler share\u00b7 llk canada, demo inflatiCJl'I nls sheep unit capm2: edu ]nc insur nls panel sirmans usa QI\u00a7. .fil!!Q: exrate ivre21 nls 11ane l 2 .w 11town cattle ivreg2 oil spuri'ous vacan ces fair \ufffd olympics sterling vacation cespro figureC-3 korea oram:ie stockton \u20221ar ch10 fi ori.d a learn oscar stockton2' vec chard food liquor fil: stockton96 vote \ufffd lon1 \ufffd sumlus \u20221ote2 cloth Right-click on the file name. Select Save Target As. A Save As dialog box pops up. Locate the folder you want to save your file in by using the arrow-down located at the extreme right of the Save in window or browsing through the list of folders displayed below it. Finally, select Save. Once the download of the file 1s completed, a Download complete window pops up. Choose Close. r Do\ufffdnlmid complete ----- \ufffd\ufffd1(ill Do \"'nload Complete food. d\ufffdt Ii-om . Downloaded: Download to: Transrer rate: VllW'tl . pr:m::i!'iesafernrnxne trirn . rnm 960 bytes ir:i Lsec C:ipocuments<1nd 961:l bytes{Sec Setti ... \\food.dat oaose\u00b7this dialmg max whien downlo\ufffdd .completes. .Open ] [ Open Foldlec l [ Clo\"e ti] Start Excel. Select the Office Button on the upper left comer of the Excel window, then Open. 16 Chapter 1 Navigate to the location of the data file. Make sure you have selected All Files in the Files of Type window. Select you food.dat file and then select Open . . -- Open Fili:'s of!;ype: Look\ufffd: IAll Files{'*,'\"') Iii:::! DATA 11\u00b7\ufffd What begins is a Windows \"Wizard\" that will take you through 3 steps to import the data into Excel. Our ASCII data files are neatly lined up in columns with no commas or anything else separating the columns. Select Fixed width, and then Next. Text I mport W izard - r:I)\ufffd Step 1 of 3 The Text Wizard has determined that your data is Delimited. If this.i;-,\u00b7mrrect, \u00b7choose Next) or ch\ufffdose the data type that best describes your \ufffdata. Original data type S:hoose the file type that best describe\ufffd your data: 0 Q_\ufffdlimited - Characters such as commas or tabs separate each fi.eld: \u00aefh\ufffd\ufffd\u00b7\ufffd\u00b7\u00b7_cii\ufffd.\\F1 - Fields are aligned in colum.ns with spaces between each field. Start import at IDW: I !-\"I .... 1____ File \ufffdigin: _ \/ - \u00b7 4.37 : OEM United States Preview-of-File C:\\data\\econ4630\\food-.dat . l . 115.ZZ 3.69 z 135. 98 4 .. 39 3 4 119-. 31 ll4. 9oS 4. 75 6_0_3, 5 lB_'I_ 05 Pr.e'View of Data file 12: 47 __ [ Cancel <Bad' !::!ext > I [ E_inish In the next step the data are previewed. By clicking on the vertical black line you could adjust the column width, but there is no need most of the time. For neatly arrayed data like ours, Excel can determine where the columns end and begin. Select Next again. Introduction to Excel ; r Tert ------ - Import Wiz.ard - Step -\ufffd=- \u00b7-- 2 of J 17 \ufffd 11:] \ufffd This s,ITeen leiB \"lf-Plil set 'fieh:l 'titdttioi (rn'lumn flreaks}. Lin ef; with <ir:ro1111s To CREATE To DELEliE a a signify a rnlumnbreak. '.break line, dick <it 1he desired position. br\u00b7eak line, double click on the hne. To MO\\IE a t:Preak line, dick and drag it .. Data .._reVie'l'll -40 30 1Hi_2:! .3 _ 6 9 135._:<l\u00b7S 4-39 11:9.34 4.7Ei 11'4. S\u2022o& 6. 03 1.87. 05 12.47 SU 60 7tl \ufffdI -1 - I \ufffdI Cam:el l [ <\ufffdck l \ufffd-:\u00b7 \u00b7\u00b7.\u00b7 -\ufffd\ufffd:it_>\u00b7_ \u00b7\u00b7\ufffd [ ] EJnish In the third and final step Excel permits you to format each column, or in fact to skip a column. In our case you can simply select Finish. r i - Text ------ Import Wizard - Step 3 of 3 -\ufffd. l1JL8.J ThlssITeen lets JIOLil .select eac:h -rnlumru -and :set the Data Fo\ufffdmat. column dara funnai: @ \u00a7erJeral Ore\u00b7xt O Q.ate.1 j'-1\"1-'o--v_ _,,,v,,J,,, __ \"General' cooverTii rn.1meric 11aliles ill numliers, d<1te v<11ues. ID d11tEs, and all r\u00b7emair;iing values. to :text. [ !!_dvanced . . .. ] 0 1Do. mit [mpcrt column (skip) Data: g_re view .... , =I 'L3'9 4_7\ufffd \u00b7 fi_ 03 vj 12.47 \ufffdI This step concludes the process and now the data is in a worksheet named food. 18 Chapter 1 II I A I B 1 115.22 3.69 2_ B5.98 4.39 4.75 - 3 119.\ufffd4 4 114.:96 6.03 - 5 187.05 12.47 - 1 .. \ufffd \ufffd \ufffd1 I food I<\" \ufffd .\u2022 Rl\"aily Next, you need to save your food data in an Excel File format. To do that, select the Office Button, Save As, and finally Excel Workbook. :: \ufffdoeel W\u00abkboolt Save the ffle as an El( (el Workboafc \u00b7 ts Enel M.acrn-Eniib!ed Wadl:bcmk. Savoe the workbook lrt !he-XML-ba5oed andi macr.a-e\u00b7nabred me farm.at. \u2022 E:x<:el _!!in.a'IY Worllboo \ufffd Save the workbook In a. b l n aryfl.feformat Ol!lfol!l1lzed far '1a1t load ing .and s.avin\ufffd. A Save As dialog box pops up. Locate the folder you want to save your file in by using the arrow-down located at the extreme right of the Save in window or browsing through the list of folders displayed below it. .Sa 11e ln: ! g9 My_Daruments Excel has automatically given a File name, food.xlsx, and specify the file format in the Save as type window, Excel Workbook (*.xlsx). File []3!11E\ufffd Save as\u00b7 type: All you need to do is select [food xlsx , ;:::===== ::: Exc:el Workbook ('\".xis:() I 11 Save. \ufffdave .\ufffd From this point you are ready to analyze the data. This completes our introductory Chapter. The rest of this manual is designed to supplement your readings of Principles ofEconometrics, 4e. We will walk you through the analysis of examples found in the text, using Excel 2007. We would like to be able to replicate most of the plots of data and tables of results found in your text. CHAPTER 2 The Simple Linear Regression Model CHAPTER OUTLINE 2.1 Plotting the Food Expenditure Data 2.4.2 Random Number Generation 2.4.3 The LINEST Function 2.1.1 Using Chart Tools 2.1.2 Editing the Graph 2.4.4 Repeated Sampling 2.1.2a Editing the Vertical Axis 2.5 Variance and Covariance of b1 and b2 2.1.2b Axis Titles 2.6 Nonlinear Relationships 2.1.2c Gridlines and Markers 2.6.1 A Quadratic Model 2.6.1a Estimating the Model 2.1.2d Moving the Chart 2.2 Estimating a Simple Regression 2.6.1b Scatter Plot of Data with Fitted 2.2.1 Using Least Squares Estimators' Formulas Quadratic Relationship 2.6.2 A Log-Linear Model 2.2.2 Using Excel Regression Analysis Routine 2.3 Plotting a Simple Regression 2.6.2a Histograms of PRICE and 2.3.1 Using Two Points ln(PR\/CE) 2.3.2 Using Excel Built-in Feature 2.6.2b Estimating the Model 2.3.3 Using a Regression Option 2.6.2c Scatter Plot of Data with Fitted 2.3.4 Editing the Chart 2.4 Expected Values of b1 and b2 Log-Linear Relationship 2.7 Regression with Indicator Variables 2.4.1 Model Assumptions 2.7.1 Histograms of House Prices 2.7.2 Estimating the Model In this chapter we estimate a simple linear regression model of weekly food expenditure. We also illustrate the concept of unbiased estimation. In the first section, we start by plotting the food expenditure data. 2.1 PLOTTING THE FOOD EXPENDITURE DATA Open the Excel file food. Save it as POE Chapter 2. Compare the values you have in your worksheet to the ones found in Table 2.1, p. 49 of Principles of Econometrics, 4e. The second part of Table 2.1 shows summary statistics. You can 19 20 Chapter 2 compute and check on those by using Excel mathematical functions introduced in Chapter 1, if you would like. Select the Insert tab located next to the Home tab. Select A2:B41. In the Charts groups of commands select Scatter, and then Scatter with only Markers. The result is: 40\u00b7 35 \u2022 30 \u2022 25 20 \u2022.series1 15 \u2022 \u2022 10 5 0 0 lOIJ 200 300 4-0U 500 60G 700 Each point on this Scatter chart illustrates one household for which we have recorded a pair of values: weekly food expenditure and weekly income. This is very important. We chose Scatter chart because we wanted to keep track of those pairs of values. For example, the point highlighted below illustrates the pair of values (187.05, 12.47) - .... -:\u00b7 found in row 6 of your table. .. 40 ' \ufffd5 \u2022 ... 6:0 ... 25 \u2022 .... .. -- I .... .. : \u2022\u2022\u2022 \ufffd ...... #\"\u2022 ,. \u2022\u2022 \u2022 . ..... . 2:0 '15 .I'\\. _\"t 10 Serier 1 Point \"187 \u2022seriesl .... - ' I . 1>5000\u00b73 \"1 [1!87.050003, 12.47] I 0 I 0 100 200. 30.Q 400 son 500 I 700 - I When we select two columns of values to plot on a Scatter chart, Excel, by default, represents values from the first vertical axis. column on the horizontal axis and values from the second column on the So, in this case, the expenditure values are illustrated on the horizontal axis and income values on the vertical axis. Indeed, you can see that the scale of the values on the The Simple Linear Regression Model 21 horizontal axis corresponds to the one of the food expenditure values in column A, and the scale of the values on the vertical axis corresponds to the one of the income values in column B. We actually would like to illustrate the food expenditure values on the vertical axis and the income values on the horizontal axis-opposite of what it is now. By convention, across disciplines, the variable we monitor the level of (the dependent variable) is illustrated on the vertical axis (Y-variable ). And by convention, across disciplines, the variable that we think might explain the level of the dependent variable is illustrated on the horizontal axis (X-variable). In our case, we think that the variation of levels of income across households might explain the variation of levels of food expenditure across those same households. That is why we would like to illustrate the food expenditure values on the vertical axis and the income values on the horizontal axis. X= Income 2.1.1 Using Chart Tools If you look up on your screen, to the right end of your tab list, you should notice that Chart Tools are now displayed, adding the Design, Layout, and Format tabs to the list. The Design tab is open. (If, at any time, the Chart Tools and its tabs seem to disappear, all you need to do is to put your cursor anywhere in your Chart area, left-click, and they will be made available again.) \ufffd Ta_ \u00b7 \u00b7a_ \ufffd \ufffdi Ch a rt- Microsoft Excel Vlew Add-ms \ufffd\ufffd\ufffd- 1 - \ufffd Auobat DeiTgin [;iyo.ut Format Chart SlylH Go to the Data group of commands, to the left, and select the Select Data button. Swit\ufffdn Select Row\/CO\u00b7IUrtll!l Datot'( D.ata \ufffd 22 Chapter 2 A Select Data Source dialog box pops up. Select Edit. ' 11]\ufffd Select Datil Source llf@ll!\u00b7MRll Cbart Qata range: rr==1 [ \ufffd S\ufffditch,RowfColumn Le!jel'ld Entries \ufffder,ies) \ufffd\ufffd\ufffd=>'!\"'=='=\ufffd=rr \ufffd\ufffd\ufffd\ufffd----:---. [ '\u00a7l Md )I CT? E:irut J[ X ;B;emove \u00b0() Seriesl JI 'It I ]\ufffd Horizontal ' (\u00a7_ateljory) Axis Labels :r\/\ufffd, 115.220001. l:J.5.979996 119 .. 339996 114.959999 187 .. 050003 [ !::!)dden and Empty Cells I OK IJ [ Cancel In the Edit Series dialog box, highlight the text from the Series X values window. Press the Delete key on your keyboard. Select B2:B41. Highlight and delete the text from the Series Y values window. Select A2:A41. Select OK. -- - - Edit s\ufffdries _Series aame: _\ufffd [i] _, c__ ________ .Series \ufffdvalues: ifimiiim m1iiq,iio1:ii 1.\u2022\u2022'41:rl l!ii .11rli 111,-----ji] -- ri ii \ufffda. [1) \ufffd :' [dit Series s..., m Range Series\ufffd \\lalrues:: = iu. 22000 i, I\ufffd=_Sh _ e_e t_1!_8_$_2:$8_ _$4_i i3... \ufffd = .3 .. 69, 4.39, 4.... iJ I Canrn \ufffd[iJ ___ \u00b7Series Y values: =Sheetl!S8$2 :$8$4 1\n\nett'lang\ufffd\n\n\ufffd-------\ufffd\n\nSeries 'i \\lalues::\n\n'-------------\ufffd\n--'\nOK\n\n\ufffdL8]\n\n----\n\nSeries o.ame:\n\n\ufffd[i]\n\n\ufffdl=_Sh _ e_e t_11\ufffd\n$A_$2_:: _\n$A_S_ 4 1___ ] OK \ufffd =\u00b7 3 .. 69, 41.39, 4.... l15.220001, t)l 1 13.. , Cancel l The Select Data Source dialog box reappears. Select OK again. You have just told Excel that income are the X-values, and food expenditure are the Y-values-not the other way around. The result is: 7{)0 600 500 400 \u2022 + + 300 \u2022\u2022 \u2022seriesl \u2022 200 100 0 () 2() 30 40 The Simple Linear Regression Model 23 2.1.2 Editing the Graph Now, we would like to do some editing. We do not need a Legend, since we have only one data series. Our expenditure values do not go over 600, so we can restrict our vertical axis scale to that. We definitely would like to label our axes. We might want to get rid of our Gridlines, and change the Format of our data series. Finally, we would like to move our chart to a new worksheet. Select the Layout tab. On the Labels group of commands, select Legend and None to delete the legend. \ufffd\ufffdila\ufffdT\ufffdolt De\ufffd 1 [;J l\"i:l \ufffd lib] lil Chart ta.yo;!) InleT Fermat Axi\u00b7s Ltgen<11 nt1e1. \ufffd Data \ufffdLabel\ufffd T Data 1able. Labers \ufffd\ufffd rl r-. Non<' Tomi offfle!)rnd \ufffd 11 2.1.2a Editing the Vertical Axis Select the Axes button on the Axes group of commands. Go to Primary Vertical Axis, and select More Primary Vertical Axis Options. Show Axis fn !lBllons Display \ufffd.xls with numbers 'e\ufffdresente:d in Billions Show Ax[s with lo-g Seal\ufffd Primali)' J:!oria:ontal Axis l'rima11Y Yertica U A1ds ' i>isplay Axis scale \u2022 1 \u2022 \u2022 - u5ing a. tog 10 based 1--Mo\u2022\ufffd Prt\"\"'ry Vert<caJ Al!is Optiorn ... A Format Axis dialog box pops up. Change the Maximum value illustrated on the axis from Auto to Fixed, and speci fy 600. \ufffdIBJ format Axis [!xis Op'ticng l Axis Optiom Number Min'imum: Fill Maximum\ufffd une Color Une 5tyle \u00ae.A uto 0 EiKed Q A!!t:o 0 f.[xed Major-uriit: @Auto Q MiMr unit: @ Aut:Q. 0 Fix!l_d R\ufffded I\" foS\ufffdO.O 1100 110 ' o a Next select Alignment, and use the arrow-down in the Text direction window to select Rotate all text 270\u00b0. I I .ABC !\\lumber Fill Line Color line St>jle Shadow J-0 f()rmat 1.. Alignment\ufffd I Alignment Te\ufffdtlay,,ut I Cente.,, I Teir! direction: IHorizonral \"-'J v \\l_erbcal \ufffdlignment: Middle C!!_\u2022tom . \"r;ge: I 4T .I i,,\ufffd I\ufffdI \u2022 rn c: Horizontal .Rotate all text 90\" Rotate all text 210\u00b0 \ufffd Stacked 24 Chapter 2 Place your cursor on the upper blue border of your Format Axis dialog box. \" Format Axis [1]['8] Left-click, hold it, and drag the box over so you can see your chart; release. Look at the vertical axis of your chart. The numbers are now displayed vertically instead of horizontally, but less of them are displayed as well: 00 00 a a v 00 0 a 00 \"' 00 0 We want to change that back. Select Close. Axis Options again. Change Number \ufffd from ------- Fill Format Axis Line .Color f Une Style Major unit Sllado\u2022fli .J,-0 Format Axis J Axis. 0 ptions Minimum: @ \ufffduto 0 Eixed Fill Maximum: Color Line Style 0 Ayto \u00ae F!xed Major unit Q Auto \u00ae Fi\ufffded Minor unit: to Fixed, ----- \u00b7 - -\ufffd Number Line Angnment Options Auto @ AutQ 0 Fix\ufffdd and specify 100. Select . Ll] rg) I J60a.o l\u00a2o.o I, 1 2.1.2b Axis Titles Labels group of commands; Title, and select Title Below Axis. Back to the select Axis Titles, go to Primary Horizontal Axis N\ufffdme Do not cd'i1pll!y\ufffdnAl<i< Title Ol\u00b7art TrtlP Axir Legenlli Titlies t& \u00b7 \u00b7 label\ufffd Dat.a Datil ta.be!'\ufffd \u00b7 Table\u00b7 I\ufffd \ufffd Prirnt\u2022ny !fori>:o\u00b0'tal !bi< TlUe \ufffd\u00b7 Prin:ui:yyentlcal Axil Title \ufffd Trtle Selow Axis Disp!ay Tiflf' belOJ\u2022W Ho ri;zontal t.xis md f\u00b0\". \ufffd re<Lze cha\u00b7rt The Simple Linear Regression Model 25 Select the generic Axis Title in the bottom of your chart and type in x =weekly income in$100.\n\ncr.:: -----------;t?\nweekly income in S10\ufffdJ\n... x=\n\n[!J-- ------------\ufffd\n\nGo back to Axis Titles, then to Primary Vertical Axis Title this time. Select Rotated Title.\n\nNone\nDo nett\nChart\nAxisc Legend Data\nDm\nTiitle\n1iit1E\u00a7 N\nLabels\ufffd Ta.hie\n\nPrimary Horizontal Axis m1e\n\n[}i;sp. \ufffda.y Rc.tt iitedl 11.Jcf,5 liitfe and' mile\n\nP1im;:11y Ye rtical \ufffdj5. Tltrt\n\n\"'S labels\n\nAili\ufffd Trtle\n\nRotated' rrtie\n\n\ufffd\n\n-\n\n\ufffd\n\ndl!1Play a.n\n\nclnart\n\n\ufffd\n\nSelect the generic Axis Title on the left of your chart and press Delete, or put your cursor on top\nof the Axis Title box, left-click, and press the Backspace key to delete the generic Axis Title.\nType in y =weekly food expenditure in $. 1:1 \u00b7-1 \ufffdI \ufffdI .,, I =1 i1I al .,, I 1111 I ... :i,1 I 111 I I 1111 I :: I I ., 1 o}\"j 2.1.2c Gridlines and Markers Back to the Axes group of commands now. Select Gridlines. Go to Primary Horizontal Gridlines, and select None. \ufffd Axes Grldttnes \ufffd\ufffd i\\xe5 \ufffdI !iii \"lilJ l l?fim a ry .t!o rilzontal Gr\ufffdd Ii roes \ufffdP1imary :\\[errtic.al GrldITne;; \ufffd M.aj'or Gr[dlirie5 Dhplay . Hmizontaf G.\ufffd icllun es for Major units \"\\ Change the Current Selection (group of commands to the far left) to Series 1 (use the arrow down button to the right of the window to make that selection). Select Format Selection. Fs ] _j.\u00b7 . \ufffd E=ornna.t Selection \ufffd \ufffd Rfid to M'atcll 'Styl\ufffd CurrentSeli:-ction \ufffdrRf'Sl l<q,, i'ormat Sell'ction\ufffd tij Reset to Matcll S:tyl\u00b7\ufffd Currenl Selection. w] 26 Chapter 2 A Format Data Series dialog box pops up. Select Marker Options. Change the Marker Type from Automatic to Built-in. Change the Type and the Size as shown below: Marker Type, 0 \ufffdbltoma1ic 0 NQne @ Buili:4n Type:\ufffd Si2e: a Next, select Marker Fill. Change it from Automatic to Solid fill. Color options pop up. Change the Color to black. Select Marker Line Color, and change it from Automatic to No line. Select Close. \u00b7\u2022\u2022!'\ufffd\" il.'11:1 .. \u2022\ufffd;\ufffd] 0 tlofill \ufffd\ufffd\ufffd::;:\ufffdtfi Series Options ll Marker OptiOMS Marlcer Fill @ 0 0 0 D Marker Fill \u00b7\ufffd 0 @ f:ic.lure or texiure fill ;i.ondfill !?r,.dientfill !:'.icture or te\ufffdture fill Markerfll Al,!toma1fc Line Color line Style \ufffd- Ab!toma1ic The result is a replica of Figure 2.6 p. 50 in N line \ufffdolid line Y:ary colors by poin.t \ufffdr..lor.: Marker Line Color \ufffd rn \ufffd Markerli\"1e Color\ufffd 0 \u00ae i;;radient line 1 11 Ay_toma1fc - Principles of Econometrics, 4e: - I' Close \ufffd\ufffdI (if it looks like some of your dots are little flowers, left-click your cursor anywhere on your screen first) .!ii I!! \" :t: .,, c 8. >< llJ .,, 0 .g ::.. ::;;: II \"' ii: II .... D D \ufffd .. , I . 0 D VI . . .. p D ... . D 0 m . D . . . . 0 \"' . . . . D 0 . \ufffd . . . . . . . . . . .. . . . . . . . . \u2022 . . I . ..... ::.. 0 0 5 10 15 x\ufffd : 20 25 30 35 40 I w\ufffdeldv inoome in$100\n....\n\n2.1.2d Moving the Chart\nGo back to the Design tab. (Remember if you don't see your Chart Tools tabs, what you need to\ndo is place your cursor in your chart area and left-click). Select the Move Chart button on the\nLocation group of commands to the far right of your screen.\n\nCh.a.rt\n\nT110!5\n\nli>esngn\n\n\ufffd:\n\nMove\n\nLayout\n\nFormat\n\nI\n\n\ufffd\ufffdrt \ufffd<\nCha\nloGJhcn;\n\nThe Simple Linear Regression Model\n\n27\n\nA Move Chart dialog box pops up. Select New sheet and give it a name like Figure 2.6. Select\n\nOK.\nChoose. where you want the dlart to be placed:\n\n\ufffd. \ufffd\n\n@'Ne:w\ufffdhe:e:t:\n\n'iF\ufffdg _ur _e _2.,6_l\n\nQ:Qbjectfn!\n\n\ufffdfs_h ee _tl\n\n___\n_\n____\n_\n\n_________\n\nOK\n\nRename Sheet\n\n] [\n\nI\n\n\ufffd\n, v\n\nCancel\n\nJ\n\n1 Data (if needed, see Section 1.4 of this manual on how to do that).\n\nWe have plotted our data, and edited our chart. Next, we want to estimate the regression line that\nbest fit the data, and add this line to the chart.\n\n2.2\n\nESTIMATING A SIMPLE REGRESSION\n\nIn this section, we are going to use two different methods to obtain the least squares estimates of\nthe intercept and slope parameters {31 and {32. Method\n\n1\n\nb1 and b2 least squares estimators' formulas. Method\n\nconsists of making use of Excel built-in\n\n2\n\nconsists of plugging in values into the\n\nregression analysis routine.\n\n2.2.1 Using Least Squares Estimators' Formulas\nThe least squares estimators are:\nb2\n\n=\n\nI(xi - x)(yi - y)\nICxi - x)2\n\n(2.1)\n(2.2)\n\nThese formulas are telling us two things:\n\n(1)\n\nwhich values we need, and\n\n(2)\n\nhow we need to\n\ncombine them to compute b1 and b2.\n(1) Which values do we need?\n\n(xi, Yi) pairs of values-they do appear explicitly in equation (2.1). We also need x\nand y, which are the sample means, or simple arithmetic averages of the xi values and Yi\nvalues-those averages appear both in equation (2.1) and equation (2.2). Note that the subscript i\nin xi and Yi keeps count of the x and y values. In other words, i denotes the ith value or ith pair\nof values. Also, x and y, are referred to as \"x-bar\" and \"y-bar\".\nWe need the\n\n28\n\nChapter\n\n2\n\n(2) How do we combine those values?\n\nThe\n\nnumerator\n\nis the sum of products; L is the Greek capital letter \"sigma\" which denotes sum.\n\nx value from its mean (xi x). The second\ncorresponding y value from its mean (yi y). The\n\nThe first term of each product is the deviation of an\nterm of each product is the deviation of the\nproducts are computed for each\nThe\n\ndenominator\n\n(xi,yJ\n\n-\n\n-\n\npair of values before they are added together.\n\nis the sum of the squared deviations from the mean, for the\n\nother words, each\n\nx\n\nx\n\nvalues only. In\n\nvalue deviation from its mean is first squared, and then all those squared\n\ndeviations values are summed.\nEquation\n\n(2.2):\n\nb1\n\n=\n\ny\n\n-\n\nb2.X\n\nThis equation tells us to multiply b2 by x, and then subtract this product from\n\ny.\n\nNote that b2\n\nmust be computed first-before b1 can be computed.\nThere is actually no magic to this. We use the food expenditure and income values we have\ncollected from our random sample of 40 households, and perform simple arithmetic operations to\ncompute the estimates the intercept and slope coefficient of our regression line.\nAs for the computation of b1 and\nknow which values are the\n\nx 's\n\nb2 itself, there is only one trick. We need to make sure we\n\nand which ones are the y' s. So, we are going to start by adding\n\nlabels to our columns of data.\nYou should be in your Data worksheet. If not, you can go back to it by selecting its tab on the\n\nSelect row 2 and insert a new row (see Section 1.4 of this manual if you need help on that). In the\nnew cell A2, type y; and in the new cell B2, type x. Right-align Al :B2.\nI\n\nA\n\nI\n\nj' jfood_exp\n'J\n_I_J\n\nB\nincome\nx\n\nNext, we need to lay out the frame of the table where we are going to store our intermediate and\nfinal computations. Type x_bar=in cell D2, y_bar=in cell D3, b2 =in cell D6, and bl=in cell\nD7.\n\nIn cell\n\nG2:J2,\n\ntype x_deviation,\n\ny_deviation,\n\n(x_dev)(y_dev),\n\nand (x_deviation)2,\n\nrespectively. (Note that you can use your Tab key, instead of moving your cursor or using the\nArrow key, to move to the next cell to your right).\n\nThe Simple Linear Regression Model\n\nD\n2\nJ.\n4\n5.\n&\n7\n8\n\nE\n\n'F\n\nG\n\nb2\nb1\n\nH\n\nI\n\nJ\n\n\u00b7\nJ:<\ufffddelliatiory_delliatior (x_dev)(y !ex deviation\n_ )2\n\nx_bar=\ny_bar-=\n\n29\n\nK\n\n=\n=\n\nBelow x_deviation we are going to compute and store the deviations of the\n\nx\n\nvalues from their\n\nmean. Below y_deviation, we are going to compute and store the deviations of they values from\ntheir mean. Below (x_dev)(y_dev), we are going to compute and store the products of the\n\nx\n\ndeviation and they deviation for each pair of values. Finally, below (x_deviation)2 we are going\nto compute and store the\n\nx\n\ndeviations squared.\n\nTo show the 2 of (x_deviation)2 as a square, place your cursor in J2, if it is not already in it.\nMove to the Formula bar to select the 2, and select the arrow to the right comer of the Font\ngroup of commands.\n\nA Format cells dialog box pops up. Select Superscript and then OK.\n\n\ufffd_nt_;\nr\nArial\n\n_________,\n\n'It Calibri (Body)\n':II'\n!\n\nle_: __\nr\nF \ufffd\ufffd nt _s cy\n\ufffd\n\ufffdiz _e:_____,\n\ufffd r\nRegular\n10\n\nliM@I\n\n\"\"'\ns,------i\n\nBold\n\n\ufffd\"\u00b7i\n\nlt.alic\n\ni\\gency FB\n\n!!\nerian\n\n.\n\ufffd\n\n':II'\n\n'Ii'\n\nBold\n\nMal Narrow\n\n9\n\nrtnlic\n\n12\n14\n\nC.ol on\n\nUnderline :\n\n,,_N-on -e -------.,.\ufffd1 1\n\nAutomatic\n\nv\n\nI\n\nD 't!i.ormal font\n\n.Effects.\n\nI g\ufffd\ufffd::\ufffdut\nOsul;i_saipt\n\nThis is a TrueType\nween.\n\nfunt.\n\nThe\n\nsame\n\nfonh'lliTI be used on both y0ur printer.and\n\nOK\ufffd\n\n[\n\nyour\n\nCancel\n\nIn cells D6 and D7 proceed to format the 2 and 1 of b2 and b1 as Subscripts instead. Bold all\nthe labels you just typed, and Align Right the ones from G2:J2. Finally, resize the width of\ncolumns G:J to accommodate the width of its labels (see Section 1.4 of this manual if you need\nhelp on that).\n\n30\n\nChapter 2\n\nNow, your worksheet should look like this one:\nD\n\nl'1P'I\n\n2 )( bar=\n- 3 y_bar=\n4\n\n__\u00a7_\n6\n\nbl=\n\n7\n\nb1 =\n\n-\n\nWe\n\nhave\n\ncomputed\n\nj\n\nI\n\nE\n\nF\n\nI\n\nI\n\nG\n\nH\n\nl\n\naverages\n\nbefore.\n\nThe\n\nI\n\nI\n\nI\n\nJ\n\n\u00b7y_devia1io11 (\ufffd_lfev'}()'\ufffddev) 1(x\ufffdd'evi11tionf I\n\n!<_:deviation\n\nI\n\n\"\n\nformula\n\nyou\n\nshould\n\nhave\n\nin\n\ncell\n\nE2\n\nis\n\n=AVERAGE(B3:B42), and the one in cell E3 is = AVERAGE(A3:A42). Compare the averages\nyou get to the sample means of Table 2.1 in\n\nPrinciples of Econometrics, 4e (p. 49); they should\n\nbe the same.\nD\n\n-1:_\n\n-\ufffd\n\nbar=\ny_bar=\n\nI\n\nx\n\nE\n19_60475\n283.5735\n\nI\n\nF\n\nI\n1t\n\nH\nG\nI\nI\n_devfatfon l..Y. de\n' viation\n\nI\nJ\nI\nlx dev)(y_d!ev) (1<_ deviati'onf_\n\n4\n\n-\n\n_j_\n6\n\nb:z=\n\n7\n\nb1\n\n-\n\n=\n\nNext, we want to compute the deviations. Think about what you are trying to compute. And then\ntype the needed formulas in G3:J3.\nYou should type =B3 - E2 in cell G3, =A3 - E3 in cell H3, =G3*H3 in cell 13, and G23A2 in\ncell J3. Here are the values you should get:\nD\n2 x-bar=\n,__\nJ y_bar=\n,_\n4\n\nI\n\nE\n\nI\n\n19.60'475\n:283.5.735--\n\nF\n\nI\n\nG\nI\nJ\nH\nI\nI\nI\nI\nx_deviation y_d'.eviation (x_\ufffdev}{y_d:ey] (x_dE:Jviaticrnf\n-15_9 1 4 7 501 -16.8_353498\n\n2679. 303845\n\n253_2792692\n\n>--\n\n2-\n\nI\n\n6 b2=\n7 b-1=\n\nI-\n\nNow, in cells G3 and H3, we gave cell references E2 and E3, where the averages are stored. Note\nthat we will need to use those averages again, and get those averages from these same exact\nlocations, to compute the deviations of the next 39 observations.\nSo, what we actually need to do is to transform these Relative cell references (E2 and E3) into\nAbsolute cell references ($E$2 and $E$3). This will allow us to copy the formula from G3:H3\n\ndown below without losing track of the fact that the values for the averages are stored in cells E2\nand E3.\nA Relative cell reference is made into an Absolute cell reference by preceding both the row and\ncolumn references by a dollar sign. Place your cursor back in cell G3 (i.e. move your mouse over\nand left-click); in the Formula bar, place your cursor before the E and insert a dollar sign (press\nthe Shift-key and the $key at the same time); move your cursor before the 2 and insert another dollar sign; place your cursor at the end of the formula and press Enter. \ufffd =B3}2 K )( .\/ \ufffdI =B3-$@\n\n'X\n\n.\/ fr\n\n=B3-$E$2l\n\nThe Simple Linear Regression Model\n\n31\n\nH3, and add the needed dollar signs there too. Now, you can select G3:J3. Select\nCopy on the Clipboard group of command. Select G4:J42, and select Paste (next to Copy). You\nGo to cell\n\nhave just copied the formulas to compute the needed deviations for the rest of the\n\n(xi, Yi) pairs.\n\nYour worksheet should look like this:\n\nI\n\nD\n\n-\n\n2\n\nI--\n\nbar=\n\nE\n1 9 60475\n\nx-\n\ny_bar\n\ufffd\n4\nb'2=\n\n7\n\nb1 =\n\nF\n\nG\n\n283.5735\n\n=\n\n5\n6\n\nI\n\n_\n\n:\n\nI\n\n1\n\nJ\n\n\ufffd68 353498\n\n2679-30\ufffd845\n\n253.2792,692'\n\n-147 5 935 03\n\n2245-598261\n\n231. 48861,91\n\n-14.8547501\n\n-164---233\ufffd03\n\n2439J)476'41\n\n'2:20 66 599\n\n13 51475 01\n\n-168_6135\n\n221!8_886121\n\n184.27363891\n\n7 13475005 -96.52349\u00a33\n\n681!.<6710199\n\n50 .9'Q4,65 828-\n\n-\n\n15 9147501\n,_\n\n-15-214!501\n\n-\n\nt\n\nH\n\n:C\ufffdd!.Y!a\ufffdt!C?:'l J\ufffdd!:'!l'!;t!\ufffdn.. J\ufffd\ufffd\ufffd\ufffdY1!\ufffdU!\ufffd'!.t Lx\ufffdd_e_v11!.'l\ufffd'!t.\n\n-\n\n-\n\n-\n\n-\n\n_\n\n_\n\n_\n\n_\n\n_\n\n3\n\nWe have everything we need to finalize the computation of b1 and b2.\n\nE6, and again think about what you need to compute b2. Recall that the\n\nleast squares estimators are:\nb2\n\n=\n\nL(Xi - .i)(yi - y)\n2\nL(xi - x)\n\n(2.1)\n\n(2.2)\n=SUM(I3:142)\/SUM(J3:J42) is the formula\nE6. The one you need in cell E7 is =E3 - E6*E2 for equation (2.2).\n\nIf you refer back to equation (2.1), you can see that\nyou need in cell\n\nYour worksheet should look like this:\n-\n\n-\n\nA\n2\n\nI\n\n-\n\nI\n\nB\n\n3\n\ny\n115.22\n\n3!69\n\n4\n\n135.98\n\n4.39\n\n119.34\n\n4.75\u00b7\n\n114.96\n\n6.031\n\n5\n\ufffd\n\n67\n\n-\n\n187.05\n\n-\n\nc\n\nx-\n\n12-47\n\n-\n\nI\n\nI\n\nD\nx\n\nbm=\n\ny_bar\n\n=\n\n-\n\n-\n\nE\n\nI\n\nF\n\n-\n\nI\n\n-\n\nG\n\nH\n\nI\n\nI\n\nj\n\n19-60475\n\nx_deviation\n\ny_deviation lx_dev')(y_d\ufffdev) 1(x_deviatio\u00b7nf\n\n283.5735\n\n-15,.9N7501\n\n-1-68.3 53498\n\n2679.303845\n\n253.279269'2\n\n-151-214 7501\n\n-147 5935 03\n\n2245_5 98251\n\n231-48861911\n\n-14.8.547501 -1'64.233503\n\n243-9.64 7641\n\n220\u2022.-66\ufffd599\n\n\u00b7-13.5747501\n\n-168_6135\n\n221! 8.8 86121\n\n184.273838 9\n\n7 13475005 -9 6_ 5234%3\n\n688:6710199\n\n50 90465828\n\n--\n\n\ufffd=\n\n10.2096:4\n\nht=\n\n83_41501\n\n-\n\n_\n\n_\n\n_\n\nIn the table above we obtain the same exact least squares estimates as those reported on p. 53 of\n\nPrinciples of Econometrics, 4e.\nThat was Method 1 of obtaining the least squares estimates of the intercept and slope parameters\n\/Ji and {32. For Method 2, we are going to use the Excel built-in regression analysis routine.\n\n2.2.2 Using Excel Regression Analysis Routine\nSelect the\n\nData tab,\n\nin the middle of your tab list. On the\n\nright of the ribbon, select\n\nData Analysis.\n\nAnalysis group of commands to the\n\nfar\n\n32\n\nChapter 2\n\nIf the Data Analysis tool does not appear on the ribbon, you need to load it first.\nSelect the Office Button in the upper left comer of your screen, Excel Options on the bottom of\nManage window at the bottom of the Excel Options dialog box, and then Go.\n. ------\n\n!\n\nExcel Options\nPopular\nFcrmurlas\nProofin.!1\n'iave\n\nExcel\n\nOptjam\n\n\ufffdX\n\nE!:it Excel\n\nI\n\nManag:e:\n\nI\n\nIn the Add-Ins dialog box, check the box in front of Analysis ToolPak. Select OK.\n\n!!dd-Ins available.:\n\n1(8!\n0 \u00b7\u2022\u00b7 mmiiij\n.. \u00b7-iiirlj\nO AnalysisTo dlPak - VB A\n\n\\\n\nD\n.I'...\n,___K\n_\nI\n-=---<\"'P\n\nNow Data Analysis should be available on the Analysis group of commands. Select it.\nA Data Analysis dialog box pops up. In it, select Regression (you might need to use the scroll up\n\nand down bar to the right of the Analysis Tools window to find it), then select OK.\n\n, Data An alysi s\n\n-\n\n\ufffdrna'lysis Tools\n'HistIJgram\nMovil]g Average\nRandom Number Gener.ation\nRank arnl Percentile\nRe\n\nESSIDn\n\n[1.JL.8]\n\ntfelP'\n\nSampling\nt-Test: Paired Two Sample filr Means\nt-Test: Two\u00b7Sample Assuming Equal Variances\nt-Test: Two-Sample Assuming Une:qual Variances\nz-Ted:Two Sam\ufffde for Means\n\nThe Regression dialog box that pops up next is very similar to the Edit Series box we\nencountered before (see Section 2.1.1). Place your cursor in the Input Y Range window, and\nselect A3:A42 to specify they-values you are working with. Similarly, place your cursor in the\nInput X Range window, and select B3:B42 to specify the x-values you are working with. Next,\n\nplace your cursor in the New Worksheet Ply window and type Regression-this is going to be\nthe name of the new worksheet where Excel regression analysis results are going to be stored.\nSelect OK.\n\nThe Simple Linear Regression Model\n\nr\n\n-\n\n-\n\nl1J \ufffd\n\n1 Re-gressfon\nlilput\n[nputJ_Range;\n\nI :$A$.3:$A$42\n\nlnput:KJRange;\n\nI '$8$3::$13$42\n\nO!oabe:ls\nD\n\n33\n\n\ufffd\n\n\ufffd\n\ufffd\n\n1jelp\n\nD \u00b7Constant is fero\n\n\ufffd%\n\nConfidem:e Level:\n\n0Ulp11I options.\n\nQ .Quj;put Rllflge:\n\n\ufffdj\nI\n\nI Regre\ufffdsionl\n\n0 New l!'JQrkslieet:pJy\ufffd\n0 New \ufffdorlibook\nReslduals-013.esiduals\nOstandardized Residuals\n\nD Re:sigual P:lotE\nD L!i:ie: RtPlots\n\nNormal Prcliabihty\nD\u00b7!'.!orrnal-'Probability Plots\n\nThe Summary Output that Excel just generated should be highlighted as shown below:\n-.,-,\n\n1\n\nA\n\nB\n\nD\n\nc\n\nI\n\nSUMMARY OUTPUT\n\nE\n\nI\n\nF\n\n:Qrr;ficarerc\n\nI\n\nF\n\nG\n\nH\n\nJ\n\nr\n\n2\n3\n\nRegression Sraiistics\n\n4\n\nMultiplB- R\n\n0.-0.204.85\n\n.5\n\nR Square\n\n0_385D_Q2\n\n&\n\nF\n\n0.368S\u00b71-6\n\n7\n\nStandard :E\n\n.89.517\n\nB\n\nObservat.io\n\n40\n\n9\niO ANOVA\ndf\n\n11\n12 R1l9ressio1\n\n1\n\nSS\n\nMS\n\n190627\n\n190627 23.78684\n\n1 3 Residual\n\n38\n\n304505.2\n\nTotal\n\n39\n\n4951'32.2\n\n14\n\nF\n\nUSE-OS\n\n8fil13_294\n\n15\n16\n\n1 7 lntemept\n\n18 X Variable\n\nCoefficienManaa'\u00abi E:m\n\nl..ower 95% UpDer 95%.ower 95. OfJipper\n\nP-vaJi.Je\n\nt\u00b7Sfat\n\n95. 09\ufffd\n\n83_41ifiQ1\n\n43_41orn\n\nB21.518\n\n()'_Qfi2182\n\n-4.4\u2022fi327\n\n1712953\n\n-4_46327\n\nHL2953\n\n10.20964\n\n2.G9326J\u00b7 4.87138:1\n\nt.95.E-05\u00b7\n\n5.972052\n\n14.44723\n\n5.972(}52\n\n14.44723\n\n-\n\n19\n20\n21\n\n22\n\nI\n\nL\n\n\u2022\u00b7\n\n1\n\nSelect the Home tab. In the Cells group of commands, select Format, and AutoFit Column\n\nWidth; this is an alternative to adjust the width of the selected columns to fit their contents.\n=\n,._\n\n:;:\n\nEB\n\nn\ufffd\nrn\n\u00b7 \ufffd\n\nn\n\nCclumn'Width ...\nAutolt=ft\n\nCoEUlll'l'li'I\n\n\ufffdef,;ult Width ...\n\nWi1dth.\ufffd\n\n34\n\nChapter\n\n2\n\nYour worksheet should now look like this:\n\n1\nf2\n3\n4\n5\n6.\n7\n1i\n9\n10\n11\n\nI\nA\nSUMMARY OUTPUT\n\nI\n\nc\n\nB\n\nD\n\nI\n\nE\n\nI\n\nF\n\nG\n\nH\n\nI\n\nRe_qression S\/:alisfics\nMulti'f1leR\nO.S20485472\nR $:quare oiSS001Z22\ufffd Adjus1ed R Squ;;ire C1.:Jliea1 sos9 Stal'ld<ird Errnr 89.51700429 OhSe.l'Vatlon s 40 AN OVA ,rJf 12 R\"J:rr:e<ssicm 13 Residual 14 Tota.I SS MS 13062\ufffd.\ufffd788 190626.9788 )0450.5.1742 8013.294058 1 38 39 F 23\".7$884'1 Q7\n\nSianificaQce f\n1.94586E-05\n\n4951'32.153\n\n15\n16\n17 lntemef11\n\nHl X Variable 1\n\nCoefficients\n\n83.4\"\"'16U0997\n\nt Stat\n1.9215779\ufffd1\n2.0!t3263461 4.BTTJB0554\n\nSlandani Error\n\n10.2095425\n\n43.4\"1016.192\n\nP-velue\n0.06,2182379\n\n1 .94586E-O\ufffd\n\nUpoer95% LDwer 95.0%. Uppe.t 95. 0%\nLower95%\n-4.46:1267721 11129.s2srr -4.4632&n21 1112952877\n5.97205221f2 14.4472328 5.972052202\n14.447.2;328\n\nThe least squares estimates are given under the Coefficients column in the last table of the\nSummary Output. The estimate for the Intercept coefficient or b1 is the first one; followed by\n\nthe estimate of the slope coefficient\n\n(X\n\nvariable 1 coefficient) or b2. The summary output\n\ncontains many other items that we will learn about shortly. For now, notice that the number of\nobservations or pairs of values,\n\n40,\n\nis given in cell BS.\n\nA convenient way to report the values for b1 and b2 is to write out the equation of the estimated\nregression line:\n\nYi\n\n=\n\n83.42\n\n+\n\n10.21xi\n\n(2.3)\n\nNow that we have the equation of our straight line, we would like to graph it. This is what we are\ndoing in the next section.\n\n2.3 PLOTTING A SIMPLE REGRESSION\nThere are different ways to draw a regression line. One way is to plot two points and draw the\nline that passes through those two points-this is the method we are going to use first. Another\nway is plot many points, and then draw the line that passes through all those points-this is the\nmethod that Excel uses in its built-in features we are going to look at next.\n\n2.3.1 Using Two Points\nWhen we draw a line by hand, on a piece of paper, using a pen and a ruler, we can use any two\npoints. We can extend our line between the points, as well as beyond the points, up and down, or\nright and left. Excel does not use a ruler. Instead, it uses the coordinates of two points to draw a\nline, and it draws the line only between them. So, to have Excel draw a line that spans over the\nwhole range of data we have, we need to choose those two points a little bit more strategically\nthan usual.\n\nThe Simple Linear Regression Model\n\n35\n\nIf you look back at your scatter chart (Figure 2.6 worksheet) or back in your table (Data\nworksheet), you can see that our\n\nx\n\nSo, we choose our first point to have an\n\nx\n\nvalue equal to\n\n0 to 35 (from 3.69 to 33.4 exactly).\n0, and our second point an x value of\n\n35.\nThe point with an\n\nx\n\nvalue of zero is our y intercept. It is the point where the line crosses the\n\nvertical axis. Its coordinates are\n\nx =\n\n0 and y\n\nFor our second point, we let\n\nx =\n\n35;\n\n=\n\nb1 or\n\nplug this\n\n(0, 83.42). This is our first point.\nvalue in equation\n\nx\n\n(2.3),\n\nand compute its\n\ncorresponding or predicted y value. We obtain:\ny\n\n=\n\n83.42 + 10.21(35)\n\nThis is our second point, with coordinates\n\n=\n\n440.77\n\n(2.4)\n\n(35, 440.77).\n\nGo back to your Data worksheet (if you are not already there). In cell Ll, type Points to graph\nregression line. In columns L and M we are going to record the coordinates of the two points we\n\nare using to draw our regression line. In cell L2, type y; in cell M2, type\n\nx.\n\nIn cell M3, type O; in\n\ncell M4, type 35. In cell L3, we actually want to record the value for our y intercept or bi, which\n\nwe already have in cell E7. So, we are going to get it from there: in cell L3, type= E7, and press\nEnter. In cell L4, we want to have the computed predicted y value from\n\n(2.4).\n\nSo we type\n\n=E7+E6*M4, and press Enter. Note that instead of typing all those cell references, you can just\n\nmove your cursor to the cells of interest as if you were actually getting the needed values-this is\na very good way to avoid typing errors. So, you would type the equal sign, move your cursor to\nE7 and left-click to select it, type the plus sign, move your cursor to cell E6 and left-click to\n\nselect it, type the asterisk, move your cursor to sell M4 and left-click to select it, and finally press\nEnter. Once you have done all of that, your worksheet should look like this:\n\nJ\n\nL\n1\n\nJ\n\nM\n\nN\n\nP'oints fo graph regre.ssion line\n\n2\n\n,_\n\ny.\n83_41601\n\nx\n\n4\n\n440.7535\n\n35\n\n,_l_\n\n..\n\n0\n\nj\n\nNote that the predicted y value we obtain in the worksheet for\nthe one we just computed in equation\n\nx =\n\n35 is slightly different than\n\n(2.4) due to rounding number differences.\n\nNow, go back to your Figure 2.6 worksheet. The data we have plotted on the chart represent one\nset or series of data. The two new pairs of values we want to add to this chart represent a second\nset or series of data.\nSelect the Design tab, then the Select data button from the Data group of commands.\n\nChart loCJh\nD\ufffdsign\n\nC'.t\n\nLaveut\n\nFormat\n\n36\n\nChapter 2\n\nIn the Legend Entries (Series) window of the Select data source dialog box, select the Add\nbutton.\n,..- _\n\n_;____ .\n\n' S.elect Data Source\nChart i;!ata range:\nThe clala '\"\"'ge is !Do comple\ufffd to be\nttie series in the-Series panel.\n\ndi'>Piayecil.\n\nlf.a new rar\n\nJP\nLegend\n\n1,\n\nEntries\n\n\u00a7eries)\n\n\ufffd[\n\n][ 'X 8,emo\ufffde ],\n\nli:? Edit\n\nSeries!\n\nPlace your cursor in the Series X values window of the Edit series dialog box, and select\n\nM3:M4 in the Data worksheet. Place your cursor in the Series Y values window (delete\nwhatever is in there), and select L3:L4 in the Data worksheet. Select OK.\n\n- rli \ufffd\n\n\ufffd\n\u00b7\n\n\ufffd dit Series\nSeries.name:\n\n[\ufffd]\n\nSeries.\ufffd valLles:\n\n:deURlitl!JF\n\n=Dara1\ufffd$3-:.$M$4 \ufffd = 0, 35 =Daral\ufffd$3:!1L$\ufffd 00 = 8,'.H1600\"997, 4.. Can(eJ GK The Select data source dialog box reappears. A second data series, Series2, was created from the selection you just specified. Select OK. - Legend Er.itries (S_erie\ufffdJ I II \\@Add Seriest II X 8,emove I \ufffd Edit I\ufffd Series2 The two points from your new series are plotted on your chart (squares below): .. .. : \"II> .5 :!! \"' .. .., ., K D 0 0 .., .. \ufffd . ..,- D D II ;=,. . . .. D \"' .. . Lil \" .. 4! J:\"\"' .. 0 D \ufffd \u00b7\u00b7. . . . 0 D . . . . . . N D ;'; II . . . . . . . . . . . . . \u2022 . . . . . . r . . D 0 5 JlO 15 20 25 SD 35 40 ,._\u2022weekly income in$100\n:\n\n\ufffd\n\n..\n\n..\n\nl\n\n-\u00b7\n\nThe Simple Linear Regression Model\n\n37\n\nNow, we need to draw a line across those two points. Go to the Layout tab. Change the Current\n\nselection (group of command to the far left) to Series 2 (use the arrow down button to the right of\nthe window to make that selection). Select Format selection.\n\n!series 2.\nChart roars\n\n\ufffdayout ts Fmmat\n\n[}esign\n\n1.\n\n\ufffd Form<>t S:l'll: 'rtior:i\n\ufffd Rrset to Matcl'.I Sfyl<\".\n\nI\n\ni\ufffd\n\nI SerHeS 2\n\nL\ufffd Fi;nma,t_SelectiCJ\ufffd\n\n\ufffd\n\n\ufffd Resetto Match Style\nCurrent 5clection\n\nC:unenlS:ele'Cllon\n\nA Format data series dialog box pops up. Select Line color and change its selection from No\nline to Solid line. Select Close.\n'\"'\n\n,11,!1.-ur.1 \u2022lm.-\ufffdw\"1...:<:J]\n\nline Color\n\nSeries.\n\n\ufffdso'ii\"cfl 1ile1\n\n0 r:-!o Line\n\nOptions.\n\nMarker Optlons\nMarker Fill\nUhe Color\n\n\ufffd;\ufffddi\ufffdtlne\n\nI\n\n0 Ay_tomatk\n\nI\n\n(;_olor;\n\nI\n\nt;d\n\nI\n\n[\ufffd T)\n\n11\n\nClose\n\n\ufffd\n\nThe result is:\n\n\ufffd\n\n.5\nE\n\n=\n:t:\n-g\n..\n\nl\n\nx\n111\n\"Ill\n..\n\n.s\nz...\n111\n111\n\n\ufffd\n\nII\n::..\n\n0\n0\nlD\n0\n0\nlf\"I\n0\n0\n\n<;!\"\n0\n\n0\n\nf\"l\n0\n0\n\nIN\n\n0\n\n0\nrl\n0\n\n0\n\n5\n\n10\n\n15\nx\n\n35\n\n40\n\n\ufffdweekly inwmie in $1!00 Note that while you need only two points to be able to draw a straight line, you can use more than two points. So we could have computed a predicted level of food expenditure for every level of income we have in our original data set, and use the 40 (xi, .Ya pairs of values as our data Series 2. This is actually what Excel does when it adds a Linear Trend Line to a Scatter chart or a Line of best Fit to Plots of data as part of the Regression Analysis routine. We are going to delete the line and two points we just added to our graph and successively look at these other two ways to plot our regression line. 38 Chapter 2 2.3.2 Using Excel Built-in Feature In the Design tab, go back to the Data group of commands, and select the Select Data button. In the Select Data Source dialog box, select Series2 and Remove. Finally select OK. Select Data Source Gflart !!!.\ufffdta range: The data nlnge is tpa mmplex the 'Series in the .Se(ies pane'I, J l:o be di\ufffdpilayed. [fa new r rnra \ufffdS\\'!']t:h Row\/C<;;fumn Chart Tool! Design\ufffd- La\ufffd\ufffdLJt form\"t To add a Linear Trend Line, select the Layout tab. Go to the Analysis group of commands, select Trendline, and then Linear Trendline. No.ne Removes the <etecte-d Tr..r1dline OJ all Layout Trendlines ili none are selerted ' 1 \ufffd Lines UpiDmwn Bars\u00b7 i!>.n\ufffdlysis Format Error Bar1 Uneatr Trend nne .Ad1'sfse1s \u2022 Your chart should look like this (see also Figure 2.8 p. 54 in Principles ...,. .! \ufffd \" \u00b7\" .., \" w.. \" OJ .., 0 .e \ufffd .a ., OJ 3 II a UneafTrendHne for the \ufffde-lected chart ser\ufffde\ufffd \"' \"i I ofEconometrics, 4e): -0 0 ID 0 0 If) . .. 0 D ... a 0 m \u00b7O 0 N a 0 rl i:-\u00b7o 0 5 10 15 x\ufffd 20 25 30 .35 40 weeklyiru:ome ini$1IDO\n\n2.3.3 Using a Regression Option\nYou can also have Excel add the Line that best Fit your data by choosing that option on the\nRegression dialog box.\n\nGo back to your Data worksheet (bottom left comer of your screen).\n\nThe Simple Linear Regression Model\n\n39\n\nSelect the Data tab, located in the middle of your tab list. Select Data Analysis on the Analysis\ngroup of commands to the far right of the ribbon. Select Regression in the Data Analysis dialog\n-----l1J (g]\n\nbox, and then OK.\n\n-\n\na _ _n a-l -.- -----:' Da-t_\nA ysi s\n\n\ufffdalysis Tools\nCovariance\n\nDescriptive Sratisties\nExponential Smoothing\n\nI\n\nF'Test Two-Sample fur \ufffdariances\nFouri:er Analysis\nHi.s\ufffdram\n\nt[elp\n\nM.:wing Average\n\nNumber Gen\ufffdation\nRank and Percentile\nRandom\n\no_al!...._,\na t!\n\n1..-Fo _\nr m_\nu a\ufffd_\ns ____\n\nReview\n\nRe\n\nAnalysis\n\nV'\n\ness1on\n\nIn the Regression dialog box, proceed as you did before, except this time, name your worksheet\nRegression and Line, and check the box in front of Line Fit Plots. Select OK.\nOutput options.\n\n0 QutputRange:\n0 New Worksheet!ely:\n0 New W.orkbook\n\n\ufffd1\nI Regression anci Line I\n\nResiduals\n\nD Residuals\nD siandar.dized Re.siduals\n\nIn addition to the Summary Output you now have a Residual Output table and a Chart in your\nnew worksheet. The Residual Output table is only partially shown below, and shown after\nAutoFitting the Column Width (see Section\n\nI\n\nA\n22 RES.IDUAL OUTPUT\n\nI\n\nB\n\nc\n\nJ\n\nI-\n\n23\n\n24\n2:5\n\nI\n\nObseNation\n\n--\n\n26\nr---\n\nPredicted Y\n\n1 121 _089590!t:\n2\n\n128.-2363405\u00b7:\n\n28\n\n3\n\n4\n\n131.9'11 ilrni\n\n.\ufffd\n\n5\n\n210.7302519'\n\n27\n\n,_._\n\nThe Predicted Y or\n\n\u00b7\n\n144_9801542:\n\nYi values\n\nR.esii:Juals\n\n-5-86% 8 9792'\n\n7 _743555458,\n\n-12.57181564\n\n2.2.2 for more details on that).\n\nX Variable l Line Fit Plot\n1000 ,----------\n\n> 50\ufffd l.,.\ufffd,\u2022mm1,wmwmm1mm,\n1,\ufffd\ufffd\ufffd\ufffd,\n\"'\nlD\nrn\n\n-.30_ 02'()1 15524\n\n,.._\n\n<;t\n\"'\nrl\n\nrn\n\n\"'\nrn\n\nlJ1\n\nr-.\n\nN\n\n..-<\n\n..-<\n\n6\n00\nori\n..-<\n\n.\nr-i\nrl\n0\n\nN\n\nui\n'l:\n....\nN\n\n0\nN\n\n<T\nN\n\n0.11\nui\nN\n\n'lD\n....\n;-;\nN\n\n\u2022Y\n\u2022\n\nlCVaria'bli\" 1\n\n23 . 6 8 024894\n\n-\n\nhave been computed for all the original observed\n\nsimilarly to the way we computed y for\n\nx\n\n=\n\nPredicte.d V\n\n35 (see Section\n\n2.3.1).\n\nxi\n\nvalues,\n\nThe least squares Residuals are defined as\n\n(2.5)\nYou can compare the Predicted Y and Residuals values reported in the Excel Residual Output\nto the ones reported in Table\nsame.\n\n2.3\n\nof\n\nPrinciples of Econometrics, 4e\n\n(p.\n\n66).\n\nThey should be the\n\n40\n\nChapter 2\n\n2.3.4 Editing the Chart\n\nColumn chart as opposed\nAxis titles are not currently\n\nNow, the chart needs a little bit of editing. For one it looks like it is a\nto a\n\nScatter\n\nChart\n\none. The scales could be changed. Finally,\n\nand\n\nChart area, and left-click, so that Chart Tools are made\navailable to you again. Select the Design tab. Go to the far left group of commands, Type, and\nselect Change Chart Type. In the Change Chart Type dialog box, select X Y (Scatter) chart,\nand then Scatters with only Markers. Finally, select OK.\nPlace your cursor anywhere in the\n\n,-\n\n-\n\nChamge Chart Type\nTemplates\n\nlltill\n\ufffd\n@\n\nChi!rt 1oor.s\n\nCo1umn\nLine\nPie\n\n\ufffd\n\nBar\n\n\ufffd\n\nArea\n\n11:1\n\n\ufffdI\n\n(Scatter)\n\nXY\n\n-\n\nThe result is:\n\nX Varj ab\ufffde 1 Line Fit Plot\n101)-0\n\nI\n\n,.. so: .\n\n-\n\n0.\n\n.\u2022'4\n\nJ!\ufffd\n\nw\n\n30\n\n\u2022 \ufffdredicted V\n\n40\n\nX \\I ariable 1\n\nNow that we have the correct chart type, we would like to draw a line through all the\n\nPredicted Y\n\npoints. Actually, since we are using those points to draw our regression line, what we want to\nshow is only the line. So, we will use the points to draw the line, and then get rid of those big\nsquare points. This way our chart won't be as busy.\ncross as shown below:\n\nPredicted Y points with your cursor.\n\nX Varfable 1 Line ! F it Plot\n\nYour cursor should turn into a fat\n\nX Variab\n\ne\n\n1 Line Fit Plot\n\n....\n\n11000\n\n,..\n\nI\nS.001-\n\n::11 \u2022\n\nSeri\n\"\n<\n\"Pmllicted \ufffd\u00b7 Poiot \"26.610001 \" 1\n(26.6100CJI, 5.0946()'71)\n\nI\nXVariable 1\n\n11 30\n\n35\n\nXVariab'le\n\n1\n\n\u2022Y\n40\n\n\u2022 Pr<edicted Y\n\nThe Simple Linear Regression Model\n\n41\n\nRight-click and select Format Data Series. A Format Data Series dialog box pops up. Select\nChange the line color to something different from the Y points.\n\nLine Color and Solid line.\n\nSelect Marker Options, and change the Marker Type from Automatic to None. Select Close.\n\nQelete\nReset to\n\n\ufffd\n\nMQtch Stylle\n\nr\n\nCha ni:i. \ufffd\u00b7 \u00b7seri es C\ufffdart T:\u00a3pe ...\n\nGfJi I\n\nJ\n\n-\n\n--\n\n:s:\ufffdlect lllata ...\n3-D\n\nSeries\n\nB_otation\n\n0\n0\n0\n0\n\nOptions\n\nMarker Option.-\n\nData. La.Q\ufffdf>\n\nMarker Flll\n\n\ufffd\n\n-\n\nFormiilt Data S-eries\n\nt!_eline\n\ufffdolidline\n\nMarker Options\n\nSeries Options\n\nMarker Type\n\n0\n\nA\ufffdtoma1ic\n\n.;;ol11r;\n\n-\n\nfmma.t Dat\"' s .. ries ...\n\n--\n\nLine Color\n\nformat Data Seri es\n\n\ufffd\n\nMarker Fill\n\nA\ufffdtoma1ic\n\n\ufffdf\ufffd\ufffd\ufffd\ufffd\ufffd\n\nLine Color\n\nThe result is:\n\nX Varmable\u00b7 1 Line Fit Plot\n10()0\n\n,_ 50:\n\n\ufffd\n--------\n\nI\n\n-\n\nI\n\n'\n\n\ufffd11\\\n20\n\n\u2022 v\n\n\u2022\n\n30\n\n-Predkted'!I\n\n)( Va rfable 1\n\nOn your chart, select the Legend with your cursor, right-click and select Delete.\n\nI\n\nX Variable 1 Line Fit Plot\n.1000\n,_\n\n500\n0\n\n1- ,J'' t\\;1\n\n0\n\n10\n\n20\n\n30\n\nQieol e\ufffdieo\n\n\ufffd\n\nReset\n\nA\n\nEont...\n\n\ufffd\n\n:S:\ufffdlect Data ...\n\n\ufffd\n\n\ufffd\n\nClilange Cnart TYJ:H' ...\n40\n\n3-n\n\nICVaria'ble\n\nto M\ufffdtch Style\n\n!I.\n\n\ufffd\n\n_E'.nt;;ilon\n\nEor_mat Legen.a...\n\nChange the Chart and Axis titles as you see fit. Below, we show you how you can change the\nChart title. You can follow a similar process to change the Axis titles.\n\nPlace your cursor in the title area and left click.\n\nX Variable 11line Fit Plot\nL>-----1Charlr T.itle,_______...,,\n1000\n\n; )-\n\n5-00\n0\n\nI\n\n0\n\n..\nHJ\n\n20\nXVariable 1\n\n30\n\n40\n\n42\n\nChapter 2\n\nSelect the generic title.\nG------------ -------------_i;i\n\nl X rVariahle ll\n\nLine Fit Plot l\n\nlch>rtTIle;\n\n\ufffd - -1T ------------ - ------0\nwoo\n>\n\n500\n0\n\nI\n\n.....\n\n\u2022\u2022, ...\n\n\u00b7\ufffd=\u00b7\u00b7\ufffd. !\n\n'\nI\n\n0\n\n\u2022\n\n10\n\n.\n\n30\n\n40\n\nX Varlab'le l\n\nYou can select any of the titles and change the Font size by going back to the Home tab. Select\nwhat you need on the Font group of commands.\n\nCalit>ri\n\n(Body)\n\nlej I\n\n!1\n\n\u2022\n\n110\n\n\ufffdA\u2022 A\ufffd]\n\n\ufffdl\ufffdl \ufffd - .A \u00b7!\nr,\n\nFnnll\n\nYou can reformat the y-axis (and\/or the x-axis) by selecting it with your cursor, right-clicking and\nselecting Format Axis.\nQ.elete\n\n.Figure,:2'.8 The fitted regression\n\n.a\n\nR\ufffds\u00b7efto Ml!tch, S:tyle\n\nA\n\nEont. ..\n\nai\n\ufffd\n\n..\n\n-\n\nChan:,ge Chart T)lpe...\n\nSS:lect\n\nData ...\n\n:3-0 ll_oh !Or>\n'\n\n40\n\nFu rm at .l!!!.ajor.-Gridli ne s ...\nw\ufffd\n\nWl'e.lily in.oome in.$100 I& Eu<fm<1tAxi1 ... [J:_ If you proceed as you did before to edit your vertical axis (see Section 2.1.2a), you should obtain the following: 'Figure2.8 The frttedl.regres:<ion To resize the whole Chart area, put your cursor over its lower border until it turns into a double cross arrow as shown below. 1\u00b7 The Simple Linear Regression Model 43 Left click, and it should turn into a skinny cross. Hold it, and drag it down until you are satisfied with the way your chart looks. Figure 2.8 The f\"itted regression ...,.. .5 \ufffd ::I -\ufffd \"Cl .. Ii x Ill \"Cl 0 Ji! ? -\"\" Ill \ufffd\" 0 0 \"' 0 a U'l 0 a <t 0 a ro a 0 N 0 a ..;\u00b7 ;:a. a 0 5 10 15 11 =wee 20 25 3() 35 40 kly lnoome il'I :$10-0\n\nYou can delete the Gridlines by first selecting them, right-clicking and then selecting Delete.\n\nFigu:re 2.g The fitted regre!l-S'icm\n\n,.\nII\n\"Cl\n0\n\nD\nD\nrn\n\n.s\n1!-\n\nO\u2022\n0\nN\n\n\ufffd\n\n0\n0\n.--i\n\n....\nIll\nII\n\n\"\n;:..\n\n_Qelete\n\nRe5i't to M;!hh- \ufffdtyle\n\noll\nLE@i\n\nChange- Cha.rt Type ...\n\n0\n\n0\n\n:m\nJI=\n\n20\n\nweeklyinoome iru$10lD 40 \ufffd - \ufffd \ufffd \ufffd S.tledi Data... 3-D _Batat1!ln ... furm af Grl d l i n, e-s ... Forma.t Axls... You can also reformat the Data Series Y by selecting the points, right-clicking and selecting Format Data Series. Then proceed as you did before to change your markers' options (see Section 2. l .2c). 44 Chapter 2 Figure 2. B The fJUed regresskm \ufffd .5 f = :1:1 .. I: 8. 0 0 \\.Cl 0 a lI'I 0 0 ... :.: Ill 0 .. Cl m .s 1=\" 0 0 \"\" \ufffd 0 0 .-i ... Qil ll >- Qe>let\ufffd .a \ufffd Reset to M\ufffdWI Sty\ufffde, Change Seri:es Ch;utT\ufffdpe... \ufffd Sgl e ct Da.ta ... 3-D B.ol:al1on CJ I{) 10 .>e= 2() 30 Acfd Data La.!?_els 40 Acfd Trl'\"ndl.lne ... weekh! ilil\u00b7oome in$!1.00\n\nI\ufffd\n\n\ufffd I\n\nEmm;;;it Data :Seuies ...\n\nofEconometrics, 4e):\n\nFigure 2 .8 The fTttedl regre:ssion\n\n\"91-\n\n.5\n!!\n\n=\n:I:\n\"Cll\nI:\n\n8.\n\n>o:\nII\n,,\nCl\n\n:f\n.2\n\nII\n\n\ufffd\n\nII\n...\n\n0\n0\nlD\n0\n0\nU1\n\n.\n\n0\n0\n<T\n\n.\n\n\\\n\n'\n\n.\n\n0\n0\nfY1\n0\n0\n\"'\n\n.\n\n.\n. .\n.. .\n\n0\n0\n.,.,\n\nIi\n\n0\n\n0\n\n10\nx=\n\n20\n\n30\n\n4'()\n\nwe\u2022eklv :in.tome in $100 In this next section we illustrate the concept of unbiased estimators. 2.4 EXPECTED VALUES OF b1 AND b2 To show that under the assumptions of the simple linear regression model, E(b2) = {32, E(b1) = {31 and we first put ourselves in a situation where we know our population and regression parameters (i.e. we know the truth). We then use the least squares regression technique to unveil the truth (which we already know). This allows us to check on the validity of the least squares regression technique, and specifically to check on the unbiasedness of the least squares estimators. The Simple Linear Regression Model 45 2.4.1 Model Assumptions First, let us restate the assumptions of the simple linear regression model (see p. 45 of Principles ofEconometrics, 4e): \u2022 The mean value of y, for each value of x, is given by the linear regression function: E(ylx) \u2022 For each value of x, the values of y = f31 + f32x (2.6) are distributed about their mean value, following probability distributions that all have the same variance: var(ylx) \u2022 = a2 (2.7) The sample values of y are all uncorrelated and have zero covariance, implying that there is no linear association among them: (2.8) x is not random and must take at least two different values. \u2022 The variable \u2022 (optional) The values of y are normally distributed about their mean for each value of x: y -N[({31 + {32x), a2] (2.9) In the specific and simplified case we are considering in this section, half of our hypothetical population of three person households has a weekly income of$1000\na weekly income of\n\n$2000 (x = 20). (x = 10), and half of it has Because we are all mighty, we know the values of our population parameters, and consequently the values of our regression parameters. Let \u00b5ylx=lO = 200, \u00b5ylx=ZO = 300, and var(ylx = 10) = var(ylx = 20) = a2 = 2500. This implies {31 = 100 and {32 = 10. The probability distribution functions of weekly food expenditure, x = 10 and an income level x = 20, y, given an income level are assumed to be Normal. They look like this: - t(vl\ufffd=10J -t(vlx=20) 46 Chapter 2 The linear relationship between weekly food expenditure and weekly income looks like the following: lJ 300 200 () Let us emphasize 10 the difference between 20 this section and Chapter 2 in Principles of Econometrics, 4e. In this section, we do know the truth. In other words, we have information all three person households that constitute our population. In Chapter 2 of Principles of Econometrics, 4e, like it is the case in real-life, you do not have that population information. You must thus rely solely on your random regarding weekly food expenditure and weekly food income on sample information to make inferences about your population. Now, as an exercise, and as a way to prove the unbiasedness of the least squares estimators, we are going to use the least square regression technique to unveil the truth. Insert a new worksheet in your workbook by selecting the Insert Worksheet tab at the bottom of your screen (or Press the Shift and Fl 1 keys). Name it Simulation. Simu lation\ufffd' i We are going to draw a random sample of 40 households from our population. Half of the sample is drawn from the first type of households, with weekly income x = 10; and half of the sample is drawn from the second type of households, with weekly income x = 20. Let us keep records of the level of weekly income for our 40 households in column A of our Simulation worksheet: in cell Al, type x 10; in cells A22:A41, record the value 20. and Right-Align it; in cells A2:A21, record the value The Simple Linear Regression Model A 47 A 1 2 20 20 3 10 20 4 10 5 10 20 6 10 7 1Q 8 10 9 rn 1.0 11 10 10 12 13 10 10 14 15 16 17 10 10 1.8 19 20 10 21 10 20 20 20 20 20 20 io 33 20 10 34 35 10 36 37 20 20 20 20 10 38 39 20 20 10 40 20 41 20 42 2.4.2 Random Number Generation We use the Random Number Generation analysis tool to draw our random sample of households. We keep record of their weekly food expenditure in column B of our Simulation worksheet: type y in Bl, and Right-Align it. 1 I J I A II B x y Select the Data tab, in the middle of your tab list. On the Analysis group of commands to the far right of the ribbon, select Data Analysis. Anal1111sc The Data Analysis dialog box pops up. In it, select Random Number Generation (you might need to use the scroll up and down bar to the right of the Analysis Tools window to find it), then select OK. \ufffdalysi,.Tools f-Test Two-Sample Fowrier Analysis Histogram Movi\ufffdverag_e for \\\/ariances DMfti@Miii.ffil\u00a7\u00a7\u00b7@M\u00b7'\u00b7!\u00b7 Rank and Per c:entile Regression Sampling t-Test: PairedT,..C>Sample for<Means t-Test: Two-Sample Assuming Equal Variances ,\ufffd [ 1YI l c:\ufffd I tfelp I vi A Random Number Generation dialog box pops up. Since we are drawing one random sample, we specify 1 in the Number of Variables window. We first draw a random samples of 20 from 48 Chapter 2 households with weekly income of x 10, = so we specify the Number of Random Numbers to be 20. For simplicity we assumed that our population of households has weekly food expenditure that is normally distributed, so this is the distribution we choose. Once you have selected Normal in the Distribution window, you will be able to specify its Parameters: for \u00b5ylx=io = 200 and its Standard deviation is .Jvar(ylx = 10) = a = x = 10, its Mean is 50. Select the Output Range in the Output options section, and specify it to be B2:B21 in your Simulation worksheet. Finally, select OK. n 'R\ufffdlldom Number Ge ner\ufffdti o_ ___ 1 \ufffd 1 \ufffdlzo Number. of\ufffdariables: rllumbff of Random Numt!ers: _ ____ \ufffd' 'Qisrnbu\u00b7tion: ffr[g] ____I_. \ufffd \ufffdI \ufffd ' [ tielp J N o _r m_aJ_ \ufffd ' -----\" Parameters M!::,an= Standard deviatior;i = \ufffd \ufffd \ufffddom S eed; Output options 0 Quljxit Range; 0 'New Worksheet.\ufffdly:\u00b7 0 New Wodcbook Repeat to draw a random sample of the Mean to \u00b5ylx=lO = 20 - Change different random sample, = 300 and the Output Range to B22:B41. ParametErs M\ufffdan= x 20. from households with weekly income of \ufffd I QutpLlt options e Qul\u00b5!Jlt'R<lnge:. 1$8$22;$6\\$41\n\nHere is the random sample that we obtained. NOTE: you will obtain a\ndue to the nature of random sampling.\n\n\ufffd\n\nThe Simple Linear Regression Model\n\nB\n\nA\n1\n\n:x\n\nB\n\nA\ny\n\n22-\n\n:m. \u00b7214.6751\n\n2\n\nHJ\n\n122.490&'\n\n23\n\n20\n\n336.57.85\n\n3\n\n11()\n\n163.1711\n\n20\n\n303.5467\n\n4\n\n11()\n\n211.0i02\n\n24\n.25\n\n5\n\n10 294.12.95\u00b7\n\n26\n\n20\n\n358.9562.\n\n6\n\n10\n\n192.9407\n\n27\n\n20\n\n278.1513\n\n1\n\n1IQ\n\n228.56.27\n\n2&\n\n20 257.9295\n\n8\n\n10\n\n223.1013\n\n291\n\n20\n\n'9\n\n1!0\n\n184.7241.\n\n30\n\n20\n\n328.9643\n\n11()\n\n10\n\n164.82\u00b7\n67\n-\n\n31\n\n20\n\n.297.1585.\n\n11\n\n10\n\n125.1754\n\n32\n\n20\n\n338.727\n\n12\n\n10\n\n274.037\n\n33\n\n20\n\n297.34.23\n\n13\n\n10 136.920'1\n\n34\n\n20\n\n201.38'94\n\n14\n\nllO\n\n190.4468\n\n35\n\n20\n\n309.4635\n\n15\n\n11()\n\n121.6272.\n\n36\n\n20\n\n305.@2.\n\n1\u00b76\n\n10\n\n202.8224\n\n37\n-\n\n20\n\n334.5588:'\n\n17\n\n10\n\n123.4H\n\n3&\n\n20\n\n2&6.24(12\n\n20 .216.4365'\n\n33.1.23.85\n\nl8\n\n10\n\n116.1414\n\n39\n\n20\n\n273.67.85'\n\n1'J\n\n10\n\n209.413.;\n\n40\n\n20\n\n318.1071\n\n20\n\n11() 152.0113'\n\n41\n\n20 .2&3.9447\n\n21\n\nllO\n\n42\n\n200.4915\n\n49\n\n2.4.3 The LINEST Function\nNext, we use the LINEST function to obtain the least squares estimates for the intercept and\nslope parameters, based on the random sample we just drew.\n\nThe LINEST function is an\n\nalternative to using the Least Squares Estimators' Formulas (see Section 2.2.1) or the Excel\nRegression Analysis Routine (see Section 2.2.2). It allows us to quickly get the least squares\nestimates for the intercept and slope parameters. For this purpose, the general syntax of the\nLINEST function is as follows:\n=\n\nLINEST(y's, x's)\n\nThe first argument of the LINEST function specifies the y values, and the second argument\nspecifies the\n\nx\n\nvalues, the least squares estimates are based on. In our case, we thus need to\n\nspecify:\n=\n\nLINEST(B2:B41,A2:A41)\n\nThe LINEST function creates a table where it stores the least squares estimates in Excel memory.\nIt first reports the slope coefficient estimate, and then the intercept coefficient estimate. So, if we\nwere to look into Excel memory, the estimates would be reported as shown below:\ncolumn 1\n\ncolumn 2\n\nrowl\nWe nest the LINEST function in the INDEX function to get the estimated coefficients, one at a\ntime. The INDEX function returns values from within a table. In the case of a table with only one\n\nrow, the INDEX function general syntax is as follows:\n\n=\n\nINDEX(table of results, column_num)\n\n50\n\nChapter 2\n\nThe first argument of the INDEX function specifies which table to get the results from. In our\ncase, this is the table of results generated by the LINEST function above. So, we replace \"table of\nresults\" by \"LINEST(B2:B41,A2:A41)\". The second argument indicates from which column of\nthe table to retrieve the result of interest to us. So, if we want to retrieve the estimate of the\nintercept coefficient, b1, from the table above, we would indicate that it can be found in column 2\nby replacing \"column_num\" by \"2\".\nWe are going to report our estimated coefficients at the bottom of our table. In cell A43,","date":"2019-10-19 09:33:24","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\": 0.34148460626602173, \"perplexity\": 10689.951545179209}, \"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-2019-43\/segments\/1570986692723.54\/warc\/CC-MAIN-20191019090937-20191019114437-00543.warc.gz\"}"}
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Please consider a tax-exempt financial donation to enable us to continue our effort to decrease homelessness and provide low-income housing to the Topeka Community.
We also accept donations of various household items that are in good condition, as well as in-kind donations of services.
If you would like to visit with us regarding a donation, please contact us.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,485
|
<?php
class Domain extends DataObject
{
const PRIMARY_KEY = 'id';
const NAME = 'name';
public static function getDatabase()
{
return '526592_crawler';
}
public static function getTable()
{
return 'domain';
}
public function getName()
{
return $this->get_field(self::NAME);
}
public function setName($name)
{
$this->set_field(self::NAME, $name);
}
}
?>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,087
|
package tehnut.resourceful.crops.client;
import net.minecraft.client.Minecraft;
import net.minecraft.client.renderer.ItemMeshDefinition;
import net.minecraft.client.renderer.block.model.IBakedModel;
import net.minecraft.client.renderer.block.model.ModelResourceLocation;
import net.minecraft.item.ItemStack;
import net.minecraftforge.client.model.ModelLoader;
import tehnut.resourceful.crops.core.RegistrarResourcefulCrops;
import tehnut.resourceful.crops.core.data.InfoOverride;
import tehnut.resourceful.crops.core.data.Seed;
import tehnut.resourceful.crops.item.ItemResourceful;
@SuppressWarnings("ConstantConditions")
public class ResourcefulMeshDefinition implements ItemMeshDefinition {
private final String baseName;
public ResourcefulMeshDefinition(ItemResourceful item) {
this.baseName = item.getBaseName();
ModelLoader.registerItemVariants(item, new ModelResourceLocation(item.getRegistryName(), "inventory"));
for (Seed seed : RegistrarResourcefulCrops.SEEDS) {
InfoOverride.ModelInfo modelInfo = seed.getOverrides().getModel(baseName);
if (modelInfo != null)
ModelLoader.registerItemVariants(item, new ModelResourceLocation(modelInfo.getPath(), modelInfo.getVariant()));
}
}
@Override
public ModelResourceLocation getModelLocation(ItemStack stack) {
if (!(stack.getItem() instanceof ItemResourceful))
return new ModelResourceLocation(stack.getItem().getRegistryName(), "inventory");
Seed seed = ((ItemResourceful) stack.getItem()).getSeed(stack);
InfoOverride.ModelInfo modelInfo = seed.getOverrides().getModel(baseName);
if (modelInfo == null)
return new ModelResourceLocation(stack.getItem().getRegistryName(), "inventory");
ModelResourceLocation location = new ModelResourceLocation(modelInfo.getPath(), modelInfo.getVariant());
IBakedModel model = Minecraft.getMinecraft().getRenderItem().getItemModelMesher().getModelManager().getModel(location);
if (model == null || model.getClass().getCanonicalName().contains("FancyMissingModel"))
return new ModelResourceLocation(stack.getItem().getRegistryName(), "inventory");
return location;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,764
|
Олександр Олексійович Байдашніков ( ) — український зоолог, малаколог, фахівець з наземних молюсків, кандидат біологічних наук (1985). Автор близько 50 наукових праць, зокрема брав участь у створенні Червоної книги України (1994 і 2009). Описав два нових для науки види наземних равликів з України.
Життєпис
1977 року закінчив Київський педагогічний інститут. Ще під час навчання у 1976 році влаштувався на роботу лаборантом у Інститут зоології АН УРСР, де працював під керівництвом відомого зоолога В. І. Монченка. Згодом був направлений останнім у цільову аспірантуру в Зоологічний інститут АН СРСР (Ленінград), де навчався протягом 1981—1984 років під керівництвом відомого радянського малаколога І. М. Ліхарева. 1985 року захистив кандидатську дисертацію на тему «Наземные моллюски Закарпатской области». З 1984 року продовжив працювати у очолюваному В. І. Монченком відділі фауни та систематики безхребетних Інституту зоології АН УРСР, до виходу на пенсію у 2015 році. Згодом повернувся до наукової роботи у відділі зоології (Зоологічний музей) Національного науково-природничого музею НАН України.
Найважливіші наукові праці
Байдашников А. А. Новый для науки наземный легочный моллюск из Восточных Карпат // Зоологический журнал. — 1985. — Т. 64, вып. 2. — С. 206—211.
Байдашников А. А. Наземные моллюски Закарпатской области и их распространение по основным ландшафтам и растительным сообществам // Труды Зоологического института АН СССР. — 1985. — 135. — С. 44–66.
Байдашников А. А. Зоогеографический состав и формирование наземной малакофауны Украинских Карпат // Зоологический журнал. — 1988. — 67 (12). — С. 1787—1797.
Байдашников А. А. Редкие наземные моллюски Украинских Карпат и пути их сохранения // Вестник зоологии. — 1989. — 3. — С. 37–41.
Байдашников А. А. Вертикальное распределение наземных моллюсков Украинских Карпат // Вестник зоологии. — 1989. — 5. — С. 55–59.
Байдашников А. А. Обзор моллюсков рода Mentissa (Gastropoda, Pulmonata) // Зоологический журнал. — 1990. — 69 (1). — С. 21–31.
Байдашников А. А. О видовой дивергенции моллюсков рода Mentissa (Gastropoda, Clausiliidae) // Вестник зоологии. — 1990. — 4. — С. 3–8.
Байдашников А. А. Восточно-европейские равнинные виды наземных моллюсков в фауне Горного Крыма // Вестник зоологии. — 1990. — 6. — С. 68–70.
Байдашников А. А. О происхождении моллюсков рода Mentissa // Вестник зоологии. — 1991. — 4. — С. 3–8.
Байдашников А. А. Наземная малакофауна Украинского Полесья. Сообщение 1. Видовой состав и связь моллюсков с растительным покровом // Вестник зоологии. — 1992. — № 4. — С. 13–19.
Байдашников А. А. Наземные моллюски (Gastropoda, Pulmonata) заповедника Кодры (Молдова) // Вестник зоологии. — 1993. — № 4. — С. 10–15.
Байдашников А. А. Наземная малакофауна Украинского Полесья. Сообщение 2. Формирование современных малакокомплексов // Вестник зоологии. — 1996. — № 3. — С. 3–13.
Байдашников А. А. Наземные моллюски (Gastropoda, Pulmonata) заказника Переладино (Подольская возвышенность) // Вестник зоологии. — 2000. — 34 (6). — С. 99–100.
Байдашников А. А. Наземные моллюски (Gastropoda, Pulmonata) заповедника «Медоборы» (Подольская возвышенность) // Вестник зоологии. — 2002. — 36 (2). — С. 73–76.
Байдашников А. А. Морфологическая связь замыкательного аппарата с формой раковины Clausiliidae (Gastropoda, Pulmonata) // Вестник зоологии. — 2003. — 37 (1). — С. 61–78.
Байдашников А. А. Морфологические предпосылки стенобионтности Clausiliidae (Gastropoda, Pulmonata) // Вестник зоологии. — 2003. — 37 (6). — С. 49–63.
Байдашников А. А. Внутривидовая изменчивость у некоторых видов Clausiliidae (Gastropoda, Pulmonata) под влиянием условий обитания // Вестник зоологии. — 2005. — 39 (5). — С. 37–47.
Байдашников А. А. Изменчивость наземных моллюсков крымского рода Mentissa (Gastropoda, Pulmonata, Clausiliidae) // Вестник зоологии. — 2006. — 40 (4). — С. 297—310.
Байдашников А. А. Внутривидовая изменчивость видов рода Vestia (Gastropoda, Pulmonata, Clausiliidae) в Украине // Вестник зоологии. — 2007. — 41 (4). — С. 291—304.
Балашов И. А., Байдашников А. А. Наземные моллюски (Gastropoda) лесостепного Приднепровья и их фитоценотическая приуроченность // Вестник зоологии. — 2010. — 44 (4). — С. 309—316.
Балашов И. А., Байдашников А. А. Наземные моллюски (Gastropoda) Винницкой области и их биотопическая приуроченность // Вестник зоологии. — 2012. — 46 (1). — С. 19-28.
Балашов И. А., Байдашников А. А. Романов Г. А., Гураль-Сверлова Н. В. Наземные моллюски Хмельницкой области (Подольская возвышенность, Украина) // Зоологический журнал. — 2013. — 92 (2). — С. 154—166.
Балашов И. А., Байдашников А. А. Наземные моллюски редколесий можжевельника высокого в Крымских горах // Зоологический журнал. — 2013. — 92 (3). — С. 257—263.
Описані види
Prostenomphalia carpathica Baidashnikov, 1985 — простеномфалія карпатська, ендемік Східних Карпат, вид занесений до Червоної книги України з 1994 року.
Mentissa velutina Baidashnikov, 1990 — ментіса оксамитова, ендемік Кримських гір.
Вшанування
1992 року, за матеріалами зібраними О. О. Байдашніковим у Карпатах, було описано новий для науки рід дрібних амфібіотичних равликів, типовий вид якого назвали на честь Олександра Олексійовича: Terrestribythinella baidashnikovi Sitnikova, Starobogatov & Anistratenko, 1992.
Примітки
Посилання та джерела
(містить коротку довідку про О. О. Байдашнікова, сс. 56-58)
Сторінка О. О. Байдашнікова у Google Scholar
Відділ фауни та систематики безхребетних Інституту зоології НАН України
Українські зоологи
Автори зоологічних таксонів
Кандидати біологічних наук України
Науковці Інституту зоології НАН України
Науковці Національного науково-природничого музею НАН України
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"redpajama_set_name": "RedPajamaWikipedia"
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By Gdl2011,
October 14, 2008 in Pan Am Games / Bids
Cuba seems to be a bit decadent to me. They were really a sports power in recent past, but since Rio 2007, we see Cuba losing positions in multi-sports events...
Do they have boxing in the Pan-Ams though? At the Olympics, they always seem to shoot up dramatically on the last few days when the boxing medals are decided.
Victor Mata 272
Victor Mata
Alignment: Chaotic Neutral
Location:MCZ-AL-BR
Interests:Olympic Games + Literature + Video Games + Food
Yes, they do. The PanAms have the same sports as the Olympics plus a few others like squash. And that's exactly what happened in Rio 2007. Cuba was oscilating between the 2nd and 3rd position - behind Brazil - in the medals table. During the second week they grabbed more golds than Brazil. But I agree they're suprisingly slow this time...
KRATK 44
KRATK
Location:Santiago de Chile
First Chilean gold. Well done Kristel
The same with Canada, at least in PanAms.
Can Brazil emerge as the second sports power in the Americas?
The reason why Brazil is doing so well is because Canada has not sent its A swimming team, but its C team. Not to make any excuses. Canada swimming prefers to use the meet for developing swimmers. If Canada had shown up with top swimmers then I would have guessed a medal count around 18-20, with a fair chunk coming off from Brazil. Add to the fact Canada's athletics team only has one medal hopeful Canada has in essence taken itself out of contention in 34+47 (81 events almost a quarter of events.)
I think it depends on the sport. In track and field the second power in the Americas are the Jamaicans with the likes of Usain Bolt, Veronica Campbell Brown and Yohan Blake. The powers in the Americas in Track and field are:
1. The United States
2. Jamaica
3. Trinidad and Tobago
And then followed by others such as Barbados, Puerto Rico, Dominican Republic, Grenada, Panama, Brazil and the Bahamas. The top two won't ever change. The other three are interchangable. Brazil fits into the latter category, although could easily vault into the top five with multiple track and field medalists.
Cuba is still somewhat a threat in track and field. Dayron Robles and Yarelis Savigne just had bad world championships. Robles could have been a world champion had he not been DQ'ed.
Brazil has one world champion at least, which is more then Canada.
And some strong relay squads.
I just wished Canada had sent their A team here. The swimmers are competing in world cups now, instead of here.
Just looked at the table for this year's FINA World Championships - I didn't realise that Brazil had come fourth on the table. Brazilian swimming's certainly soared in the past few decades.
With their drug cheat Cesar Ceilo just kidding.
DannyelBrazil 703
Proud to be Brazilian.
Location:Born in Rio de Janeiro, living in Sao Paulo
Interests:Brazil, Rio de Janeiro and to make friends around the world.
Does we also have a rivalry with Canada in Swimming??? I thought it was only about jets and beef!!!
Brazil is not a power in track and fields, but PanAms after PanAms, we are getting more and more medals.
At this same point in Rio 2007, with all home-advatage, Brazil had barely 8 gold medals and "fighting" with Cuba for 2nd position. This year we are already in 12 and counting... And Cuba is a bit behind. (A article in Brazilian media pointed this as, maybe, the best PanAms ever for Brazil, if the trend continues).
Wait and see.
At this point I would say they are tied. Brazil one three medals in Shanghai, but only one in Olympic events. Canada won three in Olympic events and had overall team depth. Hard to say.
In swimming: Canada ahead, your all-time history is better than Brazilian one.
In jets: Brazil ahead, Embraer left Bombardier eating dust in the last 10 years...
Canada defeated Argentina 1-0
The Telmex Athletics Stadium, which is set to host the Pan American Games track and field event here, has been officially opened by Pan American Sports Organisation (PASO) President Mario Vázquez Raña at a special ceremony just three days before the athletics competition begins.
Some of the sport's top athletics stars, such as Jamaica's Usain Bolt and America's Tyson Gay, will not be representing their teams in at the venue here largely due to the fact that the Games are taking place so late in the year.
After the Pan American Games, the stadium will become a multipurpose venue hosting sports including athletics and baseball, as well as music concerts.
http://www.insidethegames.biz/sports/summer/athletics/14633-telmex-athletics-stadium-opened-just-three-days-before-hosting-athletics-at-pan-american-games
Canada has qualified a synchro. swimming duet and Mo Zhang has qualified in table tennis.
The latter yes, the former no.
The Americas Team champion qualifies as the continental team AND duet. If Canada was to completely implode tomorrow in the team final, the winner (the Americans likely) would get the two berths and the Canadian team would have to go qualify for the synchro events through the test event in London, which would be highly unlikely with the Japanese, Spanish and Russians having to qualify through that event.
Oh okay, but in synchro that is unlikely.
Canada has just won its first swimming gold!
Canada screwed up in Rio. So I will not stop worrying about it until the results are in.
really? wow.
Faster are you sure?
Its wrong, they're wrong. Qualfication system is the same as Beijing, just poorly worded.
The official qualification is worded such that it could be taken two ways, that there is seperate qualifcation or that the team gets the team and duet places. Because of previous qualifcation processes and the fact that a team will not compete without also competing with a duet indicates it is the single qualification. But there is no way to say this, unless FINA issues a clarification because the best duet and team at the world championships were the same (Australia, Egypt and China)
How I love Canada!!!!!!!!!!!!!!!!!!!!!!!!!
Thanks for eliminate Argentina from the soccer tournament! =]
Brazil has just won the female Volleyball tournament over Cuba.
One more gold for Brazil, keep (for too long time) the second place in medal table!!!
olymprefer 1
olymprefer
Location:Creel, Chihuahua, Mexico
Cuba has taken over Mexico, althiugh I think this Cuban delegation is not as strong as previous ones, Brazil has to do well from now, because they are going to host the major olympic event in 2016, and they can't or better shouldn't disappoint their people, it is like a tradition, we saw that in every and each olympic games, being the host brings more medals than usual, so go Brazil.
As for Mexico I think we are doing well, Sports has never been an important topic not only for governement but for people (except during 1968 olympics), but we got talent that's a fact, I see strong contenders in diving, probably some medals in boxing, beach volleyball, and othyers, onviously pelota vasca, we did it great in artistic gymanastics.
Canada has to improve for the next games bringing a b or c team shouldn't be an excuse, they have some good athletes, and some sad absences.
USA is doing well, still the best team in the americas, but when I was in beijing I saw that China is displacing USA, maybe less budget to Sports is the reason, or maybe the budget is still high but world competitiveness has increased too.
Montreal...
Oh God! I think I've just cursed Brazil for 2016!
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#include "TTCrossFadeInFunction.h"
#include <math.h>
#define thisTTClass TTCrossFadeInFunction
#define thisTTClassName "crossFadeIn"
#define thisTTClassTags "audio, processor, function"
TT_AUDIO_CONSTRUCTOR
{
setProcessMethod(processAudio);
setCalculateMethod(calculateValue);
}
TTCrossFadeInFunction::~TTCrossFadeInFunction()
{
;
}
TTErr TTCrossFadeInFunction::calculateValue(const TTFloat64& x, TTFloat64& y, TTPtrSizedInt data)
{
y = sin(x * kTTPi * 0.5);
return kTTErrNone;
}
TTErr TTCrossFadeInFunction::processAudio(TTAudioSignalArrayPtr inputs, TTAudioSignalArrayPtr outputs)
{
TT_WRAP_CALCULATE_METHOD(calculateValue);
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
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\section{Introduction}
The B-factory experiments Belle and \mbox{\sl B\hspace{-0.4em} {\small\sl A}\hspace{-0.37em} \sl B\hspace{-0.4em} {\small\sl A\hspace{-0.02em}R}} established CP violation in the neutral and charged B meson system, and experimentally confirmed
the source of CP violation to be one single complex phase in the three-family CKM quark mixing matrix~\cite{Belle2001,BaBar2001,Belle2004c,BaBar2004c}. The Belle and \mbox{\sl B\hspace{-0.4em} {\small\sl A}\hspace{-0.37em} \sl B\hspace{-0.4em} {\small\sl A\hspace{-0.02em}R}} experiments
provide an excellent environment to study heavy-flavor decays. Most importantly the clean environment and the Lorentz boost caused by the asymmetric-energy $e^+e^-$ colliders KEKB and PEP-II,
and the quantum entanglement of neutral B mesons produced by $\Upsilon(4S)$ decays enable measurements of observables sensitive to the fundamental symmetries CP, T and CPT.
At the conference Flavor Physics \& CP Violation 2014 the current most precise measurements sensitive to the breaking of these symmetries have been presented, and compared to previous related results.
In this proceedings article the presented measurements are summarized, and references for further reading are provided.
\section{CPT Measurement by Belle}
The invariance under CPT, the combined operation of charge-conjugation C, parity-transformation P and time-reversal T, is of fundamental importance in physics. All local Lorentz-invariant
quantum field theories are assumed to conserve CPT~\cite{Lueders,Pauli,Greenberg}. A breaking of CPT symmetry would have important consequences such as the possibility for violation of
Lorentz invariance, or differences in lifetimes and masses of particles and corresponding antiparticles. Searches for the violation of CPT symmetry have been carried out before
in the neutral kaon system by the CPLEAR, KLOE and KTeV collaborations~\cite{CPLEAR2001,KLOE2006,KTeV2011}, and in the neutral B meson system by the Belle and \mbox{\sl B\hspace{-0.4em} {\small\sl A}\hspace{-0.37em} \sl B\hspace{-0.4em} {\small\sl A\hspace{-0.02em}R}} experiments~\cite{Belle2003,BaBar2004a,BaBar2006}.
An observable sensitive to CPT violation in neutral B meson mixing is the complex parameter $z$, which is related to the light ($B_{L}$) and heavy ($B_{H}$) mass eigenstates and the complex mixing parameters $p$ and $q$ by:
\begin{eqnarray}
| B_{L} \rangle \propto& p \sqrt{1 - z} | B^{0} \rangle | + q \sqrt{1 + z} | \bar{B}^{0} \rangle \nonumber \\
| B_{H} \rangle \propto& p \sqrt{1 + z} | B^{0} \rangle | - q \sqrt{1 - z} | \bar{B}^{0} \rangle \nonumber
\end{eqnarray}
In 2012, Belle performed a measurement of the CPT violating parameter $z$ based on a data sample containing $535 \times 10^6$ $B\bar{B}$ pairs collected on the $\Upsilon(4S)$~\cite{Belle2012}.
The time-dependent analysis utilizes the coherent mixing of two entangled neutral B mesons in an $\Upsilon(4S)$ event~\cite{BaBar2004a,BaBar2004b}. The measurement reconstructs one B meson either in CP eigenstates,
or in flavor-specific hadronic or semileptonic decay modes. The reconstructed CP modes $B^0 \ensuremath{\rightarrow}\xspace J/\psi K^{0}_{S}$ and $B^0 \ensuremath{\rightarrow}\xspace J/\psi K^{0}_{L}$ provide sensitivity mainly on $Re(z)$, while others are sensitive to $Im(z)$.
Besides the measurement of $Re(z)$ and $Im(z)$, the analysis extracts $\Delta \Gamma_{d} / \Gamma_{d}$ from an unbinned maximum likelihood fit of the reconstructed data containing about $560k$ events
to the full decay rate of the entangled B meson pairs in an $\Upsilon(4S)$ event. The obtained results are:
\begin{eqnarray}
Re ( z ) =& \left[ +1.9 \pm 3.7 \, (\rm{stat}) \pm 3.3 \, (\rm{syst}) \right] \times 10^{-2} \nonumber \\
Im ( z ) =& \left[ -5.7 \pm 3.3 \, (\rm{stat}) \pm 3.3 \, (\rm{syst}) \right] \times 10^{-3} \nonumber \\
\frac{\Delta \Gamma_{d}}{\Gamma_{d}} =& \left[ -1.7 \pm 1.8 \, (\rm{stat}) \pm 1.1 \, (\rm{syst}) \right] \times 10^{-2} \nonumber
\end{eqnarray}
The results are consistent with the assumption of no CPT violation, and provide the current most precise estimation of observables sensitive to CPT violation in the neutral B meson system.
\section{T Violation Measurement by \mbox{\sl B\hspace{-0.4em} {\small\sl A}\hspace{-0.37em} \sl B\hspace{-0.4em} {\small\sl A\hspace{-0.02em}R}}}
To compensate for CP violating effects the invariance under CPT of Standard Model physics processes predicts T violating phenomena.
To observe T violation by rate comparisons, initial and final states of physical processes need to be exchanged.
Performing this exchange by the reversal of reactions of unstable particles is difficult to realize in experiments. In 2012, \mbox{\sl B\hspace{-0.4em} {\small\sl A}\hspace{-0.37em} \sl B\hspace{-0.4em} {\small\sl A\hspace{-0.02em}R}} performed a measurement utilizing the coherent mixing of
two neutral B mesons in an $\Upsilon(4S)$ event
to probe for time-reversal invariance by measuring the rate differences of processes with exchanged initial and final states~\cite{BaBar2012}.
The measurement follows an idea by Banuls and Bernabeu~\cite{Banuls1999, Banuls2000} and applies a procedure called CP tagging in analogy to the flavor-tagging in standard time-dependent CP violation measurements.
This enables the inference of the CP state of one B meson at the instant of decay of the second B meson.
The approach allows the construction of proper states and the measurement of the rates of reactions related by T transformation.
For example, the transition of $\bar{B}^{0} \ensuremath{\rightarrow}\xspace B_{-}$ where $B_{-}$ denotes a CP-odd eigenstate is identified by the time-ordered final states $(l^{+}, J/\psi K^{0}_{S})$.
The related T reversed process $B_{-} \ensuremath{\rightarrow}\xspace \bar{B}^{0}$ is obtained by the final states $(J/\psi K^{0}_{L}, l^{-})$.
The measurement provides three further independent rate comparisons between $B_{+} \ensuremath{\rightarrow}\xspace B^{0}$ $(J/\psi K^{0}_{S}, l^{+})$, $\bar{B}_{0} \ensuremath{\rightarrow}\xspace B_{+}$ $(l^{+}, J/\psi K^{0}_{L})$, and
$B_{-} \ensuremath{\rightarrow}\xspace B^{0}$ $(J/\psi K^{0}_{L}, l^{+})$ and their corresponding T-conjugated transitions. Any difference in the rates of these pairs manifests the breaking of T symmetry.
Furthermore the measurement needs no assumptions about CP violation or CPT conservation as input, but measures asymmetry parameters sensitive to CP and CPT in addition to that sensitive to T violation.
For the main T violating parameters, \mbox{\sl B\hspace{-0.4em} {\small\sl A}\hspace{-0.37em} \sl B\hspace{-0.4em} {\small\sl A\hspace{-0.02em}R}} measures using a data sample containing $468 \times 10^6$ $B\bar{B}$ pairs collected on the $\Upsilon(4S)$:
\begin{eqnarray}
\Delta S^{+}_{T} =& -1.37 \pm 0.14 \, (\rm{stat}) \pm 0.06 \, (\rm{syst}) \nonumber \\
\Delta S^{-}_{T} =& 1.17 \pm 0.18 \, (\rm{stat}) \pm 0.11 \, (\rm{syst}) \nonumber
\end{eqnarray}
The result directly observes T violation with a significance of $14\sigma$, and is in agreement with the assumption of CPT conservation and CP violation.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\textwidth]{figure_HFAG_average.pdf}
\caption{Comparison and average of $A_{SL}^{s}$ and $A_{SL}^{d}$ measurements provided by the Heavy Flavor Averaging Group (HFAG)~\cite{HFAG}.}
\label{fig:HFAG_average}
\end{figure}
\section{CP Violation in Mixing Measurement by \mbox{\sl B\hspace{-0.4em} {\small\sl A}\hspace{-0.37em} \sl B\hspace{-0.4em} {\small\sl A\hspace{-0.02em}R}}}
CP violation in neutral B meson mixing provides a sensitive probe for models beyond the Standard Model. New particles might emerge in the processes mediated by box diagrams and modify the mixing asymmetry defined by~\cite{Lenz2007}:
\[
A_{SL}^{q} = \frac{ P \left( \bar{B}^{0}_{q} \ensuremath{\rightarrow}\xspace B^{0}_{q} \left(t\right) \right) - P \left( B^{0}_{q} \ensuremath{\rightarrow}\xspace \bar{B}^{0}_{q} \left(t\right) \right) }{ P \left( \bar{B}^{0}_{q} \ensuremath{\rightarrow}\xspace B^{0}_{q} \left(t\right) \right) + P \left( B^{0}_{q} \ensuremath{\rightarrow}\xspace \bar{B}^{0}_{q} \left(t\right) \right) }
= \frac{1 - |q/p|^{4}}{1 + |q/p|^{4}} \approx \frac{|\Gamma_{12}^{q}|}{|M_{12}^{q}|} \sin \phi_{q}
\]
The Standard Model predictions for the neutral $B_d$ meson system are $A_{SL}^{d} = (1.8 \pm 0.3) \times 10^{-5}$ and $\phi_{d} = (0.24 \pm 0.06)^{\circ}$~\cite{Nierste2012}. The current achievable experimental sensitivity
on $A_{SL}^{q}$ is $\mathcal{O}(10^{-3})$. Any significant deviation from 0 might point to possible beyond the Standard Model processes. An important motivation for measurements sensitive to CP violation in B mixing comes from the tension driven
by the results of the D0 experiment on the inclusive like-sign dimuon charge asymmetry~\cite{Dzero2014}.
In 2013, \mbox{\sl B\hspace{-0.4em} {\small\sl A}\hspace{-0.37em} \sl B\hspace{-0.4em} {\small\sl A\hspace{-0.02em}R}} performed a search for CP violation in $B^{0}$-$\bar{B}^{0}$ mixing using a data sample containing $468 \times 10^6$ $B\bar{B}$ pairs collected on the $\Upsilon(4S)$~\cite{BaBar2013}.
The measurement reconstructs one B meson as $B^{0} \ensuremath{\rightarrow}\xspace D^{*-} X l^{+} \nu$ by applying a partial reconstruction technique for $D^{0}$ mesons from $D^{*-} \ensuremath{\rightarrow}\xspace D^{0} \pi^{-}_{\mathrm{soft}}$ decays.
The flavor of the reconstructed B meson is inferred from the lepton charge, and the flavor of the second B meson is inferred from its decay products by employing a kaon tag.
A difficulty of the measurement is the correction for effects induced by physical and detector charge asymmetries. The strategy of the measurement is to avoid relying on control samples or Monte Carlo simulations, and to
estimate these effects directly from data by exploiting all available information from different reconstructed sub-samples accounting for electron or muon leptons, and mixed or unmixed neutral B events.
The signal fraction is estimated from a fit to distributions of the missing neutrino mass, and a yield of $(5.945 \pm 0.007) \times 10^6$ signal events obtained. The $A_{SL}^{d}$ asymmetry is obtained
by a binned four-dimensional fit to the $\Delta t$, $\sigma(\Delta t)$, $\cos(\theta_{lk})$ and $p_k$ distributions. The result of the measurement is:
\begin{eqnarray}
| \frac{q}{p} - 1 | =& ( -0.29 \pm 0.84 \, (\rm{stat}) \, ^{+1.88}_{-1.61} \, (\rm{syst}) ) \times 10^{-3} \nonumber \\
A_{SL}^{d} =& ( 0.06 \pm 0.17 \, (\rm{stat}) \, ^{+0.38}_{-0.32} \, (\rm{syst}) ) \% \nonumber \nonumber
\end{eqnarray}
The measurement provides the most precise single result on $A_{SL}^{d}$, and is in agreement with the Standard Model predictions. A comparison of results from $e^+e^-$ and hadron colliders provided by the Heavy Flavor Averaging Group is shown in Figure~\ref{fig:HFAG_average}.
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{"url":"http:\/\/www.thisweekinstupid.com\/category\/economics\/page\/2\/","text":"Categories\n\n## Why your otherwise smart professor is a socialist\n\nI saw a video the other day from the American Enterprise Institute about the morality of capitalism. Capitalism, to paraphrase, clears access to the satisfaction that comes from achieving something. Being given the same thing brings us far less happiness. Government, then, takes something from someone to whom it brings a lot of joy and gives it to someone to whom it brings very little. Further, it removes the motivation for those receiving welfare to seek the joy of production and achievement. Yuck! How can we be so heartless?\n\nIt occurred to me as I was watching that, while I\u2019ve achieved many things in my life, the American Enterprise Institute might scoff that them.\u00a0 You see, I work for big companies or worse, universities, where I do research that never makes the New York Times and won\u2019t be featured in a product next year\u2013or next decade. Even the work I do for private companies is often funded by public grants given either to my company or to our customers. Like the villains in the video, I\u2019m often not pleasing \u201ccustomers\u201d so much as the government committees who review grant applications.\n\n### Gittin\u2019 \u2018er done, collectively\n\nWhen I achieve things, I share credit with thousands of people. Can this collective achievement be as satisfying or valuable as the individual achievement described in the video? Your professor and I think so. We are used to being a small cog in an absolutely enormous machine and we recognize that some things can only be done this way. My grandfather had a tiny role in the early flights of the Space Shuttle\u2013a very big deal that improves your life whenever you turn on your GPS or check the weather report. But if you listed the contributors to that project, Grandpa would surely be buried somewhere in the back with the dolly grip and craft services. He didn\u2019t mind that at all. Whether it\u2019s better play a small part in our mission to space or a large part in a Jamba Juice franchise can certainly be debated.\n\nThese collective works we\u2019re about are far-reaching and critically important. One day we\u2019ll announce cold fusion and a cure for cancer, saving the planet and literally millions of lives. Someone who put together the last piece of the puzzle will be on the cover of Time. But behind her, there\u2019ll be legions of scientists who will sit back in their easy chairs with a self-satisfied grin, knowing they\u2019ve done good work and ever so glad they didn\u2019t take their uncle\u2019s advice to drop this ivory tower nonsense and become a day trader.\n\nCategories\n\n## Spontaneous order is always awesome\n\nAs I take aim at Friedrich Hayek, on a site called thisweekinstupid, I do it with some trepidation. Hayek was a well-spoken, skilled and innovative economist. That doesn\u2019t mean he didn\u2019t occasionally get it wrong. And in the unfortunate case I\u2019ll discuss today, Hayek is found contributing to a potent and damaging piece of stupid that characterized much of the late 20th century\u2013the cult of the invisible hand.\n\n### Pros and cons of spontaneous order\n\nIn the 1950s natural sciences like physics and especially biology began to notice that large systems made from simple parts could work together to create surprising and miraculous results. The brain is the most exciting example of this.\u00a0Although some\u00a0neurologists will\u00a0likely disagree, the dynamics of a single neuron are simple. On receiving a pulse of energy from a nearby neuron through its dendrites, it sends a pulse to other neurons through its axon. This pulse is then received by the dendrites of other neurons. No one would look at that simple system and guess that a collection of those interactions would produce human thought. That miracle of complex macrodynamics from a multiplicity of simple microsystems is what Hayek called \u201cspontaneous order.\u201d Hayek and others believed fervently in the power of spontaneous order to improve people\u2019s lives. Hayek called it a \u201cfatal conceit\u201d to imagine that a designed system could match a spontaneously ordered system for efficiency.\n\nDuring the Goldwater\/Reagan revolution, this became the justification for opposing government economic interference in almost any form. Any top-down tweaking by government moves the economy away from the spontaneous order, which is assumed to be the most efficient possible. It was also a convenient defense against the primary ideological foe of the United States\u2013the Soviet Union. To those of the Austrian school, the economic failure of the Soviet Union was definitive proof of Hayek\u2019s idea.\n\nBut on closer examination, the assumption that spontaneous order is always elegant or beneficial seems to come from nowhere, and certainly not from any of the natural sciences. As we look at other examples, we find spontaneous order is, indeed, powerful. But sometimes spontaneous order can be fatal. A herd of cattle can be thought of as a complex system made of simple parts. We could describe the behavior of cows quite simply:\u00a0Move toward\u00a0grass; avoid obstacles.\u00a0But, spurred by the wrong external stimulus, those simple dynamics can cause a stampede as one cow starts to run enticing others to run to get out of its way. Here, the order that arises spontaneously is certainly unexpected in that it does not follow in a straightforward way from the micro behavior. In this, a herd of cattle is like a snowflake or a brain or an ecosystem.\u00a0But in the case of cows, the macro behavior is not\u00a0beneficial. Although the microdynamics were about avoiding injury, the resulting stampede can cause cattle to be trampled and killed.\n\n### Spontaneously ordered transportation\n\nSo, which kind of spontaneous order is our modern economy? Here\u2019s modern-day libertarian John Stossel extolling spontaneous order\u00a0and its wisdom in leading\u00a0America away from transportation by train in favor of cars.\n\nAt last month\u2019s State of the Union, President Obama said America needs more passenger trains. How does he know? For years, politicians promised that more of us will want to commute by train, but it doesn\u2019t happen. People like their cars. Some subsidized trains cost so much per commuter that it would be cheaper to buy them taxi rides.\n\nThe grand schemes of the politicians fail and fail again.\n\nBy contrast, the private sector, despite harassment from government, gives us better stuff for less money\u2014without central planning. It\u2019s called a spontaneous order.\n\nCars may be the right answer for many communities, but\u00a0transportation innovations can be a very clear example of the failure of spontaneous order.\u00a0That is to say, the order arises, it\u2019s just not helpful. Examine the problem of electric cars. My conservative friends have posted pictures to\u00a0Twitter and Facebook of four or five completely unused car charging stations, usually at government buildings. \u201cTypical government waste,\u201d they\u2019ll say.\n\nThey think the market has spoken, and maybe it has. But the other side of the story is that the least convenient\u00a0aspect of owning an electric car is finding a place to charge it.\u00a0This certainly reduces the number of electric cars on the road. When a car buyer (one simple part in our complex system)\u00a0is shopping,\u00a0she, hypothetically, considers an electric, but\u00a0since there are no charging stations where she works, she\u00a0decides on\u00a0internal combustion.\u00a0Meanwhile,\u00a0someone at her work\u00a0proposes installing charging stations in the parking lot. They take a stroll through the parking lot and find that very few employees own electric cars. So they decide against the charging stations. And around and around we go.\u00a0More electric cars\u00a0and more charging stations might be the optimal solution, but the individual actors, pursuing their own interest, can\u2019t get there. Certainly the company can take a chance and build the charging stations hoping more employees\u00a0are enabled to buy\u00a0the electric cars they want, but that\u00a0risk undeniably reduces the chance of us getting there.\n\nFor some other examples of the inefficiencies of spontaneously ordered system, check out my post on public goods.\n\n### Lessons from simulation science\n\nIn simulating complex systems, we call this a local minimum. Very often a\u00a0complex system can find itself in a configuration that is not the global best configuration, but from which any small change looks worse. This is a local minimum. When considering electric cars, the status quo (no electric cars and no charging stations) is better than either a) some electric cars with no charging stations or b) no electric cars and some charging stations. So each individual player sees it in their interest to stay right where they are.\n\nConsider this\u00a0ball\u00a0rolling on an odd-shaped surface.\n\nThe lowest potential energy configuration for the\u00a0ball\u2013the place it \u201cwants\u201d to be\u2013is at the bottom of the valley marked 3, but in some places on the curve, point 2 for example, the ball\u00a0sees a hill on either side. It\u2019s in a \u201cstable equilibrium.\u201d If I want to move the\u00a0ball to the true lowest energy state,\u00a0it needs a push up the hill. It needs to be moved toward higher energy in order to find a better state.\n\nOur electric car economy is the same (or might be). The economy of transportation is sitting at point 2. Everyone\u2019s myopic view tells them unilateral action is wasteful. Charging stations installed at libraries and government buildings are an attempt to push us up the hill to see if we\u2019ll fall into a better global minimum. It looks like \u201ctypical government waste\u201d because we\u2019re not looking at the whole curve. All we see is the hill in from of us. It might work or it might not. Only a global view could hope to\u00a0predict. But only a fool concludes that the order found organically is always best. In game theory, this kind of stable, non-optimal state is called a Nash equilibrium after John Forbes Nash, Jr. profiled in A Beautiful Mind.\n\nSimulating large systems is what I do for a living. Hayek didn\u2019t have the benefit of huge supercomputers to predict what complex systems will do, but from my experience, assembling a system of millions of interacting parts, turning it on and expecting it to organize itself into an optimal configuration in a reasonable time without any help from me is insanity.\u00a0When we want to optimize complex systems like static fluid flows or magnetic materials, we\u00a0have to\u00a0nudge them\u00a0to pop them out of local minima or steer them speedily through what would otherwise be a slow spiral toward optimality. We try\u00a0solving pieces of the problem independently, then stitching them together. Sometimes we reset things to an alternate starting configuration and see if that leads to a better place. (The economic implications of that should keep wealthy capitalists up nights). From where I sit, to expect something as complex as a national economy to optimize its resources without any help demonstrates profound ignorance of the dynamics of complex systems.\n\nWe should neither discount the power of spontaneous order, nor place unwarranted faith in it. That\u2019d be stupid.\n\nCategories\n\n## Governor Stupid: \u201cWarren Buffett should write a check and shut up\u201d\n\nWe turn our attention to Chris Christie, who, in truth, is my kind of Republican. But nobody\u2019s perfect. Back in 2012, he repeated a familiar refrain in the GOP, although he stated it\u00a0more forcibly than some. In response to Warren Buffett\u2019s revelation that he pays a lower tax rate than his secretary, Christie said\u00a0that Buffett\u00a0should \u201cjust write a check and shut up.\u201d That is, if Buffett wants to pay more taxes, he can. What Christie resents is that Warren Buffett\u00a0wants to force others to pay more taxes to support his favorite programs.\n\nWilliam F. Buckley once said something related\n\nLiberals,\u00a0 it has been said, are generous with other peoples\u2019 money, except when it comes to questions of national survival when they prefer to be generous with other people\u2019s freedom and security.\n\nLiberals, including thisweekinstupid, disagree. Once again, an ignorance or underestimation of a basic market failure causes us to talk past one another. Government agencies, including the Department of the Interior, NOAA, the FAA, the CDC, and the SEC, \u00a0produce what are called \u201cpublic goods\u201d\u2013goods that you don\u2019t have to own to enjoy. A more thorough discussion of public goods (now featuring math!) is in our first ever thisweekinstupid appendix.\n\n### Public goods for non-economists\n\nIn brief,\u00a0Investopedia defines a public good as a product or service that one individual can consume without reducing its availability to another individual and from which no one is excluded.\u2026 National defense, sewer systems, public parks and basic television and radio broadcasts could all be considered public goods.\n\nLeft to the free market, public goods will be under-produced. That is, the dollars I voluntarily spend on public goods return a benefit spread over a large group, so, I tend to lean toward spending on private goods, the benefit of which I can secure to myself. This is rational and efficient on a personal level, but irrational and inefficient on a societal level. This is known as the \u201cfree riders\u201d problem.\n\n### An example\n\nImagine five homes in a cul-de-sac. Each of the homes is for rent by a different landlord. A local landlord\u2019s association reports that a $30 per month lawn service increases the rental value of a property by, on average,$40. Further, $100 worth of landscaping of a median in the middle of the cul-de-sac increases the rent for each of the houses by the same$40. A house with a lawn service and a landscaped median can get $55 more than without any landscaping at all. What should I as a landlord, do? Considering only myself, a lawn service for my own lawn is a good deal, so I sign up. But, if I can get everyone to chip in, landscaping the median is even better. If everyone contributes, we can get the same benefit for only$20 each. Imagine a few of us try to pull the money together for a landscaped median. But, as good libertarians, we\u2019re not going to compel anyone to contribute. \u00a0Unfortunately, one owner sees the opportunity for a free ride. If she pays only for her own lawn and the rest of us pay for the median, she can make $55 more in rent for only$30 in lawn care. The rest of us are left with a choice. Pay $25 each to landscape the median, or pay$30 to landscape our own yards. The median is still a better deal, but less so than if we had enforced a contribution, for example through home owner\u2019s association dues. \u00a0Public goods work the same way in a larger community. In the absence of public funding for arts, parks, public highways, defense, etc. the temptation to free-ride causes us to spend less than is most efficient. Failing to properly value public goods and take steps\u00a0to produce them, wastes resources. Where public goods are concerned, government spending via taxation can lead to more wealth, efficiency and happiness (more in the appendix).\n\n### Public good skepticism\n\nIntelligent conservatives know this, of course and still want to spend less than me on many government programs. The disagreement is sometimes rooted in the different value conservatives and progressives assign to public goods. Environmental issues are an easy place to see this. In discussions with Republicans and Libertarians on this topic it becomes quickly clear that, compared to myself, they underestimate the differential benefit between an intact ecosystem and one damaged by oil exploration, air pollution or deforestation. This can be either because, compared with me,\n\n1. they underestimate the ecological damage; or\n2. they undervalue the things a healthy ecosystem provides.\n\nI find that it\u2019s often a combination of these. Climate change skeptics often don\u2019t think carbon emissions cause warming to a great extent, but even if it did, they don\u2019t think warming is so bad. Connecting it to the lawn care analogy, some owners might doubt that tenants care about a landscaped median. Or they might not trust the community to choose the right landscaper and deliver the promised value. Any one of these factors may cause them to undervalue the public good and reject the cooperative solution.\n\nAnd so it goes with taxation. Compared with me, Chris Christie underestimates our connectedness. I believe that undereducated or unhealthy children and impoverished families cause great harm to us all and so education, public health care\u00a0and anti-poverty programs are a public good of great value. Further, I believe we\u2019re economically connected enough that the presence of a public safety net is a great economic benefit to the whole country, and especially to the people with assets to lose.\u00a0If we removed the public safety net, some of us would put money in the collection plate to cover rent and medical care for the poor, but to get really rich, a better strategy would be to keep your money, buy cheap apartments and rent them to the poor, collecting the charity they\u2019re given by everyone else. Free riders win again.\n\n### Role reversal\n\nThe other half of William Buckley\u2019s statement concerns the military, which is also a public good. You can\u2019t let someone opt out of funding the military because protecting every third house from the Red Dawn is just as expensive as protecting the whole country. Opters-out become free riders. But, in the case of defense spending, Republicans simply value that public good higher than I do. I think our military is big and strong enough to defend our country from any (terrestrial) enemy several times over. I regard the marginal benefit of an additional dollar in the defense budget to be zero (possibly less). So, I\u2019m quite tired of Chris Christie and William Buckley and their set being quite so generous with my money toward military contractors.\n\nArmed with an understanding of public goods and free riders, liberals are not naive enough to remove public services, cut taxes, and wait for libertarians to plow their tax savings into private programs to keep the water clean, track diseases, make sure the planes don\u2019t crash, get the mentally ill off of the street, monitor trading of securities and pay the rent and medical bills of the elderly poor. On the contrary, sensible economic thinking says the market will be far too profligate with other people\u2019s health, air quality and safety.\n\nInstead, I propose the libertarians go first. As soon as there\u2019s a private program tracking the evolution of new diseases, we can defund the CDC. When private charity clinics provide the medical care for all the poor, there\u2019ll be no need for Medicaid. And the great thing about some of these programs is that no one need \u201cgo first.\u201d Shriners Hospital provides all kinds of free medical care to children. Every surgery in a Shriners Hospital is one that doesn\u2019t have\u00a0to be done elsewhere. If the child being treated is covered by Medicaid, that\u2019s savings and deficit reduction right there. In a very organic way, private charities could take over for public services. As I\u2019ve detailed above, I have my own reasons for believing it wouldn\u2019t happen that way, but I\u2019d love to be proven wrong. And this is certainly no less reasonable than the opposite proposition\u2013defunding these programs without a viable replacement and blithely hoping one will appear.\n\nSo, Governor Christie, if you want smaller government, call up the Shriners, \u201cwrite a check, and shut up.\u201d\n\nCategories\n\n## Appendix: Public Goods\n\nOur discussion of public goods seemed incomplete without at least a little math to back it up. But no one wants to alienate the mathophobes, so we parked this tidbit in the \u201cappendix.\u201d\n\nLet\u2019s analyze a few public goods. Imagine, again,\u00a05 farmers. Last year, one farmer contracted with a beekeeper to manage a hive of bees near his field. He paid $2500 for the year. The next year, his harvest increased by$4500. A farmer in a nearby neighborhood paid for even more hives and boosted his profits even more, although the first $2500 is the most effective in this regard. But he\u2019s not the only one who benefits. Each of this neighbors also increased their crop yields. After doing some research, he discovers that, in a community such as theirs, for each of your neighbors hiring a$2500 bee hive, you can get $1500 more crop even without spending a cent on your own bees. Mathematically, let\u2019s conjecture that the value of the increased yield is $\\sqrt{\\alpha S_0 + \\sum \\beta S_i}$ and the profit from that extra spending is $\\mbox{Profit}= \\sqrt{\\alpha S_0 + \\sum \\beta S_i} \u2013 S_0$ where$latex S_0$is the amount I spend on bees and$latex S_i$are my neighbors\u2019 spending. The constant (exogenous) parameters$latex \\alpha$and$latex \\beta$control just how much benefit I get from my own and from my neighbors\u2019 spending, respectively. The reason for the square root is the idea of \u201cdiminishing marginal utility.\u201d Spending$6000 on bees is still better than spending $3000, but something less than twice as good. Your first dollar is more important than any subsequent dollar. That said, the utility curve does not have to be a square root. Now, if all of the neighbors are equally interested in their property values, they might agree to all spend the same on bees, perhaps in a written agreement. If we constrain all of$latex S_i$to be the same value, we have a formula of just one variable and can plot the profit function: From the plot, we can see that, if the neighbors all cooperate, they can each get$6000 more crop from spending just $3000 on bees. This is the most efficient level of spending. But here we run into the free rider problem. Suppose the neighbors do not cooperate and one of the neighbors does the calculation without considering anyone else. If everyone else continues spending$3000, how much should I spend to maximize my profit? When we fix $latex S_i$ at $3000 and allow just$latex S_0$to vary, the new plot looks like this: The neighbors cooperating can make a$3000 profit, but from the blue curve we\u00a0 see that the selfish neighbor can spend just $500 on bees and make a$3500 profit. In this situation, the self-interest of the individual hinders the prosperity of the community.\n\nThe situation gets even worse for goods which are more public. Perhaps the road leading between the town and the farms needs fixing and they\u2019re freestaters, so the government won\u2019t do it. They\u2019re on their own. This fix will reduce wear on the trucks that take crops to the markets, etc.. Mathematically, $latex \\alpha = \\beta$. In that case the plot looks like this:\n\nA free-rider, in this case, would have almost no incentive to contribute. That\u2019s not exactly true since even the free riders understand they\u2019re gaming the system and that others will be inclined to seek the same deal. When you incorporate this, you find that, in fact, neighbors are willing to spend just $200 on paving, leaving$3200 of profit on the table. (For more details on this aspect, check out the appendix to the appendix, Galt-ifying Public Goods.) The problem only gets worse as the community grows. If you\u2019re paving a street that benefits 20 people with the same utility function, people are willing to contribute just $47 and will miss out on more than$20,000 profit.\n\nPublic goods are not always so easily quantified and it\u2019s usually there that disagreements arise, but a little math goes a long way toward appreciating that opinions about public spending are a continuum and that pretending the market is always right (or always wrong) has real consequences.\n\nNot enough math yet? Check out the appendix to the appendix, where we ask the mathematical question, how much more efficient would John Galt\u2019s community (from the novel Atlas Shrugged) have to be to make up for refusing to subsidize public goods.\n\nCategories\n\n## The Market Will Set the Right Price for Health Care\n\nI don\u2019t suppose the interweb needs another discussion of the failings of market forces in health care markets. And yet the idea persists that setting the price of health care should be left to the unregulated market and that this will lead to maximum efficiency. The omnipotence of the market is a comforting concept.\n\nDemand-side troubles\n\nBut the market for anti-retroviral drugs bears little resemblance to golf clubs or top sirloin. At risk of sounding condescending, a typical demand curve looks like this\n\nAs the price of a good or service rises, the quantity demanded shrinks. Lots of people will buy a pair of\u00a0basketball shoes for $6. A few would pay$140.\n\nBut this doesn\u2019t often work for medical services. I recently had my appendix removed. I\u2019m told a typical appendectomy costs $25,000\u2013a lot of money no matter who you are. But whether it had cost$100 or $100,000, I still would have bought exactly one appendectomy. The demand curve for appendectomies looks more like this: There is, perhaps, some fall in demand. A 96-year-old might decline a quarter million dollar appendectomy. In some cases, there may be more than one treatment for an ailment so that one good or service may be substituted for another. But, in general, demand for health care services, especially the very expensive, life-saving treatments, tends to be highly inelastic. The quantity demanded is affected very little by the price. So, can providers of health care charge whatever they like? Supply-side salvation? Not necessarily. There may be some help on the supply side of this market. But, one reason markets are so effective and robust is the interplay of supply and demand. Without both working properly, the market can misallocate. When the supply side of the equation breaks down, for example in the case of a monopoly, we know that even with a healthy demand side, we\u2019ll run into inefficiencies. So, we already have cause to worry. A truly effective market needs both a healthy demand side and supply side. Switching to the supply side of things, if the price of a good is very high, more people will be willing to supply it. A supplier that overcharges will find herself undercut by a competitor willing to supply the good at a lower price. So, even with highly inelastic demand curves, there\u2019s an equilibrium price at the point where the supply and demand curves meet. So, it all might work out just fine, as long as actors can\u2019t manipulate the supply curve. Unfortunately, two of the easiest ways to do that are both, of necessity, highly active in health care markets. The first is patents. A patent grants a single company the exclusive right to produce a product for a certain period. A supplier with a patent cannot be undercut by a cheaper competitor. When you are the only supplier of a lifesaving procedure, the market will not place any limit on the price (although public opinion or your own morality might). For this reason, it\u2019s more useful to think about the supply curve for health care research than for particular treatments. If the payout for developing a new drug is very high, more people will be willing to do research toward that end. So, even where patents are applied, there is a functioning supply side curve at work. But in such a market, price signals can take longer to move through the market. When I need medication today, it\u2019s slim comfort to know that the exorbitant price I pay for patent-protected drugs is providing the impetus for a robust market for drug research. Another common way to move a supply curve is through licensing. Most people who treat you in the hospital are licensed, some of them very licensed, which is wonderful. It comforts me immensely that the person holding the scalpel has undergone years of training and scrutiny. But, the effect of this is to reduce the supply of doctors, nurses and other medical professionals. The FDA\u2019s approval processes provide the same type of scrutiny for medications, equipment and treatments, with the same effect. I wouldn\u2019t have it any other way, of course, but the effect of this licensing is to shift the supply curve downward, increasing the price of goods. The further the supply is reduced, the higher the price. The inelasticity of the demand curve (and also the supply curve) multiplies this effect. You can see how influence of these licensing processes could be very lucrative for suppliers. Doctors, for the most part, do important work for sincerely good reasons, but putting the AMA in charge of licensing doctors is a lot like asking the fox to guard the hen house. The tendency of almost everyone is to highly value their own work and the incentive for doctors is to limit the supply of doctors, raising their own salaries. Similarly, if pharmaceutical or medical supply companies can delay or scuttle approval for competing drugs and equipment, they also stand to make lots of money. I\u2019m not for a minute suggesting we do away with licensing of doctors or patents for drugs. Health care markets can\u2019t be effective without these things. But, perhaps it\u2019s a good idea to think hard about the markets for health care rather than blithely assuming the miraculous market will allocate everything just right. Without some advocates for consumers of health care, rising, inelastic demand will push prices out of reach and make life-saving care an unaffordable luxury. Categories ## Selfishness is Good For our first real post, we\u2019ll turn to a deeply entrenched conservative belief. You might expect this on a site called ThisCentury(and Last)InStupid. It\u2019s the idea that selfishness isn\u2019t so bad after all. In fact, we could use more of it. Occasionally, someone comes out and states this explicitly, but not usually. More often its the implied justification of cruel policy. Or it appears as an allegory as in Paul Ryan\u2019s favorite ode to greed, Atlas Shrugged. As will be familiar to many, Atlas Shrugged is a novel by uber-capitalist and self-professed narcissist Ayn Rand about a fictional America in which all the captains of industry tire of being disrespected, regulated, demonized and (most egregiously) taxed and decide to flee society to a secret mountain enclave and live together in capitalist utopia. The rest of the country, bereft of its \u201cengine\u201d grinds to a halt. Poverty and violence ensue. This vision is the engine behind Republican policies that, sometimes quite overtly, enrich the wealthy at the expense of everyone else. One fine recent example is the refusal of Republican Governors across the country to expand Medicaid in their states. Although it would cost their governments very little, and benefit the poor quite a lot, the principle of giving as little to the poor as you can get away with is so ingrained that they\u2019d rather leave the poor without health care and let them be treated in emergency rooms and the cost of their unpaid care absorbed into the premiums of the rest of the state. Or there\u2019s the fact that we can\u2019t close the \u201ccarried interest\u201d loophole by which the income of hedge fund managers is taxed as if it were long-term capital gains or dividends (on assets they don\u2019t own!). No matter how you feel about lower rates on capital gains, these are clearly income. So, you have people paying 15% on their multi-million dollar income. The people emptying their trash pay a higher rate. There are efforts at very high levels to fix this, but none very successful. Agitation against sensible policies to alleviate poverty are often accompanied by mumbled pseudo-economic arguments about how discouraging the money-making activities of the wealthy will harm us all or how giveaways to the poor create dependency. But, simmering beneath the surface of these attempts to make economics support fiscal austerity is an audacious Republican hope\u2013one buoyed by this Randian vision. The wish at the heart of modern conservatism is that selfishness, in the end, will turn out to be altruism\u2014that, when the score is all tallied, the best thing we can do for the poor is to stick it to them just as hard as we can. Thus conservative guilt is swept away by one swift stroke of Milton Friedman\u2019s pen. This miraculous moral alchemy whereby, if two wrongs don\u2019t make a right, nonetheless, two million do, is unchallenged on the Right since Adam Smith despite its manifest stupidity. This is partly because it\u2019s unstated but also because it\u2019s so damned useful. It was useful during the Cold War marriage of what today are called social and fiscal conservatism by severing the link between Christian charity and social welfare. It continues to be useful in justifying the neglect or animosity toward the social contract by the upper and middle classes today. It makes conservatism easier on the soul. Now, some conservatives will try to separate personal and public morality saying that while public welfare programs harm the poor, private programs are another matter altogether. Of course, the original source of the charity can\u2019t likely save the poor from the dependency-producing effects of free stuff. This view is likely more closely related to another pervasive dose of stupid up for discussion in a later post : \u201cThe government can\u2019t do anything right.\u201d So, you heard it here first: Selfishness isn\u2019t good. It isn\u2019t noble or necessary or even inevitable. It wrecks communities and nations and, in a cosmically ironic twist, makes it\u2019s practitioners the most miserable of all. Categories ## Appendix: Galt-ifying public goods ### See how deep the rabbit hole goes\u2026 In our first appendix, we wandered a little into the math behind public goods. That post has less algebra and more graphs. If what you read here is moving too fast, that might be a good place to start. Public goods, in brief, are goods which you don\u2019t have to own to enjoy. These are things like public parks, an army, or your neighbor\u2019s front lawn. In that post, we assumed a particular form for a utility function for public goods. $\\mbox{Utility} = \\sqrt{\\alpha S_0 + \\sum\\beta S_i}$ where$latex S_0$is the amount I spend on the good,$latex S_i$are my neighbors\u2019 spending. The parameters,$latex \\alpha$and$latex \\beta$determine how much benefit I get from my own spending and my neighbors\u2019 spending, respectively. There\u2019s more on why this specific functional form was chosen here. Other forms are certainly valid. In this model, the profit is the utility minus my expenditure,$latex S_0$$\\mbox{Profit} = \\sqrt{\\alpha S_0 + \\sum\\beta S_i} \u2013 S_0$. If an overbearing government (or home owners association, or business) enforced the same spending on public goods by all beneficiaries, we can determine the optimal spending by equating$latex S_0$and$latex S_i$and taking the derivative of profit with respect to this single variable: $\\frac{dP}{dS} = \\frac{\\alpha+(n-1)\\beta}{2\\sqrt{S(\\alpha+(n-1)\\beta)}} \u2013 1$ Here we\u2019ve introduced n, the total number of participants. Equating this derivative to zero, we find that the optimum level of spending is $S = \\frac{\\alpha + (n-1)\\beta}{4}$ for a profit of $P = \\frac{\\alpha+(n-1)\\beta}{4}$. per person. If, on the other hand, we leave it up to each individual to contribute what she thinks is best for herself, we differentiate the profit equation with respect to changes in$latex S_0$alone. $\\frac{\\partial P}{\\partial S_0} = \\frac{\\alpha}{2\\sqrt{\\alpha S_0+\\beta \\sum S_i}} \u2013 1$ Then, since all the participants are assumed to be 100% rational (and will therefore make the same decision), we equate$latex S_0$and$latex S_i$, set the derivative equal to zero and discover that spending becomes $\\frac{\\alpha^2}{4(\\alpha + (n-1)\\beta)}$ and profit is $\\frac{\\alpha}{2}-\\frac{\\alpha^2}{4(\\alpha+(n-1)\\beta)}$. In the other post on public goods,we reported that, for our simple model of paving the street leading to a cul-de-sac with 5 houses, our homeowners would only be willing, on their own, to spend$2 on paving rather than the ideal $50, which reduced their profit from$50 to $18 each. Now, let\u2019s introduce a \u201cGalt factor,\u201d G, representing the increased productivity of workers in John Galt\u2019s mountain enclave due to the absence of moochers. How high would the Galt factor have to be for the Galt society to have more efficient lawn care than us regular shmoes? Both$latex S_0$and$latex S_i$should be multiplied by G since the labor and other input in the Galt society is that much more potent. So, we multiply all$latex S_0$and$latex S_i$by G in the profit for the non-cooperative profit and equate that to the (unmodified) profit for the cooperative system. $\\frac{G\\alpha}{2} \u2013 \\frac{G\\alpha^2}{4(\\alpha+(n-1)\\beta)} = \\frac{\\alpha+(n-1)\\beta}{4}$ Solving for G, we find that the Galt factor necessary for the Galt society to operate more efficiently than a \u201cmoocher\u201d society which recognizes and subsidizes the public good in question is $\\frac{(\\alpha+(n-1)\\beta)^2}{\\alpha(\\alpha+2(n-1)\\beta)}$ Here are some sample results using numbers from the beekeeping and road paving examples in the last post. In the gardening example, most of the benefit of keeping up my lawn goes to me. Only a little goes to my neighbors which is why$latex \\alpha$is greater than$latex \\beta$in the first two rows. The final 3 rows are based on a good that\u2019s completely public\u2013everyone benefits equally no matter who paid. Last post we used the example of paving the road that leads to the cul-de-sac. Since everyone enjoys the benefits of the spending equally,$latex \\alpha=\\beta$.$latex \\alphalatex \\betalatex nlatex G$80 10 5 1.17 80 10 100 7.0 40 40 5 3.27 40 40 20 10.7 40 40 100 50.7 We could even simplify the above expression by constructing the \u201cpublic-ness\u201d of a good, P. $P = \\frac{(n-1)\\beta}{\\alpha+(n-1)\\beta}$ Public-ness ($latex P$) is the fraction of the total benefit that goes to the community as whole. The Galt factor in terms of$latex P$is $G= \\frac{1}{1-P^2}$ Now, if we had any reason to believe our model this would be a very useful formula. Should a particular service be left to the market or paid for with public funds? Just figure P, make a guess as to how much more efficient your private market would be and compare. For goods that are completely shared, P is just$latex 1- 1\/n$with$latex n$being the size of the community. For a cul-de-sac of 5, the Galt factor must be 25\/9 = 2.78. For a small town of 10,000, P = .9999 and G = 100,000,000. So, for this adorably naive model the Galt society could likely manage to outdo us in gardening, as long as their cul-de-sacs didn\u2019t get much bigger than 5 houses, but when it comes to truly public goods consumed by hundreds of people (like a public park or a security guard at your apartment complex) they\u2019ll be hard pressed to accomplish in one hour what takes the rest of us a full work week. For goods shared by hundreds of thousands of people, the Galters might as well not try. In many cases, at least some of the benefit can be secured exclusively for the benefit of the producer and her customers. For example, a road can be built with toll booths or the radio station can interrupt your music with commercials. These actions are effectively changing$latex \\alpha$and$latex \\beta$. These efforts almost always lead to a reduction in both (i.e. it\u2019s less useful to everyone), but also reduces$latex P\\$ making the private option more attractive and enticing people to invest in production. If we assume that private systems are more efficient than public ones, we can even reach the conclusion that adding obstacles (like commercials, patents and toll booths) can boost efficiency.\n\nIt\u2019s fair to say that a model this crude isn\u2019t likely to be accurate to any degree, but it gives an idea of some of the qualitative factors that effect evaluation of public goods. Too often our political discussion never gets past pithy platitudes. When we say, \u201cprivate markets are more efficient,\u201d it\u2019s important to specify, even with a crude estimate, just how much more efficient. Liberals should not pretend the Galt factor is 1, but conservatives should not pretend it\u2019s infinite. And, we should recognize that neither private nor public production is the right solution for all good in all markets. Math matters.\n\nDo you have corrections or comments? How would you improve the model? Can you think of some goods for which the utility functions could be measured? Send me an email.","date":"2020-07-03 14:30:39","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\": 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\": 0.2981274127960205, \"perplexity\": 2192.904979182377}, \"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-2020-29\/segments\/1593655882051.19\/warc\/CC-MAIN-20200703122347-20200703152347-00097.warc.gz\"}"}
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<h2 title="Uses of Class com.rcextract.minecord.event.MinecordEvent" class="title">Uses of Class<br>com.rcextract.minecord.event.MinecordEvent</h2>
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<th class="colFirst" scope="col">Package</th>
<th class="colLast" scope="col">Description</th>
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<tbody>
<tr class="altColor">
<td class="colFirst"><a href="#com.rcextract.minecord">com.rcextract.minecord</a></td>
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<td class="colFirst"><a href="#com.rcextract.minecord.event">com.rcextract.minecord.event</a></td>
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<td class="colFirst"><a href="#com.rcextract.minecord.event.channel">com.rcextract.minecord.event.channel</a></td>
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<td class="colFirst"><a href="#com.rcextract.minecord.event.server">com.rcextract.minecord.event.server</a></td>
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<h3>Uses of <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a> in <a href="../../../../../com/rcextract/minecord/package-summary.html">com.rcextract.minecord</a></h3>
<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="Use table, listing methods, and an explanation">
<caption><span>Methods in <a href="../../../../../com/rcextract/minecord/package-summary.html">com.rcextract.minecord</a> with type parameters of type <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a></span><span class="tabEnd"> </span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
</tr>
<tbody>
<tr class="altColor">
<td class="colFirst"><code><E extends <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a>><br>E</code></td>
<td class="colLast"><span class="typeNameLabel">InternalManager.</span><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/InternalManager.html#getLatestRecord-java.lang.Class-">getLatestRecord</a></span>(<a href="https://docs.oracle.com/javase/8/docs/api/java/lang/Class.html?is-external=true" title="class or interface in java.lang">Class</a><E> clazz)</code> </td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code><E extends <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a>><br>E</code></td>
<td class="colLast"><span class="typeNameLabel">InternalManager.</span><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/InternalManager.html#getOldestRecord-java.lang.Class-">getOldestRecord</a></span>(<a href="https://docs.oracle.com/javase/8/docs/api/java/lang/Class.html?is-external=true" title="class or interface in java.lang">Class</a><E> clazz)</code> </td>
</tr>
<tr class="altColor">
<td class="colFirst"><code><E extends <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a>><br><a href="https://docs.oracle.com/javase/8/docs/api/java/util/List.html?is-external=true" title="class or interface in java.util">List</a><E></code></td>
<td class="colLast"><span class="typeNameLabel">InternalManager.</span><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/InternalManager.html#getRecords-java.lang.Class-">getRecords</a></span>(<a href="https://docs.oracle.com/javase/8/docs/api/java/lang/Class.html?is-external=true" title="class or interface in java.lang">Class</a><E> clazz)</code> </td>
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<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="Use table, listing methods, and an explanation">
<caption><span>Methods in <a href="../../../../../com/rcextract/minecord/package-summary.html">com.rcextract.minecord</a> that return <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a></span><span class="tabEnd"> </span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
</tr>
<tbody>
<tr class="altColor">
<td class="colFirst"><code><a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a></code></td>
<td class="colLast"><span class="typeNameLabel">InternalManager.</span><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/InternalManager.html#getLatestRecord--">getLatestRecord</a></span>()</code> </td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code><a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a></code></td>
<td class="colLast"><span class="typeNameLabel">InternalManager.</span><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/InternalManager.html#getOldestRecord--">getOldestRecord</a></span>()</code> </td>
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<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="Use table, listing methods, and an explanation">
<caption><span>Methods in <a href="../../../../../com/rcextract/minecord/package-summary.html">com.rcextract.minecord</a> that return types with arguments of type <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a></span><span class="tabEnd"> </span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
</tr>
<tbody>
<tr class="altColor">
<td class="colFirst"><code>static <a href="../../../../../com/rcextract/minecord/Recordable.html" title="interface in com.rcextract.minecord">Recordable</a><<a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a>></code></td>
<td class="colLast"><span class="typeNameLabel">Minecord.</span><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/Minecord.html#getRecordManager--">getRecordManager</a></span>()</code>
<div class="block">Gets the record manager.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code><a href="https://docs.oracle.com/javase/8/docs/api/java/util/List.html?is-external=true" title="class or interface in java.util">List</a><<a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a>></code></td>
<td class="colLast"><span class="typeNameLabel">InternalManager.</span><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/InternalManager.html#getRecords--">getRecords</a></span>()</code> </td>
</tr>
</tbody>
</table>
<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="Use table, listing methods, and an explanation">
<caption><span>Methods in <a href="../../../../../com/rcextract/minecord/package-summary.html">com.rcextract.minecord</a> with parameters of type <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a></span><span class="tabEnd"> </span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
</tr>
<tbody>
<tr class="altColor">
<td class="colFirst"><code>void</code></td>
<td class="colLast"><span class="typeNameLabel">InternalManager.</span><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/InternalManager.html#record-com.rcextract.minecord.event.MinecordEvent-">record</a></span>(<a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a> event)</code> </td>
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<h3>Uses of <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a> in <a href="../../../../../com/rcextract/minecord/event/package-summary.html">com.rcextract.minecord.event</a></h3>
<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="Use table, listing subclasses, and an explanation">
<caption><span>Subclasses of <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a> in <a href="../../../../../com/rcextract/minecord/event/package-summary.html">com.rcextract.minecord.event</a></span><span class="tabEnd"> </span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Class and Description</th>
</tr>
<tbody>
<tr class="altColor">
<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/event/RankCreateEvent.html" title="class in com.rcextract.minecord.event">RankCreateEvent</a></span></code> </td>
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<h3>Uses of <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a> in <a href="../../../../../com/rcextract/minecord/event/channel/package-summary.html">com.rcextract.minecord.event.channel</a></h3>
<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="Use table, listing subclasses, and an explanation">
<caption><span>Subclasses of <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a> in <a href="../../../../../com/rcextract/minecord/event/channel/package-summary.html">com.rcextract.minecord.event.channel</a></span><span class="tabEnd"> </span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Class and Description</th>
</tr>
<tbody>
<tr class="altColor">
<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/event/channel/ChannelCreateEvent.html" title="class in com.rcextract.minecord.event.channel">ChannelCreateEvent</a></span></code> </td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/event/channel/ChannelEvent.html" title="class in com.rcextract.minecord.event.channel">ChannelEvent</a></span></code> </td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/event/channel/ChannelSetNameEvent.html" title="class in com.rcextract.minecord.event.channel">ChannelSetNameEvent</a></span></code> </td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/event/channel/ChannelSwitchEvent.html" title="class in com.rcextract.minecord.event.channel">ChannelSwitchEvent</a></span></code>
<div class="block"><span class="deprecatedLabel">Deprecated.</span> </div>
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<h3>Uses of <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a> in <a href="../../../../../com/rcextract/minecord/event/server/package-summary.html">com.rcextract.minecord.event.server</a></h3>
<table class="useSummary" border="0" cellpadding="3" cellspacing="0" summary="Use table, listing subclasses, and an explanation">
<caption><span>Subclasses of <a href="../../../../../com/rcextract/minecord/event/MinecordEvent.html" title="class in com.rcextract.minecord.event">MinecordEvent</a> in <a href="../../../../../com/rcextract/minecord/event/server/package-summary.html">com.rcextract.minecord.event.server</a></span><span class="tabEnd"> </span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Class and Description</th>
</tr>
<tbody>
<tr class="altColor">
<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/event/server/ServerCreateEvent.html" title="class in com.rcextract.minecord.event.server">ServerCreateEvent</a></span></code>
<div class="block">Represents the creation of a <a href="../../../../../com/rcextract/minecord/Server.html" title="class in com.rcextract.minecord"><code>Server</code></a>.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/event/server/ServerEvent.html" title="class in com.rcextract.minecord.event.server">ServerEvent</a></span></code>
<div class="block">Represents an event of a <a href="../../../../../com/rcextract/minecord/Server.html" title="class in com.rcextract.minecord"><code>Server</code></a>.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../com/rcextract/minecord/event/server/ServerLockEvent.html" title="class in com.rcextract.minecord.event.server">ServerLockEvent</a></span></code> </td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>class </code></td>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,009
|
Madeleine Sahely
A passionate, hardworking and versatile professional, Madeleine Sahely began her journey performing in the arts at an early age. Based in Melbourne, she is a committed actor and singer who has made the most out of every role and is driven towards a future within the performing arts.
Madeleine has recently furthered her experience in the performing arts in the Victorian opera, with her being selected as a member of their youth ensemble. Whilst involved within this program, she is continually exposed to the incredible experiences and lessons which professional productions bring, which enable her to become a better performer and artist in her future.
Madeleine is also involved in the Hart Theatre Company where she is able to showcase her talents in acting, singing and dancing in varied productions. In order to improve her acting abilities, Madeleine is also currently training with NIDA's Acting on Screen program, where she portrays a strong and consistent presence throughout the course.
Madeleine has a profound, yet unique passion for performing and is determined to continue evolving her craft as an artist. She is eager to showcase her love for this art and is deeply excited about any opportunities that may come her way.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,978
|
A complete match-by-match statistical record of every game played by the England national team from the first ever international match against Scotland in 1872 through to the end of 2017. Includes full line-ups for both teams, scores, scorers, attendance and venue. Published in 2017.
Size: 156mm x 234mm – 177 pages.
Special Offer – order all five of our British Isles International Complete Records for just £50.00. View our five British Isles Complete Records here.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,763
|
class java::util::concurrent::ConcurrentHashMap$EntrySet : public ::java::util::AbstractSet
{
public: // actually package-private
ConcurrentHashMap$EntrySet(::java::util::concurrent::ConcurrentHashMap *);
public:
::java::util::Iterator * iterator();
jboolean contains(::java::lang::Object *);
jboolean remove(::java::lang::Object *);
jint size();
void clear();
public: // actually package-private
::java::util::concurrent::ConcurrentHashMap * __attribute__((aligned(__alignof__( ::java::util::AbstractSet)))) this$0;
public:
static ::java::lang::Class class$;
};
#endif // __java_util_concurrent_ConcurrentHashMap$EntrySet__
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 135
|
Garchomp, Koffing, Corsola, and more!
Sanei Boueki is back with another All Star Collection wave! This wave includes 12 regular plushies, 3 medium size plushies, 2 big cushions, and 3 pouches.
Everything will be released in mid-November 2018.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,056
|
{"url":"https:\/\/en.wikipedia.org\/wiki\/Period-doubling_bifurcation","text":"# Period-doubling bifurcation\n\nIn mathematics, a period doubling bifurcation in a discrete dynamical system is a bifurcation in which a slight change in a parameter value in the system's equations leads to the system switching to a new behavior with twice the period of the original system. With the doubled period, it takes twice as many iterations as before for the numerical values visited by the system to repeat themselves.\n\nA period doubling cascade is a sequence of doublings and further doublings of the repeating period, as the parameter is adjusted further and further.\n\nPeriod doubling bifurcations can also occur in continuous dynamical systems, namely when a new limit cycle emerges from an existing limit cycle, and the period of the new limit cycle is twice that of the old one.\n\n## Examples\n\n### Logistic map\n\nBifurcation diagram for the logistic map. It shows the attractor values, like ${\\displaystyle x_{*}}$ and ${\\displaystyle x'_{*}}$, as a function of the parameter ${\\displaystyle r}$.\n\nConsider the following simple dynamics: ${\\displaystyle x_{n+1}=rx_{n}(1-x_{n})}$ where ${\\displaystyle x_{n}}$, the value of ${\\displaystyle x}$ at time ${\\displaystyle n}$, lies in the ${\\displaystyle [0,1]}$ interval and changes over time according to the parameter ${\\displaystyle r\\in (0,4]}$. This classic example is a simplified version of the logistic map.\n\nFor ${\\displaystyle r}$ between 1 and 3, ${\\displaystyle x_{n}}$ converges to the stable fixed point ${\\displaystyle x_{*}=(r-1)\/r}$. Then, for ${\\displaystyle r}$ between 3 and 3.44949, ${\\displaystyle x_{n}}$ converges to a permanent oscillation between two values ${\\displaystyle x_{*}}$ and ${\\displaystyle x'_{*}}$ that depend on ${\\displaystyle r}$. As ${\\displaystyle r}$ grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period-doublings culminate at ${\\displaystyle r\\approx 3.56995}$ from where more complex regimes appear. As ${\\displaystyle r}$ increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near ${\\displaystyle r=3.83}$. See figure.\n\nIn the interval where the period is ${\\displaystyle 2^{n}}$ for some positive integer ${\\displaystyle n}$, not all the points actually have period ${\\displaystyle n}$. These are single points, rather than intervals. These points are said to be in unstable orbits, since nearby points do not approach the same orbit as them. See Sharkovskii's theorem.\n\n### Logistical map for a modified Phillips curve\n\nBifurcation diagram for the modified Phillips curve.\n\nConsider the following logistical map for a modified Phillips curve:\n\n${\\displaystyle \\pi _{t}=f(u_{t})+b\\pi _{t}^{e}}$\n\n${\\displaystyle \\pi _{t+1}=\\pi _{t}^{e}+c(\\pi _{t}-\\pi _{t}^{e})}$\n\n${\\displaystyle f(u)=\\beta _{1}+\\beta _{2}e^{-u}\\,}$\n\n${\\displaystyle b>0,0\\leq c\\leq 1,{\\frac {df}{du}}<0}$\n\nwhere\u00a0:\n\n\u2022 ${\\displaystyle \\pi }$ is the actual inflation\n\u2022 ${\\displaystyle \\pi ^{e}}$ is the expected inflation,\n\u2022 u is the level of unemployment,\n\u2022 ${\\displaystyle m-\\pi }$ is the money supply growth rate.\n\nKeeping ${\\displaystyle \\beta _{1}=-2.5,\\ \\beta _{2}=20,\\ c=0.75}$ and varying ${\\displaystyle b}$, the system undergoes period doubling bifurcations, and after a point becomes chaotic, as illustrated in the bifurcation diagram on the right.\n\nBifurcation from period 1 to 2 for complex quadratic map\n\n## Period-halving bifurcation\n\nPeriod-halving bifurcations (L) leading to order, followed by period doubling bifurcations (R) leading to chaos.\n\nA period halving bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with half the period of the original system. A series of period-halving bifurcations leads the system from chaos to order.","date":"2019-08-22 21:07:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 33, \"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\": 0.8604824542999268, \"perplexity\": 461.99877552824563}, \"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\/1566027317359.75\/warc\/CC-MAIN-20190822194105-20190822220105-00225.warc.gz\"}"}
| null | null |
Натали Лартио Нику () е мексиканска продуцентка на теленовели.
Биография
Натали Лартио е родом от град Мексико, но е от френски произход. Започнала е кариерата си като продуцент на теленовели, произведени от съпруга ѝ Салвадор Мехия.
Кариера
Изпълнителен продуцент
Полетът към победата (2017)
Път към съдбата (2016)
Котката (2014)
Необуздано сърце (2013)
Рафаела (2011)
Море от любов (2009/10)
Внимавай с ангела (2008)
Перегрина (2005/6)
Невинната ти (2004/5)
Продуцент
Тъмна орис (2003/4)
Любов и омраза (2002)
Прегърни ме много силно (2000/1)
Росалинда (1999)
Узурпаторката (1998)
Есмералда (1997)
Директор продукция
Мария Мерседес (1992)
La pícara soñadora (1991)
Награди и номинации
Награди TVyNovelas (Мексико)
Награди People en Español
Външни препратки
Натали Лартио в IMDB
Мексикански телевизионни продуценти
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,390
|
{"url":"https:\/\/socratic.org\/questions\/if-an-object-is-dropped-how-fast-will-it-be-moving-after-falling-25-m","text":"# If an object is dropped, how fast will it be moving after falling 25 m?\n\nFeb 4, 2016\n\nAssuming no air resistance, it will be moving at $v = \\sqrt{2 a d} = \\sqrt{2 \\cdot 9.8 \\cdot 25} = \\sqrt{490} = 22.1$ $m {s}^{-} 1$\n\n#### Explanation:\n\nThe expression for how fast an object will be travelling after covering a certain distance while accelerating is:\n\n${v}^{2} = {u}^{2} + 2 a d$\n\nWhere:\n\n$v$ = final velocity $\\left(m {s}^{-} 1\\right)$\n$u$ = initial velocity $\\left(m {s}^{-} 1\\right)$\n$a$ = acceleration $\\left(m {s}^{-} 2\\right)$\n$d$ = distance $\\left(m\\right)$\n\nIn this case, we assume that the initial velocity is $0$: the question says the object was 'dropped', not 'thrown'.\n\nWe can leave out the $u$ term and take the square root of both sides to make $v$ the subject:\n\n$v = \\sqrt{2 a d}$\n\nSubstituting in our values, including the acceleration due to gravity:\n\n$v = \\sqrt{2 \\cdot 9.8 \\cdot 25} = \\sqrt{490} = 22.1$ $m {s}^{-} 1$\n\nWe have assumed that there is no air resistance acting, or that we can neglect it because it is much smaller than the gravitational force.","date":"2019-12-11 08:37:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 17, \"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\": 0.7490343451499939, \"perplexity\": 307.9743338524706}, \"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-2019-51\/segments\/1575540530452.95\/warc\/CC-MAIN-20191211074417-20191211102417-00005.warc.gz\"}"}
| null | null |
\section{Introduction}
A power-bounded linear operator $T$ on a Banach space $X$ is called \emph{mean ergodic}
if $\lim_{n\to \infty} A_nx$ exists for
every $x \in X$. Here $A_n := n^{-1}\sum_{j=1}^{n-1}T^j$ are the Ces\`aro averages of the operator.
The classical mean ergodic theorem (see \cite[\S 2.1 Theorems 1.1 and 1.3]{krengel1985})
characterizes mean ergodic operators as follows.
\begin{thm}\label{t.classicerg}
Let $T$ be a power-bounded operator on a Banach space $X$. The following are equivalent.
\begin{enumerate}[(i)]
\item $T$ is mean ergodic.
\item $A_n x$ has a $\sigma(X,X^*)$-cluster point for all $x \in X$.
\item $\mathrm{fix} (T)$ separates $\mathrm{fix} (T^*)$, i.e.\ for all $0\neq x \in \mathrm{fix} (T^*)$
there exists a $x^* \in \mathrm{fix} (T^*)$ such that $\dual{x}{x^*}_*\neq 0$.
\item $X= \mathrm{fix} (T) \oplus \overline{\mathrm{rg}}^{\|\cdot\|}(I-T)$.
\end{enumerate}
\end{thm}
There are countless extensions of Theorem \ref{t.classicerg} to more general situations.
These include weakening the assumption of power-boundedness, considering more general
semigroups than the discrete semigroup $\{T^j\,:\, j \in \mathds{N}\}$,
considering means other than the Ces\`aro averages and replacing the Banach space $X$ with a locally convex space $(X, \tau)$,
see e.g.\ \cite{eberlein1949, nagel1973, sato1978}.
An overview of these results and further references can be found in~\cite{krengel1985}.
Mean ergodic theorems for semigroups on locally convex spaces with additional assumptions are treated in~\cite{albanese2012, albanese2011}.
Even if the underlying space is a Banach space, it is not always reasonable to expect strong
convergence of the means with respect to the norm topology.
An important example arises in the study of ergodic properties of Markov processes.
Here, one works on the Banach space $\mathscr{M}(E)$
of bounded measures on the Borel $\sigma$-algebra of a Polish space $E$ or on the
subset $\mathscr{P}(E)$ of probability measures.
Even though in some exceptional cases one obtains convergence of Ces\`aro averages
(or even the semigroup itself) in the total variation norm \cite{seidler1997},
it is more natural to consider convergence in the weak topology induced by the bounded, continuous functions $C_b(E)$.
Unfortunately, it seems that one cannot treat this situation with a mean ergodic theorem on locally convex
spaces $(X, \tau)$. The reason for this is that the known results require that the means be equicontinuous
with respect to $\tau$, see \cite{albanese2012,albanese2011,eberlein1949, sato1978}. If $\tau$ is
the weak topology $\sigma (\mathscr{M}(E), C_b(E))$, equicontinuity seems a rather strong assumption
which is not satisfied in interesting examples.
The literature on weak Ces\`aro-convergence of Markov semigroups is rather extensive.
Let us mention \cite{kps2010, sw2012,wh2011b, wh2011}.
However, a characterization of mean ergodicity in the spirit of Theorem \ref{t.classicerg} is still missing.
\medskip
It is the purpose of the present article to fill this gap. We will work in the framework
of norming dual pairs introduced in \cite{kunze2009, kunze2011}
and consider simultaneously two semigroups which are related to each other via duality.
From the point of view of applications to Markov semigroups this is rather natural,
as associated with a Markov process there are two semigroups dual to each other. The first
acts on the space of bounded measurable functions on the state space $E$ (or a subspace thereof such as $C_b(E)$)
and corresponds to the Kolmogorov backward equation and the second acts on the space of bounded measures on $E$
and corresponds to the Kolmogorov forward equation (or Fokker-Planck equation).
Throughout, we allow general (in particular also noncommutative) semigroups and means -- even though our main
interest lies in Ces\`aro averages of one-parameter semigroups in discrete or continuous time -- and study convergence
of the means in the weak topologies induced by the dual pair.
In our first main result (Theorem \ref{t.erg}), we show that in this general situation the statements corresponding
to (i) and (ii) in Theorem \ref{t.classicerg} are equivalent and imply the statements corresponding to (iii) and (iv).
We also provide counterexamples to show that in general (the statements corresponding to) (iii) does not imply (iv)
and neither (iii) nor (iv) imply (i) and (ii).
\smallskip
Afterwards, we focus on the more special situation of Markovian semigroups on the norming dual pair
$(C_b(E), \mathscr{M}(E))$.
Besides others, our main assumption in this more special situation is a condition which is slightly weaker than
the \emph{e-property} which played an important role in \cite{kps2010, sw2012}.
Under that assumption we prove in Theorem \ref{t.eerg} that the statements corresponding
to (i) -- (iv) in Theorem \ref{t.classicerg} for the semigroup on $\mathscr{M}(E)$ are all equivalent.
Moreover, if the semigroup on $\mathscr{M}(E)$ is mean ergodic with respect to $\sigma (\mathscr{M}(E), C_b(E))$,
then also the semigroup on $C_b(E)$ is mean ergodic even with respect
to a topology finer than $\sigma (C_b(E), \mathscr{M}(E))$, namely the strict topology.
Considering semigroups on $(C_b(E),\mathscr{M}(E))$ rather than on the single Banach space $\mathscr{M}(E)$
makes our assumption natural, in fact, it is necessary for the convergence we obtain.
\medskip
This article is organized as follows. In Section \ref{s.ndp} we recall some basic definitions and results
about norming dual pairs. In Section \ref{s.as}, we introduce the notion of an ``average scheme'' which will
act as our means. Afterwards, we take up our main line of study.
First, we analyze convergence of average schemes on general norming dual pairs in Section \ref{s.convergence},
then the convergence of average schemes on $(C_b(E), \mathscr{M}(E))$ under additional assumptions in
Section \ref{s.eproperty}. The concluding Section \ref{s.examples} contains our Counterexamples.
\section{Norming dual pairs}\label{s.ndp}
A \emph{norming dual pair} is a triple $(X,Y, \dual{}{})$ where $X$ and $Y$ are Banach spaces and $\dual{}{}$ is a
duality between $X$ and $Y$ such that
\[
\|x\| = \sup\{|\dual{x}{y}|: y \in Y,\|y\| \leq 1\}
\quad\mbox{and}\quad \|y\| = \sup\{|\dual{x}{y}|: x \in X,\|y\| \leq 1\} \, .
\]
Identifying $y$ with the linear functional $x \mapsto \dual{x}{y}$, we see that $Y$ is isometrically isomorphic
with a norm closed subspace of $X^*$, the norm dual of $X$, which is norming for $X$. If the duality paring is understood, we will
briefly say that $(X,Y)$ is a norming dual pair.\medskip
Let us give some examples of norming dual pairs. If $X$ is a Banach space with norm dual $X^*$, then $(X,X^*)$ and thus,
by symmetry, also $(X^*,X)$ is a norming dual pair with respect to the canonical duality $\dual{}{}_*$.
If $(E, \Sigma)$ is a measurable space, we write $B_b(E)$ for the space of bounded, measurable functions
on $(E, \Sigma)$ and $\mathscr{M}(E)$ for the space of complex measures on $(E, \Sigma)$.
The space $B_b(E)$ is endowed with the supremum norm and the space $\mathscr{M}(E)$ is endowed with the
total variation norm. Then $(B_b(E), \mathscr{M}(E))$ is a norming dual pair with respect to the duality
\[
\dual{f}{\mu} := \int_E f\, d\mu\, .
\]
If $E$ is a Polish space, i.e.\ a topological space which is metrizable through a complete, separable
metric, and $\Sigma$ is the Borel $\sigma$-algebra, then also $(C_b(E), \mathscr{M}(E))$ is a norming dual pair.
For the easy proofs of these facts we refer to \cite[Section 2]{kunze2011}.\medskip
In what follows, we will be interested in locally convex topologies which are \emph{consistent (with the duality)}.
We recall that a locally convex topology $\tau$ on $X$ is consistent if $(X,\tau)' = Y$, i.e.\ every $\tau$-continuous
linear functional $\varphi$ on $X$ is of the form $\varphi (x) = \dual{x}{y}$ for some $y\in Y$. Of particular importance
are the \emph{weak topologies} $\sigma (X,Y)$ and $\sigma (Y,X)$ associated with the dual pair. To simplify notation,
in what follows we will write $\sigma$ for the $\sigma (X,Y)$ topology on $X$ and $\sigma'$ for the $\sigma (Y,X)$ topology
on $Y$. We will write $\rightharpoonup$, resp.\ $\rightharpoonup'$, to indicate convergence with respect to $\sigma$, resp.\ $\sigma'$.
\medskip
If $\tau$ is a topology on $X$, we write $\mathscr{L} (X,\tau)$ for the algebra of $\tau$-continuous linear operators
on $X$. We write $\mathscr{L} (X)$ shorthand for $\mathscr{L} (X,\|\cdot\|)$. By \cite[Prop 3.1]{kunze2011}, $\mathscr{L} (X,\sigma)$
is a subalgebra of $\mathscr{L} (X)$ which is closed in the operator norm. Moreover, identifying $Y$ with a closed
subspace of $X^*$, an operator $S\in\mathscr{L} (X)$ belongs to $\mathscr{L} (X,\sigma)$ if and only if its norm adjoint $S^*$ leaves $Y$
invariant. In that case, the $\sigma$-adjoint of $S$, denoted by $S'$, is precisely $S^*|_Y$ and $\|S\|=\|S'\|$.\medskip
Let us give a description of the operators in $\mathscr{L} (X,\sigma)$ in the case where $X= B_b(E)$ or
$X= C_b(E)$ is in canonical duality with $Y= \mathscr{M}(E)$.
We recall that a \emph{bounded kernel}
on a measurable space $(E,\Sigma)$ is a mapping $k: E\times \Sigma \to \mathds{C}$ such that (i) $k(x,\cdot)$ is a
complex measure on $(E, \Sigma)$ for all $x \in E$, (ii) $k(\cdot, A)$ is $\Sigma$-measurable for all $A \in \Sigma$
and (iii) $\sup_{x\in E}|k|(x,E) < \infty$, where $|k|(x,\cdot)$ denotes the total variation of $k(x,\cdot)$.
A linear operator $S$ on a closed subspace $X$ of $B_b(E)$ is called a \emph{kernel operator (on $X$)} if there
exists a bounded kernel $k$ on $(E,\Sigma)$ such that
\[
(Sf)(x) = \int_Ef(y)k(x,dy), \quad \mbox{for all}\, f \in X.
\]
Note that if $X$ is $\sigma(B_b(E), \mathscr{M}(E))$-dense in $B_b(E)$, then $k$ is uniquely determined by $S$. In
this case, $S$ has a unique extension to $B_b(E)$ and its $\sigma$-adjoint is given by
\[
(S'\mu)(A) = \int_E k(x,A)\, d\mu (x)\quad \forall\, \mu \in \mathscr{M}(E)\, .
\]
It was seen in \cite[Prop 3.5]{kunze2011} that on the norming dual pair $(X,\mathscr{M}(E))$, where
$X=B_b(E)$ or, if $E$ is Polish and $\Sigma$ is the Borel $\sigma$-algebra, $X=C_b(E)$, an operator $S \in \mathscr{L} (X)$
belongs to $\mathscr{L} (X,\sigma)$ if and only if it is a kernel operator.
\section{Average Schemes}\label{s.as}
For a family $\mathscr{S}$ of linear operators on a vector space $X$ we denote by
\[ \mathrm{fix}(\mathscr{S}) \mathrel{\mathop:}= \bigcap_{S\in\mathscr{S}} \ker(I-S) \]
its \textit{fixed space} and by
\[ \mathrm{rg}(I-\mathscr{S}) \mathrel{\mathop:}= \{ x-Sx : x\in X,\, S\in\mathscr{S} \}\]
the range of $I-\mathscr{S}$.
Moreover, for $x\in X$ we define
\[ \mathrm{co} (\mathscr{S}x) \mathrel{\mathop:}= \biggr\{ \sum_{k=1}^n a_k S_k x : a_k \geq 0,\,
\sum_{k=1}^n a_k = 1,\, S_k \in \mathscr{S}, \, n\in \mathds{N} \biggr\},\]
the convex hull of the orbit of $x$ under $\mathscr{S}$.
A family $\mathscr{S}$ containing the identity is called
a \emph{semigroup} if $ST\in \mathscr{S}$ for all $S, T\in \mathscr{S}$.
Inspired by \cite{eberlein1949} we make the following definition.
\begin{defn}
\label{d.as}
Let $(X,Y)$ be a norming dual pair. An \emph{average scheme on $(X,Y)$} is a pair
$(\mathscr{S},\mathscr{A})$, where $\mathscr{S}\subset\mathscr{L}(X,\sigma)$ is a semigroup
with adjoint $\mathscr{S}' \mathrel{\mathop:}= \{ S' : S\in \mathscr{S}\}$
and $\mathscr{A}=(A_\alpha)_{\alpha\in\Lambda}\subset \mathscr{L}(X,\sigma)$ is a net of
$\sigma$-continuous operators such that the following assertions
are satisfied.
\begin{enumerate}[(AS2)]
\item[(AS1)] There exists $M>0$ such that $\Vert A_\alpha \Vert \leq M$ for all $\alpha\in\Lambda$.
\item[(AS2)] $A_\alpha x \in \overline{\mathrm{co}}^\sigma(\mathscr{S}x)$
and $A'_\alpha y \in \overline{\mathrm{co}}^{\sigma'}(\mathscr{S}'y)$
for all $\alpha \in \Lambda$, $x\in X$ and $y\in Y$.
\item[(AS3)] For every $S\in\mathscr{S}$ and all $x\in X$ and $y\in Y$ one has that
\[ \lim_{\alpha} A_\alpha(S-I)x = \lim_{\alpha} (S-I)A_\alpha x = 0 \]
and
\[\lim_{\alpha} A'_\alpha(S'-I)y = \lim_{\alpha} (S'-I)A'_\alpha y = 0 \]
in the norm topology of $X$ and $Y$ respectively.
\end{enumerate}
\end{defn}
We should point out that our terminology is somewhat different from that in \cite{eberlein1949}. In the language
of Eberlein, the net $A_\alpha$ would be called a \emph{system of almost invariant integrals} and a semigroup
$\mathscr{S}$ possessing such a system would be called \emph{ergodic}.
Moreover, we should note that there is no equicontinuity assumption for the
averages $A_\alpha$ with respect to $\sigma$ or with respect to any other \emph{consistent} topology. Instead,
we assume in (AS1) equicontinuity only with respect to the (in general not consistent) norm topology. On the
other hand, in (AS3), we assume convergence in the norm topology, which is a stronger assumption than $\sigma$-convergence
(and also than convergence with respect to a consistent topology on $X$.)
\begin{rem}
\label{r.fixedpoint}
We will frequently make use of the following observation.
If $(\mathscr{S},\mathscr{A})$ is an average scheme on a norming dual pair $(X,Y)$ and
$x\in \mathrm{fix}(\mathscr{S})$, then $\overline{\mathrm{co}}^\sigma(\mathscr{S}x) = \{ x\}$ and hence, by (AS2),
$A_\alpha x = x$ for all $\alpha \in \Lambda$.
\end{rem}
We now give some typical examples of average schemes. Throughout, $(X,Y)$ denotes a norming dual pair.
\begin{example}\label{ex.cesaro}
Let $\mathscr{S} \mathrel{\mathop:}= \{ S^k : k\in \mathds{N}_0\}$ be a semigroup that consists of powers of a single
operator $S\in \mathscr{L}(X,\sigma)$ and denote by
\[ A_n \mathrel{\mathop:}= \frac{1}{n} \sum_{k=0}^{n-1} S^k \quad (n\in\mathds{N})\]
its \emph{Ces\`aro averages}. Assume that $\lim_{n\to\infty} \frac{1}{n}S^n x=0$ for all $x\in X$ and
that $\lim_{n\to\infty}\frac{1}{n}(S')^ny =0$ for all $y \in Y$. Moreover, assume that there
exists $M>0$ such that $\Vert A_n \Vert < M$ for all $n\in\mathds{N}$, i.e.\ $S$ is \emph{Ces\`aro bounded}.
Both assumptions are satisfied if $S$ is \emph{power-bounded}, i.e.\ $\sup_{n\in \mathds{N}} \|S^n\| < \infty$.
Clearly, (AS1) and (AS2) are satisfied. As for (AS3), we have
\[ \lim_{n\to\infty} A_n (S-I)x = \lim_{n\to\infty} \frac{1}{n} (S^n-I)x = 0 \text{ for all }x\in X\]
and, similarly, $\lim_{n\to\infty} A'_n (S'-I)y = 0$ for all $y\in Y$.
Thus,
$(\mathscr{S},(A_n)_{n\in\mathds{N}})$ is an average scheme.
\end{example}
\begin{example}\label{ex.abel}
We again consider $\mathscr{S} \mathrel{\mathop:}= \{ S^k : k\in \mathds{N}_0\}$ for an operator $S \in \mathscr{L} (X, \sigma)$.
If $S$ has spectral radius $r(S) = \lim_{n\to\infty} \|S^n\|^{\frac{1}{n}} \leq 1$,
then for $r \in [0,1)$ the series $\sum_{k=0}^\infty r^kS^k$ converges in
operator norm and thus represents an element of $\mathscr{L} (X,\sigma)$. We denote by
\[ A_rx := (1-r)\sum_{k=0}^\infty r^kS^kx \quad (r \in [0,1))\]
the \emph{Abel averages} of $S$. If $M :=\sup_{0\leq r< 1} \|A_r\| < \infty$, then $S$ is called \emph{Abel bounded}.
Note that power-bounded operators are Abel bounded.
For an Abel bounded operator $S \in \mathscr{L} (X,\sigma)$,
the pair $(\mathscr{S}, (A_r)_{r\in [0,1)})$ is an average scheme.
Indeed, (AS1) is clear. As for (AS2) we see that $A_r = \lim_{n\to\infty} \frac{1-r}{1-r^{n+1}}\sum_{k=0}^n
r^kS^k$ in operator norm. Hence $A_rx$ belongs even to the norm closure of $\mathrm{co}(\mathscr{S} x)$. For the
$\sigma$-adjoint, one argues similarly. It remains to verify (AS3). So that end, note that
\[
\|A_rSx - A_rx\| = (1-r)\|x-A_rSx\| \leq (1-r)\big[1 + M\|S\|\big]\|x\| \to 0
\]
as $r \uparrow 1$. On $Y$, one argues similarly.
\end{example}
\begin{example}
\label{ex.integrable}
Let $\mathscr{S} \mathrel{\mathop:}= \{ S(t)\,:\, t \geq 0\} \subset \mathscr{L} (X,\sigma)$
be an integrable semigroup on $(X,Y)$, cf.\ \cite{kunze2011}. This means that $S(0)$ is the identity on $X$ and
for $t,s \geq 0$, we have $S(t+s) = S(t)S(s)$. Moreover, there exists $M \geq 1$ and $\omega \in \mathds{R}$ such that
$\|S(t)\| \leq Me^{\omega t}$ for all $t \geq 0$. Finally, for all $x\in X$ and $y\in Y$ the function $t \mapsto
\dual{S(t)x}{y}$ is measurable and for some (equivalently, all) $\lambda$ with $\Re\lambda > \omega$ there
exists an operator $R(\lambda) \in \mathscr{L} (X,\sigma)$ such that
\[
\dual{R(\lambda)x}{y} = \int_0^\infty e^{-\lambda t}\dual{S(t)x}{y}\, dy, \quad \mbox{for all}\,\, x \in X, y \in Y.
\]
It follows from \cite[Thm 5.8]{kunze2011} that if $\mathscr{S}$ is an integrable semigroup, then for every $t >0$
there exists an operator $A_t \in \mathscr{L} (X,\sigma)$ such that
\[
\dual{A_tx}{y} = \frac{1}{t}\int_0^t\dual{S(s)x}{y}\, ds\, .
\]
We call the semigroup $\mathscr{S}$ Ces\`aro bounded if $M:=\sup_{t>0}\|A_t\| < \infty$. If $\mathscr{S}$ is an integrable,
Ces\`aro bounded semigroup such that $\frac{1}{t}S(t)x \to 0$ and $\frac{1}{t}S(t)'y \to 0$
as $t\to \infty$ for arbitrary $x \in X$ and $y \in Y$, then $(\mathscr{S}, (A_t)_{t>0})$ is an average scheme.
(AS1) is clear and (AS2) is a consequence of the Hahn-Banach theorem on the locally convex spaces $(X,\sigma)$ resp.\
$(Y,\sigma')$, cf.\ the end of the proof of Theorem 4.4 in \cite{kunze2011}. As for (AS3), we note that for $t>0$ and $s\geq 0$
we have
\[
A_tS(s) - A_t = \frac{s}{t}(I - S(t))A_s = \frac{s}{t}A_s(I-S(t))
\]
as is easy to see using the semigroup law. Consequently, for every $x \in X$ we have
$\|A_tS(s)x - A_tx\| \leq s M(\|x\|t^{-1} + \|t^{-1}S(t)x\|) \to 0$ as $t \to \infty$. On $Y$, one argues similarly.
In particular, $(\mathscr{S},(A_t)_{t >0})$ is an average scheme whenever the integrable semigroup $\mathscr{S}$ is bounded.
\end{example}
Concerning the last example, let us note that if $\mathscr{S}$ is an integrable semigroup, then the operators
$R(\lambda)$ form a pseudo resolvent, hence, there is a unique, possibly multivalued operator $\mathscr{G}$ with
$R(\lambda ) = (\lambda -\mathscr{G})^{-1}$, the \emph{generator of $\mathscr{S}$}.
In this case, as a consequence of \cite[Prop 5.7]{kunze2011}, $\mathrm{fix} (\mathscr{S}) = \mathrm{ker}\mathscr{G} =
\{ x \in X\,:\, (x,0) \in \mathscr{G}\}$.
For more information
about integrable semigroups and their generators, we refer to \cite{kunze2011}.
\section{Convergence of average schemes}\label{s.convergence}
We start with the definition of weak ergodicity.
\begin{defn}
We say that an average scheme $(\mathscr{S},\mathscr{A})$ on a norming
dual pair $(X,Y)$ is \emph{weakly ergodic},
if the $\sigma$-limit of $(A_\alpha x)$ exists for every $x\in X$ and the
$\sigma'$-limit of $(A_\alpha'y)$ exists for every $y\in Y$.
\end{defn}
In the mean ergodic theorem on norming dual pairs we need a slightly stronger version of assertion (ii)
of Theorem~\ref{t.classicerg}. This is due to the fact that the strategy for the proof differs from
the classical one since not all assertions corresponding to (i) -- (iv) are equivalent in our situation.
We use the following terminology.
\begin{defn}
\label{d.compactnet}
We say that a net $(x_\alpha)_{\alpha\in\Lambda}$ in a topological space $E$ \emph{clusters} if
every subnet of $(x_\alpha)$ has a cluster point, i.e.\ it has a convergent subnet.
\end{defn}
A net clusters whenever the set of its elements is relatively compact.
However, if a net $(x_\alpha)_{\alpha\in\Lambda}$ clusters,
one cannot infer that the set $\{ x_\alpha : \alpha\geq\alpha_0 \}$ is
relatively compact for some $\alpha_0 \in \Lambda$.
For a sequence, these two properties are equivalent, which is probably well-known.
However, we were not able to find a reference and hence present the short proof
for the sake of completeness.
\begin{lem}
\label{l.seqcompact}
Let $(x_n)$ be a sequence in a topological vector space $(X,\tau)$
such that every subsequence of $(x_n)$ has a convergent subnet.
Then the set $\mathscr{M}\mathrel{\mathop:}= \{ x_n : n\in\mathds{N}\}$ is relatively compact.
\end{lem}
\begin{proof}
In view of \cite[\S 5.6(2)]{koethe1969}, it suffices to show that $\mathscr{M}$ is totally bounded.
Assume the converse. Then there exists an open neighborhood of the origin $U$ and a subsequence
$y_k \mathrel{\mathop:}= x_{n_k}$ such that
\[ y_k \not\in \bigcup_{j=1}^{k-1} (U+y_j)\]
for all $k\in\mathds{N}$. By assumption, $(y_k)$ contains a convergent subnet
$(y_{k(\alpha)})_{\alpha\in A}$
whose limit we denote by $y$.
Now, we choose a circled neighborhood of the origin $V$ such that $V+V\subset U$,
which exists by \cite[\S 15.1(3)]{koethe1969}.
Then there is a $\beta\in A$ such that $y_{k(\alpha)} \in V+y$ for all $\alpha\geq \beta$
and hence, $y \in V+y_{k(\beta)}$. This implies that
\[ y_{k(\alpha)} = y_{k(\alpha)}-y+y \in V+y \subset V+V+y_{k(\beta)} \subset U+y_{k(\beta)}
\subset \bigcup_{j=1}^{k(\beta)}( U+y_j).\]
for all $\alpha\geq \beta$. Since $\{k(\alpha):\alpha\in A\}$ is cofinal in $\mathds{N}$,
this is in contradiction to
the construction of the sequence $(y_k)$. Hence, $\mathscr{M}$ is relatively compact.
\end{proof}
The following is the main result of this section.
\begin{thm}\label{t.erg}
Let $(\mathscr{S},\mathscr{A})$ be an average scheme on a norming dual pair $(X,Y)$.
Then the following are equivalent:
\begin{enumerate}[(i)]
\item\label{t.erg.1} The average scheme $(\mathscr{S},\mathscr{A})$ is weakly ergodic.
\item\label{t.erg.2} For every $x \in X$ the net $(A_\alpha x)$ clusters in $(X,\sigma)$ and
for every $y \in Y$ the net $(A'_\alpha y)$ clusters in $(Y,\sigma')$.
\end{enumerate}
If these equivalent conditions are satisfied, then
\begin{enumerate}[(i)]
\setcounter{enumi}{2}
\item\label{t.erg.3} The fixed spaces $\mathrm{fix} (\mathscr{S})$ and $\mathrm{fix} (\mathscr{S}')$ separate each other.
\item\label{t.erg.4} We have $X= \mathrm{fix} (\mathscr{S}) \oplus \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$ and
$Y= \mathrm{fix} (\mathscr{S}') \oplus \overline{\mathrm{span}}^{\sigma'}\mathrm{rg}(I-\mathscr{S}')$.
\item\label{t.erg.5} The operator $P$, defined by $Px \mathrel{\mathop:}= \sigma\,\mbox{-}\,\lim_{\alpha} A_\alpha x$ belongs to
$\mathscr{L} (X,\sigma)$ and the $\sigma$-adjoint $P'$ of $P$ is given by
$P'y =\sigma'\,\mbox{-}\,\lim_{\alpha} A'_\alpha y$ for all $y\in Y$.
Moreover, $P$ is the projection onto $\mathrm{fix} (\mathscr{S})$ along $\overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$, $P'$
the projection onto $\mathrm{fix}(\mathscr{S}')$ along $\overline{\mathrm{span}}^{\sigma'}(I-\mathscr{S}')$
and $PS=SP=P$ for all $S\in\mathscr{S}$.
\end{enumerate}
\end{thm}
For a weakly ergodic average scheme $(\mathscr{S},\mathscr{A})$,
the operator $P$ from (\ref{t.erg.5}) is called the \emph{ergodic projection}.
Note that the ergodic projection $P$
is uniquely determined by the semigroup $\mathscr{S}$ and
independent of the averages $\mathscr{A}$.
We prepare the proof of Theorem \ref{t.erg} through a series of lemmas.
\begin{lem}\label{l.1}
Let $X$ be a Banach space and $\mathscr{S}\subset \mathscr{L}(X)$ be semigroup of bounded operators on $X$.
Moreover, let $(A_\alpha)_{\alpha \in\Lambda}\subset\mathscr{L}(X)$ be a net and let $x\in X$ be
such that
\[ \lim_{\alpha} (S-I)A_\alpha x = 0 \text{ for all }S \in \mathscr{S}.\]
Assume that $Z \subset X^*$ separates points in $X$ and $S^*Z \subset Z$ for all $S\in\mathscr{S}$.
Then every $\sigma (X,Z)$-cluster point of $(A_\alpha x)$ belongs to $\mathrm{fix} (\mathscr{S})$.
\end{lem}
\begin{proof}
Fix $x \in X$ and let $w$ be a $\sigma (X,Z)$-cluster point of $(A_\alpha x)$.
Let $S\in\mathscr{S}$.
We have
\[
Sw-w = (S-I)(w-A_\alpha x) + (S-I) A_\alpha x
\]
for all $\alpha \in \Lambda$
and, by assumption, $(S-I) A_\alpha x \to 0$ in norm and hence also with respect to $\sigma(X, Z)$.
Now fix $z \in Z$. Given $\varepsilon>0$, we find $\alpha_0$ such that
$\vert \applied{(S-I)A_\alpha x}{z} \vert < \varepsilon$ for all $\alpha \geq \alpha_0$.
Since $S^* z \in Z$ and since
$w$ is an $\sigma (X,Z)$-cluster point of $(A_\alpha x)$, we find some $\beta \geq \alpha_0$ such that
\[
|\applied{(S-I)(w-A_\beta x)}{z}| = |\applied{w-A_\beta x}{S^*z-z}| \leq \varepsilon\, .
\]
This implies that $|\applied{Sw-w}{z}| \leq 2\varepsilon$. Since $\varepsilon>0$ was arbitrary, it follows that
$\applied{Sw-w}{z} =0$
and thus, since $z \in Z$ was arbitrary, $w = Sw$.
\end{proof}
\begin{lem}\label{l.norm}
Let $(\mathscr{S},\mathscr{A})$ be an average scheme on a norming dual pair $(X,Y)$.
Then
\[
X_0 \mathrel{\mathop:}= \{ x \in X \,:\, \lim_{\alpha} A_\alpha x\text{ exists w.r.t.\ }\Vert\cdot\Vert\,\}
\]
is a norm-closed subspace of $X$ and invariant under the action of $\mathscr{S}$.
Moreover, the sum $\mathrm{fix}(\mathscr{S})+\overline{\mathrm{span}}^{\Vert\cdot\Vert}\mathrm{rg}(I-\mathscr{S})$
is direct and
$\mathrm{fix} (\mathscr{S}) \oplus \overline{\mathrm{span}}^{\Vert\cdot\Vert}\mathrm{rg}(I-\mathscr{S})\subset X_0$.
Finally, $P_0 x\mathrel{\mathop:}= \Vert\cdot\Vert\,\mbox{-}\,\lim_{\alpha} A_\alpha x$
defines a bounded operator on $X_0$ which is a projection onto $\mathrm{fix}(\mathscr{S})$
with $\overline{\mathrm{span}}^{\Vert\cdot\Vert}\mathrm{rg}(I-\mathscr{S})\subset \ker P_0$.
\end{lem}
\begin{proof}
Let $x_k \in X_0$ and $\lim x_k = x$ with respect to $\|\cdot\|$.
We denote by $P_0x_k$ the limit of $A_\alpha x_k$ for every fixed $k\in \mathds{N}$.
Then $P_0x_k \in \mathrm{fix} (\mathscr{S})$ by Lemma~\ref{l.1} and we have
\[
\|P_0x_k - P_0x_l\| = \lim_{\alpha} \|A_\alpha (x_k-x_l)\| \leq M \|x_k-x_l\|
\]
for all $k,l \in \mathds{N}$,
where $M$ is such that $\|A_\alpha \|\leq M$ for all $\alpha\in\Lambda$.
Since $(x_k)$ is a Cauchy sequence, so is $(P_0x_k)$.
Thus, $P_0x_k \to \bar{x}$ for some $\bar{x}$ which belongs to $\mathrm{fix}(\mathscr{S})$ as the latter is closed.
A $3\varepsilon$-argument shows that $A_\alpha x \to \bar{x}$. It follows that $X_0$ is closed and $P_0x=\bar{x}$.
We have seen that $\|P_0\|\leq M$ and $P_0 X_0 \subset \mathrm{fix} (\mathscr{S})$.
Conversely, $\mathrm{fix} (\mathscr{S})\subset P_0 X_0$ since $A_\alpha x \equiv x = P_0 x$
for $x \in \mathrm{fix} (\mathscr{S})$.
Hence, $P_0 X_0 = \mathrm{fix}(\mathscr{S})$ and $P_0$ is a projection.
The $\mathscr{S}$-invariance of $X_0$ follows from (AS3).
By the definition of an average scheme, $\lim_{\alpha} A_\alpha x = 0$ for all $x\in \mathrm{rg}(I-\mathscr{S})$.
In view of the uniform boundedness of the operators $A_\alpha$, this remains true for
$x \in \overline{\mathrm{span}}^{\Vert\cdot\Vert}\mathrm{rg}(I-\mathscr{S})$.
Since $A_\alpha x \to x$ for $x\in\mathrm{fix} (S)$, it follows that the sum of $\mathrm{fix} (S)$
and $\overline{\mathrm{span}}^{\|\cdot\|}\mathrm{rg}(I-\mathscr{S})$ is direct and that
$\mathrm{fix} (S) \oplus \overline{\mathrm{span}}^{\|\cdot\|}\mathrm{rg}(I-\mathscr{S}) \subset X_0$.
\end{proof}
\begin{lem}\label{l.direct}
Let $(\mathscr{S},\mathscr{A})$ be an average scheme on a norming dual pair $(X,Y)$.
If $\mathrm{fix} (\mathscr{S})$ separates $\mathrm{fix} (\mathscr{S}')$, then
$\mathrm{fix} (\mathscr{S}) + \mathrm{rg}(I-\mathscr{S})$ is $\sigma (X,Y)$-dense in $X$.
If $\mathrm{fix} (\mathscr{S}')$ separates $\mathrm{fix} (\mathscr{S})$, then the sum
$\mathrm{fix} (\mathscr{S}) + \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$ is direct.
\end{lem}
\begin{proof}
Assume that $\mathrm{fix} (\mathscr{S})$ separates $\mathrm{fix} (\mathscr{S}')$. Let $y \in Y$ be such that
$\applied{x}{y} = 0$ for all $x \in \mathrm{fix} (\mathscr{S}) \oplus \mathrm{rg}(I-\mathscr{S})$.
Then, in particular, $0= \applied{x-Sx}{y} = \applied{x}{y-S'y}$ for all $x \in X$ and $S\in\mathscr{S}$.
Since $X$ separates $Y$, it follows that $y = S'y$ for all $S\in\mathscr{S}$, i.e.\ $y \in \mathrm{fix} (\mathscr{S}')$.
Moreover, $\applied{x}{y} =0$ for all $x \in \mathrm{fix} (\mathscr{S})$.
By assumption, this implies $y=0$. It now follows from the
Hahn-Banach theorem, applied on the locally convex space $(X,\sigma)$,
that $\mathrm{fix} (\mathscr{S}) + \mathrm{rg}(I-\mathscr{S})$ is $\sigma (X,Y)$-dense in $X$.
Now assume that $\mathrm{fix} (\mathscr{S}')$ separates $\mathrm{fix} (\mathscr{S})$. Since every $y\in\mathrm{fix}(\mathscr{S}')$
vanishes on $\mathrm{rg}(I-\mathscr{S})$, it also vanishes on $\overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$ by linearity and
continuity. Thus, $\applied{x}{y}=0$ for all $x \in \mathrm{fix} (\mathscr{S}) \cap \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$
and $y\in\mathrm{fix}(\mathscr{S}')$.
As $\mathrm{fix}(\mathscr{S}')$ separates $\mathrm{fix}(\mathscr{S})$, it follows that $0$ is the only element of
$\mathrm{fix} (\mathscr{S}) \cap \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$.
\end{proof}
\begin{lem}
\label{l.convergence}
Let $(\mathscr{S},\mathscr{A})$ be an average scheme on a norming dual pair $(X,Y)$ and
$\tau$ be a locally convex topology on $X$ finer than $\sigma$. Let
\[ X_1 \mathrel{\mathop:}= \{ x\in X : (A_\alpha x)_{\alpha\in\Lambda} \text{ clusters in } (X,\tau) \}.\]
If $\mathrm{fix} (\mathscr{S}')$ separates $\mathrm{fix} (\mathscr{S})$, then
$\tau\,\mbox{-}\,\lim_\alpha A_\alpha x \in X$ exists for all $x\in X_1$.
\end{lem}
\begin{proof}
Applying Lemma~\ref{l.norm} to the average scheme $(\mathscr{S}',\mathscr{A}')$ on $(Y,X)$, we find that
\[Y_0 \mathrel{\mathop:}= \{y \in Y\,:\, \|\cdot\|\,\mbox{-}\, \lim_{\alpha} A'_\alpha y\,\,\mbox{exists}\}\]
is a norm-closed subspace of $Y$ that contains $\mathrm{fix}(\mathscr{S}')\oplus \mathrm{rg}(I-\mathscr{S}')$.
Moreover, there exists an operator $R_0 \in \mathscr{L} (Y_0)$ such that
$\|\cdot\|\,\mbox{-}\, \lim_{\alpha}A_\alpha'y = R_0y$ for all $y\in Y_0$.
As $\mathrm{fix}(\mathscr{S}')$ separates $\mathrm{fix}(\mathscr{S})$, Lemma~\ref{l.direct} yields that $Y_0$ is $\sigma (Y,X)$-dense in $Y$.
Hence, we may identify $X$ with a subspace of $Y_0^*$. Doing so, it follows that
$\sigma (Y_0^*, Y_0)\,\mbox{-}\, \lim_{\alpha} A_\alpha x = R_0^*x$ for all $x \in X$.
Let us fix $x \in X_1$ and choose an arbitrary subnet of $(u_\beta)$ of $(A_\alpha x)$.
By assumption, $(u_\beta)$ has a $\tau$-cluster point $\bar{x}\in X$.
Since $\bar{x}$ is also a $\sigma (Y_0^*,Y_0)$-cluster point of $(A_\alpha x)$, we infer that $\bar{x} = R_0^*x$.
Thus, every subnet of $(A_\alpha x)$ has a subnet converging to $R_0^*x \in X$ in $(X,\tau)$.
This implies that $\tau\,\mbox{-}\,\lim_{\alpha} A_{\alpha}x = R_0^*x$ for all $x\in X_1$.
\end{proof}
Now, we have the tools at hand to prove Theorem~\ref{t.erg}.
\begin{proof}[Proof of Theorem~\ref{t.erg}]
The implication (\ref{t.erg.1}) $\Rightarrow$ (\ref{t.erg.2}) is trivial,
so assume that (\ref{t.erg.2}) holds.
Let us verify (\ref{t.erg.3}) first.
Since $Y$ is norming for $X$, given $x \in \mathrm{fix} (\mathscr{S})$, $x\neq 0$,
we find $y \in Y$ such that $\applied{x}{y} = a \neq 0$.
By assumption, $A'_\alpha y$ has a $\sigma (Y,X)$-cluster point $z$ which, by Lemma~\ref{l.1}, is an
element of $\mathrm{fix} (\mathscr{S}')$.
Since
\[ \applied{x}{A'_\alpha y} = \applied{A_\alpha x}{y} = \applied{x}{y}=a\]
for all $\alpha\in\Lambda$ it follows that $\applied{x}{z} = a \neq 0$.
Hence, $\mathrm{fix} (\mathscr{S}')$ separates $\mathrm{fix} (\mathscr{S})$. Interchanging
the roles of $X$ and $Y$, it follows that $\mathrm{fix} (\mathscr{S})$ separates $\mathrm{fix} (\mathscr{S}')$.
Now, Assertion (\ref{t.erg.1}) follows immediately from Lemma~\ref{l.convergence} applied to the
average schemes $(\mathscr{S},\mathscr{A})$ and $(\mathscr{S}',\mathscr{A}')$ and the weak topologies
$\sigma(X,Y)$ and $\sigma(Y,X)$, respectively.\smallskip
We continue with the verification of Assertion (\ref{t.erg.4}). By (\ref{t.erg.3}) and Lemma~\ref{l.direct},
the sums
\[ \mathrm{fix} (\mathscr{S}) + \overline{\mathrm{span}}^\sigma\mathrm{rg}(I-\mathscr{S}) \text{ and }
\mathrm{fix} (\mathscr{S}') + \overline{\mathrm{span}}^{\sigma'}\mathrm{rg}(I-\mathscr{S}')\] are direct and dense in $X$ with respect to
$\sigma$, resp.\ dense in $Y$ with respect to $\sigma'$.
Let $x\in X$ and $\bar{x} \mathrel{\mathop:}= \lim_\alpha A_\alpha x \in \mathrm{fix}(\mathscr{S})$. Since
$x-\mathrm{co}(\mathscr{S}x) \subset \mathrm{span} \,\mathrm{rg}(I-\mathscr{S})$, we have that
$x-A_\alpha x \in \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$ for all $\alpha\in\Lambda$ and, consequently,
\begin{equation}\label{eq.rangeconv}
x - \bar{x} = \sigma\,\mbox{-}\,\lim_{\alpha} (x- A_\alpha x) \in \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S}).
\end{equation}
This shows that $X=\mathrm{fix}(\mathscr{S})\oplus \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$ and analogously
we deduce that $Y= \mathrm{fix}(\mathscr{S}')\oplus \overline{\mathrm{span}}^{\sigma'} \mathrm{rg}(I-\mathscr{S}')$.
\medskip
In order to verify Assertion (\ref{t.erg.5}), we consider the operators $P$ and $R$,
defined by $Px \mathrel{\mathop:}= \sigma\,\mbox{-}\,\lim_\alpha A_\alpha x$
and $Ry \mathrel{\mathop:}= \sigma\,\mbox{-}\,\lim_\alpha A'_\alpha y$.
Since $\sup\{ \Vert A_\alpha \Vert : \alpha\in\Lambda\}<\infty$ and $X$ and $Y$ are norming for each other,
it follows that $P\in\mathscr{L}(X)$ and $R\in \mathscr{L}(Y)$.
Moreover, for $x\in X$ and $y\in Y$ we have
\[
\applied{Px}{y} = \lim_{\alpha}\applied{A_\alpha x}{y}
= \lim_{\alpha}\applied{x}{A_\alpha'y} = \applied{x}{Ry}
\]
This implies that $P^*Y = RY \subset Y$, hence $P \in \mathscr{L} (X, \sigma)$.
As fixed points are invariant under $\mathscr{A}$, it follows from Lemma~\ref{l.1} that
$P$ is a projection with $\mathrm{rg} P = \mathrm{fix}(\mathscr{S})$ and, by (AS3),
$\mathrm{span}\,\mathrm{rg}(I-\mathscr{S})\subset \ker P$. Since $P$ is $\sigma$-continuous
and $\ker P$ is closed, $\overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})\subset \ker P$.
The converse inclusion follows from \eqref{eq.rangeconv}.
Interchanging the roles of $X$ and $Y$, we see that $P'=R$ is the projection onto $\mathrm{fix}(\mathscr{S}')$
along $\overline{\mathrm{span}}^{\sigma'}(I-\mathscr{S}')$.
In view of $SPx = Px$ and $P(S-I)x = \lim_\alpha A_\alpha (S-I)x = 0$ for all $x\in X$ and $S\in\mathscr{S}$,
$P$ commutes with every operator in $\mathscr{S}$.
\end{proof}
Theorem \ref{t.erg} is symmetric in $X$ and $Y$,
in that for every statement concerning $X$ resp.\ $\mathscr{A}$, there
is a corresponding statement about $Y$ resp.\ $\mathscr{A}'$. This symmetry is crucial.
Indeed, in the case where $Y=X^*$, the norm-bounded net $(A_\alpha x)$ is always relatively $\sigma$-compact and hence
$\sigma$-clusters. However, it does not necessarily $\sigma$-converge as the example of the left shift
on $\ell^\infty$ shows.
Moreover, even if $A_\alpha x$ $\sigma$-converges for all $x \in X$, one cannot deduce
$\sigma'$-con\-ver\-gence of $A_\alpha'y$ for all $y \in Y$ and hence no weak ergodicity of the average scheme,
see Example~\ref{ex:Pnotcontinuous}.
\smallskip
Comparing Theorem~\ref{t.erg} with the classical mean ergodic theorem on Banach spaces, an immediate question
is whether assertions (\ref{t.erg.1}) and (\ref{t.erg.2}) are also equivalent
with (\ref{t.erg.3}) and (\ref{t.erg.4}). This is not the case in general, as Examples~\ref{ex:noconvergence} and \ref{ex.nodecomp} show.
However, some weaker results hold true, which are stated in the following proposition.
\begin{prop}\label{p.weakerg}
Let $(\mathscr{S},\mathscr{A})$ be an average scheme on a norming dual pair $(X,Y)$.
\begin{enumerate}[(a)]
\item\label{i.weakerg.a} Suppose that
$\mathrm{fix} (\mathscr{S})$ and $\mathrm{fix} (\mathscr{S}')$ separate each other.
If $\mathrm{fix} (\mathscr{S})$ (hence also $\mathrm{fix} (\mathscr{S}')$) is finite dimensional, then
\begin{equation}\label{eq.direct}
X = \mathrm{fix} (\mathscr{S}) \oplus \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S}) \quad\mbox{and}\quad
Y = \mathrm{fix} (\mathscr{S}') \oplus \overline{\mathrm{span}}^{\sigma'}\mathrm{rg}(I-\mathscr{S}').
\end{equation}
\item\label{i.weakerg.b} Now assume that \eqref{eq.direct} holds and let $P$ denote the projection onto
$\mathrm{fix} (\mathscr{S})$ along $\overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$.
Then $P \in \mathscr{L} (X, \sigma)$, $P'$ is the projection onto $\mathrm{fix}(\mathscr{S}')$ along
$\overline{\mathrm{span}}^{\sigma'}\mathrm{rg}(I-\mathscr{S}')$ and
\[
\sigma (X, Y_0)\,\mbox{-}\,\lim_{\alpha}A_\alpha x = Px\qquad\mbox{and}\qquad
\sigma(Y,X_0)\,\mbox{-}\,\lim_{\alpha}A_\alpha 'y = P'y
\]
for all $x\in X$ and $y\in Y$, where
\[ X_0 \mathrel{\mathop:}= \mathrm{fix} (\mathscr{S}) \oplus \overline{\mathrm{span}}^{\Vert\cdot\Vert}\mathrm{rg}(I-\mathscr{S})
\quad\mbox{and}\quad
Y_0 \mathrel{\mathop:}= \mathrm{fix} (\mathscr{S}') \oplus \overline{\mathrm{span}}^{\Vert\cdot\Vert}\mathrm{rg}(I-\mathscr{S}').\]
Moreover, $\mathrm{fix}(\mathscr{S})$ and $\mathrm{fix}(\mathscr{S}')$ separate each other.
\end{enumerate}
\end{prop}
\begin{proof}
(a) By Lemma~\ref{l.direct}, the sum $\mathrm{fix} (\mathscr{S}) \oplus \overline{\mathrm{span}}^\sigma\mathrm{rg}(I-\mathscr{S})$
is direct and $\sigma$-dense in $X$.
Since $\mathrm{fix} (\mathscr{S})$ is finite dimensional, the sum is $\sigma$-closed by \cite[\S15.5(3)]{koethe1969}.
It follows that the sum equals $X$.
Similarly one sees that $Y = \mathrm{fix} (\mathscr{S}') \oplus \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S}')$.\medskip
(b) For $x\in X$ we have $x= Px + (I-P)x \in \mathrm{fix} (\mathscr{S}) \oplus \overline{\mathrm{span}}^\sigma\mathrm{rg}(I-\mathscr{S})$.
As $A_\alpha x \equiv x$ for fixed points $x$, to prove $A_\alpha x \to Px$ with respect to
$\sigma (X, Y_0)$, it suffices to show that $\lim_\alpha \applied{A_\alpha x}{y} =0$ for
$x \in \overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$ and $y \in Y_0$.
To that end, first observe that for $w=(I-S)v\in \mathrm{rg}(I-\mathscr{S})$ and $y \in \mathrm{fix} (\mathscr{S}')$ we have that
\[ \applied{w}{y} = \applied{(I-S)v}{y} = \applied{v}{(I-S')y} = 0,\]
i.e.\ $y$ vanishes on $\mathrm{rg}(I-\mathscr{S})$ and hence on
$\overline{\mathrm{span}}^\sigma \mathrm{rg}(I-\mathscr{S})$ by continuity.
Now let
$y =z-S'z\in \mathrm{rg}(I-\mathscr{S}')$. Then
\[
\lim_\alpha \applied{A_\alpha x}{y} = \lim_\alpha \applied{(I-S)A_\alpha x}{z} = 0
\]
by (AS3).
In view of the uniform boundedness of $(A_\alpha)$, property (AS1), it follows that
$\lim_\alpha \applied{A_\alpha x}{y} = 0$ for $y \in \overline{\mathrm{span}}^{\Vert\cdot\Vert}\mathrm{rg}(I-\mathscr{S}')$, too.
Altogether, we have proved that
\[ \sigma (X,Y_0)\,\mbox{-}\,\lim_{\alpha} A_\alpha x=Px.\]
It is easy to see that $P$ is norm-closed, hence it is
bounded by the closed graph theorem.
By interchanging the roles of $X$ and $Y$, one sees that
$\sigma(Y, X_0)\,\mbox{-}\,\lim_{\alpha} A_\alpha'y = Ry$ where $R\in \mathscr{L} (Y)$ is the projection onto
$\mathrm{fix}(\mathscr{S}')$ along $\overline{\mathrm{span}}^{\sigma'}\mathrm{rg}(I-\mathscr{S}')$.
Now let $x \in X$ and $y \in Y$ be given. Since $Px=P^2x\in \mathrm{fix}(\mathscr{S})\subset X_0$
and $Ry=R^2y\in \mathrm{fix}(\mathscr{S}') \subset Y_0$, it follows that
\begin{align*}
\applied{Px}{y} & = \applied{P^2x}{y} = \lim_{\alpha} \applied{A_\alpha Px}{y}
= \lim_{\alpha} \applied{Px}{A_\alpha'y}\\
&= \applied{Px}{Ry}= \lim_{\alpha}\applied{A_\alpha x}{Ry}
=\lim_\alpha \applied{x}{A_\alpha' Ry}= \applied{x}{Ry}.
\end{align*}
This shows that $P^*Y\subset Y$ and $P'=P^*|_Y =R$. In particular, $P \in \mathscr{L} (X, \sigma)$.
In order to prove the final assertion, let $x\in \mathrm{fix}(\mathscr{S})$, $x\neq 0$, and $y\in Y$
such that $\applied{x}{y} \neq 0$. Then
\[ 0\neq \applied{x}{y} = \applied{Px}{y} = \applied{x}{P'y} \]
shows that $\mathrm{fix}(\mathscr{S}')$ separates $\mathrm{fix}(\mathscr{S})$.
Interchanging the roles of $X$ and $Y$ finishes the proof.
\end{proof}
Example~\ref{ex.nodecomp} shows that in Part (a) of Proposition~\ref{p.weakerg} the assumption
that the fixed spaces are finite dimensional cannot be omitted.
\section{Average Schemes on $(C_b(E),\mathscr{M}(E))$}\label{s.eproperty}
Throughout this section, we fix a Polish space $E$ and work on the norming dual
pair $(C_b(E), \mathscr{M}(E))$, which seems to be the most interesting norming dual pair for applications.
We will impose additional conditions on the average scheme $(\mathscr{S}, \mathscr{A})$ such that
assertions (i) -- (iv) of Theorem \ref{t.erg} are equivalent. In view of Proposition \ref{p.weakerg}(b), (iv) implies (iii), so the question is whether
(iii) implies (i).
Examples \ref{ex:noconvergence} and \ref{ex.nodecomp}
show that this is not true without additional assumptions.
In this section we break the symmetry between $X=C_b(E)$ and $Y=\mathscr{M}(E)$
by imposing additional assumptions on the semigroup on the function space. Under these assumptions we can show that
the assertions of Theorem \ref{t.erg} concerning the (adjoint)
semigroup on the space of measures are all equivalent.
\medskip
We start by recalling the definition of the strict topology on $C_b(E)$.
Denote by $\mathscr{F}_0(E)$ the space of all bounded functions $f$ on $E$ that vanish at infinity,
i.e.\ for every $\varepsilon>0$ there is a compact set
$K\subset E$ such that $\vert f(x)\vert < \varepsilon$ for all $x\in K$.
The \emph{strict topology} $\beta_0$ on $C_b(E)$ is the locally convex topology generated by the set of
seminorms $\{ q_\varphi : \varphi \in \mathscr{F}_0(E)\}$ where $q_\varphi(f) \mathrel{\mathop:}= \Vert \varphi f\Vert_\infty$.
The strict topology is consistent with the duality, i.e.\ $(C_b(E),\beta_0)'=\mathscr{M}(E)$,
see \cite[Thm 7.6.3]{jarchow1981}, and
it coincides with the compact open topology on norm bounded subsets of $C_b(E)$ \cite[Thm 2.10.4]{jarchow1981}.
Moreover, it is the Mackey topology of the dual pair $(C_b(E),\mathscr{M}(E))$ \cite[Thm 4.5, 5.8]{sentilles1972}, i.e.\
it is the finest locally convex topology on $C_b(E)$ which yields $\mathscr{M}(E)$ as a dual space.
In particular, $\mathscr{L}(C_b(E),\sigma) = \mathscr{L}(C_b(E),\beta_0)$, see \cite[21.4(6)]{koethe1969}.
\medskip
Now, we formulate and discuss the main condition we impose on an average scheme throughout this section.
Let $d$ be a complete metric $d$ that generates the topology of $E$
and denote by $\mathrm{Lip}_b(E,d)$ the space of all bounded Lipschitz continuous functions on $E$
with respect to $d$.
We assume the average scheme $(\mathscr{S},\mathscr{A})$ to satisfy the following.
\begin{hyp}
\label{c.betastar}
For every $f\in \mathrm{Lip}_b(E,d)$ the net $(A_\alpha f)_{\alpha\in\Lambda}$ clusters in $(C_b(E),\beta_0)$.
\end{hyp}
A priori, this requirement depends on the choice of the metric $d$.
However, Hypothesis \ref{c.betastar} is necessary for the assertion of
Theorem \ref{t.eerg} which, in turn, is independent of the metric $d$.
Hence, under the other assumptions of Theorem \ref{t.eerg},
Hypothesis \ref{c.betastar} holds for some metric $d$ if and
only if it holds for every complete metric on $E$ that generates its topology.
Let us fix such a metric $d$ for the rest of this section.
\medskip
Let us compare Hypothesis \ref{c.betastar} with Assertion (\ref{t.erg.2}) of Theorem \ref{t.erg}.
Instead of assuming that $(A_\alpha f)$ clusters in $(C_b(E),\sigma)$ for every $f\in C_b(E)$,
we require that the net $(A_\alpha f)$ clusters with respect to the finer topology $\beta_0$,
but only for those functions $f$ with some additional regularity, namely for Lipschitz functions.
At first sight, Hypothesis \ref{c.betastar} seems to be rather technical. However, as already mentioned, it
is necessary for Theorem \ref{t.eerg} and, important from the point of view of applications,
it is implied by both the strong Feller property and the e-property, which are well-known assumptions
in the study of ergodic properties of Markov chains and semigroups cf. \cite{kps2010, sw2012}.
Let us discuss these relationships, starting with the e-property,
before continuing with our general theory.\medskip
A family $\mathscr{T} \subset \mathscr{L}(C_b(E))$ is said to have the
\emph{e-property} if the orbits $\{Tf:T \in \mathscr{T}\}$ are equicontinuous for all $f \in \mathrm{Lip}_b(E,d)$,
i.e.\ for all $x \in E$ and $\varepsilon >0$ there exists a $\delta>0$
such that $|Tf(x) - Tf(y)| \leq \varepsilon$ for all $T\in\mathscr{T}$ whenever $d(x,y) < \delta$.
A net $(T_i)_{i\in I} \subset \mathscr{L}(C_b(E))$ is said the have the \emph{eventual e-property} if
there exists a $j \in I$ such that $\{ T_i : i\geq j\}$ has the e-property.
An average scheme $(\mathscr{S},\mathscr{A})$ is said to have the \emph{(eventual) e-property}
if $(A_\alpha)_{\alpha \in \Lambda}$ has the (eventual) e-property.
\medskip
As an instructive example, consider the shift semigroup $\mathscr{S} = (S(t))_{t\geq 0}$ on $(C_b(\mathds{R}), \mathscr{M}(\mathds{R}))$,
given by $S(t)f(x) = f(t+x)$. Then $\mathscr{S}$ has the e-property (we will see below that this implies that
also every average scheme $(\mathscr{S},\mathscr{A})$ has the e-property), since for
$f \in \mathrm{Lip}_b(\mathds{R}, |\cdot|)$ we have
\[
|S(t)f(x) - S(t)f(y)| = |f(t+x) - f(t+y)| \leq L |x-y|
\]
for all $x,y \in \mathds{R}$, where $L$ is the Lipschitz constant of $f$. Note, however, that
this does \emph{not} imply that the orbit $S(t)f$ is equicontinuous for all $f \in C_b(\mathds{R})$.
Before giving further examples, let us prove the mentioned result concerning the e-property of $\mathscr{S}$ and
that of $\mathscr{A}$.
\begin{lem}
\label{l.semieprop}
Let $\mathscr{S} \subset \mathscr{L}(C_b(E),\sigma)$ be a semigroup which has the e-property.
Then every average scheme $(\mathscr{S},\mathscr{A})$ has the e-property.
\end{lem}
\begin{proof}
Let $f\in\mathrm{Lip}_b(E,d)$, $x\in E$ and $\varepsilon>0$. By assumption, there exists $\delta>0$ such that
$\vert (Sf)(x) - (Sf)(y) \vert <\varepsilon$ for all $S\in\mathscr{S}$ whenever $d(x,y)<\delta$.
Fix such a $y$.
By (AS2), given $\alpha \in \Lambda$
we find a function $g_y := \sum_{k=1}^n a_k S_kf \in \mathrm{co}(\mathscr{S}f)$
such that $| \dual{A_\alpha f -g_y}{\delta_x - \delta_y}| \leq \varepsilon$. It follows that
\begin{align*}
|A_\alpha f (x) - A_\alpha f(y)| & \leq |\dual{A_\alpha f -g_y}{\delta_x -\delta_y}| + |\dual{g_y}{\delta_x -\delta_y}|\\
& \leq \varepsilon + \sum_{k=1}^n a_k |S_kf(x) - S_kf(y)| \leq 2 \varepsilon\, .
\end{align*}
Since $\varepsilon$ and $\alpha$ where arbitrary, $\{A_\alpha f : \alpha \in \Lambda\}$ is equicontinuous.
Thus, $(\mathscr{S},\mathscr{A})$ has the e-property.
\end{proof}
Example \ref{ex:noconvergence} shows that if $\mathscr{S}$ does not have the e-property, there might be $\mathscr{A}$
and $\tilde{\mathscr{A}}$ such that $(\mathscr{S},\mathscr{A})$ has the
e-property whereas $(\mathscr{S}, \tilde{\mathscr{A}})$ does not have the e-property.\medskip
\begin{rem}
\label{r.equicontcompact}
In view of the Arzel\`a-Ascoli theorem \cite[Thm 3.6]{khan1979} and (AS1),
an average scheme $(\mathscr{S},\mathscr{A})$ has the (eventual) e-property
if and only if for every $f\in \mathrm{Lip}_b(E,d)$ the set
$\{A_\alpha f: \alpha \in \Lambda\}$ (resp.\ $\{A_\alpha f: \alpha \geq \alpha_0\}$)
is relatively $\beta_0$-compact, equivalently,
the set is relatively compact in the compact-open topology.
Thus, an average scheme with eventual e-property satisfies Hypothesis \ref{c.betastar}.
\end{rem}
We next discuss the strong Feller property which also implies Hypothesis \ref{c.betastar}.
Let us recall that an operator $T \in \mathscr{L} (C_b(E), \sigma)$ is a kernel operator and thus has a unique extension to
an operator on $B_b(E)$.
The operator $T$ is said to be \emph{strong Feller} if this extension maps $B_b(E)$ to $C_b(E)$. It is called \emph{ultra Feller}, if
the extension maps bounded subsets of $B_b(E)$ to equicontinuous subsets of $C_b(E)$.
\begin{prop}
Let $\mathscr{S}\subset \mathscr{L}(C_b(E),\sigma)$ be a semigroup such that some
operator $S\in\mathscr{S}$ is strong Feller. Then every
average scheme $(\mathscr{S},\mathscr{A})$ satisfies Hypothesis~\ref{c.betastar}.
\end{prop}
\begin{proof}
Let $f\in C_b(E)$ and $(\mathscr{S},\mathscr{A})$ be an average scheme.
Since $S$ is strong Feller, the operator $S^2\in\mathscr{S}$ is ultra Feller \cite[\S 1.5]{revuz1975}.
Hence, by (AS1) and the Arzel\`a-Ascoli theorem \cite[Thm 3.6]{khan1979}, the set
$\{ S^2 A_\alpha f : \alpha \in \Lambda \}$ is relatively $\beta_0$-compact. Thus,
every subnet of $(A_\alpha f)$ has a subnet $(A_{\alpha(\beta)} f)_{\beta\in J}$
such that $(S^2 A_{\alpha(\beta)})_{\beta\in J}$ converges in $(C_b(E),\beta_0)$.
By (AS3),
\[ S^2 A_{\alpha(\beta)}f - A_{\alpha(\beta)}f \to 0 \]
in norm and thus with respect to the strict topology which is coarser.
\end{proof}
\medskip
Now we return to our main line of study.
Besides Hypothesis \ref{c.betastar}, we will impose further assumptions. First of all, we assume
that $\mathscr{S}$ is a \emph{Markovian semigroup}, i.e.\ every operator
$S \in\mathscr{S}$ is Markovian by which we mean that $S$ is positive and $S\mathbbm{1} =\mathbbm{1}$.
If $k_S$ denotes the kernel associated with $S$, this is equivalent with the requirement that $k_S(x,\cdot)$ is a
probability measure for all $x \in E$ and $S \in\mathscr{S}$.
Since the applications we have in mind for our theory concern transition semigroups of Markov chains or Markov
processes, this assumption is rather natural.
We will call an average scheme $(\mathscr{S}, \mathscr{A})$
\emph{Markovian} if $\mathscr{S}$ is Markovian.
It follows from (AS2) that this implies that every operator $A_\alpha$ is Markovian, too.
Second, we assume that the directed index set $\Lambda$ of the averages $(A_\alpha)_{\alpha\in\Lambda}$
in this section has a cofinal subsequence.
This holds, for instance, in the classical situations
of Examples \ref{ex.cesaro}, \ref{ex.abel} and \ref{ex.integrable}.
\medskip
We start with two auxiliary lemmas on $\beta_0$-equicontinuous operators.
Let us recall that a family $\mathscr{T}$ of linear operators on a locally convex
space $(X,\tau)$ is called \emph{$\tau$-equicontinuous}, if for every
$\tau$-continuous seminorm $p$, there exists a $\tau$-continuous seminorm $q$ such that
$p(Tx)\leq q(x)$ for every $T\in \mathscr{T}$ and $x\in X$.
In the case where $(X,\tau) = (C_b(E),\beta_0)$, we have the following characterization
of $\beta_0$-equicontinuous operators, see \cite[Prop 4.2]{kunze2009}.
A family $\mathscr{T}\subset \mathscr{L}(X,\beta_0)$ is $\beta_0$-equicontinuous
if and only if for every compact set $K\subset E$ and every $\varepsilon>0$ there exists a compact set
$L\subset E$ such that $\vert p_T \vert (x, E\backslash L) \leq \varepsilon$
for all $x\in K$ and $T\in \mathscr{T}$ where $p_T$ denotes the kernel of $T$.
In what follows we will use that a family of Borel measures on $E$ is relatively $\sigma'$-compact
if and only if it is tight and uniformly bounded in the variation norm,
cf.\ Theorems 8.6.7. and 8.6.8. of \cite{bogachev2007}.
Here, a family $\mathscr{F}\subset\mathscr{M}(E)$ of Borel measures is called \emph{tight} if for all $\varepsilon>0$
there exists a compact set $K\subset E$ such that $\vert \mu\vert(E\backslash K)<\varepsilon$ for all $\mu\in\mathscr{F}$.
\begin{lem}
\label{l.beta0equicont}
Let $\{ T_j : j\in J \} \subset \mathscr{L}(C_b(E),\sigma)$ be a family of Markovian operators
that has the e-property.
Suppose that the family $\{ p_j (x, \cdot ) : j \in J \}$ is
tight for all $x\in E$ where $p_j$ denotes the kernel associated with $T_j$.
Then the operators $\{ T_j : j \in J\}$ are $\beta_0$-equicontinuous.
\end{lem}
\begin{proof}
Let $K\subset E$ be compact and $\varepsilon>0$. By assumption, for every $x\in E$ there exists
a compact set $L_x \subset E$ such that $p_j (x, E\backslash L_x ) \leq \varepsilon$ for all $j\in J$.
We denote by
\[L_x^\varepsilon \mathrel{\mathop:}= \{ x\in E : \mathrm{dist}(x, L_x) < \varepsilon \} \]
the open $\varepsilon$-neighborhood of $L_x$ and define
\[ f_x(y) \mathrel{\mathop:}= \frac{\mathrm{dist}(x,E\backslash L_x^\varepsilon)}{\mathrm{dist}(x,L_x)+\mathrm{dist}(x,E\backslash L_x^\varepsilon)}.\]
Then $f_x$ is Lipschitz continuous and $\mathds{1}_{L_x} \leq f_x \leq \mathds{1}_{L_x^\varepsilon}$.
Since the family $(T_j)$ has the e-property, there exist $\delta_x >0$ such that
\[ \vert (T_j f_x)(x) - (T_j f_x)(y) \vert < \varepsilon\]
for all $j\in J$ whenever $d(x,y) < \delta_x$. By the compactness of $K$, we find
$x_1,\dots, x_n \in E$ such that
\[ K \subset \bigcup_{k=1}^n \{ y\in E: d(x_k,y) < \delta_{x_k} \}.\]
Let $L\mathrel{\mathop:}= \cup_{k=1}^n L_{x_k}$ and $f(y)\mathrel{\mathop:}= \max\{f_{x_1}(y),\dots,f_{x_n}(y)\}$.
Since $L^\varepsilon = \cup_{k=1}^n L_{x_k}^\varepsilon$ we have $\mathds{1}_{L} \leq f \leq \mathds{1}_{L^\varepsilon}$.
Now fix an arbitrary $x\in K$ and choose $k\in \{1,\dots,n\}$ such that $d(x,x_k) < \delta_{x_k}$.
Then we have
\begin{align*}
p_j(x,L^\varepsilon) &\geq \int_E f(y) p_j(x,\mathrm{d} y) = (T_j f)(x)
\geq (T_j f_{x_k})(x_k) -\varepsilon \\
&= \int_E f_{x_k} (y) p_j(x_k,\mathrm{d} y) - \varepsilon \geq p_j(x_k,L_{x_k})-\varepsilon \geq 1-2\varepsilon
\end{align*}
for every $j\in J$. It follows from \cite[Thm 3.2.2]{ethier1986} that
the family
\[\{p_j(x, \cdot): j\in J,\, x\in K\}\]
is tight.
Since $K\subset E$ was arbitrary, the operators $T_j$ are $\beta_0$-equicontinuous by \cite[Prop 4.2]{kunze2009}.
\end{proof}
\begin{lem}
\label{l.orbitsequicont}
Let $\{T_j : j\in J \}\subset \mathscr{L}(C_b(E),\beta_0)$ be a $\beta_0$-equicontinuous
family of operators with e-property.
Then $\{ T_j f : j\in J\}$ is equicontinuous for all $f\in C_b(E)$.
\end{lem}
\begin{proof}
Since $\mathrm{Lip}_b(E,d)$ is a subalgebra of $C_b(E)$ which separates the points of $E$,
it follows from the Stone-Weierstrass theorem
that $\mathrm{Lip}_b(E,d)$ is dense in $(C_b(E),\beta_0)$, see \cite[Theorem 11]{fremlin1972}.
Fix $f\in C_b(E)$ and $x_n,\, x\in E$ with $\lim x_n = x$. We show that $(T_j f)(x_n)$
converges to $(T_j f)(x)$, uniformly in $j\in J$.
This proves the equicontinuity of $\{ T_j f : j\in J\}$ in $x$.
Consider the compact set $K\mathrel{\mathop:}=\{x\}\cup \{x_n : n\in\mathds{N} \}$ and the associated seminorm
$p(h) \mathrel{\mathop:}= \sup \{ \vert h(x) \vert : x\in K\} = \|h\varphi\|_\infty$ for $\varphi = \mathbbm{1}_K$.
Since the family $(T_j)$ is $\beta_0$-equicontinuous,
there exists a $\beta_0$-continuous seminorm $q: C_b(E) \to [0,\infty)$
such that $p(T_j h) \leq q(h)$ for all $h\in C_b(E)$ and $j\in J$.
Now given $\varepsilon>0$, pick $g\in \mathrm{Lip}_b(E,d)$ such that
$q(f-g) \leq \varepsilon$. Since the family $\{ T_j g : j\in J\}$ is equicontinuous,
there exists $n_0 \in\mathds{N}$ such that
\[ \vert (T_j g)(x_n) - (T_j g)(x) \vert \leq \varepsilon\]
for all $n\geq n_0$ and all $j\in J$. This implies that
\[
\vert (T_j f)(x_n) - (T_j f)(x) \vert
\leq 2q(f-g) + \vert (T_j g)(x_n) - (T_j g)(x) \vert \leq 2\varepsilon + \varepsilon
\]
for all $n\geq n_0$ and $j\in J$.
\end{proof}
Now we prove the main result of this section.
\begin{thm}\label{t.eerg}
Let $(\mathscr{S},\mathscr{A})$ be a Markovian average scheme on $(C_b(E),\mathscr{M}(E))$
that satisfies Hypothesis \ref{c.betastar} and
suppose that there exists an increasing cofinal sequence in $\Lambda$. Then the following assertions
are equivalent.
\begin{enumerate}[(i)]
\item $(\mathscr{S},\mathscr{A})$ is weakly ergodic and $\beta_0\,\mbox{-}\,\lim_\alpha A_\alpha f = P f$
for all $f\in C_b(E)$ where $P$ is the ergodic projection.
\item For every $x\in E$ the net $(A'_\alpha \delta_x)$ has a $\sigma'$-cluster point.
\item $\mathrm{fix}(\mathscr{S}')$ separates $\mathrm{fix}(\mathscr{S})$.
\item $\mathscr{M}(E) = \mathrm{fix}(\mathscr{S}') \oplus \overline{\mathrm{span}}^{\sigma'} (I-\mathscr{S}')$.
\end{enumerate}
\end{thm}
\begin{proof}
By Theorem \ref{t.erg}, (i) implies (ii) -- (iv).
Moreover, the implication (iv) $\Rightarrow$ (iii) is part of the proof of Proposition \ref{p.weakerg}
and (iii) follows from (ii) as in the proof of Theorem~\ref{t.erg}.
Hence, it remains to prove that (iii) implies (i) to complete the proof.
\smallskip
Let us assume that
$\mathrm{fix}(\mathscr{S}')$ separates $\mathrm{fix}(\mathscr{S})$. We denote by
$(\alpha_n) \subset \Lambda$ an increasing cofinal sequence. As
$(A_\alpha f)$ clusters in $(C_b(E),\beta_0)$, Lemma \ref{l.convergence} yields that
\[ \beta_0\,\mbox{-}\,\lim_\alpha A_\alpha f = \beta_0\,\mbox{-}\,\lim_{n\to\infty} A_{\alpha_n} f\in C_b(E) \]
exists for all $f\in \mathrm{Lip}_b(E,d)$.
Fix a non-negative measure $\mu \in \mathscr{M}(E)$. Then the scalar sequence
$\applied{f}{A'_{\alpha_n}\mu}=\applied{A_{\alpha_n}f}{\mu}$
converges as $n\to\infty$ for all $f\in \mathrm{Lip}_b(E,d)$.
Thus, by \cite[Cor 8.6.3]{bogachev2007}, the family $\{ A_{\alpha_n}' \mu : n\in\mathds{N}\}$
is tight and Prohorov's theorem \cite[Thm 8.6.2]{bogachev2007} implies that, passing to a subsequence,
$(A'_{\alpha_n}\mu)_{n\in\mathds{N}}$ converges weakly to some measure $\tilde\mu \in \mathcal{M}(E)$.
Altogether, this shows that
\[ \lim_\alpha \applied{f}{A'_\alpha \mu} = \lim_{\alpha} \applied{A_\alpha f}{\mu}
= \lim_{n\to\infty} \applied{A_{\alpha_n}f}{\mu} = \lim_{n\to\infty} \applied{f}{A'_{\alpha_n}\mu}
= \applied{f}{\tilde\mu}\]
for all $f\in \mathrm{Lip}_b(E,d)$.
Since $E$ is separable, the set $\mathrm{Lip}_b(E,d)$ is convergence determining by \cite[Prop 3.4.4]{ethier1986},
hence it follows that $\sigma'\,\mbox{-}\,\lim_\alpha A'_\alpha \mu = \tilde\mu$.
Decomposing a general measure in positive and negative part yields that
$\sigma'\,\mbox{-}\, \lim_\alpha A'_\alpha \mu \in \mathscr{M}(E)$ exists for every $\mu\in\mathscr{M}(E)$.
By Hypothesis \ref{c.betastar} and Lemma \ref{l.seqcompact},
the set $\{A_{\alpha_n} f : n\in\mathds{N}\}$ is relatively $\beta_0$-compact for
every $f\in \mathrm{Lip}_b(E,d)$, which implies by the Arzel\`a-Ascoli theorem
(cf.\ Remark \ref{r.equicontcompact})
that the family $\{A_{\alpha_n} : n\in\mathds{N}\}$ has the e-property.
Now, we infer from Lemma \ref{l.beta0equicont}
that the operators $\{ A_{\alpha_n} : n\in\mathds{N}\}$ are $\beta_0$-equicontinuous.
By Lemma \ref{l.orbitsequicont} this implies that the orbits $\{A_{\alpha_n} f : n\in\mathds{N}\}$
are equicontinuous for all $f\in C_b(E)$.
Using the Arzel\`a-Ascoli theorem again, it follows that
$\{A_{\alpha_n} f : n\in\mathds{N}\}$ is relatively $\beta_0$-compact for all $f\in C_b(E)$.
Now we conclude from Theorem \ref{t.erg} that the average scheme $(\mathscr{S},(A_{\alpha_n}))$
is weakly ergodic with an ergodic projection $P \in \mathscr{L}(C_b(E),\sigma)$.
Since $(\alpha_n)$ was arbitrary and $P$ does not depend on the averages $(A_{\alpha_n})$,
even $(\mathscr{S},\mathscr{A})$
is weakly ergodic.
Finally, the $\beta_0$-convergence of $(A_\alpha f)$ follows from Lemma \ref{l.convergence}.
\end{proof}
\begin{rem}
Assume that $\Lambda = \mathds{N}$ or $\Lambda = [0, \infty)$ in their natural ordering and that $\alpha \mapsto A_\alpha f$ is
continuous as a map with values in $(C_b(E), \beta_0)$. If $\Lambda = \mathds{N}$, this is always the case, for $\Lambda = [0, \infty)$
this is true for the Ces\`aro averages $A_t$ of a semigroup $\mathscr{S} =(S(t))_{t \geq 0}$ on $(C_b(E), \mathscr{M}(E))$
which has $\beta_0$-continuous orbits.
If $(\mathscr{S}, \mathscr{A})$ is weakly mean ergodic, and $A_\alpha f$ converges to $Pf$ with respect to $\beta_0$,
then the means $\{A_\alpha : \alpha \in \Lambda \}$ are $\beta_0$-equicontinuous. Indeed, in this case the function
$F: \Lambda \cup \{\infty\} \to \mathscr{L} (C_b(E), \sigma)$, defined by $F(\alpha ) = A_\alpha$ for $\alpha \neq \infty$ and
$F(\infty) = P$ is strongly $\beta_0$-continuous, whence the equicontinuity follows from \cite[Lemma 3.8]{kunze2009}.
We are thus in the situation of mean ergodic theorems on locally convex spaces \cite{eberlein1949, sato1978}. Note, however, that
in Theorem \ref{t.eerg} we do not a priory assume $\beta_0$-equicontinuity since, as the example of the shift semigroup
shows, Hypothesis \ref{c.betastar} alone does not imply equicontinuity of the averages.
\end{rem}
\begin{rem}
It is immediate that in the situation of Theorem \ref{t.eerg} if $(\mathscr{S}, \mathscr{A})$ is weakly mean ergodic, then
any average scheme $(\mathscr{S}, \mathscr{B})$ which satisfies Hypothesis \ref{c.betastar} is also weakly mean ergodic.
\end{rem}
\section{Counterexamples}\label{s.examples}
We conclude this article with a collection of examples that illustrate that the results obtained in
Section \ref{s.convergence} are optimal.
Our first example shows that even if $A_\alpha x$ $\sigma$-converges for all $x\in X$, it can happen that
on $Y$ the averages $A'_\alpha y$ do not $\sigma'$-converge for some $y\in Y$.
\begin{example}
\label{ex:Pnotcontinuous}
Consider the norming dual pair $(\ell^1, c_0)$ and the power-bounded operator $S: \ell^1 \to \ell^1$, defined by
$S: (x_1, x_2, x_3, \ldots) \mapsto (x_1 + x_2, x_3, \ldots)$. Then the adjoint operator is given by
$S^* (y_1, y_2, \ldots) = (y_1, y_1, y_2, \ldots)$.
In particular, $S^*c_0 \subset c_0$ so that $S \in \mathscr{L} (\ell^1, \sigma)$. Since
\[
S^n(x_1,x_2,\ldots) = (\sum_{j=1}^n x_j, x_{n+1}, x_{n+2}, \ldots)
\]
clearly $\sigma$-converges to $(\sum_{j=1}^\infty x_j, 0,0, \ldots)$, the $\sigma$-limit of the Ces\`aro averages
\[ A_n \mathbf{x} = \frac{1}{n} \sum_{k=0}^{n-1} S^k \mathbf{x} \]
exists for all $\mathbf{x} \in \ell^1$.
However, in this situation the
ergodic projection $P : \mathbf{x} \mapsto (\sum_{j=1}^\infty x_j, 0, 0, \ldots)$ does not respect the duality, i.e.\
$P \not\in \mathscr{L} (\ell^1, \sigma)$. Indeed, for $\mathbf{x} \in \ell^1$ and $\mathbf{y} \in \ell^\infty = (\ell^1)^*$ we have
\[
\applied{P\mathbf{x}}{\mathbf{y}} =y_1\sum_{j=1}^\infty x_1 = \applied{\mathbf{x}}{y_1\mathbbm{1}}
\]
whence $P^*c_0 \not\subset c_0$. By Theorem \ref{t.erg}, the
$\sigma'$-limit of the adjoint averages $A'_n \mathbf{y}$ does not exist for some $\mathbf{y} \in c_0$.
\end{example}
Our next example shows that if $(\mathscr{S},\mathscr{A})$ is an average scheme so that both $X$ and $Y$ can be
decomposed as in (\ref{t.erg.4}) of Theorem \ref{t.erg}, the average scheme is not necessarily weakly ergodic.
In fact, we present two different averages $\mathscr{A}$ and $\tilde{\mathscr{A}}$ for the same semigroup $\mathscr{S}$
such that $(\mathscr{S},\mathscr{A})$ is weakly ergodic whereas for $(\mathscr{S},\tilde{\mathscr{A}})$ only the weaker
convergence properties of Proposition \ref{p.weakerg} (\ref{i.weakerg.b}) hold.
\begin{example}\label{ex:noconvergence}
We consider the set $E = \mathds{Z}\cup \{\infty\}$, where every point in $\mathds{Z}$ is isolated, whereas the neighborhoods of the
extra point $\infty$ are exactly the sets which contain a set of the form $\{ n, n+1, \ldots\}\cup\{\infty\}$ for
some $n\in\mathds{Z}$. Note that $E$ is homeomorphic with $\{-n\,:\, n\in\mathds{N}\} \cup \{1-\frac{1}{n}\,:\, n\in\mathds{N}\} \cup\{1\}$
endowed with the topology inherited from $\mathds{R}$, thus $E$ is Polish.
We work on the norming dual pair $(C_b(E),\mathscr{M}(E))$. Note that a function $f : E \to\mathds{R}$ is continuous
if and only if $\lim_{n\to\infty}f(n) = f(\infty)$ and that $\mathscr{M}(E) = \ell^1(E)$. Consider the
semigroup
$\mathscr{S}:= \{S^n: n\in \mathds{Z}\}$, where
\[
(Sf)(k) = f(k+1)\quad \mbox{for } k \in\mathds{Z}\qquad \mbox{and}\qquad (Sf)(\infty) = f(\infty)\, .
\]
Then $\mathrm{fix} (\mathscr{S}) = \{c\mathbbm{1}_E: c \in \mathds{R}\}$ and $\mathrm{fix} (\mathscr{S}') = \{ c \delta_\infty: c \in\mathds{R}\}$.
In particular, the fixed spaces separate each other and are finite dimensional so that, as a consequence of
Proposition~\ref{p.weakerg} (a), we have
\[
C_b(E) = \mathrm{fix} (\mathscr{S}) \oplus \overline{\mathrm{span}}^\sigma \mathrm{rg} (I -\mathscr{S})
\quad\mbox{and}\quad
\mathscr{M}(E) = \mathrm{fix} (\mathscr{S}') \oplus \overline{\mathrm{span}}^{\sigma'} \mathrm{rg} (I -\mathscr{S}').
\]
We now consider the average schemes $A_n$ and $\tilde{A}_n$, defined by
\[
A_nx \mathrel{\mathop:}= \frac{1}{n}\sum_{k=0}^{n-1}S^kx\quad\mbox{and}\quad
\tilde{A}_nx \mathrel{\mathop:}= \frac{1}{n}\sum_{k=0}^{n-1}S^{-k}x.
\]
That $A_n$ and $\tilde{A}_n$ are indeed average schemes is proved as in Example~\ref{ex.cesaro}.
Defining $Y_0 := \mathrm{fix} (\mathscr{S}') \oplus \overline{\mathrm{span}}^{\|\cdot\|}\mathrm{rg} (I-\mathscr{S}')$, it
follows from Proposition~\ref{p.weakerg} (b) that
\[\sigma (C_b(E), Y_0) \,\mbox{-}\,\lim_{n\to\infty}A_nf = \sigma (C_b(E),Y_0)\,\mbox{-}\,\lim_{n\to\infty}\tilde{A}_nf
= Pf = f(\infty)\mathbbm{1}_E
\]
for all $f\in C_b(E)$.
Actually, using that $f(n) \to f(\infty)$ as $n\to \infty$ for all $f \in C_b(E)$, it is easy to see that
$A_nf \to f(\infty)\mathbbm{1}_E$ pointwise for all $f \in C_b(E)$, hence with respect to $\sigma (C_b(E),\mathscr{M}(E))$.
However, $\tilde A_n f$ does not $\sigma(C_b(E),\mathscr{M}(E))$-converge to $f(\infty)\mathds{1}_E$ for some $f\in C_b(E)$.
Indeed, for $f := \mathbbm{1}_{\mathds{N}\cup{\{}\infty{\}}} \in C_b(E)$
the sequence $\tilde{A}_nf$ converges pointwise to the function $\mathbbm{1}_{{\{}\infty\}}$ which is not continuous.
This shows that in Proposition~\ref{p.weakerg} we cannot expect better convergence than with respect to
$\sigma (X,Y_0)$. On the other hand, this example also shows that even if both $X$ and $Y$ have ergodic decompositions,
it can depend on the average scheme how strong the convergence to the ergodic projection is.
Let us also note that the average scheme $A_n$ has the e-property, whereas $\tilde{A}_n$ does not have the e-property.
For the e-property, the only point of interest is the point $\infty$, as all other points of $E$ are isolated. First note
that in this case $C_b(E) = \mathrm{Lip}_b(E,d)$. Given $f \in C_b(E)$ and $\varepsilon >0$, we find $n_0$ such that
$|f(n) - f(\infty)| \leq \varepsilon$ for all $n\geq n_0$. Consequently, we also have $|S^kf(n) - S^kf(\infty)| = |f(n+k) - f(\infty)|
\leq \varepsilon$ for all $n\geq n_0$ and all $k \geq 0$. Thus $|A_kf(n) - A_kf(\infty)| \leq k^{-1}\sum_{j=0}^{k-1}
|f(n+j) -f(\infty)| \leq \varepsilon$ for all $k \in \mathds{N}$ and all $n\geq n_0$, i.e.\ $\{A_kf:k\in\mathds{N}\}$ is equicontinuous.
On the other hand, $\tilde{A}_n$ cannot have the e-property, since in this case it would follow from Theorem~\ref{t.eerg}
that $\tilde{A}_nf \to Pf$ pointwise, which was seen to be wrong above.
\end{example}
We have seen in Proposition \ref{p.weakerg} (\ref{i.weakerg.a}) that if $\mathrm{fix}(\mathscr{S})$ and $\mathrm{fix}(\mathscr{S}')$ separate
each other and are finite dimensional, then both $X$ and $Y$ can be decomposed as in (\ref{t.erg.4}) of Theorem \ref{t.erg}.
Our last example shows that this is not true for infinite dimensional fixed spaces.
\begin{example}
\label{ex.nodecomp}
In the following we construct a positive,
contractive and $\sigma$-continuous operator $S$ on the norming dual pair
$(C_b(E),\mathscr{M}(E))$ such that, for $\mathscr{S}:=\{S^n:n\in\mathds{N}_0\}$, we have
\[ \mathscr{M}(E)\not= \mathrm{fix}(\mathscr{S}') \oplus \overline{\mathrm{rg}}^\sigma(I-\mathscr{S}')\]
while the fixed spaces of $\mathscr{S}$ and $\mathscr{S}'$ separate each other.
For $n\in\mathds{N}$ let $K_n\mathrel{\mathop:}= \{0,\dots,n\} \times \{1/n\}$ and $K_0 \mathrel{\mathop:}= \mathds{N}_0\times \{0\}$.
On the set $E \mathrel{\mathop:}= \bigcup_{n\in\mathds{N}_0} K_n$
endowed with the topology inherited from $\mathds{R}^2$,
we consider the continuous mapping $\varphi : E\to E$, given by
\[ \varphi((k,1/n)) \mathrel{\mathop:}= \begin{cases}
((k+1), 1/n ) & n\in\mathds{N},\, k\in\{0,\dots,n-1\}\\
(0,1/n) & n\in\mathds{N},\, k=n \end{cases} \]
and $\varphi(k,0) \mathrel{\mathop:}= (k+1,0)$ for all $k\in\mathds{N}$. Thus on each $K_n$ the map $\varphi$ shifts to the right and for
$n \neq 0$ the point $(n,\frac{1}{n})$ is mapped to $(0,\frac{1}{n})$, see Figure~\ref{fig.omega}.
\begin{figure}[h!]
\setlength{\fboxsep}{10pt}
\fbox{\includegraphics[width=0.5\textwidth]{omega2.pdf}}
\caption{Transformation of $E$ by the action of $\varphi$}
\label{fig.omega}
\end{figure}
Let $S$ denote the induced operator on $C_b(E)$, defined as $Sf \mathrel{\mathop:}= f\circ \varphi$.
It is easy to see that
\[ \mathrm{fix}(\mathscr{S}) = \mathrm{fix}(S) = \Big\{ \sum_{n=0}^\infty a_n\mathbbm{1}_{K_n}\,:\,
\lim_{n\to\infty}a_n = a_0\Big\}
\]
On the other hand,
\[ \mathrm{fix}(\mathscr{S}') = \mathrm{fix}(S') = \Big\{ \sum_{n=1}^\infty a_n\zeta_n\,:\,
(a_n) \in \ell^1\Big\},
\]
where $\zeta_n$ denotes counting measure on $K_n$ with the normalization $\zeta_n(K_n)=1$.
It thus follows that the fixed spaces separate each other.
We now show that
\[ \delta_0 \not\in \mathrm{fix} (\mathscr{S}') \oplus \overline{\mathrm{span}}^{\sigma'}\mathrm{rg}(I-\mathscr{S}').\]
Aiming for a contradiction, let us assume that there exists a sequence $(a_n)\in \ell^1$ and
a net $(\mu_\alpha)_\alpha \subset \mathrm{span}\, \mathrm{rg} (I-\mathscr{S}')$, $\sigma'$-converging to $\mu \in \mathscr{M}(E)$, such that
\[ \delta_0 = \sum_{n=1}^\infty a_n \zeta_n + \mu.\]
Since $\mathbbm{1}_{K_n}$ is a fixed
point of $\mathscr{S}$ and $\mu_\alpha$ belongs to $\mathrm{span} \,\mathrm{rg} (I-\mathscr{S}')$, we have
$\dual{\mathbbm{1}_{K_n}}{\mu_\alpha} = 0$
for all $n\in\mathds{N}$ and all $\alpha$. Thus also $\dual{\mathbbm{1}_{K_n}}{\mu} = 0$. It follows that
\[ 0 = \applied{\mathds{1}_{K_n}}{\delta_0} = a_n \zeta_n(K_n) + \mu(K_n) = a_n\]
for all $n\in\mathds{N}$ and hence $\delta_0 = \mu$. Since $\mathbbm{1}_E \in \mathrm{fix}{\mathscr{S}}$, the contradiction
\[ 1 = \dual{\mathbbm{1}_E}{\delta_0} = \lim_{\alpha} \dual{\mathbbm{1}_E}{\mu_\alpha} = 0\]
follows.
\end{example}
\bibliographystyle{abbrv}
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"redpajama_set_name": "RedPajamaArXiv"
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\section{Introduction} \label{s:intro}
The study of self-organizing dynamics is a very active research subject and a
vast literature is focused on models describing alignment or
agreement/disagreement phenomena caused by the interaction between agents, see
\cite{castellano2009statistical} for instance.
There exist various strategies, such as the approach based on cellular
automata allowing a continuum of states obeying to a probabilistic interaction
law \cite{weisbuch2003interacting}, or models describing the time evolution of
probability densities, see \cite{bou-sal, tos06}.
We consider here the viewpoint of dynamical systems and study
first-order consensus dynamics, following the works of Hegselmann and Krause
\cite{heg-kra, kra}. We recall moreover that there are other celebrated
consensus models, for example the alignment second-order models due to
Vicsek {\it et al.} \cite{vicsek1995novel, vic-zaf-2012} and to Cucker
and Smale \cite{cuc-sma, cuc-sma-2}, see also \cite{ajm-bel-gib}.
Many studies of the Hegselmann-Krause model were recently implemented, as in \cite{bic-ko-zua, has21, jab-mot, mot-tad, pao21}. We point out \cite{weber2020bounded, web-the-mot}, where the authors proposed graph-theory related ideas to develop their theory. Second-order models were also studied in \cite{degond2008continuum, ha2009simple, jab-mot, mot-tad-2011, shen2007cucker}. There are of course natural questions arising on the control on these models, which led, among other works to \cite{altafini2013consensus,ayd-cap-etal, blondel2009krause, caponigro2013sparse, caponigro2015sparse, olshevsky2009convergence, piccoli2015control2, wongkaew2015control}. The models were also embedded in more sophisticated ones, as in \cite{barbaro2016phase}, or allowed to establish some models hierarchy in asymptotic analysis studies, \textit{e.g.} \cite{carrillo2014derivation, carrillo2010asymptotic, degond2013macroscopic, ha2008particle}. The production of accurate numerical strategies for the quantitative study of these models has been the
goal of, for example, \cite{albi2013binary}.
\paragraph{First-order linear consensus model in finite dimension.}
The Hegselmann-Krause model allows to describe the time evolution of a set of
time-depending state variables $y_i:\mathbb R_+\to\mathbb R$, $1\leq i \leq N$. Each state
variable evolves with respect to time, according to the following first-order
differential system, which quantifies the modifications induced by the
interactions between all state variables of the system. For any
$1\leq i,j\leq N$, let $\sigma_{ij}\geq 0$ be the interaction frequency of the agent
$i$ with the agent $j$. The differential system is then written as
\begin{equation} \label{e:krause}
\dot y_i(t) = \sum_{j=1}^N \sigma_{ij} \left( y_j(t) - y_i(t)
\right), \qquad t\in\mathbb R_+, \qquad 1\leq i \leq N.
\end{equation}
The right-hand side of \eqref{e:krause} stands for binary interactions between
agents. Due to the form of the system, the values of $\sigma_{ii}$, $1\leq i \leq
N$, can be arbitrarily chosen since they do not influence the dynamics. Equation~\eqref{e:krause} is supplemented with initial conditions
\begin{equation} \label{e:initconddiscrete}
y_i(0)=y_i^{\mathrm{in}}, \qquad 1\leq i \leq N,
\end{equation}
where $y_i^{\mathrm{in}} \in \mathbb R$, $1\leq i \leq N$.
The linear finite-dimensional Cauchy problem
\eqref{e:krause}--\eqref{e:initconddiscrete} has of course a unique global solution.
The state variables have many possible interpretations. For example, we can
consider a population composed of $N$ agents and that, for all $i$, $y_i$
represents the position of the agent $i$. In this case,
Equations~\eqref{e:krause}--\eqref{e:initconddiscrete} describe a situation of
\textsl{herding}. More generally, this model belongs to the category of
\textsl{consensus systems} because of its stabilization properties in large
time.
The properties of the system are sensitive with respect to the values of
$\sigma_{ij}$ as well as the methods of proof. In particular, if the system is
\textit{symmetric}, \textit{i.e. } when $\sigma_{ij}=\sigma_{ji}$ for any $1\leq i,j\leq N$, or when
$\sigma_{ij}$ only depends on either $i$ or $j$, the mathematical study of
\eqref{e:krause}--\eqref{e:initconddiscrete} can be widely simplified.
This explains why these assumptions, though restrictive, are popular.
Equation~\eqref{e:krause} can be rewritten in a matrix form. Let us consider
the matrices $\sigma=(\sigma_{ij})_{1\leq i,j \leq N}\in \mathbb R^{N\times N}$ and
$A\in \mathbb R^{N\times N}$ such that
\begin{equation*}
A_{ij}= \sigma_{ij}~~\mbox{ if } i\neq j, \qquad
A_{ii}=-\sum_{k\neq i} \sigma_{ik}.
\end{equation*}
If we denote by $\operatorname{diag}(z)$ the diagonal matrix where the nonzero coefficients are given by the coordinates of a vector $z\in\mathbb R^N$, and if we set $e=(1,\dots,1)^\intercal \in \mathbb R^N$, then we can write
\begin{equation}
\label{eq:Adiscr}
A=\sigma-\operatorname{diag}(\sigma e).
\end{equation}
In fact, $A$ can be seen as an arbitrary $N\times N$ real matrix whose
off-diagonal coefficients are nonnegative, and such that the sum of coefficients
of any of its rows is zero. With these notations, the linear problem
\eqref{e:krause}--\eqref{e:initconddiscrete} is written, with the state
$$
y:\mathbb R_+\to\mathbb R^N, ~t\mapsto
\begin{pmatrix} y_1(t) \\ \vdots \\ y_N(t) \end{pmatrix},$$
as
\begin{equation} \label{e:pbdimfinie}
\dot y(t) = A y(t), \qquad y(0)=
\begin{pmatrix}
y_1^{\mathrm{in}} \\ \vdots \\ y_N^{\mathrm{in}}
\end{pmatrix}:=y^{\mathrm{in}}.
\end{equation}
This matrix form allows to highlight the difficulties induced by the
non-symmetry of $A$ when investigating the large-time behavior of solutions
to \eqref{e:pbdimfinie}.
\paragraph{Why the usual ``$\boldsymbol{L^2}$ theory'' cannot be applied in the non-symmetric case.}
The proof of convergence towards consensus is easy to establish in the symmetric case by using the standard Euclidean approach, denoting by $\langle\cdot,\cdot\rangle$ the usual Euclidean scalar product in $\mathbb R^N$ and by $|\cdot|$ the associated Euclidean
norm. The classical way to prove convergence towards equilibrium of a dynamical
system consists in studying the time decay of $|y-y_{\mbox{\tiny eq}}|$,
where $y_{\mbox{\tiny eq}} \in\ker A$ is the expected consensus.
Let us first find out the value of $y_{\mbox{\tiny eq}}$. We shall prove below that $\ker A$ is the one-dimensional space spanned by $e=(1,\dots,1)^\intercal \in \mathbb R^N$ (see Proposition\,\ref{t:poids}). When $A$ is self-adjoint, it is straightforward to prove that $t\mapsto \sum_i y_i(t)$ is constant. Indeed, its derivative equals
$\langle\dot y(t), e\rangle=\langle Ay(t), e\rangle= \langle y(t),A e\rangle=0$.
Therefore $y_{\mbox{\tiny eq}}=\frac1N(\sum_iy_i^{\mathrm{in}}) e$.
Besides, one has
\begin{equation*
\frac12 \frac\mathrm{d}{\mathrm{d} t} |y(t)-y_{\mbox{\tiny eq}}|^2 =
\langle A(y(t)-y_{\mbox{\tiny eq}}),y(t)-y_{\mbox{\tiny eq}}\rangle.
\end{equation*}
Since, thanks to the Gershgorin circle theorem, for any eigenvalue $\mu$ of $A$, there exists $i$ such that
\begin{equation} \label{e:gershgorin}
\Big|\mu+\sum_j \sigma_{ij}\Big| \leq \sum_j \sigma_{ij},
\end{equation}
it is clear that, apart from $0$, all eigenvalues of $A$ have a negative real part. Consequently, if $A$ is self-adjoint, then the system is dissipative and we can prove the convergence of $y$ towards $y_{\mbox{\tiny eq}}$ in large time, with an explicit exponential rate given in terms of nonzero eigenvalues of $A$.
In contrast, the quantity $\langle Ay,y\rangle$ may be positive for some
$y\in\mathbb R^N$ when $A$ is not self-adjoint. For instance, if we choose, for
any $j$, $\sigma_{ij}=1$ when $i<N$ and $\sigma_{Nj}=\sigma_N\neq1$, it is
easy to check that $\langle Ay,y\rangle>0$, with
$y=(1,\cdots,1,\gamma)^\intercal$, for some values of $\gamma\in\mathbb R$. It is of
course not contradictory with the the fact that all eigenvalues of $A$ have
nonpositive real parts.
Therefore, the study of the non-symmetric linear Hegselmann-Krause model cannot
rely on a standard variance-based strategy. In \cite{jab-mot, mot-tad-2011}, the authors
have developed a \emph{$L^\infty$ theory}, or hybrid theories which are at least
partially based on $L^\infty$ tools. Although the latter approach is remarkable, the
$L^2$ framework remains more convenient to investigate stability and robustness properties of the system, or to design asymptotically stabilizing controls, see \cite{weber2020bounded, web-the-mot}.
\paragraph{First-order consensus model in infinite dimension.}
In this work, we also tackle the problem generalized to set of a continuum of agents. The corresponding model describes the time evolution of a continuum of agents labelled by a continuous variable lying in an open bounded subset $\Omega\subset \mathbb R^d$, $d\geq 1$. Without loss of generality, we assume that $|\Omega|=1$ (Lebesgue measure of $\Omega$). The space $\mathbb R^d$ is endowed with the standard Euclidean norm $|\cdot|$.
Let $\sigma\in L^\infty(\Omega^2)$ the generalized nonnegative interaction function, and define $S\in L^\infty(\Omega)$ related to $\sigma$ through
\begin{equation} \label{e:defS}
S(x)=\int_{\Omega} \sigma(x,x_*)\, \mathrm{d} x_*, \qquad \mbox{for a.e.}~x\in\Omega.
\end{equation}
The unknown is now a function $y:\Omega\times\mathbb R_+ \to \mathbb R$
which evolves according to the dynamics
\begin{equation}\label{e:diminfinie}
\frac{\partial y}{\partial t}(x,t) =
\int_\Omega \sigma(x,x_*) ( y(x_*,t)-y(x,t) ) \, \mathrm{d} x_*,
\end{equation}
for $x\in\Omega$ and $t\geq 0$, with initial condition
\begin{equation}\label{e:diminfinieinitcond}
y(\cdot,0) = y^{\mathrm{in}},
\end{equation}
where $y^{\mathrm{in}}\in L^2(\Omega)$ is given.
The previously defined interaction matrix $A$ becomes a linear
interaction operator $A:L^2(\Omega)\to L^2(\Omega)$ defined, for every
$z\in L^2(\Omega)$, by
\begin{equation}\label{d:operateurA}
(Az)(x) = \int_\Omega \sigma(x,x_*) ( z(x_*)-z(x) ) \, \mathrm{d} x_*, \qquad
\mbox{for a.e.}~x\in\Omega.
\end{equation}
Since $\sigma$ lies in $L^\infty$ and $\Omega$ is bounded, $A$ is
bounded. Eventually, we emphasize that $\sigma$ is not assumed to have any
symmetry properties, so that $A$ is not self-adjoint in general.
The operator $A$ only acts on the labelling variable $x\in\Omega$. Hence, the Cauchy problem \eqref{e:diminfinie}--\eqref{e:diminfinieinitcond} can be written as
\begin{equation} \label{e:pbdiminfinie}
\frac{\partial y}{\partial t} = A y, \qquad y(\cdot,0) = y^{\mathrm{in}}.
\end{equation}
Since $A$ is bounded, \eqref{e:pbdiminfinie} has a unique global solution $t\mapsto e^{tA}y^{\mathrm{in}}$, which lies in $C^1(\mathbb R_+;L^2(\Omega))$. We also have a maximum principle on $y$. To recover this result, we first need to introduce the (Hilbert-Schmidt hence compact) operator $K:L^2(\Omega)\to L^2(\Omega)$ of kernel $\sigma$ defined by
\begin{equation}\label{e:defK}
(K z)(x) = \int_\Omega \sigma(x,x_*) z(x_*) \, \mathrm{d} x_*, \qquad
\mbox{for a.e.}~x\in\Omega,
\end{equation}
and the (bounded) multiplication operator $M_S:L^2(\Omega)\to L^2(\Omega)$ by $S$, \textit{i.e. } $M_S=S\,\mathrm{Id}$ where $\mathrm{Id}$ is the identity on $L^2(\Omega)$. Then it is possible to write \eqref{d:operateurA} under the form, to be related to \eqref{eq:Adiscr},
\begin{equation*
A = K - M_S = K-S\,\mathrm{Id}= K-(Ke)\,\mathrm{Id},
\end{equation*}
where $e$ is the constant function equal to $1$, and \eqref{e:diminfinie} as
$$\frac{\partial y}{\partial t}+M_S y = Ky.$$
Using $(x,t)\mapsto y(x,t)e^{S(x)t}$ and the fact that $K$ obviously
preserves nonnegativity, we conclude that the solution of \eqref{e:diminfinie}
remains nonnegative almost everywhere if its initial datum is nonnegative, and
that $y$ remains bounded between the essential infimum and supremum of
$y^{\mathrm{in}}$.
The article is structured as follows. In the next section, we state our theorems of convergence to consensus. The proof of those results starts with preliminary results, mainly of geometric nature, in Section~\ref{s:AAstarv} and is concluded in Section~\ref{s:convcons}. Then, in Section~\ref{s:lyap}, we present further results: the time-discrete version of the finite-dimensional model, the link between our models, noting that the finite-dimensional model can be seen as the infinite-dimensional one where the counting measure is used instead of the Lebesgue measure in the integrals, and some arguments related to Lyapunov functionals. Eventually, in Section~\ref{s:num}, we describe some numerical simulations.
\section{Main results} \label{s:main}
In what follows, $X$ will denote the state space, which is a Hilbert space endowed with its scalar product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$. This space will be either $\mathbb R^N$ or $L^2(\Omega)$ endowed with their usual scalar product. With both finite and infinite dimension notations, we have $Ae=0$. Besides, it will also be convenient to denote by $W$ the Banach space where $\sigma$ lies, either $\mathbb R^{N\times N}$ or $L^\infty(\Omega^2)$.
\subsection{Strong connectivity}
Our work relies on graph-theory-related assumptions on $\sigma$ already discussed in \cite{web-the-mot} when $X=\mathbb R^N$.
\subsubsection*{In finite dimension}
We associate to $(\sigma_{ij})$ the directed graph $G$ (see for instance \cite[Chapter~10]{bon-mur}), whose vertices are $1$, $2$, $\ldots$, $N$, and which has an edge from $i$ to $j$ when $\sigma_{ij}>0$. This concept of directed graph allows to handle the heterogeneity of the reciprocal influence of the agents. In this setting, the agents are the vertices and the matrix $A$ is linked to the edges between two given vertices. More precisely, when an entry of $A$ is zero, there is no direct interaction between the corresponding agents and when an entry of $A$ is positive, the corresponding agents are directly connected. We recall that $G$ is \emph{strongly connected} if, for any pair $(i,j)$ with $i\neq j$, there exists a a finite set of arcs, called a path, joining $i$ to $j$ in $G$, \textit{i.e. } there exists a sequence $(i_0, \dots, i_r)$, $r\in\mathbb N^*$, of distinct indices satisfying
$$i_0=i, \quad i_r=j, \qquad \sigma_{i_{k}i_{k+1}}>0, \quad 0\leq k\leq r-1.$$
Usually, the strong connectivity definition states that, for any pair $(i,j)$ with $i\neq j$, there exist a path joining $i$ to $j$ \textit{and} a path joining $j$ to $i$, to differ from the weak connectivity notion, for which the previous \textit{and} is replaced by \textit{or}.
The strong connectivity assumption can be interpreted as a constraint on the links between the agents of the system: for instance, when $G$ is strongly connected, any pair $(i,j)$ of individuals of the population can interact directly (if $\sigma_{ij}>0)$, or indirectly through other individuals (when $\sigma_{ij}=0$ but there is a path between $i$ and $j$).
\subsubsection*{In infinite dimension}
The same notion of directed graph is extended in the infinite-dimensional setting in the following way.
The vertices of the directed graph $G$ associated to $\sigma\in L^\infty(\Omega^2)$ are chosen as the Lebesgue points $x$ of $\sigma$ in $\Omega$, \textit{i.e. } the ones such that $x_*\mapsto \sigma(x,x_*)$ is defined almost everywhere in $\Omega$. Then, for any vertices $x_1$, $x_2$ such that $x_1\neq x_2$, we say that $(x_1,x_2)$ is an arc if $x_2 \in \mathrm{ess\,supp\,} \sigma(x_1,\cdot)$.
The directed graph $G$ is strongly connected if both following properties hold:
\begin{enumerate}
\item For any Lebesgue points $(x,x_*)$ with $x\neq x_*$, there exists a path joining $x$ to $x_*$ in $G$, \textit{i.e. } there exist two-by-two distinct Lebesgue points $x_0$, ..., $x_r$, $r\in\mathbb N^*$ such that
$$x_0=x, \quad x_r=x_*, \qquad x_{k+1} \in \mathrm{ess\,supp\,} \sigma(x_k,\cdot), \quad 0\leq k\leq r-1.$$
\item Recalling that $S$ is defined by \eqref{e:defS}, we have
\begin{equation} \label{e:hypSdelta}
\delta := \operatorname{ess\,inf\,} S >0,
\end{equation}
\end{enumerate}
The first property is exactly the one which defined a strongly connected directed graph in finite dimension. The second one, which means that (almost) every agent can interact with a significant continuum of agents in $\Omega$, measured thanks to $\delta$, is clearly satisfied in finite dimension, without further assumption. Indeed, for any $i$, there necessarily exists $j\neq i$ such that $\sigma_{ij}>0$. This ensures that the term $\sum_j \sigma_{ij}$ corresponding to $S$ is positive for any $i$. In fact, that second property is also directly satisfied if, for instance, $\sigma$ is continuous on $\bar\Omega^2$ and satisfies the first property.
\subsection{Main results}
Under this strong connectivity assumption, we can now identify the consensus value in terms of an eigenvector $v$ of $A^*$, for which we prove the existence and positivity properties that were only assumed in \cite{olsab-mur}. We also recover the $L^2$-convergence towards consensus obtained in finite dimension in \cite{web-the-mot}, extend it to the infinite-dimensional model, and provide the sharp convergence rate in both cases.
We state hereafter our two main results, valid in finite and infinite dimensions, and we start by providing the proper consensus value.
\begin{theorem} \label{t:poids}
Assume that the graph associated to $\sigma$ is strongly connected. Then there exists a unique $v\in\ker A^*$ such that $v>0$ and $\langle v,e\rangle =1$, and the weighted mean of any solution $y$ to \eqref{e:pbdimfinie} or \eqref{e:pbdiminfinie} defined by $\bar y^v=\langle y(t),v\rangle\,e$ is constant with respect to time.
\end{theorem}
Note that, if $\sigma$ is symmetric, then $A$ is self-adjoint, and $v=e/\|e\|^2$. Let us also point out that $v$ was also defined in \cite{olsab-mur}, but the fact that $v>0$ was only assumed, not proved. The weighted mean $\bar y^v$ is the value of the consensus to which any solution to \eqref{e:pbdimfinie} or \eqref{e:pbdiminfinie} converges in large time.
\begin{theorem} \label{t:convcons}
Assume that the directed graph associated to $\sigma$ is strongly connected. Let $y:\mathbb R_+\to X$ solving \eqref{e:pbdimfinie} or \eqref{e:pbdiminfinie}. Then there exists $\rho>0$ such that, for any $\varepsilon\in (0,\rho)$, there exists $M_\varepsilon>0$ satisfying
$$\|y(t)-\bar y^v\| \leq M_\varepsilon \|y^{\mathrm{in}}-\bar y^v\| e^{(-\rho+\varepsilon)t}, \qquad \forall t\geq0.$$
\end{theorem}
The convergence rate $\rho$ was already exhibited in \cite{web-the-mot} when $X=\mathbb R^N$, as being $|\operatorname{Re} \lambda_2|$, where $\lambda_2$ is the eigenvalue of $A$ whose real part is the highest one, apart from $0$. Note that, if $A$ is a symmetric matrix, $|\operatorname{Re} \lambda_2|$ is the so-called Fiedler number, which measures the strong connectivity of the graph associated to $(\sigma_{ij})$, see \cite{olsab-mur}. We shall prove that the sharp value of $\rho$ is
$$\rho=\mathrm{s}(A_2),$$
where $A_2:\operatorname{im} A \to \operatorname{im} A$ is the homoeomorphism defined by $A_2 z=Az$ for every $z\in \operatorname{im} A$ (indeed, we shall see that $\operatorname{im} A = \operatorname{im} A^2$), and $\mathrm{s}(A_2)<0$ is the spectral bound of $A_2$.
\begin{remark} \label{r:partialconnect}
If the strong connectivity property is not satisfied on the whole set of agents, but only on subsets of a disjoint partition of that set, we recover a clustering effect and the optimal exponential convergence rate can be computed in the same way as explained above.
\end{remark}
\paragraph{Strategy of proof.}
Let us briefly explain the key arguments of our proof of Theorems~\ref{t:poids}--\ref{t:convcons}. We start by checking in Proposition~\ref{p:ptykerA-kerAstar} that the null spaces of $A$ and $A^*$ are one-dimensional, and that $\ker A$ is generated by $e$, which is defined by \eqref{e:def-e}. Then we prove that any nonzero element $y\in\ker A^*$ is either positive or negative (meaning that $y>0$ or $y<0$ almost everywhere when $X=L^2(\Omega)$ or that all coordinates of $y$ are nonzero and have the same sign when $X=\mathbb R^N$). To prove this fact, we perform a deformation of the model: we define a homotopy path joigning the non-symmetric interaction function (or matrix) $\sigma$ to a symmetric one, and we show that, along this path, elements of $\ker A^*$ always keep the same sign. This ensures the existence and uniqueness of a positive weight $v\in\ker A^*$ as in Proposition~\ref{p:poidsrappel}, which is used to compute the consensus value $\bar y^v$. Moreover, we define a Hilbert structure on $X$, equivalent to the standard one, involving a scalar product and a norm weighted by $v$. Proposition~\ref{p:directsums} then clarifies the geometric context of our convergence result. Indeed, in order to prove Theorem~\ref{t:convcons}, we also need to introduce the $v$-orthogonal projector $\pi$ on $\operatorname{im} A$ and the homeomorphism $A_2:\operatorname{im} A\to\operatorname{im} A$, $z\mapsto Az$. In finite dimension, the matrix $A_2$ is Hurwitz, which is enough to conclude, and incidentally recover the well-known property related to the Fiedler number when $A$ is symmetric. In infinite dimension, we perform a thorough study of the discrete and essential spectra of the operators $A$ and $A_2$, beginning with Proposition~\ref{p:spectreA}, where the Hilbert structure weighted by $v$ is crucial, since it allows to work in the appropriate setting, with respect to $v$. The sharp exponential rate is then obtained as $|\mathrm{s}(A_2)|$, where $\mathrm{s}(A_2)<0$ is the spectral bound of $A_2$.
\section{Properties of $\boldsymbol{A}$ and $\boldsymbol{A^*}$, definition of the weight} \label{s:AAstarv}
This section is dedicated to the preliminary results required to prove Theorems~\ref{t:poids}--\ref{t:convcons}. The reader can focus on the statement (and skip the proof) of the various propositions below, before getting to Section~\ref{s:convcons}.
Let us first write the expression of $A^*$, which is the same notation for the transposed matrix of $A$ when $X=\mathbb R^N$ and the adjoint operator of $A$ when $X=L^2(\Omega)$. We have, for any $z\in\mathbb R^N$,
\begin{equation*
(A^*z)_i = \sum_j \sigma_{ji} z_j- \Big(\sum_j \sigma_{ij}\Big)z_i,\qquad
1\leq i\leq N,
\end{equation*}
and, for any $z\in L^2(\Omega)$,
\begin{multline*
A^*z(x) = \int_\Omega \sigma(x_*,x) z(x_*)\,\mathrm{d} x_*- \left(\int_\Omega \sigma(x,x_*)\,\mathrm{d} x_*\right)z(x) \\
=\int_\Omega \sigma(x_*,x) z(x_*)\,\mathrm{d} x_*- S(x)z(x),\qquad
\mbox{for a.e.}~x\in\Omega.
\end{multline*}
Consequently, if $z\in\ker A^*$, we have either, when $X=\mathbb R^N$,
\begin{equation*
\Big(\sum_j \sigma_{ij}\Big)z_i = \sum_j \sigma_{ji} z_j, \qquad 1\leq i\leq N,
\end{equation*}
or, when $X=L^2(\Omega)$,
\begin{equation}\label{e:ptySker}
S(x) z(x) = \int_\Omega \sigma(x_*,x) z(x_*)\,\mathrm{d} x_*, \qquad \mbox{for a.e.}~x\in\Omega.
\end{equation}
Let us emphasize that the previous equality implies that $\ker A^*$ is continuously embedded in $L^\infty(\Omega)$. Indeed, consider $z\in \ker A^*$. From \eqref{e:hypSdelta}--\eqref{e:ptySker}, we have, for almost every $x\in\Omega$,
$$|z(x)|=\frac{1}{S(x)} \left|\int_\Omega \sigma(x_*,x) z(x_*)\,\mathrm{d} x_*\right| \leq
\frac{\|\sigma\|_{L^\infty}}{\delta} \|z\|_{L^2(\Omega)}.$$
We can now proceed with the preliminary propositions leading to the proof of Theorems~\ref{t:poids}--\ref{t:convcons}. We start by recalling that
\begin{equation} \label{e:def-e}
e=(1,\dots,1)^\intercal ~~ \mbox{if }X=\mathbb R^N, \qquad e=1 ~~ \mbox{if }X=L^2(\Omega).
\end{equation}
\subsection{Null spaces of $\boldsymbol{A}$ and $\boldsymbol{A^*}$}
\begin{proposition} \label{p:ptykerA-kerAstar}
The following properties hold:
\begin{enumerate}[(i)]
\item $\ker A=\ker A^2$ is a one-dimensional subspace of $X$ spanned by $e$,
\item $\ker A^*=\ker (A^*)^2$ is a one-dimensional subspace of $X$,
\item $0$ is a simple eigenvalue of both $A$ and $A^*$.
\end{enumerate}
\end{proposition}
Note that we can also obtain that $\dim \ker A=1$ thanks to the Perron-Frobenius theorem, as in \cite{web-the-mot}.
\begin{proof} Proposition~\ref{p:ptykerA-kerAstar} is proved hereafter in a unified way, not depending on the fact that we work in a finite-dimensional setting or not. However, its proof may require some arguments based on dimension-related arguments on $X$.
\subsubsection*{Step 1 -- $\boldsymbol{\ker A}$ and $\boldsymbol{\ker A^*}$ are finite-dimensional subspaces of $\boldsymbol{X}$.}
Since it is obvious when $X=\mathbb R^N$, let us focus on the infinite-dimensional setting. The fact that $\sigma \in L^\infty(\Omega^2)$ also implies that the operator $K$ given by \eqref{e:defK} is compact $L^2(\Omega)\to L^2(\Omega)$ as a Hilbert-Schmidt operator with kernel $\sigma \in L^2(\Omega^2)$. Consequently, thanks to the assumptions on $S$, $M_S^{-1}K$ is compact $L^2(\Omega)\to L^2(\Omega)$. Hence, $M_S^{-1}K - \mathrm{Id}$ satisfies the Fredholm
alternative. In particular,
\begin{equation} \label{e:fredhMSKs}
\dim \ker (M_S^{-1}K-\mathrm{Id}) = \dim \ker (M_S^{-1}K-\mathrm{Id})^*)<+\infty.
\end{equation}
Then we observe that
\begin{equation} \label{e:AsigAsig*}
A=M_S(M_S^{-1}K-\mathrm{Id}), \qquad A^*=(M_S^{-1}K-\mathrm{Id})^*M_S,
\end{equation}
since $M_S$ is self-adjoint. Equalities~\eqref{e:AsigAsig*} ensure
that
$$\ker A = \ker (M_S^{-1}K-\mathrm{Id}), \qquad M_S(\ker A^*)= \ker (M_S^{-1}K-\mathrm{Id})^*,$$
which, together with \eqref{e:fredhMSKs}, imply
$$\dim \ker A = \dim \ker A^*<+\infty.$$
\subsubsection*{Step 2 -- $\boldsymbol{\ker A = \span e}$}
We already noticed that $e\in\ker A$. Let us now prove that any $y\in\ker A$ belongs to $\span e$. This property mostly follows from the strong connectivity of the graph associated to $\sigma$.
\medskip
When $X=L^2(\Omega)$, we have to prove that $y$ is constant almost everywhere in $\Omega$. Modifying $y$ if necessary on a zero-measure subset of $\Omega$, we can choose $x_0\in\Omega$ such that $y(x_0)=\operatorname{ess\,\,sup\,} u$, which is finite, since $y$ lies in $L^\infty(\Omega)$. It is also possible to choose $x_0$ as a Lebesgue point of $\sigma$, \textit{i.e. } so that $x_*\mapsto \sigma(x_0,x_*)$ is defined almost everywhere in $\Omega$. Consider now $\xi\in\Omega$ some Lebesgue point for both $y$
and $\sigma$ at the same time, for which we intend to prove that $y(\xi)=y(x_0)$. By strong connectivity, there exist pairwise distinct elements of $\Omega$, $(x_1,\ldots,x_K)$, $K\in\mathbb N^*$, which are Lebesgue points for both $\sigma$ and $y$ such that $x_{K}=\xi$, and for any $0\leq k<K$,
$$|\mathrm{ess\,supp\,} \sigma(x_k,\cdot)|>0, \qquad x_{k+1} \in \mathrm{ess\,supp\,} \sigma(x_k,\cdot).$$
Besides, because of \eqref{e:ptySker}, for any $0\leq k<K$,
$$y(x_k)\int_\Omega \sigma(x_k,x_*)\,\mathrm{d} x_* = \int_\Omega \sigma(x_k,x_*)y(x_*)\,\mathrm{d} x_*.$$
For $k=0$, the previous equality implies that
$$\sigma(x_0,x_*) (y(x_0)-y(x_*))=0, \qquad \mbox{for a.e.}~x_*\in \Omega,$$
and then $y(x_*)=y(x_0)$ for almost every $x_* \in \mathrm{ess\,supp\,} \sigma(x_0,\cdot)$. In particular, it holds for $x_*=x_1$, \textit{i.e. } $y(x_1)=y(x_0)=\operatorname{ess\,\,sup\,} u$. The conclusion $y(\xi)=y(x_{K})=\dots=y(x_0)$ is then straightforward by induction. This ensures that $y$ is equal to its essential supremum almost everywhere.
\medskip
When $X=\mathbb R^N$, the proof follows the same idea. We provide it for completeness. Denote by $i_0$ an index such that $y_{i_0}=\max_i y_i$. Choose then an arbitrary index $i^*\neq i_0$. By strongly connectivity, there exist pairwise distinct indices $(i_k)_{1\leq k\leq K}$, $K\in\mathbb N^*$, such that $i_K=i^*$ and $\sigma_{i_ki_{k+1}}>0$ for any $k$. Consequently, for any $k$, we have
$$\sum_{j\neq i_{k+1}} \sigma_{i_kj}(y_{j}-y_{i_k}) +
\sigma_{i_ki_{k+1}}(y_{i_{k+1}}-y_{i_k})=0, $$
which immediately implies that $y_{i_{k+1}}=y_{i_k}$ by induction. Hence,
$y_{i_0}=y_{i^*}$ for any $i^*$, which ensures that $y=y_{i^*} e$.
\subsubsection*{Step 3 -- $\boldsymbol{\ker A = \ker A^2}$}
There is only one non-trivial inclusion, to be proved again by strong connectivity. Let $y\in \ker A^2$, so that $Ay \in \ker A$. Thanks to Step~1, there exists $\nu \in\mathbb R$ such that $Ay=\nu e$.
When $X=L^2(\Omega)$, the previous equality yields
$$
\int_\Omega \sigma(x,x_*)\,(y(x_*)-y(x))\,\mathrm{d} x_* = \nu, \qquad \mbox{for a.e.}~x\in \Omega.
$$
Modifying $y$ if necessary on a zero-measure subset of $\Omega$, we can choose a Lebesgue point $x\in\Omega$ for $\sigma$ such that $y(x)=\operatorname{ess\,\,sup\,} y$. Hence, $\nu\leq 0$. In the same way, we get $\nu\geq 0$. Consequently, $\nu=0$ and $y\in \ker A$.
When $X=\mathbb R^N$, we have
$$
\sum_j \sigma_{ij}(y_j-y_i)=\nu, \qquad 1\leq i\leq N,
$$
and we successively choose $i$ as an index such that $y_i=\max_j y_j$ and $y_i=\min_j y_j$ to obtain $\nu=0$ and $y \in \ker A$.
\subsubsection*{Step 4 -- Conclusion}
The fact that $0$ is a simple eigenvalue of $A$ is a direct consequence of Steps~2--3. The adjoint $A^*$ of $A$ naturally inherits all the properties of $A$ previously proved: $\dim \ker A^*=1$, $\ker A^* =\ker (A^*)^2$ and $0$ is a simple eigenvalue of $A^*$.
\end{proof}
\subsection{Definition of the positive weight}
Since $\ker A^*$ is one-dimensional, let us focus on a particular vector generating $\ker A^*$ and prove the first part of Theorem~\ref{t:poids}, which we recall in the proposition below and is, in some sense, at the heart of the overall proof.
\begin{proposition} \label{p:poidsrappel}
There exists a unique $v\in \ker A^*$ such that $v>0$ and $\langle v,e\rangle =1$.
\end{proposition}
\begin{proof}
The uniqueness of $v$ is straightforward, it relies on the fact that $\dim \ker A^*=1$.
Otherwise, the proof of Proposition~\ref{p:poidsrappel} is mainly based on an homotopy argument: we use the symmetric (in fact constant) case to conclude for the non-symmetric one. Indeed, in the symmetric case, when $X=\mathbb R^N$, $v=e/N$ and, when $X=L^2(\Omega)$, $v=e/|\Omega|=e$ clearly are the only elements of $\ker A^*$ satisfying the required sign and scalar-product properties.
Denote by $M$ the constant function equal to $\|\sigma\|_{L^\infty}$ if $X=L^2(\Omega)$, and the matrix with all its coefficients equal to $\max_{i,j} \sigma_{ij}$ if $X=\mathbb R^N$, and consider
the analytic function
$$[0,1]\to W, \quad \lambda\mapsto \sigma_\lambda:=\lambda \sigma + (1-\lambda)M.$$
In the remainder of this proof, for the sake of clarity, we denote with an index
$\lambda\in[0,1]$ any matrix, operator or function built from $\sigma_\lambda$, \textit{e.g.} $A_\lambda$. Of course $A_\lambda$ inherits all the properties already known for $A$. Moreover, $\lambda\mapsto A_\lambda^*$ is analytic.
Set $F=(\span e)^\perp=(\ker A_\lambda)^\perp=\operatorname{im} A_\lambda^*$, for any $\lambda\in[0,1]$. Since $\ker A_\lambda^* = \ker (A_\lambda^*)^2$, $F\cap \ker A_\lambda^*=\{0\}$. This implies that $X=F\oplus \ker A_\lambda^*$. Let $\Pi_\lambda$ be the projector onto $\ker A_\lambda^*$ along $F$, which analytically depends on $\lambda$, see \cite[Chapter~VII, \textsection~1, Section~3, Theorem~1.7]{kato}. The function of $\lambda$ defined by $v_\lambda=\Pi_\lambda e$ meets all the required properties: it is analytic, and each $v_\lambda$ is a non-trivial element of $\ker A_\lambda^*$. For the sake of completeness, we provide hereafter a more explicit construction of $\Pi_\lambda$ and $v_\lambda$. Consider the operator $J_\lambda:F\to F$, $y\mapsto A_\lambda^*y$, which is injective since $F\cap \ker A_\lambda^*=\{0\}$. Of course, $\lambda\mapsto J_\lambda$ is analytic.
Then, for the sake of clarity, it is better to provide two different proofs depending on whether $X=L^2(\Omega)$ or $X=\mathbb R^N$.
\subsubsection*{Case 1 -- Infinite-dimensional case}
First, the operator $J_\lambda$ belongs to the Banach algebra $\mathcal B(F)$ of bounded operators on $F$. Second, it is also surjective. Indeed, since $L^2(\Omega)=\span e \oplus F$, we just have to prove that $A_\lambda^* e$ has a pre-image by $A_\lambda^*$ which lies in $F$. Let $w\neq 0$ spanning $\ker A_\lambda^*$. Using the previous direct sum, we can write $w=\alpha e +f$, for some $\alpha \in \mathbb R$ and $f\in F$. But $\alpha\neq 0$, which is proved by contradiction: if $\alpha=0$, then $\langle w,e\rangle=0$, which ensures that $e\in (\ker A_\lambda^*)^\perp=\operatorname{im} A_\lambda$. Since $\ker A_\lambda^2 = \ker A_\lambda$, that would imply that $e=0$, which is not the case. Consequently, we can write $e$ as $e=\frac 1\alpha(w-f)$ and then $A_\lambda^* e=-\frac 1\alpha A_\lambda^*f$. Thus $J_\lambda$ is surjective, and subsequently, bijective.
Furthermore, thanks to the closed graph theorem, $J_\lambda^{-1}$ also lies in $\mathcal B(F)$, and since $\lambda\mapsto J_\lambda$ is clearly analytic and $\mathcal B(F)$ is a Banach algebra, then $\lambda\mapsto J_\lambda^{-1}$ is also analytic, and so is $\lambda\mapsto\Pi_\lambda=\mathrm{Id}-J_\lambda^{-1}A_\lambda^*$. We can then conclude by setting $v_\lambda=\Pi_\lambda e$. The fact that $v_\lambda$ analytically depends on $\lambda$ is now clear. Since $e\not\in F$, $v_\lambda\neq 0$ for any $\lambda$, in $L^2(\Omega)$ and almost everywhere. Finally, we have $A_\lambda^*v_\lambda=A_\lambda^*e-A_\lambda^*e=0$.
Let us now prove that $v_\lambda$ remains positive for any $\lambda$ and almost everywhere on $\Omega$. Thanks to \eqref{e:ptySker} applied to $v_\lambda$, we can write
$$
v_\lambda(x)=\frac 1{S_\lambda(x)} \int_\Omega \sigma_\lambda(x_*,x)
v_\lambda(x_*)\,\mathrm{d} x_*, \qquad \mbox{for a.e.}~x\in\Omega.
$$
By continuity of $v$ as a function of $\lambda\in[0,1]$, $\lambda \mapsto \operatorname{ess\,inf\,} v_\lambda$ is also continuous. By contradiction, suppose that there exists $\mu \in(0,1]$ such that $\operatorname{ess\,inf\,} v_\mu=0$. As in the proof of Proposition~\ref{p:poidsrappel}, modifying $v$ if necessary on a zero-measure subset of $\Omega$, we can find $x\in\Omega$ such that $v_\mu(x)=0$. We then intend to prove that $v_\mu=0$ almost everywhere in $\Omega$, which will raise a contradiction. Consider a Lebesgue point $\xi$ for both $v_\mu$ and $\sigma_\mu$, and prove that $v_\mu(\xi)=0$. By strong connectivity, there exists $K\in\mathbb N^*$ and $x_0$, ..., $x_{K}$ Lebesgue points for $v_\mu$ and $\sigma_\mu$ satisfying $x_0=\xi$, $x_{K}=x$, and for any $0\leq k<K$,
$$|\mathrm{ess\,supp\,} \sigma(x_k,\cdot)|>0, \qquad x_{k+1} \in \mathrm{ess\,supp\,} \sigma(x_k,\cdot).$$
Besides, for any $k$, we can write
$$S_\mu(x_{k+1}) v_\mu(x_{k+1})=\int_\Omega\sigma_\mu(x_*,x_{k+1}) v_\mu(x_*)\,\mathrm{d} x_*.$$
If we choose $k=K-1$, the left-hand side of the equality becomes $0$. Since $x_{K} \in \mathrm{ess\,supp\,} \sigma(x_{K-1},\cdot)$, we can deduce that $v_\mu(x_{K-1})=0$. We conclude by descending induction on $k$ that $v_\mu(\xi)=v_\mu(x_0)=0$. This implies that $v_\mu=0$ almost everywhere, which is impossible: $v_\mu$ must span the one-dimensional space $\ker A_\mu^*$.
Thus, for any $\lambda$, $\operatorname{ess\,inf\,} v_\lambda$ remains positive, which implies, in particular, that $v_1$ has the same (positive) sign as $v_0=e$ almost everywhere.
\subsubsection*{Case 2 -- Finite-dimensional case}
The proof in this case is simpler. First, the injectivity of $J_\lambda$ implies its bijectivity. Thanks, for instance, to the formula linking $J_\lambda^{-1}$ to the cofactor matrix of $J_\lambda$, $\lambda\mapsto J_\lambda^{-1}$ is also analytic. Then $\Pi_\lambda= \mathrm{Id}-J_\lambda^{-1} A_{\lambda}^*$ and $v_\lambda=\Pi_{\lambda} e $ inherit the required analyticity property with respect to $\lambda$.
We prove by contradiction, as in Case~1, that all the coordinates of $v_\lambda$ are positive for any $\lambda\in [0,1]$. Assume that there exist an index $i^*$ and a real number $\mu \in (0,1]$ such that $(v_\mu)_{i^*}=0$. They can be chosen such that, for any $j$, and any $\lambda < \mu$, $(v_\lambda)j>0$, implying that $(v_\mu)_j\geq (v_\mu)_{i^*}=0$. It is possible to do so because $\lambda \mapsto (v_\lambda)_j$ is continuous on $[0,1]$ and $(v_0)_j=1/N>0$ for any $j$.
Let $j^*\neq i^*$, by strong connectivity, there exist pairwise different indices $i_0=j^*$, $i_1$, $\ldots$, $i_r=i^*$ such that $(\sigma_\mu)_{i_ki_{k+1}}>0$ for all $k$.
Moreover, since $A_{\mu}^* v_\mu=0$, we can write, for any $i$,
\begin{equation} \label{e:relationker}
(v_\mu)_i=\Big({\displaystyle \sum_{j\neq i} (\sigma_\mu)_{ji} (v_\mu)_j}\Big)\Big{/}\Big(\sum_{j\neq i} (\sigma_\mu)_{ij}\Big).
\end{equation}
Writing \eqref{e:relationker} for $i=i^*$, we deduce that
$$\displaystyle \sum_{j\neq i^*} (\sigma_\mu)_{ji^*} (v_\mu)_j=0,$$
and it follows that $(v_\mu)_{i_{r-1}}=0$ since $(\sigma_\mu)_{i_{r-1}i^*}>0$. Then, by finite induction, applying successively \eqref{e:relationker} to $i=i_{r-1}$, $\ldots$, $i=i_{1}$, we eventually obtain $(v_\mu)_{j^*}=0$. The integer $j^*$ being arbitrary,
it would imply that $v_\mu=0$, which is not possible.
This ends the proof of Proposition~\ref{p:poidsrappel}.
\end{proof}
\begin{remark}
It is interesting to provide an interpretation of $v$ in a social science
problem, for instance, when $y$ is an opinion vector of the population
regarding a binary question in a referendum. Assume that, for all $i$, $j$,
$\sigma_{ij}$ does not depend on $j$, \textit{i.e. } $\sigma_{ij} =\sigma_{i}$. Then
$\sigma_{i}$, which has the physical dimension of a frequency, can be seen as
the (uniform) persuasion force of the $i$-th agent on the population.
First, we note that, for any $i$, $j$, $(A^\intercal)_{ij}= \sigma_j/N
-\sigma_i \delta_{ij}$, where $\delta_{ij}=1$ if $i=j$ and $\delta_{ij}=0$
otherwise. Then $v$ satisfies, for any $i$,
$$\sigma_i v_i = \frac 1N \sum_{j=1}^N \sigma_j v_j.$$
Since the null space of $A^\intercal$ is one-dimensional, the
previous equality ensures that $v$ is collinear to the vector
$(1/\sigma_1,\dots, 1/\sigma_N)^\intercal$. Therefore, for any $i$,
the component $v_i$ of $v$ only depends on $1/\sigma_i$, and has the physical
dimension of time. More precisely, $v_i$ is proportional to the time period
in which agent $i$ interacts with the other agents in the population.
\end{remark}
\subsection{Weighted Hilbert structure on $\boldsymbol{X}$}
Proposition~\ref{p:poidsrappel} provides an element $v$ of $\ker A^*$ which satisfies $v>0$, \textit{i.e. } $v_i>0$ for any $i$ when $X=\mathbb R^N$ and $\operatorname{ess\,inf\,} v >0$ if $X=L^2(\Omega)$. This key property allows to build a new scalar product $\langle \cdot,\cdot\rangle_v$ and its associated norm $\|\cdot\|_v$ on $X$, weighted by $v$. The latter norm is equivalent to $\|\cdot\|$ because $v$ is lower and upper-bounded by positive constants. More precisely, when $X=\mathbb R^N$, we set
$$\langle y,z\rangle_v = \sum_{i=1}^N v_i y_i z_i, \qquad y,z\in\mathbb R^N,$$
and, when $X=L^2(\Omega)$,
$$\langle y,z\rangle_v = \int_{\Omega} y(x) z(x)\, v(x)\,\mathrm{d} x, \qquad y,z\in L^2(\Omega),$$
where $v(x)\,\mathrm{d} x$ is an absolutely continuous probability measure. Note that the weighted mean can then be written in terms of weighted scalar product, \textit{i.e. }
\begin{equation} \label{e:defmoypond}
\bar y^v=\langle y,v\rangle e=\langle y,e\rangle_v\, e.
\end{equation}
Noticing that $\|e\|_v=1$, $\bar y^v$ then appears as the orthogonal projection of $y$ on $\ker A$ with respect to the weighted scalar product. This leads us to identify the orthogonal complement of $\ker A$ with respect to this scalar product. In what remains, when dealing with notions related to the weighted scalar product, we shall add the index $v$, \textit{e.g.} $\perp_v$ for the orthogonality or $*_v$ for an adjoint operator with respect to the weighted scalar product. For the sake of simplicity, we shall speak about $v$-scalar product, $v$-norm, $v$-orthogonality, $v$-adjoint.
\begin{proposition} \label{p:directsums}
The following properties hold:
\begin{enumerate}[(i)]
\item $(\ker A)^{\perp_v}=\operatorname{im} A$,
\item $\ker A^{*_v} = \ker A = \span e,\quad \operatorname{im} A^{*_v} = \operatorname{im} A$,
\item $X=\ker A \stackrel{\scriptstyle\perp_v}{\oplus} \operatorname{im} A=\ker A^* \stackrel{\scriptstyle\perp}{\oplus} \operatorname{im} A$,
\item $\operatorname{im} A = \operatorname{im} A^2$,
\end{enumerate}
the direct sums in \textit{(iii)} being respectively $v$-orthogonal and orthogonal.
\end{proposition}
\begin{proof}
To obtain \textit{(i)}, let us first consider $y\in X$. Since $\langle Ay,e\rangle_v = \langle Ay,v\rangle=\langle y,A^*v\rangle=0$, then $Ay\in(\ker A)^{\perp_v}$. Conversely, if $z\in (\ker A)^{\perp_v}$, $\langle z,v\rangle=\langle z,e\rangle_v=0$, which ensures that $z\in (\ker A^*)^\perp$. Properties \textit{(ii)} are direct consequences of \textit{(i)}. The last equality in \textit{(iii)} is a well-known result which we recall here for the reader's convenience, and the previous one of course comes from \textit{(i)}. Eventually, \textit{(iv)} is straightforwardly deduced from \textit{(iii)}.
\end{proof}
\begin{remark}
Even if the previous proposition may suggest it, $A$ \textit{is not $v$-self-adjoint}. Indeed, let us consider the operator $D_v$ defined on $X$ by $D_v = \operatorname{diag} v$ if $X=\mathbb R^N$, and $D_v:z\mapsto vz$ if $X=L^2(\Omega)$. We have
$$A^{*_v}={D_v}^{-1} A^* D_v.$$
If $A$ were $v$-self-adjoint, we would have $(D_vA)^*=D_vA$, which is not true in general.
\end{remark}
\paragraph{Definition of $\boldsymbol{\pi}$ and $\boldsymbol{A_2}$.}
To conclude this section, we introduce the $v$-orthogonal projector $\pi$ on $\operatorname{im} A$, so that we can write $y-\bar y^v =\pi y$ for any $y\in X$, and the operator $A_2:\operatorname{im} A\to\operatorname{im} A$, $y\mapsto Ay$, which are well defined thanks to Proposition~\ref{p:directsums}. Indeed, since, for instance, $\ker A^*$ is stable by $A^*$, then $\operatorname{im} A= (\ker A^*)^{\perp}$ is stable by $A$. Moreover, thanks to the open mapping theorem for bounded linear operators in Banach spaces, $A_2$ is a homeomorphism of $\operatorname{im} A$.
\section{Proof of Theorems~\ref{t:poids}--\ref{t:convcons}} \label{s:convcons}
\subsection{Proof of Theorem~\ref{t:poids}}
Let us sum up the situation about the proof of Theorem~\ref{t:poids}. Proposition~\ref{p:ptykerA-kerAstar} ensures that $\ker A^*$ is a one-dimensional subspace of $X$. Then, in Proposition~\ref{p:poidsrappel}, we have proved with a homotopy argument that all elements of $\ker A^*$ have the same sign with respect to $1\leq i\le N$ or $x\in\Omega$ (almost everywhere), and that there is a unique $v\in\ker A^*$ such that $v>0$ and $\langle v,e \rangle =1$. The latter property corresponds to the first part of Theorem~\ref{t:poids}. Hence, it remains to explain why the weighted mean $\bar y^v$ given in \eqref{e:defmoypond} remains constant along any solution of $\dot y(t) = Ay(t)$. In fact, it is a straightforward consequence of the definition of $v$, since
$$\frac{\mathrm{d} \bar y^v}{\mathrm{d} t}(t)=\langle \dot y(t),v\rangle\, e=\langle y(t),A^*v\rangle\, e = 0,$$
in finite as well as in infinite dimension. The proof of Theorem~\ref{t:poids} is then completed.
\subsection{Proof of Theorem~\ref{t:convcons} in finite dimension} \label{ss:thm2dimfinie}
The finite-dimensional case has already been studied in \cite{olsab-mur, web-the-mot}. We provide the proof for completeness and to prepare for the infinite-dimensional case. Following Proposition~\ref{p:ptykerA-kerAstar}, making a change of basis if necessary, $A$ can be written in block matrices
$$A=\begin{pmatrix} 0 & 0 \\ 0 & A_2 \end{pmatrix}.$$
We point out that
$$y(t)-\bar y^v = e^{tA}(y(0)-\bar y^v) = e^{tA} \pi y(0) = e^{tA_2}\pi y(0), \qquad t\geq 0.$$
Besides, thanks to the Gershgorin circle theorem, as in \eqref{e:gershgorin}, we deduce that, apart from $0$, all eigenvalues of $A$ have a negative real part, \textit{i.e. } $A_2$ is a Hurwitz matrix. Considering that $\lambda_2$ is the second eigenvalue of $A$ (the one with the highest real part, apart from $0$), for any $\varepsilon>0$, there exists $M(\varepsilon)>0$ such that
$$\| e^{tA_2} \| \leq M(\varepsilon)\, e^{(\varepsilon+\operatorname{Re} \lambda_2)t}, \qquad t\geq 0.$$
The previous estimate gives the sharp exponential decay rate.
\subsection{Proof of Theorem~\ref{t:convcons} in infinite dimension}
We already know from Proposition~\ref{p:ptykerA-kerAstar} that $0$ is a simple eigenvalue of $A$. Let us study the whole spectrum $\mathfrak{S}(A)$ of $A$, where the notation $\mathfrak{S}$ is not to be confused with the interaction function $\sigma$. We recall that $\mathfrak{S}(A)$ is the set of complex numbers $\lambda$ such that $A-\lambda \mathrm{Id}$ is not bijective.
\begin{proposition}\label{p:spectreA}
The spectrum of $A$ satisfies
$$\mathfrak{S}(A) \subset \{z\in \mathbb C~|~\operatorname{Re} z \leq 0\}, \qquad \mathfrak{S}(A_2) =\mathfrak{S}(A)\backslash\{0\}\subset \{z\in \mathbb C~|~\operatorname{Re} z <0\}.$$
\end{proposition}
\begin{proof}
Let us introduce a complex Hilbert structure on $X$, keep the same notations for the scalar products and the adjoints as in the real case, and consider the following quadratic forms respectively defined for $y\in X$ and $z\in\operatorname{im} A$ by
\begin{equation}\label{e:quadformC}
Q(y)=\left\langle y,\frac{A+A^{*_v}}2y\right\rangle_v = \operatorname{Re} \left(\langle y,Ay\rangle_v\right), \qquad Q_2(z) = \operatorname{Re} \left(\langle z,A_2z\rangle_v\right).
\end{equation}
Of course, $Q$ and $Q_2$ coincide on $\operatorname{im} A$. The equality between $\mathfrak{S}(A_2)$ and $\mathfrak{S}(A)\backslash\{0\}$ is then straightforward with the $v$-orthogonal direct sum decomposition of $X$ as $\ker A \stackrel{\scriptstyle\perp_v}{\oplus} \operatorname{im} A$. Let us now compute $Q$. We have, for any $y\in L^2(\Omega;\mathbb C)$,
$$\langle y,Ay\rangle_v = \iint_{\Omega^2} v(x) \sigma(x,x_*) (y(x_*)-y(x)) \overline{y(x)}\,\mathrm{d} x_* \,\mathrm{d} x.$$
Applying \eqref{e:ptySker} to $v\in\ker A^*$, the previous equality can be rewritten as
$$\langle y,Ay\rangle_v =
\iint_{\Omega^2} v(x) \sigma(x,x_*) y(x_*) \overline{(y(x)-y(x_*))}\,\mathrm{d} x_* \,\mathrm{d} x.$$
Consequently, combining both previous expressions, we get
\begin{equation*
Q(y)=\operatorname{Re} \left(\langle y,Ay\rangle_v \right) = -
\frac 12 \iint_{\Omega^2} v(x) \sigma(x,x_*) |y(x_*)-y(x)|^2\,\mathrm{d} x_* \,\mathrm{d} x,
\end{equation*}
and thus $Q$ is negative semi-definite. Let us now prove that $Q(y)=0$ if and only if $y\in \ker A$.
If $Q(y)=0$, then $\sigma(x,x_*) |y(x_*)-y(x)|^2=0$ for almost all $x$ and $x_*$. Let $x_0$ a Lebesgue point for $y$. It is also possible to choose $x_0$ as a Lebesgue point of $\sigma$, \textit{i.e. } so that $x_*\mapsto \sigma(x_0,x_*)$ is defined almost everywhere in $\Omega$. Consider now $X\in\Omega$ some Lebesgue point for both $y$ and $\sigma$ at the same time, for which we intend to prove that $y(X)=y(x_0)$. By strong connectivity, there exist $i^*\in\mathbb N^*$ and $x_1,\ldots,x_{i^*}\in\Omega$ Lebesgue points for both $\sigma$ and $u$ such that $x_{i^*}=X$, and for any $0\leq k\leq i^*-1$,
$$|\mathrm{ess\,supp\,} \sigma(x_k,\cdot)|>0, \qquad x_{k+1} \in \mathrm{ess\,supp\,} \sigma(x_k,\cdot).$$
Since $\sigma(x,x_*) |y(x_*)-y(x)|^2=0$ for almost every $x$ and $x_*$, by taking $k=0$ in the previous property of $\sigma$, we get $y(x_*)=y(x_0)$ for almost every $x_* \in \mathrm{ess\,supp\,} \sigma(x_0,\cdot)$. In particular, it holds for $x_*=x_1$, \textit{i.e. } $y(x_1)=y(x_0)$. The conclusion $y(X)=y(x_{i^*})=\cdots=y(x_0)$ is then straightforward by induction. This ensures that $y$ is constant almost everywhere, \textit{i.e. } $y\in \span e=\ker A$.
Hence $A$ is dissipative and $A_2$ is strictly dissipative. We deduce from the Hille-Yosida theorem and one of its corollaries, see \cite[Chapter~1, Section~1.3, Theorem~3.1 \& Corollary~3.6]{pazy} for instance, that the resolvent set of $A$ contains the open right complex half-plan. This eventually leads to the required results of both spectra of $A$ and $A_2$ and ends the proof of Proposition~\ref{p:spectreA}.
\end{proof}
\begin{remark}
Of course, the result of Proposition~\ref{p:spectreA} also holds in finite dimension, and the associated quadratic form is then given by
$$Q(y) = -\frac 12 \sum_{i,j} v_i \sigma_{ij} |y_i-y_j|^2.$$
\end{remark}
In finite dimension, this is sufficient to conclude the proof of Theorem~\ref{t:convcons}, and we obtain $\rho=\operatorname{Re}(\lambda_2)$, as noted in Section~\ref{ss:thm2dimfinie}. But, in infinite dimension, this is not enough: Proposition~\ref{p:spectreA} implies that $A_2$ is Hurwitz, but this property is however not sufficient to ensure that the corresponding semi-group is exponentially stable. We need to apply the spectral mapping theorem (see \cite{eng-nag, pazy}) and thus carefully study the spectral properties of $A_2$. Let us start with a particular easier case.
\subsubsection*{When $\boldsymbol{S}$ is constant}
Assume that $S(x)=\int_\Omega \sigma(x,x_*)\,\mathrm{d} x_*=\delta$ for almost every $x\in\Omega$. Then $A=K-\delta \mathrm{Id}$, which ensures that $e^{tA}=e^{-\delta t}e^{tK}$.
When $\sigma$ is symmetric, the compact operator $K$ is also self-adjoint, implying that $K$ is diagonalizable with real eigenvalues. Consequently, $A_2$ is also diagonalizable, with negative eigenvalues. The convergence rate is then the Fiedler number $|\lambda_2|=-\lambda_2$.
When $\sigma$ is not symmetric, since $K$ is compact, its spectrum $\mathfrak{S}(K)$ contains $0$, and any element of $\mathfrak{S}(K)\backslash\{0\}$ is an eigenvalue with finite multiplicity. Consequently, $\mathfrak{S}(A)$ contains $-\delta$ and any element of $\mathfrak{S}(A)\backslash\{-\delta\}$ is an eigenvalue with finite multiplicity.
\subsubsection*{General case}
We already know from Proposition~\ref{p:spectreA} that $\mathfrak{S}(A) \subset \{z\in \mathbb C~|~\operatorname{Re} z \leq 0\}$. Moreover, since $A$ is a bounded operator, thanks to \cite[Corollary\,IV.1.4]{eng-nag} (for instance), $\mathfrak{S}(A)$ is also a compact subset of the closed disk centered at $0$ with radius $\|A\|$ in the complex plane.
Let us now use the fact that $A=K-M_S$ to study $\mathfrak{S}(A)$. First, since $K$ is compact, its spectrum $\mathfrak{S}(K)$ is countable, $0$ is the only possible accumulation point, and any nonzero element in the spectrum is an eigenvalue. Besides, from \cite[Proposition\,I.4.10]{eng-nag}, the spectrum of the multiplication operator $M_S$ is the essential range of $S$, \textit{i.e. }
$$
\mathfrak{S}(M_S)=\operatorname{ess\,ran} S
= \left\{\lambda\in\mathbb C~|~
\operatorname{meas}(\{x\in\Omega~|~|S(x)-\lambda|<\varepsilon\})
\neq 0\,\,\,\,\forall\varepsilon>0 \right\}.
$$
Recall that $\mathfrak{S}(A)$ is the disjoint union of the discrete $\mathfrak{S}_d(A)$ and essential $\mathfrak{S}_e(A)$ spectra of $A$ (see \cite{kato}). Because of the properties of $\mathfrak{S}(K)$, we have $\mathfrak{S}_e(A)=\mathfrak{S}_e(-M_S) \subset \operatorname{ess\,ran}(-S)$. Hence, $\mathfrak{S}_e(A)\subset\mathbb R$ and $$\sup \mathfrak{S}_e(A)\leq \sup \operatorname{ess\,ran}(-S) = -\operatorname{ess\,inf\,} S = -\delta.$$
The discrete spectrum is the set of eigenvalues of $A$, but can also be seen as the subset of isolated points $\lambda$ of $\mathfrak{S}(A)$ such that their corresponding Riesz projector $P_\lambda$ is of finite rank. Recall (see \cite[Chapter~III, \textsection~6, Section~4, Theorem~6.17]{kato}) that
$$P_\lambda = \frac 1{2i\pi}\oint_\Gamma (z\mathrm{Id}-A)^{-1}\mathrm{d} z,$$
$\Gamma$ being a simple curve in the complex plane enclosing a region where $\lambda$ is the only element of $\mathfrak{S}(A)$. The Riesz projector $P_0$ associated to the eigenvalue $0$ is $\mathrm{Id}-\pi$, which commutes with $A$. Thanks to the $v$-orthogonal direct sum decomposition from Proposition\,\ref{p:directsums}, we have
$$\mathfrak{S}_d(A_2) = \mathfrak{S}_d(A)\backslash\{0\}, \qquad \sup \operatorname{ess\,ran} A_2 \leq -\delta.$$
Moreover, for any $\varepsilon\in(0,\delta)$,
$$\mathfrak{S}_d(A_2)\cap \{z\in\mathbb C~|~-\delta+\varepsilon\leq \operatorname{Re} z<0\} =
\mathfrak{S}(A_2)\cap \{z\in\mathbb C~|~-\delta+\varepsilon\leq \operatorname{Re} z<0\}.$$
Since $\mathfrak{S}(A_2)$ is compact, there is at most a finite number of elements in $\mathfrak{S}_d(A_2)\cap \{z\in\mathbb C~|~-\delta+\varepsilon\leq \operatorname{Re} z<0\}$. Hence,
$$\sup \{\operatorname{Re} z~|~z\in\mathfrak{S}_d(A_2) \}<0.$$
Consequently, the spectral bound $\mathrm{s}(A_2)=\sup \{\operatorname{Re} z~|~z\in\mathfrak{S}(A_2) \}$ is negative. More precisely, if there is no eigenvalue of $A_2$ whose real part lies in $(-\delta,0)$, then $\mathrm{s}(A_2)=-\delta$.
Otherwise, there exists $\lambda_2\in\mathfrak{S}_d(A_2)$ such that $\mathrm{s}(A_2)=\operatorname{Re} \lambda_2$, $\lambda_2$ being one of the eigenvalues of $A_2$ with the highest real part. This allows to conclude on the exponential convergence towards consensus. Indeed, following \cite[Corollary\,IV.2.4]{eng-nag}, which is a consequence of the \textit{spectral mapping theorem} applied to the bounded operator $A_2$, $\mathrm{s}(A_2)$ coincides with the spectral growth of the semi-group generated by $A_2$. Therefore, $|\mathrm{s}(A_2)|$ is the convergence rate, and Theorem~\ref{t:convcons} is proved.
\section{\textcolor{black}{Further results}}\label{s:lyap}
In this section, we address three issues related to the previous analysis.
In the first subsection,
we consider a discrete-time version of \eqref{e:krause}--\eqref{e:initconddiscrete} (which is the linear version of the standard Hegselmann-Krause model \cite{heg-kra}) and discuss the convergence to consensus. In the second subsection, we clarify the relationships between the two problems studied in this article, namely \eqref{e:krause}--\eqref{e:initconddiscrete} and \eqref{e:diminfinie}--\eqref{e:diminfinieinitcond}: we prove that \eqref{e:diminfinie}--\eqref{e:diminfinieinitcond} can be obtained from \eqref{e:krause}--\eqref{e:initconddiscrete} by means of a rigorous limiting procedure as $N$ goes to $+\infty$. In the third subsection, we investigate the time asymptotics of both standard and $v$-weighted variances. In particular, we show that the weighted variance is an appropriate tool for studying the stability in the $L^2$-setting
\subsection{Discrete-time setting}
In this subsection, we study a discrete-time version of \eqref{e:krause}--\eqref{e:initconddiscrete}, obtained by replacing time derivatives by difference quotients. Without loss of generality, we present our results in the finite-dimensional setting, but the analysis is similar in the infinite-dimensional one.
Let $\Delta t>0$ and consider
$\Gamma=(\gamma_{ij})\in\mathbb R^{N\times N}$ such that $\Delta t \, \Gamma$ is a stochastic
matrix, \textit{i.e. } the sum of elements of each row equals $1$. We are here interested in the following problem
\begin{eqnarray}
\label{e:tempsdiscret}
y_i^{n+1} &=&\displaystyle\sum_{j=1}^N \gamma_{ij}\Delta t\,y_j^n, \quad 1\leq i\leq N, \quad n\in \mathbb N, \phantom{\int}\\
y_i^0&=& y_i^{\mathrm{in}},
\label{e:tempsdiscret_ic}
\end{eqnarray}
where $y_i^n$ denotes the state variable of agent $i$ at discrete time $n\Delta t$. This problem is in fact the original consensus model studied in \cite{heg-kra}, but also in prior works \cite{degroot, lehrer}. The link to \eqref{e:krause}--\eqref{e:initconddiscrete} is quite clear. Indeed, we can write
$$y_i^{n+1}=\sum_{j\neq i} \gamma_{ij}\Delta t\,y_j^n + \gamma_{ii}\Delta t\,y_i^n
= \sum_{j\neq i} \gamma_{ij}\Delta t\,y_j^n + \left(1-\sum_{j\neq i} \gamma_{ij}\Delta t
\right)y_i^n,
$$
which implies
$$\frac{y_i^{n+1}-y_i^{n}}{\Delta t}=\sum_{j\neq i}\gamma_{ij}(y_j^n-y_i^n).$$
Hence the time-discrete problem \eqref{e:tempsdiscret} appears as the explicit
Euler time discretization of \eqref{e:krause}. Note that the stochasticity of
$\Delta t \, \Gamma$ implies that $\Delta t$ must satisfy the stability condition
$$\max_i \Big(\sum_{j\neq i} \gamma_{ij}\Big)\Delta t \leq 1.$$
For any $n\in\mathbb N^*$, we denote
$$
y^n=
\begin{pmatrix} y_1^n \\ \vdots \\ y^n_N \end{pmatrix}
\textrm{ and }
y^{\mathrm{in}}=
\begin{pmatrix} y_1^{\mathrm{in}} \\ \vdots \\ y^{\mathrm{in}}_N \end{pmatrix}.
$$
We have the following theorem , whose proof is a variant of the strategy developed in Sections~\ref{s:AAstarv} and \ref{s:convcons}, and thus is not provided. The matrix $A$ is defined by \eqref{eq:Adiscr}.
\begin{theorem}
\label{t:poids-disc}
Assume that the graph associated to $\sigma$ is strongly connected. Then there exists a unique $v\in\ker A^*$ such that $v>0$ and $\langle v,e\rangle =1$, and the weighted mean of any solution $y^n$ to \eqref{e:tempsdiscret}--\eqref{e:tempsdiscret_ic} defined by $\bar y^v=\langle y^n,v\rangle\,e$ is constant with respect to $n$.
Moreover, there exist $\rho_*\in(0,1)$ and $M_*>0$ satisfying
$$
\|y^n-\bar y^v\| \leq M_*\, \|y^{\mathrm{in}}-\bar y^v\| \, \rho^n_*, \qquad \forall n\in\mathbb N.
$$
\end{theorem}
\subsection{Kinetic limit}
In order to study the kinetic limit of \eqref{e:krause}--\eqref{e:initconddiscrete} as $N$ goes to $+\infty$, we consider the family of (constant) functions $x_i: \ \mathbb R^+ \to \Omega$, $1\leq i\leq N$, and write \eqref{e:krause}--\eqref{e:initconddiscrete} as an artificial second-order model, which is reminiscent of the classical Cucker-Smale model \textcolor{black}{\cite{cuc-sma}}:
$$\dot x_i(t) = 0, \qquad \dot \xi_i(t) = \frac{1}{N} \sum_{j} \sigma_{ij} (\xi_j(t)-\xi_i(t)).$$
The functions $x_i$ being constant for each agent, we can re-write the interactions kernels $\sigma_{ij}$ between the agents $i$ and $j$ by introducing a continuous kernel $\sigma\ : \ \bar\Omega^2\to \mathbb R^+_*$ satisfying
$$\sigma(x_i,x_j) = N\sigma_{ij}.$$
We hence end up with
\begin{equation} \label{e:system}
\dot x_i(t)=0, \qquad \dot \xi_i(t)=\frac{1}{N} \sum_{j} \sigma(x_i,x_j) (\xi_j(t)-\xi_i(t)).
\end{equation}
\begin{theorem}
Passing to the kinetic limit when $N$ goes to $+\infty$ gives a probability measure on $\Omega^2$
$$\mu(t) = f(t,x,\xi)\, \mathrm{d} x\, \mathrm{d} \xi$$
solution of
\begin{equation} \label{e:chocolat}
\partial_t\mu + \operatorname{div}_\xi(X[\mu]\mu)=0
\end{equation}
or equivalently, $\partial_t f + \operatorname{div}_\xi(fX[f]) = 0$, where $\operatorname{div}_\xi$ is the divergence with respect to $\xi$, $X$ is the vector field defined by
$$X[\mu](x,\xi) = \iint_{\Omega^2} \sigma(x,x_*) (\xi_*-\xi) \, \frac{1}{F(x_*)} \, \mathrm{d}\mu(x_*,\xi_*),
$$
and
$$F(x) = \int_{\Omega} \mathrm{d}\mu(t,x)(\xi) = \int_\Omega f(t,x,\xi)\, \mathrm{d} \xi
$$
is the density marginal, which does not depend on $t$.
Moreover, the relationship between \eqref{e:chocolat} and the infinite-dimensional problem \eqref{e:pbdiminfinie} is that the function
$$
y(t,x) =\frac 1{F(x)}\int_{\Omega} \xi \, f(t,x,\xi)\, \mathrm{d} \xi =
\frac{\int_{\Omega} \xi \, f(t,x,\xi)\,\mathrm{d} \xi}{\int_{\Omega} f(t,x,\xi)\, \mathrm{d} \xi}
$$
is the solution of \eqref{e:pbdiminfinie} (with the corresponding initial condition).
\end{theorem}
In the statement above, we have assumed $\mu(t)$ to be absolutely continuous to simplify the expression of the vector field $X[\mu]$. But it can be generalized without difficulty, by disintegrating $\mu(t)$ with respect to its marginal. We leave the details to the reader.
\begin{proof}
Passing to the kinetic limit can be done as in the usual Cucker-Smale model: this is done in detail in \cite[Section~2.3]{piccoli2015control2}. In a few words, one passes from the kinetic model to the finite-dimensional model by taking empirical measures
$$
\mu(t) = \frac{1}{N}\sum_{i=1}^N \delta_{(x_i(t),\xi_i(t))} .
$$
Conversely, the unique solution of the kinetic equation is $\mu(t) = \Phi(t)_\sharp \mu(0)$, that is the pushforward of the initial measure under the flow $\Phi(t)$ generated by the vector field $(0,X)^\intercal$. The fact that $F$ does not depend on $t$ immediately follows by integrating \eqref{e:system} with respect to $\xi$.
Let
$$
y(t,x) = \frac 1 {F(x)}\int_{\Omega} \xi \, f(t,x,\xi)\, \mathrm{d}\xi.
$$
We deduce that $y(t)$ is solution of \eqref{e:pbdiminfinie}. Indeed,
\begin{eqnarray*}
\partial_t y(t,x) &=& \displaystyle - \frac{1}{F(x)}\int_\Omega \xi\, \operatorname{div}_\xi(f X[f])\, \mathrm{d} \xi = \frac{1}{F(x)}\int_\Omega f X[f]\, \mathrm{d} \xi \\[10pt]
&=& \displaystyle \frac{1}{F(x)}\int_\Omega f(t,x,\xi) \iint_{\Omega^2} \sigma(x,x_*) (\xi_*-\xi) \, \frac{f(t,x_*,\xi_*)}{F(x_*)} \, \mathrm{d} x_*\,\mathrm{d} \xi_* \\[10pt]
&=& \displaystyle \frac{\int_\Omega f(t,x,\xi)\, \mathrm{d} \xi}{F(x)} \int_\Omega \sigma(x,x_*) \frac{\int_\Omega v_* f(t,x_*,\xi_*)\, \mathrm{d} \xi_*}{F(x_*)}\, \mathrm{d} x_* \\[10pt]
&& \displaystyle \qquad\qquad\qquad - \frac{\int_\Omega \xi\, f(t,x,\xi)\, \mathrm{d} \xi}{F(x)} \int_\Omega \sigma(x,x_*) \frac{\int_\Omega f(t,x_*,\xi_*)\, \mathrm{d} \xi_*}{F(x_*)} \, \mathrm{d} x_* \\[10pt]
&=& \displaystyle \int_\Omega \sigma(x,x_*) ( y(t,x_*)-y(t,x) )\, \mathrm{d} x_*,
\end{eqnarray*}
and the result follows.
\end{proof}
\begin{remark}
The exponential convergence of $y(t)$ to $\bar y^v$ when $t\to +\infty$ can be interpreted in the kinetic setting by saying that
the measure $\mu(t,x)$ converges (vaguely) towards the Dirac measure $\delta_{\bar y^v}$.
\end{remark}
\subsection{Lyapunov functionals and time stabilization}
In this subsection, we present two methods to find a Lyapunov functional for our problems, allowing to recover the convergence towards consensus property with an exponential decay. The first one uses the classical Lyapunov lemma, and the second one involves a weighted variance. Eventually, we present an application of this functional to design a Jurdjevic-Quinn-type stabilizing control. It can also be used to discuss stability properties under perturbations of our system (such as nonlinearities, noises).
\subsubsection{Using the Lyapunov lemma}
In finite dimension, since $A_2$ is a Hurwitz matrix, there exists (see, for instance, \cite{Khalil}) a unique matrix $P\in \mathbb R^{(N-1)\times (N-1)}$, symmetric positive definite, defined on $\operatorname{im} A$, such that
$$PA_2+A_2^\intercal P = -\mathbb{I}_{N-1}.$$
In infinite dimension, since $A_2$ is a homeormorphism on $\operatorname{im} A$ which is strictly dissipative and generates an exponentially stable semi-group, it follows from \cite[Theorem~5.1.3]{cur-zwa} that there exists a unique bounded self-adjoint positive definite operator $P$ defined on $\operatorname{im} A$ such that
$$PA_2+A_2^* P = -\mathrm{Id}_{\operatorname{im} A},$$
In both cases, $P$ is given by
$$P=\int_0^{+\infty} e^{tA_2^*} e^{tA_2}\,\mathrm{d} t. $$
This operator $P$ induces a norm on $\operatorname{im} A$, given by $\|z\|_P=\langle z,Pz\rangle=\|P^{1/2}z\|$. We define the Lyapunov functional on $\operatorname{im} A$, for any $z\in\operatorname{im} A$, as
$$\operatorname{Var}_P(z)=\langle z,Pz\rangle.$$
We call it the variance associated with $P$. When $X=\mathbb R^N$, we can choose $\lambda_{\max}>0$ as the highest eigenvalue of $P$. When $X=L^2(\Omega)$, we notice that
$$\|z\|^2=\langle z,z\rangle = -2 \langle z, A_2^*Pz\rangle
\leq 2\|z\| \|A_2^*\|\|P^{1/2}\|\|P^{1/2}z\|, \qquad \forall z\in\operatorname{im} A.$$
This implies that, in both cases, there exists $\lambda_{\max}>0$ such that $\operatorname{Var}_P(z)\leq \lambda_{\max} \|z\|^2$ for any $z\in\operatorname{im} A$.
In order to recover the exponential convergence of a solution $y$ of \eqref{e:pbdimfinie} or \eqref{e:pbdiminfinie} towards consensus, recalling that $\bar y^v \in\ker A$ and $\pi y = y-\bar y^v\in \operatorname{im} A$, $z=\pi y$ satisfies $\dot z = \dot y = A(z+\bar y^v) = A_2z$. Then we introduce $V_P:t\mapsto \operatorname{Var}_P(\pi y(t))$, so that
$$\dot V_P =\langle\dot z, Pz\rangle + \langle z, P \dot z\rangle
= \langle z, (A_2^* P + PA_2) z\rangle=-\|z\|^2\leq -\frac 1{\lambda_{\max}} V_P,$$
which ensures the required exponential convergence. This argument suffices to prove exponential convergence, but not to obtain the sharp convergence rate stated in Theorem~\ref{t:convcons}. We mention the recent paper \cite{arn-jin-woh} for techniques to design Lyapunov functionals achieving that sharp rate.
\subsubsection{Weighted variance}
We propose an alternative variance, based on the geometric properties of our problem, and involving the weight $v$ built in Theorem~\ref{t:poids}. In the weighted scalar product framework on $X$, we define the weighted expectation
$$\operatorname{E}_v[y] = \langle y,v\rangle=\langle y,e\rangle_v= \left\{
\begin{array}{ccl}
\displaystyle \sum_i v_i y_i & \mbox{if } & X=\mathbb R^N, \\
\displaystyle \int_\Omega v(x) y(x)\,\mathrm{d} x & \mbox{if } & X=L^2(\Omega).
\end{array} \right. $$
It is clear that $\bar y^v=\operatorname{E}_v[y] e$. Then we define the weighted variance of $y\in X$ as
$$\operatorname{Var}_v y = \operatorname{E}_v\left[ (y-\operatorname{E}_v[y])^2\right] = \operatorname{E}_v[y^2]-\operatorname{E}_v[y]^2
= \|y-\bar y^v\|_v^2=\|y\|_v^2-\|\bar y^v\|_v^2=\|\pi y\|_v^2.$$
When $X=\mathbb R^N$, we have
\begin{equation*
\operatorname{Var}_v y = \sum_{i} v_i (y_i-\langle y,e\rangle_v)^2
=\sum_{i} v_i y_i^2 - \langle y,e\rangle_v^2=\frac12 \sum_{i,j} v_iv_j(y_i-y_j)^2,
\end{equation*}
and when $X=L^2(\Omega)$,
\begin{multline*}
\operatorname{Var}_v y = \int_\Omega v(x) (y(x)-\bar y^v)^2\,\mathrm{d} x=
\int_\Omega v(x) y(x)^2\,\mathrm{d} x-(\bar y^v)^2\\
=\frac12 \iint_{\Omega^2} v(x) v(x_*) (y(x)-y(x_*))^2\,\mathrm{d} x_*\,\mathrm{d} x.
\end{multline*}
Setting $V_v:t\mapsto \operatorname{Var}_v(y(t))$, where $y$ solves \eqref{e:pbdimfinie} or \eqref{e:pbdiminfinie}, this weighted variance can be used as a Lyapunov functional. Indeed, let us study the monotonicity of $V_v$. Remembering that $A^{*_v}\bar y^v=0$, we have
$$\dot V_v = 2 \langle y-\bar y^v, \dot y\rangle_v
=2 \langle y-\bar y^v, Ay\rangle_v =2\langle y,Ay\rangle_v =2Q(y)=2Q_2(\pi y),$$
where $Q$ and $Q_2$ are the real Hilbert versions of the quadratic forms with the same names defined in \eqref{e:quadformC} in a complex Hilbert structure. Recall that
$$Q(y)=\frac12\left\{ \begin{array}{ll}
\displaystyle\sum_{i,j} v_i\, \sigma_{ij} (y_j-y_i)^2 & \mbox{in finite dimension}, \\
\displaystyle\iint_{\Omega^2} v(x)\sigma(x,x_*) (y(x)-y(x_*))^2\,\mathrm{d} x_*\,\mathrm{d} x
& \mbox{in infinite dimension}.
\end{array}\right.$$
We also already know that $A$ and $A_2$ are generally not self-adjoint (except if $\sigma$ is symmetric), and respectively dissipative and strictly dissipative with respect to the weighted scalar product (but not to the standard scalar product). Dissipativity holds thanks to the strong connectivity assumption, as explained in the previous section. This ensures that the weighted variance strictly decreases if $y^\mathrm{in}\neq\bar y^v$.
Then we can obtain the convergence towards to consensus thanks to the LaSalle invariance principle, see, for instance, \cite{hal69, har91, Khalil, wal80}. In finite dimension, the convergence of $V_v$ towards $0$ is then obtained because the invariant set of the differential system in the LaSalle sense is $\ker A$. Consequently, the only possible accumulation point of the trajectory is the consensus, which implies that $V_v$ converges towards $0$ when $t$ goes to $+\infty$. In infinite dimension, we also recover the convergence of $V_v$ to $0$, provided that the orbits of the system are precompact (which is true indeed, the proof is not presented here because it does not yield anything new with respect to Theorem~\ref{t:convcons}).
\medskip
\begin{comment}
To prove that the orbits are precompact, we develop hereafter an argument inspired from \cite{web79} (see also \cite[page 37]{har91}). Thanks to the Duhamel formula, we have $e^{tA_\sigma}=e^{-tM_S}+ F(t)$, where
$$F(t)=\int_0^t e^{-\tau M_S} K_\sigma e^{(t-\tau)A_\sigma}\,\mathrm{d} \tau, \qquad t\geq 0.$$
For any $t>0$, $F(t)$ is a compact operator on $L^2(\Omega)$. Indeed, for any $t>0$ and $N\in\mathbb N^*$, if we set $t_i=it/N$ and
$$F_N(t)=\sum_{i=0}^{N-1} e^{-t_i M_S} K_\sigma\int_{t_i}^{t_{i+1}} e^{(t-\tau)A_\sigma}\,\mathrm{d} \tau,$$
we see that $F_N(t)$ is compact as an operator from $L^2(\Omega)$ to $L^2(\Omega)$, and that $(F_N(t))_{N\in \mathbb N^*}$ converges to $F(t)$ in the Banach space of the bounded operators on $L^2(\Omega)$, denoted by $\mathcal B(L^2(\Omega))$, hence $F(t)$ is compact.
We consider the orbit of $y:t\mapsto e^{tA_\sigma}y_0$ for any $y_0\in L^2(\Omega)$. Noticing that the semi-group $(e^{tA_\sigma})$ is bounded, \textit{i.e. } there exists $\mu\geq 0$ such that $\|e^{tA_\sigma}\|_{\mathcal B(L^2(\Omega))}\leq \mu$ for any $t\geq 0$, we immediately get that $\|y(t)\|_{L^2(\Omega)}\leq \mu \|y_0\|_{L^2(\Omega)}$ for any $t\geq 0$.
To obtain the relative compactness of $y(\mathbb R_+)$ in $L^2(\Omega)$, we write, for some $t_0>0$ to be fixed later,
$$y(\mathbb R_+)=\left\{ e^{tA_\sigma} y_0~|~t\geq 0\right\}=\left\{ e^{sA_\sigma} y_0~|~0\leq s\leq t_0\right\} \cup \left\{ e^{(t_0+s)A_\sigma} y_0~|~s\geq 0\right\}.$$
The first set in the right-hand side of the the previous equality is compact since $y$ is continuous and $[0,t_0]$ is compact. To handle the second set, we note that $e^{(t_0+s)A_\sigma}=(e^{-t_0M_S}+F(t_0))e^{sA_\sigma}$ and then obtain
$$\left\{ e^{(t_0+s)A_\sigma} y_0~|~s\geq 0\right\} \subset e^{-t_0M_S}(y(\mathbb R_+)) + F(t_0)(y(\mathbb R_+)).$$
Since $F(t_0)$ is compact and $y(\mathbb R_+)$ is bounded in $L^2(\Omega)$, $F(t_0)(y(\mathbb R_+))$ is relatively compact in $L^2(\Omega)$. Lastly, we prove that $e^{-t_0M_S}(y(\mathbb R_+))$ is in fact precompact by using the exponential stability of the semi-group $(e^{-tM_S})_{t\geq 0}$. Let $\varepsilon>0$. Since $S\geq \delta>0$ almost everywhere on $\Omega$, $\|e^{-t_0M_S}\|_{\mathcal B(L^2(\Omega))}\leq e^{-\delta t_0}$. Consequently, $e^{-t_0M_S}(y(\mathbb R_+))$ is bounded by $\mu \|y_0\|_{L^2(\Omega)} e^{-\delta t_0}$, and we can choose $t_0$ such that the previous bound is smaller than $\varepsilon$, which ensures the expected precompactness property.
Finally, $y(\mathbb R_+)$ is relatively compact in $L^2(\Omega)$ as the union and sum of (at least) relatively compact subsets of $L^2(\Omega)$.
\end{comment}
\begin{remark}
The variances $\operatorname{Var}_P$ and $\operatorname{Var}_v$ can be respectively expressed in terms of $P$ or $D_v$, \textit{i.e. } for any $z\in\operatorname{im} A$, we have
$$\operatorname{Var}_P(z)=\langle z,Pz\rangle, \qquad \operatorname{Var}_v(z)=\langle z,D_v z\rangle.$$
This means that, in the same way the Lyapunov lemma provides a self-ajoint positive definite operator $P$ such that $PA_2+A_2^*P=\mathrm{Id}_{\operatorname{im} A}$, we have proved here that we can find a multiplicative (diagonal) positive operator $D_v$ such that $D_v A + A^* D_v$ is dissipative (and strictly on $\operatorname{im} A$).
\end{remark}
\subsubsection{Applications}
\paragraph{Jurdjevic-Quinn stabilization.}
One of the interests of Lyapunov functionals is the possibility to speed up the convergence towards consensus by adding a control (see \cite{caponigro2017mean, caponigro2017JQ}) designed by the Jurdjevic-Quinn method (see \cite{jur-qui-78}). Let $u:\mathbb R^+\to X$ be a control function, and consider the Cauchy problem
$$\dot y(t) = Ay(t) +u(t), \qquad y(0)=y^\mathrm{in}.$$
Since $\bar y^v=(\mathrm{Id}-\pi)y$, we have
$$\frac{\mathrm{d} \bar y^v}{\mathrm{d} t} =(\mathrm{Id}-\pi)Ay+(\mathrm{Id}-\pi)u.$$
We study again $V_P:t\mapsto \langle\pi y,P\pi y\rangle$ (or $V_v$). Setting $z=y-\bar y^v$, we have
$$\dot z = Ay +u -(\mathrm{Id}-\pi)Ay-(\mathrm{Id}-\pi)u=\pi Ay +\pi u= A_2z+\pi u.$$
Then
$$\dot V_P =-\|z\|^2+2\langle \pi u,Pz\rangle \leq
-\frac 1{\lambda_{\max}}V_P+2\langle \pi u,Pz\rangle.$$
We are led to choose $u=-\alpha \pi y\in \operatorname{im} A$, $\alpha>0$, so that
$$\dot V_P \leq -\left(\frac 1{\lambda_{\max}}+2\alpha\right)V_P,$$
which then arbitrarily improves the convergence rate towards consensus.
\paragraph{Robustness under a class of nonlinear perturbations.}
Another interest of the Lyapunov functionals consists in ensuring exponential convergence under some nonlinear perturbations. Let $f:X\rightarrow X$ be a function of class $C^1$, locally bounded, satisfying $\langle e,f(y) \rangle_v = \langle e,f(y) \rangle = 0$ and the dissipativity property $\langle y,f(y) \rangle_v \leq 0$ for every $y\in X$. The first property implies that $f(X)\subset\operatorname{im} A$; in finite dimension, it means that $\sum_i f_i(y)=0$. We consider the Cauchy problem
$$\dot y(t) = Ay(t) +f(y(t)), \qquad y(0)=y^\mathrm{in}.$$
Since $\operatorname{im} A=(\ker A)^{\perp_v}$, it follows from the first property of $f$ that
$$\frac{\mathrm{d} \bar y^v}{\mathrm{d} t} = \langle Ay+f(y),e \rangle_v~e = 0 .$$
Hence the weighted mean remains constant. Then, thanks to the dissipativity property,
$$\dot V_v = 2 \langle y-\bar y^v, Ay+f(y)\rangle_v
\leq 2Q_2(\pi y)-2 \langle \bar y^v, f(y)\rangle_v = 2Q_2(\pi y),$$
which implies that the above Cauchy problem is globally well posed and that the solution converges exponentially to consensus.
\section{Numerical illustrations} \label{s:num}
In this section, we present some numerical simulations in the finite-dimensional case, where the non-symmetric interaction matrix $(\sigma_{ij})$ satisfies, or not, connectivity properties. We investigate three different situations. In the first one, the population is ``fully connected'', \textit{i.e. } all non-diagonal coefficients of $(\sigma_{ij})$ are chosen positive. The second situation fits in the one we investigated in this work, \textit{i.e. } the graph associated to $(\sigma_{ij})$ is assumed to be strongly connected. We shall then refer to a ``strongly connected'' population. In our third situation, we focus on a population in which only some subgroups of the population are ``strongly connected'', the population itself being ``partially connected''. We deal with a population of $N=100$ individuals, and we focus on the collective dynamics. For the numerical simulations, we used a standard RK4 routine to solve \eqref{e:krause}--\eqref{e:initconddiscrete}.
\begin{comment}
\subsection{Small population}
The population under study consists of $7$ agents, and their initial state is
$$\boldsymbol{y}^{\mathrm{in}}=(0.5\hspace{.3cm} 0\hspace{.3cm} 0.1\hspace{.3cm}
0.3\hspace{.3cm} 0.2\hspace{.3cm} -1\hspace{.3cm} 1)^\intercal.$$
In the first situation, the coefficients of $\Tsig$ are randomly chosen between $0.1$ and $0.9$ (the zero diagonal coefficients excepted), so that each individual has an influence on all other members of the population, for instance
$$
\Tsig = \begin{pmatrix}
0. & 0.4604 & 0.8969 & 0.7538 & 0.8285 & 0.5399 & 0.1608 \\
0.2325 & 0. & 0.1625 & 0.7950 & 0.2455 & 0.2160 & 0.2919 \\
0.5816 & 0.2832 & 0. & 0.1675 & 0.3110 & 0.7824 & 0.1987 \\
0.3104 & 0.8307 & 0.1853 & 0. & 0.2164 & 0.5976 & 0.2471 \\
0.6233 & 0.2219 & 0.8695 & 0.3079 & 0. & 0.3808 & 0.2920 \\
0.6514 & 0.7607 & 0.1037 & 0.7401 & 0.7954 & 0. & 0.4338 \\
0.6985 & 0.5307 & 0.7199 & 0.4451 & 0.5638 & 0.4214 & 0.
\end{pmatrix},
$$
which provides the approximate value of the weight, clearly not proportional to $\boldsymbol{e}$,
$$\boldsymbol{v}=(0.1136\hspace{.3cm} 0.2126\hspace{.3cm} 0.1536\hspace{.3cm}
0.1867\hspace{.3cm} 0.1383\hspace{.3cm} 0.1213\hspace{.3cm} 0.0740)^\intercal.$$
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=7.35cm]{7-bounded-history.eps} \
\includegraphics[width=7.35cm]{7-bounded-wvariance.eps}
\caption{Time evolution on $[0,1]$ of (a) each $y_i$, $1\leq i\leq 7$, (b)
$\log_{10} V$, for a small fully connected population.}
\label{f:Fig1}
\end{center}
\end{figure}
Figure\,\ref{f:Fig1}(a) shows the convergence of each $y_i$ towards the
consensus at the weighted mean, and, in Figure\,\ref{f:Fig1}(b), we observe that this convergence is exponential in time, as expected. The computed time exponential decay rate is close to $7.40$. It must be lower bounded by the real part of the Fiedler number of $-\Asig$, actually equal to $2.02$, which is the case.
\medskip
The second situation deals with a strongly connected population, with
$$
\Tsig = \begin{pmatrix}
0 & {0.1} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & {0.2} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & {0.3} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & {0.4} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & {0.5} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & {0.6} \\
{0.1} & {0.2} & 0 & 0 & 0 & 0 & 0
\end{pmatrix}.
$$
The associated weight has the approximate value
$$\boldsymbol{v}=(0.1575\hspace{.3cm} 0.2362\hspace{.3cm} 0.1575\hspace{.3cm}
0.1181\hspace{.3cm} 0.0945\hspace{.3cm} 0.0787\hspace{.3cm} 0.1575)^\intercal.$$
We see in Figure\,\ref{f:Fig2}(a) the expected convergence to consensus. We also note that the convergence is slower (time scale around $20$ instead of $1$), for the same order of nonzero coefficients of $\Tsig$. Moreover, according to Figure\,\ref{f:Fig2}(b), the exponential decay rate of convergence to the weighted mean is close to $0.409$, to be compared to the real part of the Fiedler number of $-\Asig$, which equals $0.1515$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=7.3cm]{7-connected-history.eps} \ \includegraphics[width=7.3cm]{7-connected-wvariance.eps}
\caption{Time evolution on $[0,20]$ of (a) each $y_i$, $1\leq i\leq 7$, (b)
$\log_10 V$, for a small strongly connected population.}
\label{f:Fig2}
\end{center}
\end{figure}
\medskip
The third situation is dedicated to a partially connected
population, with
$$\Tsig=\begin{pmatrix}
0 & {0.1} & {0.1} & 0 & 0 & 0 & 0 \\
{0.9} & 0 & {0.1} & 0 & 0 & 0 & 0 \\
{0.1} & {0.1} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & {0.1} & {0.1} & {0.1} \\
0 & 0 & 0 & {0.2} & 0 & {0.1} & {0.9} \\
0 & 0 & 0 & {0.1} & {0.2} & 0 & {0.1} \\
0 & 0 & 0 & {0.9} & {0.1} & {0.1} & 0
\end{pmatrix}.$$
With this choice, there are two subgroups of the
population which cannot interact with each other, agents $1$ to $3$ on the one
side, agents $4$ to $7$ on the other side.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=9cm]{7-twoclusters-history.eps}
\caption{Time evolution on $[0,15]$ of each $y_i$, $1\leq i\leq 7$, in a small
partially connected population, with two strongly connected subgroups.}
\label{f:Fig3}
\end{center}
\end{figure}
Each subgroup is led to a consensus, that is why Figure\,\ref{f:Fig3} shows the
appearance of two clusters. Of course, there is no weighted variance in this case.
\end{comment}
The initial state of the population is by means of a random sampling between $0$ and $1$.
\begin{figure}[!ht]
\begin{center}
\psfrag{State functions}{\hspace{-.2cm}{\tiny State functions}}
\psfrag{Time}{{\tiny Time}}
\psfrag{0.1}{{\tiny $\!0.1$}}
\psfrag{0.2}{{\tiny $\!0.2$}}
\psfrag{0.3}{{\tiny $\!0.3$}}
\psfrag{0.4}{{\tiny $\!0.4$}}
\psfrag{0.5}{{\tiny $\!0.5$}}
\psfrag{0.6}{{\tiny $\!0.6$}}
\psfrag{0.7}{{\tiny $\!0.7$}}
\psfrag{0.8}{{\tiny $\!0.8$}}
\psfrag{0.9}{{\tiny $\!0.9$}}
\psfrag{1}{{\tiny $\!1$}}
\psfrag{0.05}{{\tiny $0.05$}}
\psfrag{0.15}{{\tiny $0.15$}}
\psfrag{0}{{\tiny $\!0$}}
\psfrag{Logarithm of the variances}{\hspace{.75cm}{\tiny Variance}}
\psfrag{standard variance}{ {\tiny standard}}
\psfrag{weighted variance}{ {\tiny weighted}}
\includegraphics[width=7.3cm]{etats_fc_ok.eps} \hfill \includegraphics[width=7.3cm]{variances_fc_ok.eps}
\caption{Time evolution on $[0,0.15]$ of (a) each $y_i$, $1\leq i\leq 100$, (b) the standard and weighted variances (log scale), for a fully connected population.}
\label{f:Fig4}
\end{center}
\end{figure}
The first situation considers a population for which the interaction coefficients $(\sigma_{ij})$ are randomly chosen in $(0,1)$, the diagonal coefficients excepted. The computations provide a weight $v$ whose coordinates vary between $0.00830$ and $0.0118$ (to be compared to $1/N=0.01$). Figure~\ref{f:Fig4}(a) shows a very fast exponential convergence towards the weighted mean close to $0.462$. The line slope in Figure~\ref{f:Fig4}(b) is approximately $-102$. It must be compared to $2\,\,\mathrm{s}(A_2)\simeq -92.6$, since $\log V_v(t)=2\log \|y(t)-\bar y^v\|_v$. As expected, the slope is lower than $2\,\,\mathrm{s}(A_2)$, but of the same order of magnitude.
\begin{figure}[!ht]
\begin{center}
\psfrag{State functions}{\hspace{-.2cm}{\tiny State functions}}
\psfrag{Time}{{\tiny Time}}
\psfrag{0.1}{{\tiny $\!0.1$}}
\psfrag{0.2}{{\tiny $\!0.2$}}
\psfrag{0.3}{{\tiny $\!0.3$}}
\psfrag{0.4}{{\tiny $\!0.4$}}
\psfrag{0.5}{{\tiny $\!0.5$}}
\psfrag{0.6}{{\tiny $\!0.6$}}
\psfrag{0.7}{{\tiny $\!0.7$}}
\psfrag{0.8}{{\tiny $\!0.8$}}
\psfrag{0.9}{{\tiny $\!0.9$}}
\psfrag{1}{{\tiny $\!1$}}
\psfrag{500}{{\tiny $500$}}
\psfrag{1000}{{\tiny $1000$}}
\psfrag{1500}{{\tiny $1500$}}
\psfrag{2000}{{\tiny $2000$}}
\psfrag{0}{{\tiny $\!0$}}
\psfrag{Logarithm of the variances}{{\hspace{.75cm}{\tiny Variance}}}
\psfrag{standard variance}{ {\tiny standard}}
\psfrag{weighted variance}{ {\tiny weighted}}
\includegraphics[width=7.4cm]{etats_sc_ok.eps} \hfill \includegraphics[width=7.4cm]{variances_sc_ok.eps}
\caption{Time evolution on $[0,2000]$ of (a) each $y_i$, $1\leq i\leq 100$, (b)
the standard and weighted variances (log scale), for a strongly connected population.}
\label{f:Fig5}
\end{center}
\end{figure}
In the second situation, where the population is strongly connected, the interaction matrix is chosen such that $\sigma_{N1}$ and $\sigma_{i,i+1}$, for any $1\leq i\leq N-1$, are randomly chosen in $(0,1)$, all the other coefficients being zero. The coordinates of the weight $v$ are quite different from $0.01$ this time, they vary between $0.00291$ and $0.0530$. Figure~\ref{f:Fig5}(a) then shows a slower (with respect to the fully connected case) convergence towards the weighted mean, which is close to $0.532$. This slow convergence towards the weighted mean, with a slope equal to $-0.00240$, corresponds to the worst-case convergence scenario since $2\,\,\mathrm{s}(A_2)\simeq-0.00240$. Note that the standard and weighted variances have the same asymptotic behavior, but there are oscillations on the standard variance. It is not surprising: the weighted variance is indeed non-increasing, but we already pointed out that the standard variance does not satisfy monotonicity properties with respect to time.
\begin{figure}[!ht]
\begin{center}
\psfrag{State functions}{\hspace{-.5cm}{\small State functions}}
\psfrag{Time}{{\small Time}}
\psfrag{0.1}{{\tiny $\!0.1$}}
\psfrag{0.2}{{\tiny $\!0.2$}}
\psfrag{0.3}{{\tiny $\!0.3$}}
\psfrag{0.4}{{\tiny $\!0.4$}}
\psfrag{0.5}{{\tiny $\!0.5$}}
\psfrag{0.6}{{\tiny $\!0.6$}}
\psfrag{0.7}{{\tiny $\!0.7$}}
\psfrag{0.8}{{\tiny $\!0.8$}}
\psfrag{0.9}{{\tiny $\!0.9$}}
\psfrag{1.5}{{\tiny $\!1.5$}}
\psfrag{1}{{\tiny $\!1$}}
\psfrag{0}{{\tiny $\!0$}}
\includegraphics[width=9.3cm]{etats_nc_ok.eps}
\caption{Time evolution on $[0,1.5]$ of each $y_i$, $1\leq i\leq 100$, in a
partially connected population, with three strongly connected subgroups.}
\label{f:Fig6}
\end{center}
\end{figure}
As noticed in Remark~\ref{r:partialconnect}, the third situation leads to three clusters, because the populations is divided into three non-interacting subgroups, each of them being fully connected. Hence the clustering effect appears very fast again, as it is shown on Figure~\ref{f:Fig6}.
\section{Conclusion and prospects}\label{s:conc}
In this article, we studied the convergence to consensus, both in finite and infinite dimensions, for first-order non-symmetric systems, under the condition that the graph associated to $\sigma$ be strongly connected.
We identified a positive weight $v$ and checked that the corresponding $v$-weighted mean remains constant in time. We have moreover proved that the system exponentially converges to consensus, and exhibited the sharp exponential rate.
The $L^2$ approach has many advantages as it allows, for instance, to use the
Jurdjevic-Quinn approach \cite{jur-qui-78} as in \cite{caponigro2017mean, caponigro2017JQ}. In our problem, we proved that the $v$-weighted variance is a Lyapunov functional, which can be an alternative to the Lyapunov functional obtained in the framework of the standard (non-weighted) $L^2$ theory.
Our analysis paves the way to further research on the subject. We mention below some of them, without claiming to be exhaustive.
First of all, it may be interesting to study the effect of noise sources on the system, by introducing an additive noise, or by studying the system behavior when $\sigma$ is noised (\textit{i.e. } $A$ is noised around a fixed matrix). Another extension may consist in allowing $\sigma$ to be time-dependent. This hypothesis naturally leads to study models of influence sphere, with a possible loss of lower bounds and thus the emergence of local clusters. In such a case, the control of clusters is a question to be explored.
An open issue is the study of the system, when $\sigma$ depends on $|x_i-x_j|$, as in the original Hegselmann-Krause model \cite{heg-kra, kra}, especially the sharpness of the asymptotic convergence rate. Another open problem is the extension of our study to non-symmetric second-order models, such as generalized Cucker-Smale models \cite{caponigro2017JQ, cuc-sma, cuc-sma-2, has21, mot-tad}.
\bibliographystyle{abbrv}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,088
|
Q: Android how to invoke binding adapter without changing view Visibility? I am using a bindingAdapter to set the background of a view based on the state.
But the bindingAdapter works only when the view reinflates or the visibility change. Is there any way to invoke the binding without reinflating or changing the visibility?
@BindingAdapter("bgClickable")
fun bgClickable(layout: ConstraintLayout, state: String) {
when (state) {
DISCONNECTED -> {
val outValue = TypedValue()
layout.context.theme
.resolveAttribute(android.R.attr.selectableItemBackground, outValue, true)
layout.setBackgroundResource(outValue.resourceId)
}
else -> {
layout.background = null
}
}
}
View
.
.
.
<androidx.constraintlayout.widget.ConstraintLayout
android:id="@+id/layout_location"
android:layout_width="wrap_content"
app:bgClickable="@{viewModel.getCurrentStatus()}"
android:layout_height="wrap_content"
android:onClick="@{()->viewModel.redirectToExplorer()}"
app:layout_constraintBottom_toTopOf="@+id/guideline3"
app:layout_constraintEnd_toEndOf="parent"
app:layout_constraintStart_toStartOf="parent"
app:layout_constraintTop_toTopOf="parent"
app:layout_constraintVertical_bias="0.25">
<.../>
A: A common mistake with binding adapter is not to add a lifecycleOwner to the binding object and without it the changes to the underlying live data won't be notified to the binding object. so add the following in the fragment/activity class
binding.lifecycleOwner = this
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 291
|
{"url":"https:\/\/www.snapxam.com\/solver?p=%5Clim_%7Bx%5Cto-3%7D%5Cleft%28%5Cfrac%7Bx%5E4-81%7D%7Bx%2B3%7D%5Cright%29&method=0&tab=1","text":"Try NerdPal! Our new app on iOS and Android\n\n# Find the limit of $\\frac{x^4-81}{x+3}$ as $x$ approaches $-3$\n\nGo!\nGo!\n1\n2\n3\n4\n5\n6\n7\n8\n9\n0\na\nb\nc\nd\nf\ng\nm\nn\nu\nv\nw\nx\ny\nz\n.\n(\u25fb)\n+\n-\n\u00d7\n\u25fb\/\u25fb\n\/\n\u00f7\n2\n\ne\n\u03c0\nln\nlog\nlog\nlim\nd\/dx\nDx\n|\u25fb|\n\u03b8\n=\n>\n<\n>=\n<=\nsin\ncos\ntan\ncot\nsec\ncsc\n\nasin\nacos\natan\nacot\nasec\nacsc\n\nsinh\ncosh\ntanh\ncoth\nsech\ncsch\n\nasinh\nacosh\natanh\nacoth\nasech\nacsch\n\n## Basic Derivatives\n\n\u00b7 Sum Rule for Differentiation\n$\\frac{d}{dx}\\left[f\\left(x\\right)+g\\left(x\\right)\\right]=\\frac{d}{dx}f\\left(x\\right) + \\frac{d}{dx}g\\left(x\\right)$\n\u00b7 Derivative of a Constant\n$\\frac{d}{dx}\\left(c\\right)=0$\n\u00b7 Power rule for derivatives\n$\\frac{d}{dx}\\left(x^a\\right)=ax^{\\left(a-1\\right)}$\n\u00b7 Derivative of the linear function\n$\\frac{d}{dx}\\left(x\\right)=1$\n\n## Limits\n\n$\\lim_{x\\to c}\\left(ab\\right)=a\\lim_{x\\to c}\\left(b\\right)$\nSnapXam A2\n\n### beta Got another answer? Verify it!\n\nGo!\n1\n2\n3\n4\n5\n6\n7\n8\n9\n0\na\nb\nc\nd\nf\ng\nm\nn\nu\nv\nw\nx\ny\nz\n.\n(\u25fb)\n+\n-\n\u00d7\n\u25fb\/\u25fb\n\/\n\u00f7\n2\n\ne\n\u03c0\nln\nlog\nlog\nlim\nd\/dx\nDx\n|\u25fb|\n\u03b8\n=\n>\n<\n>=\n<=\nsin\ncos\ntan\ncot\nsec\ncsc\n\nasin\nacos\natan\nacot\nasec\nacsc\n\nsinh\ncosh\ntanh\ncoth\nsech\ncsch\n\nasinh\nacosh\natanh\nacoth\nasech\nacsch\n\n$\\lim_{x\\to-3}\\left(\\frac{x^4-81}{x+3}\\right)$\n\nLimits\n\n~ 0.24 s","date":"2022-08-07 19:39:27","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\": 0.4662491977214813, \"perplexity\": 10397.730446515501}, \"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-2022-33\/segments\/1659882570692.22\/warc\/CC-MAIN-20220807181008-20220807211008-00443.warc.gz\"}"}
| null | null |
\section{Introduction}
Let $(M, g)$ be a compact oriented Riemannian $n$-manifold.
We call $g$ formal if all products of harmonic forms are again harmonic.
If a compact oriented manifold admits a formal Riemaniann metric, we call it geometrically formal.
If $g$ is formal, then the space of the harmonic forms is a subalgebra of the de Rham complex of $M$ and isomorphic to the real cohomology of $M$.
By this, a geometrically formal manifold is a formal space (in the sense of Sullivan \cite{Sul}).
But the converse is not true (see \cite{Kot} \cite{KT}).
For very simple examples, closed surfaces with genus$\ge2$ are formal but not geometrically formal.
Thus we have one problem of geometrical formality of formal spaces.
Kotschick's nice work in \cite{Kot} stimulates us to consider this problem.
In this paper we prove the following theorem by using computations of the de Rham cohomology of general solvmanifolds given in \cite{K2}.
\begin{theorem}\label{MT}
Let $G=\mathbb{R}^{n}\ltimes_{\phi} \mathbb{R}^{m}$ with a semisimple action $\phi$.
Suppose $G$ has a lattice $\Gamma$.
Then $G/\Gamma$ admits an invariant formal metric.
\end{theorem}
We also study geometrical formality of low-dimensional aspherical manifolds with the virtually solvable fundamental groups.
We consider infra-solvmanifolds which are quotient spaces of simply connected solvable Lie groups by subgroups of the groups of the affine transformations of $G$ satisfying some conditions (see Section $7$ for the definition).
We classify geometrically formal compact aspherical manifolds of dimension less than or equal to $4$ with the virtually solvable fundamental groups.
\begin{theorem}
Let $M$ be a compact oriented aspherical manifold of dimension less than or equal to $4$ with the virtually solvable fundamental group.
Then $M$ is geometrically formal if and only if $M$ is diffeomorphic to a torus or an infra-solvmanifold which is not a nilmanifold.
\end{theorem}
\section{Notation and conventions}
Let $k$ be a subfield of $\mathbb{C}$.
A group $\bf G$ is called a $k$-algebraic group if $\bf G$ is a Zariski-closed subgroup of $GL_{n}(\mathbb{C})$ which is defined by polynomials with coefficients in $k$.
Let ${\bf G}(k)$ denote the set of $k$-points of $\bf G$
and ${\bf U}({\bf G})$ the maximal Zariski-closed unipotent normal $k$-subgroup of $\bf G$ called the unipotent radical of $\bf G$.
Denote $U_{n}(k)$ the group of $k$-valued upper triangular unipotent matrices of size $n$.
\section{Unipotent hull of solvable Lie group}\label{Hul}
\begin{theorem}{\rm (\cite{R})}\label{ttt}
Let $G$ be a simply connected solvable Lie group.
Then there exists a unique $\mathbb{R}$-algebraic group ${\bf H}_{G}$ with an injective group homomorphism $\psi :G\to {\bf H}_{G}(\mathbb{R}) $
so that:
\\
$(1)$ \ $\psi (G)$ is Zariski-dense in ${\bf H}_{G}$.\\
$(2)$ \ The centralizer $Z_{{\bf H}_{G}}({\bf U}({\bf H}_{G}))$ of ${\bf U}({\bf H}_{G})$ is contained in ${\bf U}({\bf H}_{G})$.\\
$(3)$ \ $\dim {\bf U}({\bf H}_{G})$=${\rm dim}\,G$(resp. ${\rm rank}\, G$).
\end{theorem}
We denote ${\bf U}_{G}={\bf U}({\bf H}_{G})$.
\begin{theorem}\label{abab} {\rm (\cite{K})}
Let $G$ be a simply connected solvable Lie group.
Then ${\bf U}_{G}$ is abelian if and only if $G=\mathbb{R}^{n}\ltimes_{\phi} \mathbb{R}^{m}$ such that the action $\phi:\mathbb{R}^{n}\to {\rm Aut} (\mathbb{R}^{m})$ is semisimple.
\end{theorem}
\section{Hodge theory}
Let $(V, g) $ be a $\mathbb{R}$ or $\mathbb{C}$-vector space of dimension $n$ with an inner product $g$.
Let $\bigwedge V =\bigoplus_{p=0} \bigwedge^{p} V$ be the exterior algebra of $V$.
We extend $g$ to the inner product on $\bigwedge V$.
Take $vol \in \bigwedge^{n} V$ such that $g(vol, vol)=1$.
We define the linear map $\ast_{g}:\bigwedge^{p} V\to \bigwedge^{n-p} V$ as:
\[v \wedge \ast_{g}\bar u=g(v,u)vol\]
Let $\{\theta_{1},\dots \theta_{n}\}$ be an orthonormal basis of $(V, g) $.
Then we have
\[\ast_{g}(\theta_{i_{1}}\wedge\dots \theta_{i_{p}})=({\rm sgn}\sigma_{IJ}) \theta_{j_{1}}\wedge\dots \theta_{j_{n-p}}\]
where $J=\{j_{1}, \cdots ,j_{n-p}\}$ is the complement of $I=\{i_{1},\dots ,i_{p}\}$ in $\{1,\dots, n\}$ and $\sigma_{IJ}$ is the permutation $\begin{pmatrix}
1 \cdots p& p+1 \cdots n \\
i_{1} \cdots i_{p}&j_{1} \cdots j_{n-p}
\end{pmatrix}
$.
Let $(M, g)$ be a compact oriented Riemannian $n$-manifold.
Let $(A^{\ast}(M), d)$ be the de Rham complex of $M$ with the exterior derivation $d$.
For $x\in M$ by the inner product $g_{x}$ on $T_{x}M$ we define the linear map $\ast_{g}:A^{p}(M)\to A^{n-p}(M)$ by
\[(\ast_{g}(\omega))_{x}=\ast_{g_{x}}\omega_{x} \]
for $\omega \in A^{p}(M)$.
Define $\delta :A^{p}(M)\to A^{p-1}(M)$ by $\delta =(-1)^{np+n+1}\ast_{g} d\ast_{g}$.
We call $\omega\in A^{p}(M)$ harmonic if $d\omega =0$ and $\delta \omega =0$.
Let ${\mathcal H}^{p}(M)$ be the subspace of $A^{p}(M)$ which consists of harmonic $p$-forms.
Let ${\mathcal H}(M)=\bigoplus{\mathcal H}^{p}(M)$.
It is known that the inclusion ${\mathcal H}(M)\subset A^{\ast}(M)$ induces an isomorphism
\[{\mathcal H}^{p}(M)\cong H^{p}(M,\mathbb{R}).\]
In general a wedge product of harmonic forms is not harmonic and so ${\mathcal H}^{p}(M)$ is not a subalgebra of $A^{\ast}(M)$.
\begin{definition}
We call a Riemannian metric $g$ formal if all products of harmonic forms are again harmonic.
We call an oriented compact manifold $M$ geometrical formal if $M$ admits a formal metric.
\end{definition}
\section{Invariant forms on solvmanifolds (proof of Theorem \ref{MT})}
Let $G$ be a simply connected solvable Lie group and $\frak{g}$ the Lie algebra which is the space of the left invariant vector fields on $G$.
Consider the exterior algebra $\bigwedge \frak{g}^{\ast} $ of the dual space of $\frak{g}$.
Denote $d:\bigwedge^{1}\frak{g}^{\ast}\to \bigwedge^{2}\frak{g}^{\ast}$ the dual map of the Lie bracket of $\frak{g}$ and $d:\bigwedge^{p}\frak{g}^{\ast}\to \bigwedge^{p+1}\frak{g}^{\ast}$ the extension of this map.
We can identify $(\bigwedge \frak{g}^{\ast}, d)$ with the left invariant forms on $G$ with the exterior derivation.
Let ${\rm Ad}: G\to {\rm Aut}(\frak{g})$ be the adjoint representation.
Representations of $G$ are trigonalizable in $\mathbb{C}$ by Lie's theorem.
We define the diagonal representation ${\rm Ad}_{s}:G\to {\rm Aut}(\frak{g}_{\mathbb{C}}) $ as the diagonal entries of a triangulation of $\rm Ad$.
Let $X_{1},\cdots ,X_{n}$ be a basis of $\frak{g}_{\mathbb{C}}$ such that ${\rm Ad}_{s}$ is represented by diagonal matrices.
Then we have ${\rm Ad}_{sg}X_{i}=\alpha_{i}(g)X_{i}$ for characters $\alpha_{i}$ of $G$.
Let $x_{1},\dots,x_{n}$ be tha dual basis of $X_{1},\dots ,X_{n}$.
We assume that $G$ has a lattice $\Gamma$.
Define the sub-DGA $A^{\ast}$ of the de Rham complex $A^{\ast}_{\mathbb{C}}(G/\Gamma)$ as
\begin{multline*}
A^{p}
=\left\langle \alpha_{i_{1}\dots i_{p}} x_{i_{1}}\wedge \dots \wedge x_{i_{p}} {\Big \vert} \begin{array}{cc}1\le i_{1}<i_{2}<\dots <i_{p}\le n,\\ {\rm the \, \, restriction} \, \, of \, \, \alpha_{i_{1}\dots i_{p}}\, \, {\rm on \, \, \Gamma \, \, is\, \, trivial}\end{array}\right\rangle
\end{multline*}
where $\alpha_{i_{1}\dots i_{p}}=\alpha_{i_{1}}\dots \alpha_{i_{p}}$.
\begin{theorem}\label{isoc}{\rm (\cite[v4. Corollary 7.6]{K2})}
The inclusion
\[ A^{\ast}\subset A^{\ast}_{\mathbb{C}}(G/\Gamma)\]
induces a cohomology isomorphism and
$A^{\ast}$ can be considered as a sub-DGA of $\bigwedge {\frak u}^{\ast}$ where ${\frak u}$ is the Lie algebra of ${\bf U}_{G}$ as in Section \ref{Hul}.
\end{theorem}
Define $g$ the Hermittian inner product as
\[ g(X_{i}, X_{j})=\delta_{ij} .\]
Since $ {\rm Ad}_{s}$ is an $\mathbb{R}$-valued representation, the restriction of $g$ on $\frak{g}$ is an inner product on $\frak{g}$.
We consider $g$ as an invariant Riemannian metric on $G/\Gamma$.
\begin{theorem}
If ${\bf U}_{G}$ is abelian, then $g$ is a formal metric on $G/\Gamma$.
\end{theorem}
\begin{proof}
By the assumption, the differential on $\bigwedge {\frak u}^{\ast}$ is $0$.
By Theorem \ref{isoc}, the derivation on $A^{\ast}$ is $0$ and we have an
isomorphism
\[ A^{\ast}\cong H^{\ast}(G/\Gamma).\]
Thus it is sufficient to show that all elements of $A^{\ast}$ are harmonic.
Let $\ast_{g}$ be the star operator.
Then for $\alpha_{i_{1}\dots i_{p}} x_{i_{1}}\wedge \dots \wedge x_{i_{p}} \in A^{p}$ we have
\[\ast_{g}(\alpha_{i_{1}\dots i_{p}} x_{i_{1}}\wedge \dots \wedge x_{i_{p}} )=({\rm sgn} \sigma_{IJ}) \bar \alpha_{i_{1}\dots i_{p}} x_{j_{1}}\wedge\dots \wedge x_{j_{n-p}}.\]
Since the restriction of $ \alpha_{i_{1}\dots i_{p}}$ on $ \Gamma $ is trivial, the image $\alpha_{i_{1}\dots i_{p}}(G)=\alpha_{i_{1}\dots i_{p}}(G/\Gamma)$ is compact and hence $\alpha_{i_{1}\dots i_{p}}$ is unitary.
Since $G$ has a lattice $\Gamma$, $G$ is unimodular (see \cite[Remark 1.9]{R}) and hence we have
\[\bar \alpha_{i_{1}\dots i_{p}} =\alpha_{i_{1}\dots i_{p}}^{-1}=\alpha_{j_{1}\dots j_{n-p}}.
\]
Hence we have
\[\bar \alpha_{i_{1}\dots i_{p}} x_{j_{1}}\wedge\dots \wedge x_{j_{n-p}}=\alpha_{j_{1}\dots j_{n-p}} x_{j_{1}}\wedge\dots \wedge x_{j_{n-p}}\in A^{n-p}\]
and thus we have $\ast_{g}(A^{\ast})\subset A^{\ast}$.
Since the derivation on $A^{\ast}$ is $0$, we have $\delta(A^{\ast})=0$.
Hence the theorem follows.
\end{proof}
By this theorem and Theorem \ref{abab}, we have Theorem \ref{MT}.
\begin{remark}
Not every invariant metric on $G/\Gamma$ in Theorem \ref{MT} is formal.
See the following example.
\end{remark}
\begin{example}
Let $H=\mathbb{R}\ltimes_{\phi}\mathbb{R}^{2}$ such that $\phi(z)(x,y)=(e^{z}x, e^{-z}y)$.
Consider $G=H\times \mathbb{R}$.
Then for some non-zero number $a\in \mathbb{R}$, $\phi(a)$ is conjugate to an element of $SL_{2}(\mathbb{Z})$, and hence
$G$ has a lattice $\Gamma=a\mathbb{Z}\ltimes_{\phi}\Gamma^{\prime}\times \mathbb{Z}$ for a lattice $\Gamma^{\prime}$ of $\mathbb{R}^{2}$.
Let $\frak{g}$ be the Lie algebra of $G$ and $\frak{g}^{\ast}$ the dual of $\frak{g}$.
The cochain complex $(\bigwedge \frak{g}^{\ast}, d)$ is generated by a basis $\{x,y,z,w\}$ such that
\[dx=-z\wedge x,\ dy=z\wedge y,\ dz=0,\ dw=-z\wedge x.\]
Consider the invariant metric $g=x^2+y^2+z^2+w^2$.
Then $z$ and $w-x$ are harmonic for $g$.
But $z\wedge (w-x)$ is not harmonic.
Thus $g$ is not formal.
\end{example}
\begin{example}\label{CCC}
Let $G=\mathbb{C}\ltimes_{\phi} \mathbb{C}^{2}$ with $\phi(z)(x,y)=(e^{z}x, e^{-z}y)$.
For some $p,q\in \mathbb{R}$, $\phi(p\mathbb{Z}+\sqrt{-1}q\mathbb{Z})$ is conjugate to a subgroup of $SL_{4}(\mathbb{Z})$ and hence we have a lattice $\Gamma=(p\mathbb{Z}+\sqrt{-1}q\mathbb{Z})\ltimes \Gamma^{\prime\prime}$ for a lattice $\Gamma^{\prime\prime}$ of $\mathbb{C}^{2}$ (see \cite{Na} and \cite{Hd}).
For any lattice $\Gamma$, $G/\Gamma$ is geometrically formal by Theorem \ref{MT}.
\begin{remark}
In \cite{BT} for some lattice of $G$ in Example \ref{CCC}, it is proved that $G/\Gamma$ is geometrically formal.
But the de Rham cohomology of $G/\Gamma$ varies according to a choice of a lattice $\Gamma$.
\end{remark}
\end{example}
\begin{example}
Let $K$ be a finite extension field of $\mathbb{Q}$ with the degree $r$ for positive integers.
We assume $K$ admits embeddings $\sigma_{1},\dots \sigma_{s},\sigma_{s+1},\dots, \sigma_{s+2t}$ into $\mathbb{C}$ such that $s+2t=r$, $\sigma_{1},\dots ,\sigma_{s}$ are real embeddings and $\sigma_{s+1},\dots, \sigma_{s+2t}$ are complex ones satisfying $\sigma_{s+i}=\bar \sigma_{s+i+t}$ for $1\le i\le t$.
We suppose $s>0$.
Denote ${\mathcal O}_{K}$ the ring of algebraic integers of $K$, ${\mathcal O}_{K}^{\ast}$ the group of units in ${\mathcal O}_{K}$ and
\[{\mathcal O}_{K}^{\ast\, +}=\{a\in {\mathcal O}_{K}^{\ast}: \sigma_{i}(a)>0 \,\, {\rm for \,\, all}\,\, 1\le i\le s\}.
\]
Define $\sigma :{\mathcal O}_{K}\to \mathbb{R}^{s}\times \mathbb{C}^{t}$ by
\[\sigma(a)=(\sigma_{1}(a),\dots ,\sigma_{s}(a),\sigma_{s+1}(a),\dots ,\sigma_{s+t}(a))
\]
for $a\in {\mathcal O}_{K}$.
We denote
\[\sigma(a)\cdot \sigma(b)=(\sigma_{1}(a)\sigma_{1}(b),\dots ,\sigma_{s}(a)\sigma_{s}(b),\sigma_{s+1}(a)\sigma_{s+1}(b),\dots ,\sigma_{s+t}(a)\sigma_{s+t}(b))\]
for $a,b\in {\mathcal O}_{K}$.
Then the image $\sigma({\mathcal O}_{K})$ is a lattice in $\mathbb{R}^{s}\times \mathbb{C}^{t}$.
Define $l:{\mathcal O}_{K}^{\ast\, +}\to \mathbb{R}^{s+t}$ by
\[l(a)=(\log \vert \sigma_{1}(a)\vert,\dots ,\log \vert \sigma_{s}(a)\vert , 2\log \vert \sigma_{s+1}(a)\vert,\dots ,2\log \vert \sigma_{s+t}(a)\vert)
\]
for $a\in {\mathcal O}_{K}^{\ast\, +}$.
Then by Dirichlet's units theorem, the image $l({\mathcal O}_{K}^{\ast\, +})$ is a lattice in the vector space $L=\{x\in \mathbb{R}^{s+t}\vert \sum_{i=1}^{s+t} x_{i}=0\}$.
By this we have a geometrical representation of the semi-direct product ${\mathcal O}_{K}^{\ast\, +}\ltimes {\mathcal O}_{K}$ as $l({\mathcal O}_{K}^{\ast\, +})\ltimes_{\phi} \sigma({\mathcal O}_{K})$ with
\[\phi(t_{1},\dots, t_{s+t})(\sigma(a))=\sigma(l^{-1}(t_{1},\dots, t_{s+t}))\cdot\sigma (a)
\]
for $(t_{1},\dots ,t_{s+t})\in l({\mathcal O}_{K}^{\ast\, +})$.
Since $l({\mathcal O}_{K}^{\ast\, +})$ and $\sigma({\mathcal O}_{K})$ are lattices of $L$ and $\mathbb{R}^{s}\times \mathbb{C}^{t}$ respectively, we have an extension $\bar \phi:L\to {\rm Aut} (\mathbb{R}^{s}\times \mathbb{C}^{t}) $ of $\phi$ and $l({\mathcal O}_{K}^{\ast\, +})\ltimes_{\phi} \sigma({\mathcal O}_{K})$ can be seen as a lattice of $L\ltimes_{\bar \phi} (\mathbb{R}^{s}\times \mathbb{C}^{t})$.
By Theorem \ref{MT}, the solvmanifold $L\ltimes_{\bar \phi} (\mathbb{R}^{s}\times \mathbb{C}^{t})/l({\mathcal O}_{K}^{\ast\, +})\ltimes_{\phi} \sigma({\mathcal O}_{K})$ is geometrically formal by Theorem \ref{CCC}.
For a subgroup $U\subset {\mathcal O}_{K}^{\ast\, +}$, we have a Lie group $L^{\prime}\ltimes_{\bar \phi} (\mathbb{R}^{s}\times \mathbb{C}^{t})$ which contains $l(U)\ltimes_{\phi} \sigma({\mathcal O}_{K})$ as a lattice.
The solvmanifold $L^{\prime}\ltimes_{\bar \phi} (\mathbb{R}^{s}\times \mathbb{C}^{t})/l(U)\ltimes_{\phi} \sigma({\mathcal O}_{K})$ is also geometrically formal by Theorem \ref{MT}.
\end{example}
\begin{example}\label{EXC}
Let $G=\mathbb{R}\ltimes_{\phi} U_{3}(\mathbb{R})$ such that
\[\phi(t)\left(
\begin{array}{ccc}
1& x& z \\
0& 1&y \\
0& 0&1
\end{array}
\right)=\left(
\begin{array}{ccc}
1& e^{t}x& z \\
0& 1&e^{-t}y \\
0& 0&1
\end{array}
\right).\]
The left-invariant forms $\bigwedge \frak{g}^{\ast}$ on $G$ is generated by $\{ e^{-t}dx,e^{t}dy, dz-xdy, dt\}$.
It is known that $G$ has a lattice $\Gamma$ (see \cite{Saw}).
By simple computations, we have $H^{1}(\frak{g}^{\ast})=\langle dt \rangle$, $\dim H^{2}(\frak{g}^{\ast})=0$ and $\dim H^{3}(\frak{g}^{\ast})=1$.
Since $G$ is completely solvable, we have $H^{\ast}(G/\Gamma, \mathbb{R})\cong H^{\ast}(\frak{g}^{\ast})$ (see \cite{Hatt}) and hence ${\mathcal H}(\frak{g})={\mathcal H}(G/\Gamma)$ where ${\mathcal H}^{\ast}(\frak{g})$ is the set of left-invarinat harmonic forms.
By $d(\bigwedge^{3}\frak{g}^{\ast})=0$, for any invariant metric $g$ on $G$, we have:
\[{\mathcal H}^{1}(\frak{g})=\langle dt\rangle , \]
\[{\mathcal H}^{2}(\frak{g})=0,\]
\[{\mathcal H}^{3}(\frak{g})=\langle (\ast_{g} dt)\rangle .\]
Thus any invariant metric on $G/\Gamma$ is formal.
Otherwise we have ${\bf U}_{G}=U_{3}(\mathbb{C})\times \mathbb{C}$ and hence this solvmanifold is different from examples of geometrically formal solvmanifold given in Theorem \ref{MT}.
\end{example}
\section{The extension of Theorem \ref{MT}}
Let $G$ be a simply connected solvable Lie group and $g$ an invariant metric which we construct in Section 5.
Denote $C_{g}$ the group of the isometrical automorphisms of $(G, g)$.
Consider $C_{g}\ltimes G$ and the projection $p :C_{g}\ltimes G\to C_{g}$.
\begin{corollary}\label{Inf}
Suppose $G=\mathbb{R}^{n}\ltimes_{\phi} \mathbb{R}^{m}$ with a semi-simple action $\phi$.
Let $\Gamma\subset C_{g}\ltimes G$ be a torsion-free discrete subgroup such that $G/\Gamma$ is compact.
Suppose $p(\Gamma)$ is finite.
Then the metric $g$ given in the last section is a formal metric on $G/\Gamma$.
\end{corollary}
\begin{proof}
Let $\Delta=\Gamma\cap G$.
Since $\Gamma/\Delta\cong p(\Gamma)$, $\Delta$ is a finite index normal subgroup of $\Gamma$ and $G/\Delta$ is compact and hence $\Delta\subset G$ is a lattice.
Denote ${\mathcal H}(G/\Gamma)$ and ${\mathcal H}(G/\Delta)$ the sets of the harmonic forms on $G/\Gamma$ and $G/\Delta$ for the metric $g$.
Since we have $A^{\ast}(G/\Gamma)=A^{\ast}(G/\Delta)^{\Gamma/\Delta}$, we have
\[{\mathcal H}(G/\Gamma)= {\mathcal H}(G/\Delta)^{\Gamma/\Delta}.\]
By Theorem \ref{MT}, ${\mathcal H}(G/\Delta)$ is closed under the wedge product, so is ${\mathcal H}(G/\Delta)^{\Gamma/\Delta}$.
Hence the corollary follows.
\end{proof}
\begin{remark}\label{BIB}
Not all cocompact discrete subgroup $\Gamma$ satisfies the assumption of the finiteness of $p(\Gamma)$.
See the following example.
\end{remark}
\begin{example}\label{INO}
Let $G=\mathbb{R}\ltimes_{\phi}\mathbb{R}^{3}$ such that $\phi(t)=\left(
\begin{array}{ccc}
e^{t}& 0& 0 \\
0& e^{t}&0 \\
0& 0&e^{-2t}
\end{array}
\right)$.
Then $G$ has no lattice (see \cite[Chapter 7]{Hil}).
Consider the metric $g=e^{-2t}dx^2+e^{-2t}dy^2+e^{4t}dz^2+dt^2$.
Then we have $C_{g}=O(2)\times O(1)$ acting as rotations and reflections on the $(x,y)$-coordinates and reflection on the $z$-coordinate.
$C_{g}\ltimes G$ admits a torsion-free cocompact discrete subgroup $\Gamma$.
Since $G\cap \Gamma$ is not a lattice of $G$, $p(\Gamma)$ is not finite.
In \cite[Chapter 8]{Hil} it is proved that $\Gamma\cong \mathbb{Z}\ltimes_{\phi}\mathbb{Z}^{3}$ and for $t\not=0$ $\phi(t)\in SL_{3}(\mathbb{Z})$ has a pair of complex conjugate eigenvalues (see \cite[Chapter 7]{Hil}).
Hence $\Gamma$ can be a lattice of a Lie group $H=\mathbb{R}\ltimes _{\phi}\mathbb{R}^{3} $ with $\phi(t)=\left(\begin{array}{ccc} e^{t}\cos c t & -e^{t}\sin c t &0\\ e^{t}\sin c t &e^{t}\cos c t&0\\
0&0& e^{-2t}
\end{array}\right)
$, and $G/\Gamma=H/\Gamma$ is geometrically formal by Theorem \ref{MT}.
\end{example}
\section{Thurston's Geometries and infrasolvmanifold}
We say that a compact oriented manifold $M$ admits a geometry $(X, g)$ if $M=X/\Gamma$ where $X$ is a simply connected manifold with a complete Riemaniann metric $g$ and $\Gamma$ is a cocompact discrete subgroup of the group $Isom_{g}(X)$ of isometries.
If $(X,g)$ is a solvable Lie group with an invariant metric $g$, we call it a solvable Lie type geometry.
We consider the following $3$-dimensional solvable Lie type geometries.\\
(3-A) $X=E^{3}= \mathbb{R}^{3}$, $g_{E^{3}}=dx^{2}+dy^{2}+dz^{2}$.\\
(3-B) $X=Nil^{3}=U_{3}(\mathbb{R})=\left\{\left(
\begin{array}{ccc}
1& x& z \\
0& 1&y \\
0& 0&1
\end{array}
\right): x,y,z\in \mathbb{R}\right\}$, $g_{Nil^3}=dx^{2}+dy^{2}+(dz-xdy)^{2}$.\\
(3-C) $X=Sol^{3}=\mathbb{R}\ltimes_{\phi}\mathbb{R}^{2}$ with $\phi(z)=\left(
\begin{array}{cc}
e^t& 0 \\
0& e^{-t}
\end{array}
\right)$, $g_{Sol^{3}}=e^{2z}dx^{2}+e^{-2z}dy^{2}+dz^{2}$.
By the theory of geometry and topology of $3$-dimensional manifolds we have the following theorem (see \cite{PS}).
\begin{theorem}\label{333}
A compact aspherical $3$-dimensional manifold with the virtually solvable fundamental group admits one of the geometries (3-A$\sim$C).
\end{theorem}
We also consider the following $4$-dimensional solvable Lie type geometries (listed in \cite{W}).\\
(4-A) $X=E^{4}=\mathbb{R}^{4}$, $g_{E^{4}}=dx^{2}+dy^{2}+dz^{2}+dt^{2}$.\\
(4-B) $X=Nil^3\times E=U_{3}(\mathbb{R})\times \mathbb{R}$, $g_{Nil^3\times E}=dx^{2}+dy^{2}+(dz-xdy)^{2}+dt^{2}$.\\
(4-C) $X=Nil^{4}=\left\{\left(
\begin{array}{cccc}
1& t& \frac{1}{2}t^{2}&z \\
0& 1&t & y \\
0& 0&1 &x \\
0&0&0&1
\end{array}
\right): x,y,z,t\in \mathbb{R}\right\}$,\\ $g_{Nil^4}=dx^2+(dy-tdz)^2+(dz-tdy+\frac{1}{2}t^2dx)^2+dt^2$\\
(4-D) $X=Sol^{3}\times E$, $g_{Sol^{3}\times E}=e^{2z}dx^{2}+e^{-2z}dy^{2}+dz^{2}+dt^{2}$.\\
(4-E) $X=Sol^{4}_{m,n}=\mathbb{R}\ltimes_{\phi}\mathbb{R}^{3}$ such that $\phi(t)=\left(
\begin{array}{ccc}
e^{at}& 0& 0 \\
0& e^{bt}&0 \\
0& 0&e^{ct}
\end{array}
\right)$, where $e^a, e^{b}, e^{c}$ are distinct roots of $X^{3}-mX^2+nX-1$ for real numbers $a<b<c$ and integers $m<n$,
$g_{Sol^{4}_{m,n}}=e^{-2at}dx^2+e^{-2bt}dy^2+e^{-2ct}dz^2+dt^2$.\\
(4-F) $X=Sol^{4}_{0}=\mathbb{R}\ltimes_{\phi}\mathbb{R}^{3}$ such that $\phi(t)=\left(
\begin{array}{ccc}
e^{t}& 0& 0 \\
0& e^{t}&0 \\
0& 0&e^{-2t}
\end{array}
\right)$, \\
$g_{Sol^{4}_{0}}=e^{-2t}dx^2+e^{-2t}dy^2+e^{4t}dz^2+dt^2$.\\
(4-G) $X=Sol_{1}^{4}=\mathbb{R}\ltimes_{\phi} U_{3}(\mathbb{R})$ such that
$\phi(t)\left(
\begin{array}{ccc}
1& x& z \\
0& 1&y \\
0& 0&1
\end{array}
\right)=\left(
\begin{array}{ccc}
1& e^{t}x& z \\
0& 1&e^{-t}y \\
0& 0&1
\end{array}
\right)$,\\
$g_{Sol^{4}_{1}}=e^{-2t}dx^2+e^{2t}dy^2+ (dz-xdy)^2+dt^2$.
Let $G$ be a simply connected solvable Lie group and $g$ an invariant metric on $G$.
We consider the affine transformation group ${\rm Aut}(G)\ltimes G$ and the projection $p:{\rm Aut}(G)\ltimes G\to{\rm Aut}(G)$.
Let $\Gamma\subset {\rm Aut}(G)\ltimes G$ be a torsion-free discrete subgroup such that $p(\Gamma)$ is contained in a compact subgroup of ${\rm Aut}(G)$ and the quotient $G/\Gamma$ is compact.
We call $G/\Gamma$ an infra-solvmanifold.
If $G$ is nilpotent, $G/\Gamma$ is called an infra-nilmanifold.
Since $\Gamma \subset Isom_{g}(G)$ does not satisfies $\Gamma\subset {\rm Aut}(G)\ltimes G$, a compact manifold with a solvable Lie type geometry is not an infra-solvmanifold in general.
Suppose $Isom_{g}(G)\subset {\rm Aut}(G)\ltimes G$.
Then for an isometry transformation $(\phi, x)\in {\rm Aut}(G)\ltimes G$, $\phi$ is an also isometry transformation.
By this, for the group $C_{g}$ of the isometrical automorphisms of $G$, we have $Isom_{g}(G)=C_{g}\ltimes G$.
Thus in the assumption $Isom_{g}(G)\subset {\rm Aut}(G)\ltimes G$, a compact manifold with a solvable Lie type geometry is an infra-solvmanifold.
It is known that for the Euclidian geometry $(E^{n}, g_{E^{n}}=dx^2_{1}+\dots+ dx_{n}^{2})$ we have $Isom_{g_{E^{n}}}=O(n)\ltimes \mathbb{R}^{n}$ and the geometries (3-A$\sim$C) satisfies $Isom_{g}(G)\subset {\rm Aut}(G)\ltimes G$(see \cite{PS}).
In \cite{Hil}, Hillman studied the structures of $Isom_{g}(G)$ of the geometries (4-A$\sim$H) and proved $Isom_{g}(G)\subset {\rm Aut}(G)\ltimes G$.
In \cite{Hil2}, Hillman proved the following theorem.
\begin{theorem}{\rm (\cite[Theorem 8]{Hil2})} \label{HILL}
A $4$-dimensional infra-solvmanifold is diffeomorphic to a manifold which admits one of the geometries (4-A$\sim$G).
\end{theorem}
\begin{remark}\label{InfB}
By Baues's result in \cite{B}, any compact aspherical manifold with the virtually solvable fundamental group is homotopy equivalent to an infra-solvmanifold $G/\Gamma$.
But for dimension$\ge 4$, there may exist a compact aspherical manifold with virtually solvable fundamental group which is not diffeomorphic to an infra-solvmanifold.
\end{remark}
\section{Geometrical formality of $3$-manifolds}
\begin{theorem}\label{33}
Let $M$ be a compact oriented aspherical $3$-manifold with the virtually solvable fundamental group.
If $M$ is a torus or not a nilmanifold, then $M$ is geometrically formal.
\end{theorem}
\begin{proof}
By Theorem \ref{333}, it is sufficient to consider the geometries (3-A$\sim$C).
In the case (3-A), by Corollary \ref{Inf} and the first Bieberbach theorem $g_{E^{3}}$ is a formal metric on $G/\Gamma$.
In the case (3-C), it is known that $C_{g}$ is isomorphic to the finite dihedral group $D(8)$ (see \cite{PS}) and hence by Corollary \ref{Inf} $g_{Sol^{3}}$ is a formal metric on $G/\Gamma$.
Suppose $(G,g)$ is in the case (3-B).
Then $C_{g}$ has two components and the identity component of $C_{g}$ is isomorphic to a circle $S^{1}$.
Let $\Delta=\Gamma\cap G$.
By Generalized Bieberbach's theorem (see \cite{Au}), $\Delta$ is a finite index normal subgroup of $\Gamma$.
Consider the projection $p: C_{g}\ltimes G \to C_{g}$.
If $p(\Gamma)$ is trivial, then $\Gamma\subset G$ is a lattice and $G/\Gamma$ is a non-toral nilmanifold and hence not formal (see \cite{H}).
Suppose $p(\Gamma) $ is non-trivial.
By Nomizu's theorem (\cite{Nom}) we have
\[H^{\ast}(G/\Delta, \mathbb{R})\cong H^{\ast}(\frak{g})\]
where $\frak{g}$ is the Lie algebra of $G$.
By this we have
\[H^{\ast}(G/\Gamma, \mathbb{R})\cong H^{\ast}(G/\Delta, \mathbb{R})^{\Gamma/\Delta}\cong H^{\ast}(\frak{g})^{\Gamma/\Delta}.
\]
In \cite[Lemma 13.1]{BG}, it is shown that a non-trivial semisimple automorphism of a nilpotent Lie algebra $\frak{g}$ acts non-trivially on $H^{1}(\frak{g})$.
Since $\Gamma/\Delta \cong p(\Gamma)$ is a nontrivial finite group,
\[H^{1}(\frak{g})^{\Gamma/\Delta}\not= H^{1}(\frak{g}).
\]
Since $\dim H^{1}(\frak{g})=2$, $\dim H^{1}(\frak{g})^{\Gamma/\Delta}=0$ or $1$.
If $ \dim H^{1}(\frak{g})^{\Gamma/\Delta}=0$, then $G/\Gamma$ is a rational homology sphere and any metric on $G/\Gamma$ is formal.
Suppose $\dim H^{1}(\frak{g})^{\Gamma/\Delta}=1$.
Then $b_{i}=1$ for any $1\le i\le 3$.
For any $1\le i\le 3$, we have
\[H^{\ast}(G/\Gamma,\mathbb{R})\cong H^{\ast}(\frak{g})^{\Gamma/\Delta}=\bigoplus_{i=1}^{3}\langle [\alpha_{i}]\rangle\]
for non-zero cohomology classes $[\alpha_{i}]\in H^{i}(\frak{g})$.
We can choose invariant harmonic forms $\alpha_{i}$, $i=1,2,3$ for the invariant metric $g$.
Then we have ${\mathcal H}(G/\Gamma)=\bigoplus_{i=1}^{3}\langle \alpha_{i}\rangle$,
Since all elements of $\bigwedge ^{3}\frak{g}^{\ast}$ are harmonic, $\alpha_{1}\wedge\alpha_{2}$ is harmonic.
For $i<j$ with $(i,j)\not=(1,2)$, we have $\alpha_{i}\wedge\alpha_{j}=0$.
Thus $g$ is a formal metric on $G/\Gamma$.
This completes the proof of the theorem.
\end{proof}
\begin{remark}
There exists a closed $3$-dimensional infra-nilmanifold which is not a nilmanifold.
By this theorem such a manifold is geometrically formal.
\begin{example}
Consider $\Gamma=\mathbb{Z}\ltimes_{\phi}\mathbb{Z}^{2}$ such that $\phi(t)=\left(
\begin{array}{cc}
(-1)^t& (-1)^t t \\
0& (-1) ^t
\end{array}
\right)$.
Then we can embed $\Gamma$ in $Isom_{g_{Nil^{3}}}( Nil^{3})$ (see \cite{K}).
By the direct computation of the lower central series, $\Gamma$ is non-nilpotent and hence $Nil^{3}/\Gamma$ is not a nilmanifold.
\end{example}
\end{remark}
\section{Aspherical manifolds with the virtually solvable fundamental groups}
\begin{theorem}\label{44}
Let $M$ be an oriented $4$-dimensional infra-solvmanifold.
If $M$ is a torus or not a nilmanifold, then $M$ is geometrically formal.
\end{theorem}
\begin{proof}
In the case (4-A), by Corollary \ref{Inf} and the first Bieberbach theorem $g_{E^{4}}$ is a formal metric on $G/\Gamma$.
In the case (4-D) (resp (4-E)), $C_{g}$ is isomorphic to the finite group $D(8)\times (\mathbb{Z}/2\mathbb{Z})$ (resp. $(\mathbb{Z}/2\mathbb{Z})^{3}$) (see \cite[Chapter 7]{Hil}) and hence $g_{Sol^{3}\times E}$ (resp. $g_{Sol^{4}_{m,n}}$) is a formal metric on $G/\Gamma$ by Corollary \ref{Inf}.
As we showed in Example \ref{INO}, in the case (4-F) $G/\Gamma$ is geometrically formal.
In the case (4-B), the group of all the orientation preserving isomorphisms is $Isom_{g_{Nil^{3}} }Nil^{3}\times \mathbb{R}$ (see \cite{Ue}, \cite{W}, or \cite{W2}).
Thus as the proof of Theorem \ref{33} for the case (3-B), if $G/\Gamma$ is a nilmanifold then $G/\Gamma$ is not formal, and if $G/\Gamma$ is an infra-nilmanifold but not a nilmanifold then $g_{Nil^{3}\times \mathbb{R}}$ is formal.
In the case (4-C), the group of all the orientation preserving isomorphisms is $Nil^{4}$ itself (see \cite{Ue}, \cite{W}, or \cite{W2}).
Thus oriented $Nil^{4}$ manifolds are only nilmanifolds and so all oriented $Nil^{4}$ manifolds are not formal.
In the case (4-G) we have $Isom_{g_{Sol^{4}_{1}}}(Sol^{4}_{1})\cong D(4)\ltimes Sol^{4}_{1}$.
For any cocompact discrete subgroup $\Gamma\subset Isom_{g_{Sol^{4}_{1}}}(Sol^{4}_{1})$, since for the projection $p:{\rm Aut} G\ltimes G \to {\rm Aut} (G)$, $p(\Gamma)\subset D(4)$ is finite, we have a subgroup $\Delta\subset \Gamma$ which is a lattice of $ Sol^{4}_{1}$ and we have ${\mathcal H} (Sol^{4}_{1}/\Gamma)= {\mathcal H} (Sol^{4}_{1}/\Delta)^{\Gamma/\Delta}$.
In Example \ref{EXC}, we showed that the metric $g_{Sol^{4}_{1}}$ on the solvmanifold $Sol^{4}_{1}/\Delta$ is formal.
Thus the metric $g_{Sol^{4}_{1}}$ on every $Sol^{4}_{1}$ manifold is formal.
Hence the theorem follows.
\end{proof}
Finally we prove:
\begin{theorem}
Let $M$ be a compact oriented aspherical manifold of dimension less than or equal to $4$ with the virtually solvable fundamental group.
Then $M$ is geometrically formal if and only if $M$ is diffeomorphic to a torus or an infra-solvmanifold which is not a nilmanifold.
\end{theorem}
\begin{proof}
It is sufficient to show that $M$ is an infra-solvmanifold if $M$ is geometrically formal.
If $\dim M\le 2$, it is obvious.
If $\dim M=3$, it follows from Theorem \ref{333}.
We consider $\dim M=4$.
As Remark \ref{InfB}, $M$ is homotopy equivalent to an infra-solvmanifold $G/\Gamma$.
It is known that the Euler characteristic $\chi(G/\Gamma)$ of an infra-solvamanifold is $0$ (see \cite[Cahpter 8]{Hil}).
Since $G/\Gamma$ is an oriented $4$-manifold, $\chi(G/\Gamma)=0$ implies $b_{1}(G/\Gamma)\not=0$.
Thus we have $b_{1}(M)\not=0$.
If $M$ is geometrically formal, we have a submersion $M\to T^{b_{1}(M)}$ (see \cite[Theorem 7]{Kot}) and hence $M$ is a fiber bundle over a torus $T^{b_{1}(M)}$.
Now we suppose that $M$ is a compact oriented aspherical manifold of dimension $4$ with the virtually solvable fundamental group.
By the exact sequence of homotopy groups associated by the fiber bundle, the fiber of $M\to T^{b_{1}(M)}$ is a compact aspherical manifold of dimension less than or equal $3$ with the virtually solvable fundamental group, and hence it is an infra-solvmanifold.
Thus $M$ is a fiber bundle whose fiber is an infra-solvmanifold and base space is a torus.
By \cite[Theorem 7]{Hil2}, $M$ is diffeomorphic to an infra-solvmanifold with the fundamental group $\pi_{1}(M)$.
\end{proof}
\ \\
{\bf Acknowledgements.}
The author would like to express many thanks to the referee for his careful reading of the earlier version of manuscript with several important remarks, which lead to many improvements in the revised version.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Dirt Bag
Meek Mill Has Been Released From Prison on Bail
Hannah Gold
Meek Mill was granted bail on Tuesday following the Pennsylvania Supreme Court's order that the rapper be released from prison.
Last week, Philadelphia's district attorney's office argued that Mill's 2008 conviction on drug and firearm charges should be overturned, and questioned the credibility of his arresting officer.
Mill said in a statement on Tuesday:
"Although I'm blessed to have the resources to fight this unjust situation, I understand that many people of color across the country don't have that luxury and I plan to use my platform to shine a light on those issues. In the meantime, I plan to work closely with my legal team to overturn this unwarranted conviction and look forward to reuniting with my family and resuming my music career."
Michael Rubin, co-owner of the Philadelphia 76ers and one of Meek's longtime friends, told Page Six that the rapper is attending the team's game on Tuesday night. Rubin has admirable follow-through.
[TMZ]
Kanye West has, by certain metrics, been getting a lot of shit done lately. He's tweeting, he's tweeting about stuff he's doing or will be doing, and he's reportedly parting ways with much of his management team, including several lawyers and his manager, Scooter Braun.
Also, just wanted to make sure you all saw this:
Come to think of it, here's every guy I've ever talked to at a party:
Counterpoint: Damian Lewis hardly looks like Rob Ford from the chin down. [Vulture]
Kourtney Kardashian is now an advocate for stricter regulation of makeup products. She also released new makeup products today? [The Cut]
Former Miss America (crowned 2005) and current OBGYN Deirdre Downs married her girlfriend, attorney Abbott Jones. [AP]
Ed Westwick's girlfriend is still a fan of Ed Westwick. [NYDN]
contributing writer, nights
BartlebyThesScrivener
Bret Easton Ellis is trash.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,219
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The history of gender fluidity of the Japanese
By Kyota Ko|2020-06-17T14:18:33+00:00December 29th, 2019|Categories: Japanese history, Japanese people, Japanese thought|Tags: gender, Japanese culture, Japanese history, Japanese people, Japanese philosophy, sex|
Japanese boy bands cast young boys with feminine facial features and hairless skin, and girls are crazy about them. Legendary Kabuki actors throughout its 400-year history are all known for pulling off the sexiest female roles, and grabbed the heart [...]
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 5,635
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title: Deployment methods
authors: aortega
wiki_title: Deployment methods
wiki_revision_count: 1
wiki_last_updated: 2014-06-25
---
# Deployment methods
## Packstack Use Case
Packstack is primarily a proof-of-concept (PoC) tool intended to be used in two different scenarios: All in one deployments with and without additional compute nodes. Due to its PoC nature, Packstack makes a number of simplifications to reduce the number of potential configuration parameters, which streamlines the installation process.
Packstack supports the following components: Nova, Keystone, Glance, Cinder, Ceilometer, Horizon, Swift, Heat, and Neutron. On top of it, Packstack will also support Sahara and Marconi.
The number of supported network set-ups have been reduced to two options: Neutron with ML2 and VxLAN, and Nova-network.
Packstack supports two AMQP brokers: RabbitMQ and Qpid. However it assumes the same message broker will be used for all services.
Finally, Packstack supports the deployment of simple, non-high availability configurations of OpenStack components.
## Foreman Use Case
TBC
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,879
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\section{Introduction}
Theoretical predictions for future experiments are necessary for
determining the kinematic regions of validity of the Standard Model (SM). Such
predictions depend on constants which must be determined from past
experiments since these quantities are otherwise uncalculable, either
because no theory exists which can determine them from more
fundamental parameters, or because the solutions of the current theory
are insufficient to determine them from the SM parameters.
Quantum Chromodynamics (QCD), the theory of the strong interaction and
one of the theories that make up the SM, is required in the description of
processes involving hadrons. The best tool for solving QCD to perform such descriptions is
perturbation theory. However, perturbative QCD (pQCD) can only
describe the high energy components of the cross section, while a
process will contain low energy components if a hadron is in the
initial state or is observed in the final state. Fortunately, from the
Factorization Theorem, the low and high energy scale components of
such processes can be separated. The low energy components are
universal and so can be used to make predictions. Since they cannot
yet be reliably calculated from QCD, they must be extracted from
experimental data.
The pQCD description of data involving the inclusive production of
hadrons requires fragmentation functions (FFs), which form the low
energy components of such processes and describe the inclusive
emission of a hadron from a quark or gluon (parton) for every momentum
fraction. One reason FFs are important is that
model independent predictions of LHC cross sections in which
a hadron is detected in the final state depend on them.
There are many theoretical obstacles to the extraction of
FFs from data: The Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) \cite{dglap}
evolution equation for FFs is only known
to next-to-leading order (NLO), and is furthermore unreliable
at small and possibly even intermediate momentum fractions of the emitted
parton, where the only reliable determination of FFs is via the
Modified Leading Logarithm Approximation (MLLA) \cite{Albino:2004yg}. Despite these
problems, FFs at intermediate to large momentum fractions obtained
from fits to data now yield compatible results with other data sets \cite{Kniehl:2000fe}.
Much precise data from $e^+ e^-$ colliders now exists for the
production of the three lightest charged hadrons, which are the pion
($\pi^{\pm}$), kaon ($K^{\pm}$) and proton ($p/\overline{p}$). In much
of this data, the observed hadron is identified as one of these
particles, and the emitting parton is identified as either a gluon,
light ($u$, $d$ and $s$) quark, $c$ quark or $b$ quark, which allowed
for a precise determination of the corresponding individual FFs in
Refs.\ \cite{Kniehl:2000fe,Kretzer:2000yf} \cite{footnote1}.
However, the individual light quark FFs
could only be extracted by making reasonable physical assumptions.
Since this analysis, the OPAL Collaboration has presented light
flavour separated measurements on light charged hadron production
\cite{Abbiendi:1999ry} for the $e^+ e^-$
centre-of-mass (CM) energy $\sqrt{s}=M_Z$, allowing for the first time the
extraction of flavour dependent FFs of light quarks. In this OPAL analysis, high energy
mesons ($\pi^{\pm}$, $K^{\pm}$ and $K_S^0$) and baryons ($p/\overline{p}$ and $\Lambda$)
were identified in the large $Z$ boson decay data sample and used as tagging products. In addition,
high momentum $e^{\pm}$, $\mu^{\pm}$ and $D^{* \pm}$ particles and identified
bottom events were used to measure heavy flavour backgrounds in the above meson and
baryon sample. As suggested in Ref.\ \cite{Field:1976ve} and precisely studied in
a recent analysis by the SLD Collaboration \cite{Abe:1998zs}, these
high energy particles give information about the original quark. For more details
see the OPAL work \cite{Abbiendi:1999ry}, where it is explained how the Collaboration
measured the probability
$\eta_a^h(x_p,s)$ for a quark flavour $a$ to develop into a jet containing the particle $h$ with
a momentum fraction $x$ larger than $x_p=2p_h/\sqrt{s}$.
Since the valence structure of the proton is $uud$, knowing the difference between the
individual light flavour FFs, in particular for
$u$ and $d$ quarks into $K^{\pm}$, is very much needed for predicting the
inclusive cross sections for the productions of these hadrons in
collisions involving protons, such as $ep$, $pp$ and $p\overline{p}$
collisions. For example, results from the inclusive production of hadrons in
$pp$ collisions provide the
baseline to which one compares heavy-ion collision results in order to
determine the properties of the hot quark-gluon plasma \cite{d'Enterria:2004rp}.
Tests presented in Ref.\ \cite{Kniehl:2000hk} of the KKP FFs in the process
$p +\overline{p}\rightarrow h^{\pm}+X$, where $h^{\pm}$ are light
charged hadrons, were generally successful, as was a recent check of
the pion FFs by comparison to $p+p\rightarrow \pi^0 +X$ data
(taking $\pi^0=\frac{1}{2}\left(\pi^+ +\pi^-\right)$) from the
PHENIX Collaboration \cite{Adler:2003pb} at RHIC.
However, it is likely that the inaccuracy on the information on the $u$, $d$ and $s$ quark FFs
canceled out due to the superimposition of the
hadrons in $h^{\pm}$.
In this paper, we update the analysis of Ref.\ \cite{Kniehl:2000fe} by
including the data of Ref.\ \cite{Abbiendi:1999ry} in the fit to obtain for the first
time a phenomenological determination of the individual light quark
FFs for each light charged hadron species. Since we do not impose those
physical assumptions on the light quark FFs that were used in Ref.\
\cite{Kniehl:2000fe} in our calculation of the cross sections used for
the fit, the other FFs extracted in this fit are also more reliable.
In Section \ref{formalism}, we summarize the basic theoretical tools used in our calculations for the fit.
In Section \ref{method} we justify specific choices for our fit such as the data used and
the FF parameterization. Our results are then presented in Section \ref{results},
and finally in Section \ref{conclusions} we present our conclusions. The details
of the longitudinal cross section calculation are given in Appendix \ref{app}.
\section{Formalism \label{formalism}}
The optimal way to determine FFs is to fit them to measurements of the processes
$e^+ +e^- \rightarrow (\gamma,Z)\rightarrow a +\overline{a}\rightarrow h+X$,
where $a$ is the tagged quark,
$h$ is a detected hadron and $X$ is the remaining unobserved part of
the final state. In a typical experiment the hadron is only
detected if its species $h$ belongs to a specified set of hadron species $S_H$ and the species of
the tagged quark $a$ belongs to a set of flavours $S_A$. Writing the CM momentum of the
observed hadron as $x\sqrt{s}/2$, the data for
such a process are typically presented as
\begin{equation} F^{S_H}_{S_A}(x,s)
=\frac{\sum_{a\in S_A,\ h\in S_H}\frac{d\sigma_a^h}{dx}(x,s)}
{\sum_{a\in S_A}\sigma_a(s)}.
\label{genformofdata}
\end{equation}
The total cross section $\sigma_a$ is given to NLO by
\begin{equation}
\sigma_a(s)=\sigma_0(s) N_c Q_a(s) \left(1+2a_s(s)\right),
\end{equation}
where $\sigma_0=4\pi\alpha^2/(3s)$ is the leading order (LO) cross section for the
process $e^+ +e^- \rightarrow \gamma \rightarrow
\mu^+ +\mu^-$, $N_c$ is the number of colours and $a_s(\mu^2)
=\alpha_s(\mu)/(2\pi)$. $Q_a(s)$ is the effective electroweak charge of quark $a$ \cite{Eidelman:2004wy}.
From the Factorization Theorem, the higher twist component of the
differential cross section in Eq.\ (\ref{genformofdata}) is of $O(1/\sqrt{s})$ or less
and may and will be neglected in this paper, while the leading twist component is
obtained by convoluting the corresponding high energy partonic cross
sections with the FFs $D_a^h(y,M_f^2)$, where $y$
is the fraction of the momentum of parton $a$ taken away by the produced hadron $h$ and $M_f$
is the factorization scale. This may be written concisely by taking $y=x/z$, in which case
\begin{equation}
\begin{split}
&\frac{d\sigma^h_a}{dx}(x,s)=
\int_x^1 \frac{dz}{z}\Bigg[
\frac{d\sigma^a_{a,{\rm NS}}}{dz}\left(z,s,M_f^2\right)
D_a^h\left(\frac{x}{z},M_f^2\right)\\
&+\sum_b \frac{d\sigma^b_{a,{\rm PS}}}{dz}\left(z,s,M_f^2\right)
D_b^h\left(\frac{x}{z},M_f^2\right)
+\frac{d\sigma^g_a}{dz}\left(z,s,M_f^2\right)
D_g^h\left(\frac{x}{z},M_f^2\right)
\Bigg],
\end{split}
\label{XSfromFFs}
\end{equation}
where, for the emission of a hadron $h$ from a quark $a$, the non-singlet
partonic cross section $d\sigma^a_{a,{\rm NS}}/dz$ contains only and all those contributions from
diagrams in which the quark line connected to the electroweak vertex and the quark line
emitting the hadron $h$ are the same, while the pure singlet
partonic cross section $d\sigma^a_{b,{\rm PS}}/dz$ contains all other contributions.
Since the $Z$ boson only splits into a quark $a$ and its antiquark $\overline{a}$, each
partonic cross section is proportional to $Q_a$, and thus may be written
\cite{footnote2}
\begin{equation}
\begin{split}
\frac{d\sigma^a_{a,{\rm NS}}}{dz}\left(z,s,M_f^2\right)
=&\sigma_0 (s) Q_a(s)C_{\rm NS}\left(z,a_s(s),\ln \frac{M_f^2}{s}\right),\\
\frac{d\sigma^b_{a,{\rm PS}}}{dz}\left(z,s,M_f^2\right)
=&\sigma_0 (s) \frac{Q_a(s)}{n_f}
C_{\rm PS}\left(z,a_s(s),\ln \frac{M_f^2}{s}\right)\ {\rm and}\\
\frac{d\sigma^g_a}{dz}\left(z,s,M_f^2\right)=&2\sigma_0 (s)
Q_a(s)C_g\left(z,a_s(s),\ln \frac{M_f^2}{s}\right),
\end{split}
\end{equation}
where the $C_i$ are the perturbatively calculable coefficient functions.
$n_f$ is the number of active quark flavours.
For the choice $M_f^2=s$, the $C_i(z,a_s(s),0)=C_i(z,a_s(s))$ for the unpolarized (i.e.\
summed over transverse and longitudinal components)
cross section are given to NLO by \cite{Altarelli:1979kv}
\begin{eqnarray}
C_{\rm NS}(z,a_s)&=& \delta(1-z)+a_s C_F
\Bigg[\left(\frac{2\pi^2}{3}-\frac{9}{2}\right)\delta(1-z)
-\frac{3}{2}\left[\frac{1}{1-z}\right]_+ \nonumber \\
&& +(1+z^2)\left[\frac{\ln (1-z)}{1-z}\right]_+
+1+2\frac{1+z^2}{1-z}\ln z +\frac{3}{2}(1-z)\Bigg],\\
C_{\rm PS}(z,a_s)&=&O(a_s^2)\ \ \ {\rm and}\\
C_g(z,a_s)&=&a_s C_F \left[
\frac{1+(1-z)^2}{z}\left(\ln (1-z)+2\ln z\right)-2\frac{1-z}{z}\right].
\end{eqnarray}
Note that the pure singlet contribution only enters at NNLO.
In contrast, in the longitudinal cross section
\begin{equation} F^{S_H}_{L,S_A}(x,s)
=\frac{\sum_{a\in S_A,\ h\in S_H}\frac{d\sigma_{L,a}^h}{dx}(x,s)}
{\sum_{a\in S_A}\sigma_a(s)},
\label{genformofLdata}
\end{equation}
there is a contribution from the pure singlet sector at NLO, while the
gluon FF enters at LO (see Appendix \ref{app}).
It is clear that we only apply electroweak theory to LO. We can therefore easily see that
Eq.\ (\ref{XSfromFFs}) for the cross section when quark $a$ is tagged
is a physical observable, since it can be obtained
by differentiating the untagged cross section (Eq.\ (\ref{XSfromFFs}) with $a$ summed over all
flavours) with respect to $\ln Q_a$, where $Q_a$
is the effective electroweak charge of quark $a$ discussed above.
Therefore the tagged cross section is formally
independent of the factorization and renormalization scales and
schemes, as it must be to qualify as an observable.
For $M_f^2\neq s$, the coefficient functions will contain terms of the form
$a_s^n(s) \ln^p \frac{M_f^2}{s}$, where $p=n,n-1,...$, which will spoil the convergence
of the series unless $M_f^2=O(s)$. Thus, in order to be able to describe data over a large
range in $s$, the dependence of the FFs on $M_f^2$ must be known. Fortunately this can
be calculated using the DGLAP equation,
\begin{equation}
\frac{d}{d \ln M_f^2} D_a^h(z,M_f^2)=\sum_{b=g,q}
\int_z^1 \frac{dy}{y}P_{ab}\left(\frac{z}{y},a_s(M_f^2)\right)
D_b^h(y,M_f^2),
\label{DGLAPeq}
\end{equation}
where the $a\rightarrow b$ splitting functions $P_{ab}$
are perturbatively calculable, and are known to NLO.
Therefore, in the calculation of cross sections
it is sufficient to know the FFs at just one factorization
scale $M_f=M_0$.
The DGLAP equation is however not valid when $z$ is small, since due to soft gluon emission
the $P_{ag}(z,a_s)$ contain terms which behave in the limit $z\rightarrow 0$ like
$(a_s^n/z) \ln^{2n-1-m}z$, where
$m=1,...,2n-1$ labels the class of terms (finite terms which behave like $a_s^n$ when
integrated over the range $0<z<1$
are classified as $m=2n$), and are therefore unreliable in this limit. This implies that
the cross section cannot be reliably calculated at small $x$, and the FFs $D_a^h(z,M_0^2)$
cannot be fitted at small $z$. In this case a description
of the data requires an alternative approximation
such as the MLLA, which is beyond the scope of this paper.
Dependence on the factorization scale is introduced
in the usual way. Specifically, the FFs are
evolved to $M_f^2=k_f s$, where $k_f$ is a constant which is taken
to be equal to 1 for the main fit, and 1/4 and 4 in two further fits to determine
the theoretical errors on fitted parameters. We counter-balance this $k_f$ dependence at NLO
using the result (where the
$x$ dependence, integrals, discrete labels, sums and charges have been removed for brevity)
\begin{equation}
C(a_s(s),\ln k_f)=C(a_s(s),0)-\ln k_f C (a_s(s),0) P (a_s(s)).
\label{coeffwithkfnotone}
\end{equation}
Dependence on the renormalization scale $\mu$ is introduced by choosing $\mu^2=k s$, where $k$
is a constant chosen to obey $k=k_f$. At NLO, this amounts to replacing $a_s(s)$ in the
coefficient functions with $a_s(k s)$.
The fastest and most accurate way of calculating a cross section is
in Mellin space, defined by the transformation
\begin{equation}
F^{S_H}_{S_A}(n,s)=\int_0^1 dx x^{n-1} F^{S_H}_{S_A}(x,s),
\end{equation}
since convolutions such as that in Eq.\ (\ref{XSfromFFs}) become simple products.
In particular, Eq.\ (\ref{DGLAPeq}) becomes
\begin{equation}
\frac{d}{d \ln M_f^2}D_a^h(n,M_f^2)=\sum_{b=g,q} P_{ab}(n,a_s(M_f^2))
D_b^h(n,M_f^2),
\end{equation}
which can be solved analytically
order by order. The cross section in $x$ space can then be obtained numerically
via the inverse Mellin transform,
\begin{equation}
F^{S_H}_{S_A}(x,s)=\frac{1}{2\pi i}\int_C dn\ x^{-n} F^{S_H}_{S_A}(n,s),
\label{invmeltran}
\end{equation}
where $C$ is a contour in Mellin space from ${\rm Im} (n)=-\infty$ to ${\rm Im} (n)=\infty$, which
passes to the right of all poles.
Predictions for data averaged over an $x$-bin in the range $x_l<x<x_h$ are calculated from the formula
\begin{equation}
\langle F^{S_H}_{S_A}\rangle (x_l,x_h,s)=\frac{1}{x_h-x_l}\int_{x_l}^{x_h}dx
F^{S_H}_{S_A}(x,s).
\end{equation}
This integral over $x$ can be done analytically in
Eq.\ (\ref{invmeltran}),
\begin{equation}
\langle F^{S_H}_{S_A}\rangle (x_l,x_h,s)=\frac{1}{x_h-x_l}
\frac{1}{2\pi i}\int_C dn \frac{x_h^{1-n} -x_l^{1-n}}{1-n} F^{S_H}_{S_A}(n,s),
\label{xintinvmelltrans}
\end{equation}
giving a further advantage for
working in Mellin space that no extra numerical integration is required to obtain
$x$-bin averaged cross sections.
The light flavour separated data in Ref.\ \cite{Abbiendi:1999ry} may be interpreted as the probability
for a tagged quark flavour $a$ to inclusively emit a hadron of type $h$ with momentum greater than
$x_p \sqrt{s}/2$, in which case the corresponding
theoretical result for such data may be calculated from the formula
\begin{equation}
\eta^h_a(x_p,s)=\int_{x_p}^{1}dx F^{\{h\}}_{\{a\}}(x,s)=
(1-x_p)\langle F^{\{h\}}_{\{a\}}\rangle (x_p,1,s),
\label{calcofetas}
\end{equation}
and we note that for this expression the $\eta^h_a(x_p,s)$ constrain the
FFs at large momentum fraction even more than the $F^{S_H}_{S_A}(x,s)$.
However, the experimental definition of the $\eta^h_a$ is a little more subtle.
For a given number $N_a$ of $e^+e^-$ annihilation events in which a quark $a$ is tagged,
the number $N_{a\rightarrow h}$ of times that
an {\it event hemisphere}, defined to be the two regions separated
by the plane perpendicular to the thrust axis for each event,
contains a particle $h$ with $x>x_p$ is determined. Therefore,
at LO, where $a$ and $\overline{a}$ are never in the same hemisphere,
$\eta_a^h (x_p,s)$ is given by the integral over $D_a^h(x,s)$ in the range $x_p<x<1$,
and this result is consistent with Eq.\ (\ref{calcofetas}). At NLO
the quark $a$ can emit a gluon which in turn emits the hadron $h$ according to the
gluon FF $D_g^h$ (see Eq.\ (\ref{XSfromFFs})).
In the measurement of $\eta^h_a(x_p,s)$, processes in which the gluon is
in the opposite hemisphere from the quark $a$ that emitted it are excluded.
However, such processes contribute to Eq.\ (\ref{calcofetas}). Fortunately,
such events in which the gluon is emitted with a large angle with respect to the quark $a$ are very
rare and should contribute very little both
to Eq.\ (\ref{calcofetas}) and the measured $\eta^h_a$.
\section{Method \label{method}}
In this Section we describe our method for obtaining FFs from data.
As in Ref.\ \cite{Kniehl:2000fe}, where a detailed discussion of all available
data sets is given which
will not be repeated here, we use identified hadron data with and without flavour
separation from DELPHI \cite{Abreu:1998vq} and SLD \cite{Abe:1998zs},
and identified hadron data without flavour separation from
ALEPH \cite{Buskulic:1994ft} and TPC \cite{Aihara:1988fc}.
In addition, we use identified hadron data with flavour separation
from TPC \cite{Aihara:1986mv}, which was used in Ref.\
\cite{Kretzer:2000yf} but not in Ref.\ \cite{Kniehl:2000fe}.
Furthermore, for the first time we also include the light
flavour separated measurements of quark tagging probabilities from the
OPAL Collaboration \cite{Abbiendi:1999ry}.
However, we exclude unidentified hadron data since, although such data is accurate, it is
typically contaminated with charged particles other than the
$\pi^{\pm}$, $K^{\pm}$ and $p/\overline{p}$. Such data was used in
Ref.\ \cite{Kniehl:2000fe}, leading to consistent results. However, since in
this analysis we aim for more reliable FFs, we use only hadron species
separated measurements. We also exclude data for which $x_l<0.1$, since
the prediction for the cross section is unreliable in this region as a result of
the logarithms from soft gluon emission mentioned in Section \ref{formalism}.
After fitting, we then compare
cross sections calculated from our FFs and $\alpha_s(M_Z)$ with
the unidentified hadron data with flavour separation
from TPC \cite{Aihara:1986mv}, with and without flavour separation
from ALEPH \cite{Buskulic:1995aw,PadillaAranda:1995wi}, DELPHI \cite{Abreu:1998vq} and
OPAL \cite{Ackerstaff:1998hz}, without flavour separation from
SLD \cite{Abe:1998zs}, the unidentified
hadron gluon-tagged three-jet data from ALEPH \cite{Barate:1998cp} and OPAL \cite{Abbiendi:1999pi}
and the identified hadron tagging probabilities
with heavy quark flavour separation from OPAL \cite{Abbiendi:1999ry}. The latter
data set is not included in the fit since the heavy quark FFs are much better
constrained by the larger quantity and quality of heavy quark-tagged data from DELPHI, SLD and TPC.
In all data, correlation effects between data points are not yet known, and therefore
in the calculation of
the covariance matrix for the $\chi^2$ to be minimized, we fix the off-diagonal
elements to zero, i.e.\ we assume that the data points are uncorrelated and that the
error on each one is given by its statistical and systematic errors added in quadrature.
Note that this common deficiency in published data limits the
reliability of results obtained from analyzing them.
All theoretical quantities are calculated to NLO in the $\overline{\rm MS}$ scheme.
For our main fit, we evolve the FFs from $M_f=M_0$ to
$M_f=\sqrt{s}$, and vary the scales as described in Section
\ref{formalism} to determine the theoretical errors on $\alpha_s(M_Z)$.
We take $M_0=\sqrt{2}$ GeV for $a=u,d,s,g$. As $M_f$ is increased
from $M_0$ to $\sqrt{s}$, the number of
flavours used in the evolution of the FFs and the strong coupling is first set to $n_f=3$ and only
the light quark and gluon FFs are non zero until $M_f=m(\eta_c)=2.9788$ GeV, where the charm FF is
set equal to its initial distribution and included in the set of FFs to be evolved, and the number
of flavours is taken to be $n_f=4$. The bottom FF is treated in the same way,
being introduced when $M_f=m(\Upsilon)=9.46037$ GeV.
Both flavour thresholds are respectively twice the pole masses of these two heavy quarks,
and therefore perturbative matching conditions are required at NLO. Rather than
implementing this matching explicitly, we define our heavy quark FFs to be the complete ones,
not just the intrinsic FFs, which means the matching term, dependent
on the gluon FF, is absorbed into them.
Our FFs are summed over hadrons which are of the same species but
opposite charges, and averaged over quark and antiquark. We do not
consider cross sections which depend on the difference between
quark and antiquark FFs summed over any given set of emitted hadrons, although it must
be noted that this difference is zero when this set contains a sum over charges, by
charge conjugation invariance.
Since we use accurate data at 29 GeV and $91.2$ GeV, we are in a
position to extract the parameter $\alpha_s(M_Z)$, the quantity
which determines the running of $a_s(\mu^2)$. We therefore
free this parameter in our fit.
The matching on $a_s(\mu^2)$ is implemented by determining
$\Lambda_{\overline{\rm MS}}^{(5)}$ from $\alpha_s(M_Z)$, and then using it to determine
$\Lambda_{\overline{\rm MS}}^{(4)}$ and $\Lambda_{\overline{\rm MS}}^{(3)}$ from the NLO relations given in
Ref.\ \cite{Marciano:1983pj} (these were checked using the results of
Ref.\ \cite{Chetyrkin:1997sg}).
We choose the usual parameterization
\begin{equation}
D_a^h(x,M_0^2)=N x^{\alpha} (1-x)^{\beta}
\label{param}
\end{equation}
for each of our FFs. In Mellin space, the FFs are then proportional to
$\Gamma(n+\alpha)/\Gamma(n+\alpha+\beta+1)\simeq 1/(n+\alpha)$ for $n\simeq -\alpha$,
and this behaviour persists
even after evolution and convolution with coefficient functions.
For such behaviour, the numerical evaluation of Eq.\ (\ref{xintinvmelltrans}) is
best performed with the integration variable $0\leq t\leq 1$ and contour defined through
\begin{equation}
n=c+\frac{3}{3+\ln\frac{1}{x_l}}+\frac{1}{1-x_l}+\frac{2(1-i)\ln t}{\ln\frac{1}{x_l}},
\label{nfromt}
\end{equation}
where the real constant $c$ is chosen such that the contour lies to the right
of all poles,
since as $t \rightarrow 1$ the integrand in the integral over $t$ becomes a finite
constant, while as $t \rightarrow 0$ the integrand vanishes like
$\exp ((-2(1-i)\ln t / \ln (1/x_l))\ln x)$. As a result of the
second and third term in Eq.\ (\ref{nfromt}),
the intersection of the contour with the ${\rm Im}(n=0)$ line goes from $n=1$
to $n=\infty$ as $x$ goes from 0 to 1. This approximately follows the saddle point \cite{RDBall}
of the integrand,
thus ensuring the contour is close to the contour of steepest descent, which gives the fastest
convergence of the integral.
In Ref.\ \cite{Kniehl:2000fe}, no data was used which could allow
for the difference between the $d$ and $s$ FFs to be determined.
(The FFs for the $u$ can be determined since its electroweak charge is
different to that of $d$ and $s$.) The authors
constrained this difference by imposing
the valence quark structure at all momentum fractions and SU$(3)$
invariance, giving the relations
\begin{equation}
\begin{split}
D_u^{\pi^{\pm}}(x,M_0^2)&=D_d^{\pi^{\pm}}(x,M_0^2),\\
D_u^{K^{\pm}}(x,M_0^2)&=D_s^{K^{\pm}}(x,M_0^2)\ {\rm and}\\
D_u^{p/\overline{p}}(x,M_0^2)&=2D_d^{p/\overline{p}}(x,M_0^2).
\end{split}
\label{lqconstraints}
\end{equation}
Such constraints can be implemented by fixing the parameters $N$, $\alpha$ and $\beta$ of the FFs
on the right hand sides to be equal to those of the FFs on the left hand
side, with the exception that the parameter $N$ of $D_u^{p/\overline{p}}$ must be fixed to twice
the value of that of $D_d^{p/\overline{p}}$
\cite{footnote3}.
With such conditions on the parameterization, a good fit to the data used was obtained.
The first line in Eq.\ (\ref{lqconstraints}) also follows from SU$(2)$
isospin invariance, and is therefore expected to be accurate
\cite{Gronau:1973gc}. Indeed, the approximate result $\eta_d^{\pi^{\pm}}=\eta_u^{\pi^{\pm}}$
implied by this relation is found to hold within
$2\%$ for $x_p\geq 0.2$. However, the second line in Eq.\
(\ref{lqconstraints}) is expected to be strongly violated since the
$s$ quark has a significantly larger mass than the $u$ quark. Already
in 1977, Field and Feynman \cite{Field:1976ve} assumed that due to the
larger mass of $s$ quarks, the $\overline{s} \rightarrow K^+$ transition
should happen more frequently than the $u \rightarrow K^+$ one because
less energy is needed for the creation of a $u\overline{u}$ pair from the
vacuum than for a $s\overline{s}$ pair. This is measured by the suppression
factor $\gamma_s$ of strange quarks, which is known from various
strange/non-strange hadron production rates to be around
$\gamma_s\simeq 0.3$. (For a compilation, see Ref.\ \cite{Knowles:1995kj}.)
The third line in Eq.\ (\ref{lqconstraints}), assumed earlier also in
Ref.\ \cite{Baier:1979tp}, can also be justified for $x\rightarrow 1$ by the valence
ratios and dimensional counting powers \cite{Jones:1978he}. Indeed, in the OPAL
analysis of Ref.\ \cite{Abbiendi:1999ry}, the ratio $\eta_d^{p/\overline{p}}/\eta_u^{p/\overline{p}}$
is consistent with 0.5 for all $x_p \geq 0.2$, but only inside the rather large errors.
However, decays from heavier baryons such as $\Lambda$ or $\Delta$ resonances might change this ratio.
Furthermore, within the LUND string model \cite{Andersson:1983ia} the actual value of the ratio
$\eta_d^{p/\overline{p}}/\eta_u^{p/\overline{p}}$
at large $x_p$ would be a direct measure of the size of the
suppression of diquarks with spin 1 relative to those with spin 0,
since Fermi-Dirac statistics requires a $uu$ diquark to have angular
momentum $L=1$. In summary, all relations in Eq.\ (\ref{lqconstraints}), particularly
the last two, may be violated to a possibly relevant degree, but in any case
since we will use the data of Ref.\ \cite{Abbiendi:1999ry} in our analysis,
we shall not impose any relations between the light quark flavour FFs.
\section{Results \label{results}}
In this Section we report the results obtained from the fit described in
Section \ref{method}. We obtain
\begin{equation}
\alpha_s(M_Z)=0.1176^{+0.0053}_{-0.0067}
{\rm [exp]}^{+0.0007}_{-0.0009}{\rm [theo]}=
0.1176^{+0.0053}_{-0.0068}.
\label{resforalphas}
\end{equation}
This is equivalent to the result
$\Lambda_{\overline{\rm MS}}^{(5)}=221\pm 74{\rm [exp]}_{-10}^{+9}{\rm [theo]}$ MeV.
The experimental errors are obtained by varying $\alpha_s(M_Z)$, keeping all other parameters fixed,
until $\chi^2_{\rm DF}$ increases by unity.
The theoretical errors, determined using the method described in Section
\ref{formalism}, turn out to be negligible relative to the experimental ones,
most likely because the $x$ range of the data used is very limited.
The second result in Eq.\ (\ref{resforalphas}), whose upper and lower errors are obtained by
adding the upper and lower errors respectively of both sources in quadrature,
is consistent with the KKP result \cite{Kniehl:2000cr} of $\alpha_s(M_Z)=0.1170^{+0.0058}_{-0.0073}$
(which includes the theoretical error).
In Table \ref{pars}, we show the values of the remaining, FF parameters obtained from the fit.
Since $N$ and $\beta$ are highly correlated and the large $x$ data generally
has the largest errors, for some FFs these two parameters are large. However,
over the range $0.1\leq x \leq 1$, all FFs are of similar order in magnitude.
Also shown in Table \ref{pars} is the symmetrized propagated experimental error on each parameter.
This quantity is the average of the two resulting errors
obtained by varying the parameter, keeping the other parameters fixed,
until $\chi^2$ increases by 1 from its minimum value. The
correlated errors
between the parameters are expected to be of a
similar order of magnitude to the purely statistical errors shown.
Note that these results show no obvious consistency with Eq.\ (\ref{lqconstraints}).
With the inclusion of correlation effects in the data, a deeper investigation into parameter
errors would be worthy.
We obtained $\chi^2_{\rm DF}=1.15$, indicating an
overall good description of the data.
The resulting $\chi^2_{\rm DF}$ values for the OPAL light quark
tagging probabilities from Ref.\ \cite{Abbiendi:1999ry}
are shown in Table \ref{chiDFforlqetas}. The description of the data in which $K^{\pm}$ or
$p/\overline{p}$ is detected is excellent, except for the process
$d\rightarrow p/\overline{p}$. For this and the $\pi^{\pm}$ data, which has the highest accuracy,
a fit without the data points at $x_p=0.2$
results in all values of $\chi^2_{\rm DF}$ being around
unity, although the resulting FFs from that fit are not considerably different to those from
the main fit. This data, together with
the corresponding theoretical curves calculated from our FF set (labeled AKK), and with the curves
from the sets of Ref.\ \cite{Kniehl:2000fe} (labeled KKP) and
Ref.\ \cite{Kretzer:2000yf} (labeled Kretzer), are shown in Fig.\ \ref{fig1}. We see
that for the $s,d\rightarrow K^{\pm}$ transitions, the corresponding AKK curves are in good
agreement with the data while the KPP and Kretzer curves strongly disagree. The Kretzer set
fails to lead to a decent description
for the $\eta^{\pi}_d$ data, but otherwise all $\pi^{\pm}$ data is well described
by all three sets. Our set and the KKP set lead to a good description of the $p/\overline{p}$
data (which were not used in the determination of the Kretzer set).
Fig.\ \ref{fig7} shows the heavy quark tagging probabilities, which were not used in the fit,
together with the corresponding theoretical curves from the same FF sets as were used in
Fig.\ \ref{fig1}.
In Table \ref{chiDFforhqetas} we list the corresponding $\chi^2_{\mathrm{DF}}$ values. Clearly
these values are unacceptably high.
In order to check that this was not a result
of the inadequacy of our parameterization to allow for a description of both
small $x$ and large $x$ data (since, as discussed around Eq.\ (\ref{calcofetas}),
the OPAL quark tagging probabilities provide
more constraints on the FFs at large $x$), we performed three new fits which
included the heavy quark tagging probabilities, the first being otherwise similar to the main fit,
the other two having the following differences:
For the second fit, the quark FFs were modified by
multiplying the right hand side of Eq.\ (\ref{param}) with $(1+\gamma x)$, with
$\gamma$ different for each quark FF and fixed to zero for the gluon FF, and each $\gamma$
was included in the set of free parameters to be fitted.
In the third fit, all $x_l<0.2$ data were excluded. No significant improvement to
the description of the heavy quark tagging probabilities was obtained in all three fits.
We therefore assume that this discrepancy is caused by the inclusion of
large angle gluon emission effects in Eq.\ (\ref{calcofetas}), as described at the end of Section
\ref{formalism}. However, since we have sufficient data to constrain
the heavy quark FFs, we will not pursue this problem further in this paper.
All remaining values of $\chi^2_{\rm DF}$ from data used in the fit are listed in Table \ref{allchis}.
Each of these lie around or below unity.
Since an excellent fit is obtained to DELPHI, SLD and TPC heavy quark-tagged data,
we conclude that our fitted heavy quark FFs are reliable even though
using them in Eq.\ (\ref{calcofetas}) leads to a poor description of
the OPAL heavy quark tagging probabilities. Since the DELPHI, SLD and TPC
light quark-tagged data is well fitted with the light quark
tagging probabilities, Eq.\ (\ref{calcofetas}) is sufficient
for describing the latter data.
The values of $\chi^2_{\rm DF}$ for the data to be used for comparison, which were
discussed at the beginning of Section \ref{method}, are also shown.
The serious disagreement with the ALEPH
\cite{Buskulic:1995aw,PadillaAranda:1995wi}
and OPAL \cite{Ackerstaff:1998hz} data found here was also found in Ref.\ \cite{Kniehl:2000fe},
where it was argued that this data has a sizeable contribution from
charged particles other than the three lightest charged hadrons. For the ALEPH
data without flavour separation, this argument is supported by the fact
that the data for charged hadron production significantly overshoots the sum
of the hadron identified data.
In Figs.\ \ref{fig2} -- \ref{fig5}, we show all these normalized differential cross section
data used for fitting and for comparison,
together with the corresponding theoretical curves from the fit.
The TPC flavour separated data \cite{Aihara:1986mv}, particularly the $uds$ quark-tagged data, lie
far from their theoretical predictions. However, it must be understood that these data are rather old
compared to the rest of the data used in the fit. At any rate, using them
has not affected the overall quality of the fit since their errors are large, which
explains why their $\chi^2_{\rm DF}$ values in Table \ref{allchis} are not too far from unity.
Qualitatively, at least, the rise in the calculated cross section at low $x$ for decreasing $\sqrt{s}$
is confirmed by the TPC data, as was first noted in Ref.\ \cite{Kretzer:2000yf}.
These figures show that the only TPC data which
can significantly constrain $\alpha_s(M_Z)$ are the $\pi^{\pm}$ and $K^{\pm}$ identified data
shown in Fig.\ \ref{fig2}. In Fig.\ \ref{fig6}, we show the gluon-tagged three-jet data together with the
theoretical curves for $D_g(x,4 E_{\rm jet}^2)$. The resulting $\chi^2_{\mathrm{DF}}$ values
shown in the last two lines of Table \ref{allchis} are
very high, but it must be kept in mind that the theoretical calculation is only correct at LO, and
the gluon is only determined at LO.
In Ref.\ \cite{Kniehl:2000fe}, where this data was used in the fit,
this identification was made only because the gluon FF is
much less constrained by the remaining data than the quark FFs.
In Fig.\ \ref{fig8}, we compare the longitudinal cross section with the data
without flavour separation from
ALEPH \cite{Buskulic:1995aw}, DELPHI \cite{Abreu:1997ir} and OPAL \cite{Akers:1995wt}
and for light and $b$ quark separation from DELPHI \cite{Abreu:1997ir}.
The $x$ space coefficient functions of the longitudinal cross section
are given in Ref.\ \cite{Rijken:1996vr}. However, since our cross sections
are calculated in Mellin space, we calculate the Mellin transform of these
quantities as detailed in Appendix \ref{app}. (An alternative procedure would be to evolve the FFs
in Mellin space as before, and perform the convolution of the coefficient functions
with the evolved FFs in $x$ space. However, this procedure is numerically very slow.)
In the unpolarized cross sections used in our fit, the gluon FF for each hadron enters only at NLO
and so is only determined to LO in our analysis, while it
enters at LO in the longitudinal cross section, for which a gluon FF determined to NLO
is therefore required. Thus the curves in Fig.\ \ref{fig8}
are not completely NLO, but serve to determine the quality of our gluon FF. The agreement
is excellent for the ALEPH and OPAL data, and good for the DELPHI data. Our curves are
also very similar to those obtained in Ref.\ \cite{Kniehl:2000fe}, where the LO curves
from these authors' LO analysis are also shown. These latter curves do not agree with
the ALEPH and OPAL data as well as the NLO ones. Thus treating the LO gluon FF obtained
from their and our fits as NLO results in no loss of consistency in this case.
Finally, we compare cross sections calculated using our FFs
for particle production in proton-(anti)proton initiated processes with experimental data.
Such processes are highly dependent on the individual light quark flavour FFs, due
to the partonic structure of the proton.
We use the coefficient functions for the processes
$a+ b \rightarrow c + X$, where $a$, $b$ and $c$ denote partons,
to NLO as calculated in Ref.\ \cite{ACGG}. We convolute these with our evolved
FFs for parton $c$, and the evolved CTEQ6M parton distribution functions \cite{Stump:2003yu}
for $a$ and $b$.
Since our fitted result of $\Lambda_{\overline{\rm MS}}^{(5)}=221$ MeV is very similar to the
result of $\Lambda_{\overline{\rm MS}}^{(5)}=226$ MeV obtained in Ref.\ \cite{Stump:2003yu}, we use
the former result in the calculation of $a_s(\mu^2)$. We take $M_f^2=k p_T^2$. The cross section
at $x_T=2p_T/\sqrt{s}$ depends on the FFs for the whole region $x_T <z<1$.
Since we do not (reliably) determine the FFs below $z=0.1$ and/or $M_f=M_0$, we take them
in this region to be equal to their values at this point. Graphically, we found no discernible
difference between the resulting predictions and those obtained when
the FFs in this region were fixed to zero.
Firstly, we calculate the invariant differential cross section for inclusive $\pi^0$ production for
the process $p+ p \rightarrow \pi^0 +X$ as measured by PHENIX at $\sqrt{s}=200$ GeV
in Ref.\ \cite{Adler:2003pb}. For this we assume the relation
\begin{equation}
D_a^{\pi^0}(x,M_f^2)=\frac{1}{2}D_a^{\pi^{\pm}}(x,M_f^2)
\end{equation}
to be true, which follows from SU(2) flavour symmetry for pions
(see Ref.\ \cite{Binnewies:1995pt}). Here, $D_a^{\pi^0}$ is the average of the FFs
for the processes $a,\overline{a}\rightarrow \pi^0$. (Recall $D_a^{\pi^{\pm}}(x,M_f^2)$
is also averaged over $a$ and $\overline{a}$, but summed over $\pi^+$ and $\pi^-$.)
The results are shown in Fig.\ \ref{fig9} for $k=$1/4, 1 and 4,
together with the PHENIX data. In addition, we also compare the cross section calculated from
the FFs obtained in Ref.\ \cite{Kniehl:2000fe}. For $p_T>7$ GeV,
the curve for $k=1$ lies closer to the centre of the data than the KKP curve does.
Secondly, we calculate the invariant differential cross section for inclusive $K^0_S$ production for
the process $p+p \rightarrow K^0_S +X$ as preliminarily measured by STAR at
$\sqrt{s}=200$ GeV \cite{MH}\cite{footnote4}, and for
the process $p+\overline{p} \rightarrow K^0_S +X$ as measured by UA1 at
$\sqrt{s}=630$ GeV in Ref.\ \cite{Bocquet:1995jq}. For this we assume the relation
\begin{equation}
D_a^{K^0_S}(x,M_f^2)=\frac{1}{2}D_b^{K^{\pm}}(x,M_f^2)
\label{K0SfromKcharged}
\end{equation}
to be true, where $b=u,d$ if $a=d,u$, otherwise $b=a$.
Eq.\ (\ref{K0SfromKcharged}) follows from SU(2) flavour symmetry for kaons
(see Ref.\ \cite{Binnewies:1995pt}), and is confirmed by the fact that
the OPAL measurements in Ref.\ \cite{Abbiendi:1999ry} for the production of
$K^0_S$ and $K^{\pm}$ mesons agree within their errors.
The predictions are shown in Fig.\ \ref{fig10}, in a format
similar to Fig.\ \ref{fig9}. For $p_T>1.5$ GeV, the $k=1$ curve agrees better
with the STAR data than the KKP curve. This disagreement in the latter case was observed in
Ref.\ \cite{MH}. However, for the older UA1 data our predictions differ considerably over the whole range,
although they are consistent with the data within the theoretical errors for $p_T>4.5$ GeV, while
the KKP curve gives good agreement.
Since the most important difference between our analysis and
that of Ref.\ \cite{Kniehl:2000fe} is the inclusion of the OPAL tagging
data in our fit, we conclude that this agreement of the KKP curve is accidental.
\section{Conclusions \label{conclusions}}
This work is an update of the KKP analysis \cite{Kniehl:2000fe}, the
main difference being that the OPAL results on light quark tagging
probabilities have been used to phenomenologically constrain the
individual light quark FFs for the first time. We find that the
inclusion of this data in the fit makes an important difference to the
description of the $d,s\rightarrow K^{\pm}$ transitions.
Light flavour separated FFs are essential for making predictions
for inclusive cross sections in which there is at least one proton in the initial state
and one light hadron in the final state (or more than one, in which case
other non perturbative quantities are also required for subprocesses in which
multiple hadrons are emitted from
a single parton). Such cross sections will be measured, for example, at the LHC.
In addition, we have included the flavour separated TPC data \cite{Aihara:1986mv} at $\sqrt{s}=29$ GeV,
but such data makes little difference to the fit.
We have excluded all charged data to be confident that none of the data sets used were contaminated
with charged particles other than the three lightest charged hadrons.
However, good agreement with much of the available charged hadron data, in particular
that from DELPHI and SLD, was achieved.
We point out that although our gluon FF for each hadron has been formally determined to LO only,
treating it as NLO leads to good agreement with the measured longitudinal cross sections in
the literature. Finally, relative to the KKP predictions, we obtain
with our FFs a shift towards
the PHENIX data for the
invariant differential cross section for inclusive $\pi^0$ production and
towards the STAR data for the
invariant differential cross section for inclusive $K^0_S$ production.
A determination of $\alpha_s(M_Z)$ has been performed. We have also calculated the theoretical error
and find it to be negligible relative to the experimental error.
We obtain $\alpha_s(M_Z)=0.1176^{+0.0053}_{-0.0068}$, which agrees with the
Particle Data Group's world average of $\alpha_s(M_Z)=0.1187\pm 0.002$ \cite{Eidelman:2004wy}.
In order to make predictions, our fitted FFs over the range $0.1<z<1$ and $M_0<M_f<200$ GeV
can be obtained from the FORTRAN routines at \verb!http://www.desy.de/~simon/AKK2005FF.html!,
which are calculated using cubic spline
interpolation on a linear grid in $(z,\ln M_f^2)$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,452
|
Manaye Misganaw, Ethiopian Biodiversity Institute, Addis Ababa, Ethiopia
Girma Mengesha, Wondo Genet College of Forestry and Natural Resources, Shashemene, Ethiopia
Tesfaye Awas, Ethiopian Biodiversity Institute, Addis Ababa, Ethiopia
Received: Aug. 12, 2020; Accepted: Aug. 25, 2020; Published: Sep. 14, 2020
DOI: 10.11648/j.ajbio.20200805.11 View 217 Downloads 121
Pollination is one of a valuable ecosystem services in the maintenance of biodiversity and ensures the survival of plant species. Therefore, Insect pollinators' diversity and their role in the ecosystem are not sufficiently recorded; thus, conducting assessment of their diversity and roles helps to recognize the economic and ecological value of insect pollination, and potential impacts of the loss of insect pollinators. Therefore, the overall aim of this study was to assess and identify insect pollinators' diversity and frequently visited plant species in cropland and natural habitat of the study area. Transect sampling and direct field observation was used to collect data. The abundance of insect pollinators from the three study sites were sampled systematically using two transects one along the Shrubland and the other on farmland habitat. A total of 60 transect sample plots 30 in the farmland and 30 in the Shrubland habitats were observed in the study areas. A total of 34 insect pollinator species were identified. The most frequently recorded insect pollinator was Apis mellifera in Shrubland (60.4%) and farmland (67.3%). Insect diversity of the Shrubland was higher (H'=1.72) than farmland (H'=1.514). Similarly, evenness was higher in the Shrubland (J'=0.5485) as compared to farmland (J'=0.4974) which is somehow even distribution in both habitats. To understand the most visited plants by insect pollinators 40 wild plants and 4 crop species were identified. Among the sampled plants Crassocephalum macropappurn was the most frequently visited plant by different insect pollinators while Guizotia abyssinica was the most frequently visited among the sampled crops. The study has shown occurrence of diverse insect pollinators and plant species visited by insect pollinators as function of ecosystem services in the area.
ApisMellifera, Ecosystem Service, Frequently Visited Plants, Insect Diversity, Pollinators Conservation
Manaye Misganaw, Girma Mengesha, Tesfaye Awas, Diversity of Insect Pollinators in Gozamin District of Amhara Region, Ethiopia, American Journal of BioScience. Vol. 8, No. 5, 2020, pp. 123-131. doi: 10.11648/j.ajbio.20200805.11
Callicott JB, Crowder LB, Mumford K (1999). Current Normative Concepts in Conservation. Conservation Biology 13: 22–35.
Kluser S, Peduzzi P (2007). Global Pollinator Decline: A Literature Review, UNEP/GRID Europe, UNEP.
Klatt B. K, Holzschuh A, Westphal C, Clough Y, Smit I, Pawelzik E, Tscharntke T (2014). Bee pollination improves crop quality, shelf life and commercial value. Proc. R. Soc. B 281: 20132440. http://dx.doi.org/10.1098/rspb.2013.2440.
Roubik DW (1995). Pollination of Cultivated Plants in the Tropics. Rome, Italy: Food and Agriculture Org., UN.
African Pollinator Initiative (API) (2006). Pollinators and pollination: A resource book for policy and practice. ISBN: 1-86849-310-5. pp. 1-52.
Peterson W, John SR, Ellsworth D, Colopy M (2015). Pollinators and Rights-of-Way. pp. 36-39.
Ollerton J, Winfree R, Tarrant S (2011). How many flowering plants are pollinated by animals? Oikos, 120: 321–326.
Patil HS, Duhoon SS (2006). Self-Incompatibility, Male Sterility and Pollination Mechanism in Niger {Guizotia abyssinica (L. F.) CASS.}. A review. All India Coordinated Research Project on Niger, Zonal Agricultural Research Station, India: 113-121.
Intergovernmental Science Policy platform on Biodiversity and Ecosystem Services (IPBES) (2016). Summary for Policymakers of the Assessment Report on Pollinators, Pollination and Food Production. Pp 10.
Winfree R, Ramiro A, Diego P, Vazquez, Gretchen L, Marcelo A, Aizen (2009). A meta-analysis of bees' response to anthropogenic disturbance. University of Californi, Berkeley, California 94720-3114 USA. Ecology, 90 (8): 2068–2076.
Bauer DM, Wing IS, (2011). The Macroeconomic Cost of Catastrophic Pollinator Declines, Department of Geography and Environment, Boston University. USA. pp.13.
Kearns CA, Inouye DW, Waser NM, (1998). Endangered Mutualism: The Conservation of Plant-pollinator Interaction. Annu. Rev. Ecol. 29: 83-112.
Goulson DG, Lye C, Darvill B (2008). Decline and conservation of bumble bees. Annual Review of Entomology 53: 191–208.
Vanbergen AJ, Insect Pollinators Initiative (IPI) (2013). Threats to an ecosystem service: pressures on pollinators. Frontiers in Ecology and the Environment, doi: 10.1890/120126
Ulrika S (2016). The impact of forest on pest damage, pollinators and pollination services in an Ethiopian agricultural landscape. Stockholm University, Sweden, ISBN 978-91-7649-354-0.
Bianchi F, Booij CJ, Tscharntke T (2006). Sustainable pest regulation in agricultural landscapes: a review on landscape composition, biodiversity and natural pest control. Proceedings of the Royal Society B-Biological Sciences, 273: 1715-1727.
Macfadyen S, Hopkinson J, Parry H, Neave MJ, Bianchi F, Zalucki MP (2015). Early-season movement dynamics of phytophagous pest and natural enemies across a native vegetation-crop ecotone. Agriculture Ecosystems and Environment, 200: 110-118.
Haimanot R, Zemede A, Ensermu K (2015). Contribution of Traditional Farmers for Medicinal Plant Conservation on the Farming Site in Gozamin District, Amhara Region, Ethiopia. International Journal of Life Sciences 4: 24-35.
Gozamin District Finance and Economic Development office (GDFED) (2015). The second five year Growth and transformation plan. Integrated Budget Planning and development case team (unpublished). Pp.4-5.
Hennig EI, (2011). Plant Diversity Effects on Plant-Pollinator Interactions in Urban and Agricultural Settings. University Duisburg-Essen. pp. 10-80.
Ward K. D, Cariveau E, May M, Roswell M, Vaughan N, Williams R, Winfree R, Isaacs, Gill K. (2014). Streamlined Bee Monitoring Protocol for Assessing Pollinator Habitat. Portland, OR: The Xerces Society for invertebrate Conservation. pp. 16.
Food and Agriculture Organization (FAO) (2011). Protocol To Detect And Assess Pollination Deficits In Crops: A Handbook For Its use. Pollination service for Sustainable Agriculture. Field manual, Rome Italy. pp 20-50.
Magurran AE (2004). Measuring Biological Diversity. Blackwell Publishing. pp. 106-110.
Adams J (2009). Species Richness Patterns in the Diversity of Life. Springer-Praxis Books in Environmental Sciences Subject Advisory Editor, ISBN 978-3-540-74277-7. pp. 1-20.
Stojnić SM, Andrić A, Józan Z, Vujić A (2012). Pollinator diversity (Hymenoptera and Diptera) in semi-natural habitatsin Serbia during summer. University of Novi Sad, Serbia 2 H-7453 Mernye, Hungary.
James ET (2004). Backyard Beekeeping. Alabama Cooperative Extension System. ISBN 0-9722580-6-X. pp. 25.
Winfree R, Neal M, William, Hannah G, John S, Ascher, Kremen C (2008). Wild bee pollinators provide the majority of crop visitation across land-use gradients in New Jersey and Pennsylvania, USA. Princeton University, Princeton, NJ 08544, USA. Journal of Applied Ecology, 45: 793–802.
Zahoor MK., Suhail A, Iqbal J, Zulfaqar Z, Anwar M (2003). Biodiversity of Noctuidae (Lepidoptera) in Agro-Forest Area of Faisalabad, University of Agriculture, Government of Punjab, Pakistan Int. J. Agri. Biol. 5: 560–563.
Fichtl R, Admasu A (1994). Honeybee Flora of Ethiopia. Margraf, Verlag, Weikersheim, Germany. ISBN 3-8236-1234-4. pp. 45.
Park M, Bryan D, John L, David B, Mace V, Jolie D, Edwin R, Arthur A (2012). Wild Pollinators of Eastern Apple Orchards and How to Conserve Them. Cornell University, The Xerces Society.
Haftom G, Alemayehu T (2014). Effect of honey bee (Apis mellifera) pollination on seed yield and yield parameters of Guizotia abyssinica (L. F), Apiculture and sericulture case team, Mekelle Agricultural Research center. 9: 3687-3691.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,505
|
/*
* Xero Payroll AU
* This is the Xero Payroll API for orgs in Australia region.
*
* The version of the OpenAPI document: 2.2.4
* Contact: api@xero.com
*
* NOTE: This class is auto generated by OpenAPI Generator (https://openapi-generator.tech).
* https://openapi-generator.tech
* Do not edit the class manually.
*/
package com.xero.models.payrollau;
import java.util.Objects;
import com.fasterxml.jackson.annotation.JsonProperty;
import io.swagger.annotations.ApiModelProperty;
import java.util.UUID;
import com.xero.api.StringUtil;
/** Account */
public class Account {
StringUtil util = new StringUtil();
@JsonProperty("AccountID")
private UUID accountID;
@JsonProperty("Type")
private AccountType type;
@JsonProperty("Code")
private String code;
@JsonProperty("Name")
private String name;
public Account accountID(UUID accountID) {
this.accountID = accountID;
return this;
}
/**
* Xero identifier for accounts
*
* @return accountID
*/
@ApiModelProperty(
example = "c56b19ef-75bf-45e8-98a4-e699a96609f7",
value = "Xero identifier for accounts")
public UUID getAccountID() {
return accountID;
}
public void setAccountID(UUID accountID) {
this.accountID = accountID;
}
public Account type(AccountType type) {
this.type = type;
return this;
}
/**
* Get type
*
* @return type
*/
@ApiModelProperty(value = "")
public AccountType getType() {
return type;
}
public void setType(AccountType type) {
this.type = type;
}
public Account code(String code) {
this.code = code;
return this;
}
/**
* Customer defined account code
*
* @return code
*/
@ApiModelProperty(example = "420", value = "Customer defined account code")
public String getCode() {
return code;
}
public void setCode(String code) {
this.code = code;
}
public Account name(String name) {
this.name = name;
return this;
}
/**
* Name of account
*
* @return name
*/
@ApiModelProperty(example = "General expenses", value = "Name of account")
public String getName() {
return name;
}
public void setName(String name) {
this.name = name;
}
@Override
public boolean equals(java.lang.Object o) {
if (this == o) {
return true;
}
if (o == null || getClass() != o.getClass()) {
return false;
}
Account account = (Account) o;
return Objects.equals(this.accountID, account.accountID)
&& Objects.equals(this.type, account.type)
&& Objects.equals(this.code, account.code)
&& Objects.equals(this.name, account.name);
}
@Override
public int hashCode() {
return Objects.hash(accountID, type, code, name);
}
@Override
public String toString() {
StringBuilder sb = new StringBuilder();
sb.append("class Account {\n");
sb.append(" accountID: ").append(toIndentedString(accountID)).append("\n");
sb.append(" type: ").append(toIndentedString(type)).append("\n");
sb.append(" code: ").append(toIndentedString(code)).append("\n");
sb.append(" name: ").append(toIndentedString(name)).append("\n");
sb.append("}");
return sb.toString();
}
/**
* Convert the given object to string with each line indented by 4 spaces (except the first line).
*/
private String toIndentedString(java.lang.Object o) {
if (o == null) {
return "null";
}
return o.toString().replace("\n", "\n ");
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,219
|
Q: Симметрическая разность Симметрическая разность wiki
Нужно найти симметрическую разность массивов.
Написал функцию сравнения двух массивов, в которой объединяю массивы в один и в цикле ищу повторяющиеся значения. При нахождении удаляю.
function sym() {
var args = Array.prototype.slice.call(arguments);
var result = compareTwoArray(arguments[0], arguments[1]);
if (arguments.length > 2) {
for (var i = 2; i < arguments.length; i++) {
// console.log('result = ' + result);
result = compareTwoArray(result, arguments[i]);
}
}
return result;
}
function compareTwoArray() {
var args = Array.prototype.slice.call(arguments);
var result = [];
var newArr = args.reduce(function(prev, curr) {
return prev.concat(curr);
})
for (var i = 0; i < newArr.length; i++) {
var count = 0;
for (var j = i+1; j < newArr.length; j++) {
if (newArr[i] === newArr[j]) {
count += 1;
newArr.splice(j, 1);
j -= 1;
}
}
if (count === 0) {
result.push(newArr[i]);
}
}
return result;
}
console.log(sym([1, 2, 3], [3, 1, 5]));
// [2, 5]
console.log(sym([1, 1, 3], [4, 6]));
// [1, 3, 4, 6]
console.log(sym([1, 1, 2, 5], [2, 2, 3, 5]));
// [1, 3]
console.log(sym([1, 1, 2, 5], [2, 2, 3, 5], [3, 4, 5, 5]));
// [1, 4, 5]
console.log(sym([1, 2, 5], [2, 3, 5], [3, 4, 5]));
// [1, 4, 5]
Не знаю, как обрабатывать повторяющиеся значения.
В одном случае его нужно удалить.
[1, 2, 3]
[3, 1, 5]
[2, 5] - должен получиться
[2, 5] - получается с моим кодом
В другом оставить.
[1, 1, 3]
[4, 6]
[1, 3, 4, 6] - должен получиться
[3, 4, 6] - получается с моим кодом
В другом и оставить(1) и удалить(2).
[1, 1, 2, 5]
[2, 2, 3, 5]
[1, 3] - должен получиться
[3] - получается с моим кодом
A: В яваскрипте есть класс Set, который в данном случае можно использовать в виде множества, в итоге запись получится почти по формуле
В примере ниже использовался spread оператор, и rest-параметры
function sym(...arrs) {
return arrs.reduce((acc, arr, i) => {
var accSet = new Set(acc), // множество из элементов первого массива
arrSet = new Set(arr); // множество из элементов второго массива
return [...accSet].filter(a => !arrSet.has(a)) // элементы первого множества без элементо второго
.concat( // объединение
[...arrSet].filter(a => !accSet.has(a)) // элементы второго множества без элементов первого
);
}, []);
}
console.log(sym([1, 2, 3], [3, 1, 5]));
// [2, 5]
console.log(sym([1, 1, 3], [4, 6]));
// [1, 3, 4, 6]
console.log(sym([1, 1, 2, 5], [2, 2, 3, 5]));
// [1, 3]
console.log(sym([1, 1, 2, 5], [2, 2, 3, 5], [3, 4, 5, 5]));
// [1, 4, 5]
console.log(sym([1, 2, 5], [2, 3, 5], [3, 4, 5]));
// [1, 4, 5]
A: Решение для двух массивов. Хотя 3 и больше массивов можно обрабатывать ступенчато: сначала два первых, потом результат первой обработки и 3 массив и так далее.
function sum(arr1, arr2) {
var tmp = arr1.concat(arr2),
result = [],
value, sum;
for (var i = 0; i < tmp.length; i++) {
var value = tmp[i];
if (result.indexOf(value) == -1) {
sum = 0;
if (arr1.indexOf(value) != -1) {
sum++;
}
if (arr2.indexOf(value) != -1) {
sum++;
}
if (sum == 1) {
result.push(value);
}
}
}
return result;
}
console.log(sum([1, 2, 3], [3, 1, 5]));
// [2, 5]
console.log(sum([1, 1, 3], [4, 6]));
// [1, 3, 4, 6]
console.log(sum([1, 1, 2, 5], [2, 2, 3, 5]));
// [1, 3]
A: Как вариант еще можно сделать так:
function sym() {
var diff = [];
// Допускаем, что симметричная разность пустого массива - пустой массив,
// а симметричная разность одного массива - разность пустого массива
// и этого массива равна самому этому массиву.
// Также симметрическая разность равна объединению минус пересечение,
// поэтому...
[].forEach.call(arguments, function(arg) {
var sum = diff.concat(arg);
// ...объединяем массивы...
diff = sum.filter(function(item) {
// ...оставляем только те элементы, которые не найдены хотя бы
// в одном из массивов...
return (-1 === arg.indexOf(item) || -1 === diff.indexOf(item));
});
});
// ...возвращаем уникальные элементы
// (иначе sym([1, 1, 2, 5], [2, 2, 3, 5]) будет [1, 1, 3])
return diff.filter(function(item, index, self) {
return self.indexOf(item) === index;
});
}
console.log(sym());
console.log(sym([1, 2, 3]));
console.log(sym([1, 2, 3], [3, 1, 5]));
console.log(sym([1, 1, 3], [4, 6]));
console.log(sym([1, 1, 2, 5], [2, 2, 3, 5]));
console.log(sym([1, 1, 2, 5], [2, 2, 3, 5], [3, 4, 5, 5]));
console.log(sym([1, 2, 5], [2, 3, 5], [3, 4, 5]));
A: Основывая на описании:
Симметрическая разность двух множеств
Вот пример для двух множеств (докрутить большее количество множеств, думаю, догадаетесь как):
function getDiff(a, b){
let result = [],
pointer = -1;
if(!Array.isArray(a) || !Array.isArray(b))
throw new Error(`Both arguments must be arrays!`);
a.forEach(e => {
if((pointer = b.indexOf(e)) !== -1)
b.splice(pointer, 1);
else if(result.indexOf(e) === -1)
result.push(e);
});
b.forEach(e => {
if((pointer = a.indexOf(e)) !== -1)
a.splice(pointer, 1);
else if(result.indexOf(e) === -1)
result.push(e);
});
return result;
}
// Пример с вики
console.info(getDiff([1, 2, 3, 4, 5], [3, 4, 5, 6, 7]));
// Ваши примеры
console.info(getDiff([1, 2, 3], [3, 1, 5]));
console.info(getDiff([1, 1, 3], [4, 6]));
console.info(getDiff([1, 1, 2, 5], [2, 2, 3, 5]));
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,487
|
\section{Introduction}
Over the past ten years, cold atomic gases have gradually become a
widely employed and highly tunable tool for testing new ideas in
many areas of quantum physics: quantum phase transitions
(Bose-Einstein condensation, Fermi degenerate gases, Mott-Hubbard
transition)~\cite{bec,fermi,mott-hubbard}, quantum
chaos~\cite{chaos}, applications in metrology~\cite{hsurm},
disordered systems~\cite{cbsat,thierry} to cite a few. In the
latter case, cold atomic vapors act as dilute gases of randomly
distributed atoms multiply scattering an incident monochromatic
laser light. In this case, the scattered light field exhibit a
speckle-like structure due to (multiple) interference between all
possible scattering paths. The key point is that the disorder
average is insufficient to erase all interference effects. This
gives rise to weak or strong localization effects in light
transport depending on the strength of
disorder~\cite{Houches,AkkerMon}. A hallmark of this coherent
transport regime is the coherent backscattering (CBS) phenomenon:
the average intensity multiply scattered off an optically thick
sample is up to twice larger than the average background in a
small angular range around the direction of backscattering,
opposite to the incoming light~\cite{cbs}. This interference
enhancement of the diffuse reflection off the sample is a
manifestation of a two-wave interference. As such, it probes the
coherence properties of the outgoing light~\cite{photon}. The CBS
effect in cold atomic gases has been the subject of extensive
studies in the weak localization regime, both from theoretical and
experimental points of view~\cite{cbsatoms}. In particular,
modifications brought by atoms, as compared to classical
scatterers, for light transport properties (mean-free path,
coherence length, CBS enhancement factor) have been highlighted.
They are essentially due to the quantum internal atomic
structure~\cite{internal,cbsB}.
Another interesting feature of atoms is their ability to display a
nonlinear behavior: the scattered light is no more proportional to
the incident one. This leads to a wide variety of phenomena, like
pattern formation, four-wave mixing, self-focusing effects,
dynamical instabilities,
\emph{etc}~\cite{boyd,prl72GMP,praDHGC,prl85SM}. For a weak
nonlinearity, introducing an intensity-dependent susceptibility is
enough to properly describe these effects, including quantum
properties~\cite{boyd,pra70WGDM,facteur3}, \emph{e.g.} the Kerr
effect (intensity dependence of the refractive index) can be
obtained with a $\chi^{(3)}$ nonlinearity. However, when the
incident intensity is large enough, and this is easily achieved
with atoms, perturbation theories eventually fail and a full
nonlinear treatment is required. For a single two-level atom, the
solution is usually given by the so-called optical Bloch (OB)
equations. Together with the quantum regression theorem, they
allow for a complete description of the spectral properties of the
fluorescence light~\cite{Cohenrouge}. In particular, these
equations show that the atomic nonlinear behavior is intrinsically
linked to the quantum nature of the electromagnetic field. More
specifically, as opposed to classical nonlinear scatterers, the
radiated light exhibits quantum fluctuations characterized by
peculiar time correlation properties. They define a power
spectrum, known as the Mollow triplet, emphasizing inelastic
scattering processes at work in the emission
process~\cite{pr188M,Cohenrouge,ZG}.
However, even if all these aspects are well understood in the case
of a \emph{single} atom exposed to a strong monochromatic
field~\cite{Cohenrouge}, the situation changes dramatically in the
case of a large number of atoms where a detailed analysis
including both quantum nonlinear properties and coherence effects
is still lacking. Until now, the nonlinear coupling between the
atoms and the quantum vacuum fluctuations is either included in a
perturbative scheme~\cite{facteur3,Wellens_long}
or simply described by a classical
noise~\cite{pra46YMC,pra46YMC2,pra51YC,pra52DPGC,pra56B}. In the
dilute regime $\lambda \ll R$ where the light wavelength $\lambda$
is much less than the average particle separation $R$,
one expects the quantum fluctuations to
reduce the degree of coherence of the scattered light.
This will alter not only propagation parameters (mean-free path,
refraction index), but also weak localization corrections to
transport, and the CBS enhancement factor, which is
related to the coherence properties of the scattered
light field~\cite{thierry,photon}. We want here to stress that,
even beyond interference and weak localization phenomena, any
transport property which may be influenced by saturating the
atomic transition deserves a special and necessary study on its
own. The most striking systems falling in this category where both
nonlinear and disordered descriptions are intimately interwoven
are coherent random lasers \cite{cao} where interference
effects lead to localized light modes inside the disordered medium,
comparable to resonator eigenmodes in standard lasers.
Even if, in this case, one
would require an active ({\em i.e.} amplifying) medium, a key
point is the understanding of the mutual effects between multiple
interferences and nonlinear scattering.
In the present paper, we will focus on the rather simple case of
two atoms in vacuum. Our aim is threefold: (i) firstly to properly
calculate quantum correlations between pairs of atoms as a crucial
step towards a better understanding of the physical mechanisms at
work; (ii) secondly to implement a method allowing for a simple
incorporation of frequency-dependent propagation effects; (iii)
finally to understand, in the CBS situation, the modifications
brought by the (quantum) nonlinearity to the interference
properties. We hope that these points, once mastered, can provide
an efficient way to produce realistic computer models to simulate
real experiments. Point (i) alone could easily be solved using the
standard OB method~\cite{pra45VA,prl94SMB}. But the latter almost
becomes useless regarding point (ii), since frequency-dependent
propagation leads to complicated time-correlation functions. From
a numerical point of view, it also leads to such large linear
systems of coupled equations that its practical use is limited up
to only a few atoms, very far from a real experimental situation.
For these reasons, we will rather use the quantum Langevin method
for our purposes. This method not only solves points (i) and (ii),
but also leads to a simple explanation of point (iii), through a
direct evaluation of the quantum noise spectrum. Note however
that, in the absence of any effective medium surrounding the two
atoms, and as long as only the numerical results are concerned
(but not the physical interpretation), the quantum Langevin
approach is completely equivalent to solving the multi-atoms
optical Bloch equations like in~\cite{pra45VA,prl94SMB}.
This paper divides as follows: in section~\ref{oneatom}, the
notations are defined and the quantum Langevin approach is
explained for the single atom case. In section~\ref{twoatoms}, the
method is adapted to the case where two atoms are weakly coupled
by the dipole interaction. The validity and relevance of the
method is controlled by a comparison with a direct calculation
using OB equations. Then, in the CBS configuration, numerical
results for different values of the laser intensity and detuning
are presented and discussed. In particular, possible reasons for
the reduction of the enhancement factor are put forward.
\section{Single two-level atom case}
\label{oneatom}
\subsection{Time-domain approach}
We consider an atom with a zero angular momentum electronic ground
state ($J_g=0$) exposed to a monochromatic light field. The light
field frequency $\omega_L$ is near-resonant with an optical dipole
transition connecting this ground state to an excited state with
angular momentum $J_e=1$. The angular frequency separation between
these two states is $\omega_0$ and the natural linewidth of the
excited state is $\Gamma$. We will denote hereafter by $\delta =
\omega_L-\omega_0$ the laser detuning. The ground state is denoted
by $|0\,0\rangle$ while the excited states are denoted by
$|1\,m_e\rangle$, with $m_e=-1,0,1$ the Zeeman magnetic quantum
number. As we assume no magnetic field to be present throughout this
paper, the excited state is triply degenerate.
In the Heisenberg picture, this two-level atom is entirely
characterized by the following set of 16 time-dependent
operators:
\begin{equation}
\Pi^g=|0\,0\rangle\langle 0\,0| \quad ; \quad
\Pi^e_{m_e\,m'_e}=|1\,m_e\rangle\langle 1\,m'_e| \quad ; \quad
\mathcal{D}^+_{m_e}=|1\,m_e\rangle\langle 0\,0| \quad ; \quad
\mathcal{D}^-_{m_e}=|0\,0\rangle\langle 1\,m_e|
\end{equation}
The atomic operators obey the completeness constraint
\begin{equation}
\label{constraint}
\openone=\Pi^g + \Pi^e
\end{equation}
where $\Pi^g$ and $\Pi^e = \sum_{m_e} \Pi^e_{m_e\,m_e}$ are the
ground and excited state atomic population operators.
The full atom-field Hamiltonian $\mathcal{H}$ is the sum of the
free atom Hamiltonian $\mathcal{H}_A = \hbar\omega_0 \Pi^e$, of
the free quantized field Hamiltonian $\mathcal{H}_F =
\sum_{\textbf{k},\boldsymbol{\epsilon} \perp \textbf{k}}
\hbar\omega_\textbf{k} \, a^\dag_{\textbf{k}\boldsymbol{\epsilon}}
a_{\textbf{k}\boldsymbol{\epsilon}}$ and of the dipolar
interaction $\mathcal{V} = - \mathbf{d} \cdot (\mathbf{E}_L +
\mathbf{E}_V)$ between the atomic dipole $\mathbf{d}$, the
classical laser field $\mathbf{E}_L$ and the quantum
electromagnetic vacuum field $\mathbf{E}_V$. Performing the usual
approximations of quantum optics, \emph{i.e.} neglecting
non-resonant terms (rotating wave approximation) and assuming
Markov-type correlations between the atomic operators and the
vacuum field, one obtains the quantum Langevin equations
controlling the time evolution of any atomic observable
$\mathcal{O}$ in the rotating frame~\cite{pra46YMC,Cohenrouge}:
\begin{equation}
\label{langevin1}
\frac{d\mathcal{O}}{dt}=i\delta_L[\mathcal{O},\Pi^e]
-\frac{i}{2}\sum_q(-1)^q[\mathcal{O},\mathcal{D}^+_q]\Omega^{L+}_{-q}(\textbf{R})
-\frac{i}{2}\sum_q[\mathcal{O},\mathcal{D}^-_q]\Omega^{L-}_{q}(\textbf{R})
-\frac{\Gamma}{2}\left(\mathcal{O}\Pi^e+\Pi^e\mathcal{O}\right)
+\Gamma\sum_q\mathcal{D}^+_q\mathcal{O}\mathcal{D}^-_q+
\mathcal{F}_{\mathcal{O}}(\textbf{R},t),
\end{equation}
where $\Omega^{L+}_q$ (resp. $\Omega^{L-}_q$) are the components
of the Rabi frequency of the positive (resp. negative) frequency
parts of the incident laser beam, i.e.
$\hbar\mathbf{\Omega}=-d\mathbf{E}$ where $d$ is the
dipole strength.
Finally $\mathcal{F}_{\mathcal{O}}(t)$ is the Langevin force
depicting the effects of the quantum fluctuations of the vacuum
electromagnetic field and reads as follows:
\begin{equation}
\label{langevinforce}
\mathcal{F}_{\mathcal{O}}(t)=-\frac{i}{2}\sum_q(-1)^q
[\mathcal{O},\mathcal{D}^+_q] \Omega^{0+}_{-q}(\textbf{R},t)
-\frac{i}{2}\sum_q\Omega^{0-}_{q}(\textbf{R},t)[\mathcal{O},\mathcal{D}^-_q],
\end{equation}
where $\Omega^{0+}(\textbf{R},t)$ is the vacuum Rabi field
operator
\begin{equation}
\mathbf{\Omega}^{0+}(\textbf{R},t)=-\frac{2id}{\hbar}\sum_{\textbf{k},
\boldsymbol{\epsilon}\perp\textbf{k}}\mathcal{E}(\omega)
\boldsymbol{\epsilon}\,a_{\textbf{k}\boldsymbol{\epsilon}}(t_0)
e^{i\textbf{k}\cdot\textbf{R}-i(\omega-\omega_L)(t-t_0)}
\end{equation}
with $t_0$ an initial time far in the past. From the preceding
expression, one can calculate the time correlation functions of
the vacuum field~\cite{Cohengris}:
\begin{equation}
(-1)^q[\Omega^{0+}_{-q}(\textbf{R},t),\Omega^{0-}_{q'}(\textbf{R},t')]=
4\Gamma\delta_{q\,q'}f(t-t'),
\end{equation}
where $f(\tau)$ in a function centered around $\tau=0$, whose
width $\tau_c$ is much smaller than any characteristic atomic
timescale (i.e. $\tau_c\ll \omega_0^{-1}\ll\Gamma^{-1}$) and whose
time integral is equal to unity. Thus, hereafter, $f(\tau)$ will
be safely replaced by a $\delta$-function: $f(\tau) \to
\delta(\tau)$.
The time evolution for the expectation values is obtained by
averaging over the initial density matrix $\sigma(t_0)$, i.e.,
$\langle\mathcal{O}(t)\rangle=\mathrm{Tr}(\mathcal{O}(t)\sigma(t_0))$.
Since the atom and the vacuum field are supposed to be decoupled
initially, $\sigma(t_0)$ is simply
$\sigma_{at}(t_0)\otimes|0\rangle\langle 0|$ ($|0\rangle$ being
the vacuum field state). Because of the normal ordering, one
immediately gets:
\begin{equation}
\langle\mathcal{F}_{\mathcal{O}}(t)\rangle=0,
\end{equation}
and the time correlation functions of the Langevin forces:
\begin{equation}
\langle\mathcal{F}_{\mathcal{O}}(t)\mathcal{F}_{\mathcal{O}'}(t')\rangle=
-\Gamma\left\langle\sum_q [\mathcal{O}(t),\mathcal{D}^+_q(t)]
[\mathcal{O}'(t'),\mathcal{D}^-_q(t')]\right\rangle \delta(t-t').
\label{corr_Langevin}
\end{equation}
The physical picture of the quantum Langevin approach is to
represent quantum fluctuations by a fluctuating force acting on
the system, in analogy with the usual Brownian motion. Not
surprisingly, this leads to a diffusive-like behavior of
expectation values. More precisely, because of the
$\delta$-function in Eq.~(\ref{corr_Langevin}), we can set $t'=t$
for the atomic operators and we finally obtain in the stationary
regime $t \gg t_0$:
\begin{equation}
\langle\mathcal{F}_{\mathcal{O}}(t)\mathcal{F}_{\mathcal{O}'}(t')\rangle=
\frac{\Gamma}{4} \, D_{\mathcal{O}\,\mathcal{O}'}\, \delta(t-t'),
\end{equation}
where $D$ is a matrix of diffusion constants depending only on the stationary
values of the atomics operators. The stationary hypothesis also
results from the fact that these correlation functions only depend
on the time difference $t-t'$.
From this, it is possible to prove that the quantum regression
theorem applies~\cite{CR92,Cohenrouge}, allowing for the
calculation of two-times correlation functions of the atomic
operators and of their expectation values. From their Fourier
transforms, one can obtain the spectrum of the radiated light.
But, for the reasons mentioned in the introduction, we will
explain how these properties can be obtained in a much simpler way
by directly translating the Langevin equations in the Fourier
domain~\cite{CR92}.
\subsection{Frequency-domain approach}
First, because of the constraint~\eqref{constraint}, only 15
atomic operators are actually independent. More specifically, we
will use the following set, denoted by the column vector
$\mathbf{X}$:
\begin{equation}
\textbf{X}\left\{
\begin{aligned}
\Pi^z_{m_e}&=\frac{1}{2}\left[\Pi^e_{m_e\,m_e}-\Pi^g\right]\\
\Pi^e_{m_e\,m'_e}&=|1\,m_e\rangle\langle 1\,m'_e|\qquad m_e\neq m'_e\\
\mathcal{D}^+_{m_e}&=|1\,m_e\rangle\langle 0\,0|\\
\mathcal{D}^-_{m_e}&=|0\,0\rangle\langle 1\,m_e|
\end{aligned}\right..
\label{def_X}
\end{equation}
The Langevin equations for $\mathbf{X}$ then formally read as follows:
\begin{equation}
\label{Langevin2}
\frac{d}{dt}\mathbf{X}(t)=M\mathbf{X}(t)+\mathbf{L}+\mathbf{F}(t),
\end{equation}
where $M$ is a time-independent matrix depending on the laser Rabi
frequency $\Omega^{L\pm}$, $\mathbf{L}$ is a constant vector
scaling with $\Gamma$ and $\mathbf{F}(t)$ is a vector
characterizing the Langevin forces at work on the atom (for
simplicity, we have dropped the explicit position dependence). The
stationary expectation values are then simply given by:
\begin{equation}
\label{stationary}
\langle \mathbf{X}\rangle=-M^{-1}\mathbf{L}.
\end{equation}
Using Kubo's notations, the Fourier transforms of the different
quantities are defined as follows:
\begin{equation}
\begin{aligned}
f[\Delta]&=\int dt f(t)e^{i\Delta t}\\
f(t)&=\int \frac{d\Delta}{2\pi} f[\Delta]e^{-i\Delta t},
\end{aligned}
\end{equation}
leading to the Langevin equations in the frequency domain:
\begin{equation}
\label{Langevinfreq}
\left(-i\Delta\openone-M\right)\mathbf{X}[\Delta]=2\pi\delta[\Delta]\mathbf{L}+
\mathbf{F}[\Delta].
\end{equation}
Introducing the Green's function
$G[\Delta]=\left(-i\Delta\openone-M\right)^{-1}$, the solution of
the preceding equations simply reads:
\begin{equation}
\label{sollanfre}
\mathbf{X}[\Delta]=G[\Delta]\left(2\pi\delta[\Delta]\mathbf{L}+
\mathbf{F}[\Delta]\right).
\end{equation}
Using $G[0]=-M^{-1}$ and (\ref{stationary}), this solution
separates into a non-fluctuating part $\mathbf{X}_L[\Delta]$ and a
fluctuating (frequency-dependent) part $\mathbf{X}_F[\Delta]$:
\begin{equation}
\left\{
\begin{aligned}
\mathbf{X}_L[\Delta]&=2\pi\delta[\Delta]\langle\mathbf{X}\rangle\\
\mathbf{X}_F[\Delta]&=G[\Delta]\mathbf{F}[\Delta]
\end{aligned}\right..
\end{equation}
From the linearity of the Fourier transform, we still have
$\langle\mathbf{F}[\Delta]\rangle=\mathbf{0}$ implying
$\langle\mathbf{X}_F[\Delta]\rangle=\mathbf{0}$. The time
correlation functions for the Langevin force components, Eq.~(\ref{corr_Langevin}), become:
\begin{equation}
\label{corr11} \langle
\textbf{F}_p[\Delta']\textbf{F}_q[\Delta]\rangle=2\pi\delta[\Delta'+\Delta]D_{pq}.
\end{equation}
where the $2\pi\delta[\Delta'+\Delta]$ function is a direct
consequence of the time-translation invariance, \emph{i.e.} that
we calculate the correlation functions in the stationary regime.
This implies that the correlation function for the components of
$\mathbf{X}_F$ in the frequency domain are:
\begin{equation}
\langle \big(\mathbf{X}_F[\Delta']\big)_p \,
\big(\mathbf{X}_F[\Delta]\big)_q \rangle = 2\pi
\delta[\Delta+\Delta'] \, \big(G \, D \, ^t\!G \big)_{pq}
\end{equation}
where the superscript $t$ means matrix transposition.
The field radiated at frequency $\Delta$ by the atom at a distance
$r\gg\lambda$ (far-field regime) reads as follows:
\begin{equation}
\label{propvac}
\Omega^+_q[\Delta]=-\frac{3}{2}\Gamma\,\mathcal{P}^{\textbf{r}}_{qq'}
\,\mathcal{D}^-_{q'}[\Delta]\frac{e^{ikr}}{kr},
\end{equation}
where we use implicit sum over repeated indices and where
$\mathcal{P}^{\textbf{r}}$ is the projector onto the plane
perpendicular to vector $\textbf{r}$:
\begin{equation}
\mathcal{P}^{\textbf{r}}_{qq'}=
\bar{\boldsymbol{\epsilon}}_{q}\mathcal{P}^{\textbf{r}}\boldsymbol{\epsilon}_{q'}
=\bar{\boldsymbol{\epsilon}}_{q}
\left(\openone-\frac{\textbf{r}{}^{\;\;t}\!\textbf{r}}{r^2}\right)
\boldsymbol{\epsilon}_{q'}=\delta_{qq'}-(-1)^q\frac{\textbf{r}_{-q}
\textbf{r}_{q'}}{r^2},
\end{equation}
where the bar denotes complex conjugation and where
$(\textbf{r}{}^{\;\;t}\!\textbf{r})$ is a dyadic tensor.
The correlation functions $\langle \Omega^{-}_{q'}[\Delta']
\Omega^{+}_q[\Delta]\rangle$ of the light emitted by the atoms is then
proportional to $\langle \mathcal{D}^{+}_{q'}[\Delta']
\mathcal{D}^{-}_q[\Delta]\rangle$ and read as follow:
\begin{equation}
\langle \Omega^{-}_{q'}[\Delta']\Omega^{+}_q[\Delta]\rangle\propto
(2\pi)^2\delta[\Delta]\delta[\Delta']\langle
\mathcal{D}^{+}_{q'}\rangle\langle\mathcal{D}^{-}_q\rangle
+2\pi\delta[\Delta'+\Delta]\sum_{p'p}G_{i'p'}(\Delta')G_{ip}(\Delta)D_{p'p},
\end{equation}
where the index $i$ (resp. $i'$) corresponds to
$\mathcal{D}^{-}_q$ (resp. $\mathcal{D}^{+}_{q'}$). The
non-fluctuating part gives rise to a spectral component of the
emitted light at exactly the incident laser frequency and is thus
naturally called the \textit{elastic} part. The fluctuating part
gives rise to the \textit{inelastic} Mollow triplet
spectrum~\cite{pra5M}, whose properties (position and width of the
peaks) are given by the poles of $G[\Delta]$, \emph{i.e.} by the
complex eigenvalues of $M$. Actually, we simply recover the
results of the quantum regression theorem, which states that the
atomic time correlation functions evolve with the same equations
than the expectation values
$\dot{\langle\mathbf{X}\rangle}=M\langle\mathbf{X}\rangle+\mathbf{L}$~\cite{Cohenrouge,pr188M}.
\section{Two-atom case}
\label{twoatoms}
\subsection{Optical Bloch equations}
\label{equivalence_ob}
We now consider two isolated atoms, located at fixed positions
$\textbf{R}_1$ and $\textbf{R}_2$. Defining
$\textbf{R}=\textbf{R}_2-\textbf{R}_1 = R \, \textbf{u}$ (with
$R=|\textbf{R}|$ and $\textbf{u}$ the unit vector joining atom 1
to atom 2), we assume the far-field condition $R\gg\lambda$ to
hold. We also assume that $R$ is sufficiently small for the light
propagation time $R/c$ to be much smaller than any typical atomic
timescales $(\Gamma^{-1},\delta^{-1},\Omega_L^{-1}$). In this
regime, all quantities involving the two atoms
are to be computed at the same time $t.$
The contribution of the atom-atom dipole interaction in the
Langevin equation for any
atomic operator $\mathcal{O}$ reads:
\begin{equation}
\label{langevin12}
{\left.\frac{d\mathcal{O}}{dt}\right|}_{\text{dip.}}=i\frac{3\Gamma}{4}
\left\{\left(\left[\mathcal{O},\mathcal{D}^{1+}_q\right]
\mathcal{P}^{\textbf{R}}_{qq'}\mathcal{D}^{2-}_{q'}+
\left[\mathcal{O},\mathcal{D}^{2+}_q\right]\mathcal{P}^{\textbf{R}}_{qq'}
\mathcal{D}^{1-}_{q'}\right)\frac{e^{ikR}}{kR}
+\left(\mathcal{D}^{1+}_q\mathcal{P}^{\textbf{R}}_{qq'}
\left[\mathcal{O},\mathcal{D}^{2-}_{q'}\right]+
\mathcal{D}^{2+}_q\mathcal{P}^{\textbf{R}}_{qq'}
\left[\mathcal{O},\mathcal{D}^{1-}_{q'}\right]\right)
\frac{e^{-ikR}}{kR}\right\}.
\end{equation}
In the OB equations, the two-atom system is entirely described by
the set of 256 operators $X_{ij}$ made of all possible products
$X_i^{1}X_j^{2}$. The stationary expectation values $\langle
X_{ij}\rangle$ are then obtained as solutions of a linear system
resembling equation~\eqref{stationary}. This is the
approach used in~\cite{prl94SMB}, where such optical Bloch equations
are solved.
Since the two atoms are far enough from each other, the
electromagnetic field radiated by one atom onto the other can be
treated as a perturbation with respect to the incident laser
field. More precisely, the solutions $\langle X_{ij}\rangle$ can
be expanded up to second order in powers of $g$ and $\bar{g}$:
\begin{equation}
\langle X_{ij}\rangle=\langle X_{ij}\rangle^{(0)}+ g\,\langle
X_{ij}\rangle^{(g)}+\bar{g}\,\langle X_{ij}\rangle^{(\bar{g})}+
g\bar{g}\,\langle X_{ij}\rangle^{(g\bar{g})}+ g^2 \,\langle
X_{ij}\rangle^{(gg)} + \bar{g}^2 \,\langle
X_{ij}\rangle^{(\bar{g}\bar{g})}
\end{equation}
where the complex coupling constant $g$ is:
\begin{equation}
g=i\frac{3\Gamma}{2}\frac{\exp{(ikR)}}{kR}
\end{equation}
In fact, it will be shown below that both terms in $g^2$ and
$\bar{g}^2$ give a vanishing contribution to the coherent
backscattering signal.
As explained in the introduction, this approach has two drawbacks:
(i) the solutions obtained in this way are global and, thus, do
not provide a simple understanding of the properties of the
emitted light; (ii) when the two atoms are embedded in a medium
whose susceptibility strongly depends on the frequency, the field
radiated by one atom onto the other at a given time $t$ now
depends on the atomic operators of the first atom at earlier times
(since retardation effects become frequency dependent). Time
correlation functions in the dipole interaction then explicitly
show up.
\subsection{Langevin approach}
The Langevin equations for the two sets of atomic operators
$\mathbf{X}^{\alpha}$, with $\alpha=1,2$, read formally:
\begin{equation}
\dot{\mathbf{X}}^{\alpha}=M^{\alpha}\mathbf{X}^{\alpha}+\mathbf{L}+
\mathbf{F}^{\alpha}+
gT^{q+}\mathbf{X}^{\alpha}\mathcal{P}^{\textbf{R}}_{qq'}
\mathcal{D}^{\beta-}_{q'}+\bar{g}\mathcal{D}^{\beta+}_q
\mathcal{P}^{\textbf{R}}_{qq'}T^{q'-}\mathbf{X}^{\alpha},
\end{equation}
where $\beta$ denotes the other atom and where $T^{q\pm}$ are
$15\times15$ matrices defined by
$\left[X_i,\mathcal{D}^{\pm}_q\right]=\pm 2T^{q\pm}_{ij}X_j$.
Taking the Fourier transform of these equations, one gets:
\begin{equation}
\label{langevinalpha}
\mathbf{X}^{\alpha}[\Delta]=G^{\alpha}[\Delta]
\left(2\pi\delta[\Delta]\mathbf{L}+\mathbf{F}^{\alpha}[\Delta]\right)
+gG^{\alpha}[\Delta]T^{q+}\mathcal{P}^{\textbf{R}}_{qq'}
\left(\mathbf{X}^{\alpha}\otimes\mathcal{D}^{\beta-}_{q'}\right)[\Delta]
-\bar{g}G^{\alpha}[\Delta]\mathcal{P}^{\textbf{R}}_{qq'}T^{q'-}
\left(\mathcal{D}^{\beta+}_q\otimes\mathbf{X}^{\alpha}\right)[\Delta],
\end{equation}
where $\otimes$ is the convolution operator:
\begin{equation}
\left(A\otimes B\right)[\Delta]=\frac{1}{2\pi}\iint d\Delta_1d\Delta_2
\delta[\Delta_1+\Delta_2-\Delta]A[\Delta_1]B[\Delta_2].
\end{equation}
Introducing, for simplicity, the following notations:
\begin{equation}
\left\{\begin{aligned}
\mathbf{X}^{\alpha^{(0)}}[\Delta]&=G^{\alpha}[\Delta]
\left(2\pi\delta[\Delta]\mathbf{L}+\mathbf{F}^{\alpha}[\Delta]\right)\\
G^{\alpha^+_q}[\Delta]&=G^{\alpha}[\Delta]T^{q'+}
\mathcal{P}^{\textbf{R}}_{q'q}\\
G^{\alpha^-_q}[\Delta]&=G^{\alpha}[\Delta]T^{q'-}
\mathcal{P}^{\textbf{R}}_{qq'}
\end{aligned}\right.,
\end{equation}
equation~\eqref{langevinalpha} becomes:
\begin{equation}
\label{langevin2}
\mathbf{X}^{\alpha}[\Delta]=\mathbf{X}^{\alpha^{(0)}}[\Delta]+
gG^{\alpha_q^+}[\Delta]\left(\mathbf{X}^{\alpha}\otimes
\mathcal{D}^{\beta-}_{q}\right)[\Delta]-
\bar{g}G^{\alpha_q^-}[\Delta]\left(\mathcal{D}^{\beta+}_q\otimes
\mathbf{X}^{\alpha}\right)[\Delta],
\end{equation}
from which one gets the expansion in power of $g$ and $\bar{g}$ (up to
$g\bar{g}$) for
the atomic operators:
\begin{equation}
\label{opexp}
\begin{aligned}
X_i^{\alpha}[\Delta]&=X_i^{\alpha^{(0)}}[\Delta]
+gG_{ij}^{\alpha^+_q}[\Delta] \bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]
-\bar{g}G_{ij}^{\alpha^-_q}[\Delta]
\bigl(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
X_j^{\alpha^{(0)}}\bigr)[\Delta]\\
&-g\bar{g}\biggl\{G_{ij}^{\alpha^+_q}[\Delta]\left(X_j^{\alpha^{(0)}}
\otimes G^{\beta^-_p}_{\mathcal{D}_q^-j'}
\left(\mathcal{D}^{{\alpha+}^{(0)}}_p \otimes
X^{\beta^{(0)}}_{j'}\right)\right)[\Delta]
+G_{ij}^{\alpha^+_q}[\Delta]\left(G^{\alpha^-_p}_{jj'}
\left(\mathcal{D}^{{\beta+}^{(0)}}_p\otimes
X_{j'}^{\alpha^{(0)}}\right)\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\right)[\Delta]
\biggr.\\
&\phantom{+g\bar{g}}\biggl.\quad+G_{ij}^{\alpha^-_q}[\Delta]
\left(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
G^{\alpha^+_p}_{jj'}\left(X_{j'}^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_p\right)\right)[\Delta]
+G_{ij}^{\alpha^-_q}[\Delta]\left(G^{\beta^+_p}_{\mathcal{D}_q^-j'}
\left(X_{j'}^{\beta^{(0)}}\otimes
\mathcal{D}^{{\alpha-}^{(0)}}_p\right)\otimes
X_j^{\alpha^{(0)}}\right)[\Delta] \biggr\}.
\end{aligned}
\end{equation}
Two-body terms expansions, obtained from Eq.~\eqref{opexp}, read
as follows:
\begin{equation}
\begin{aligned}
\label{corrggb} {X}_{i'}^{\beta}[\Delta']{X}_i^{\alpha}[\Delta]&=
{X}_{i'}^{\beta^{(0)}}[\Delta']{X}_i^{\alpha^{(0)}}[\Delta]\\
&+g\biggl\{{X}_{i'}^{\beta^{(0)}}[\Delta']
G_{ij}^{\alpha^+_q}[\Delta] \bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]
+G_{i'j'}^{\beta^+_q}[\Delta'] \bigl(X_{j'}^{\beta^{(0)}}\otimes
\mathcal{D}^{{\alpha-}^{(0)}}_q\bigr)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\biggr\}\\
&-\bar{g}\biggl\{{X}_{i'}^{\beta^{(0)}}[\Delta']
G_{ij}^{\alpha^-_q}[\Delta]
\bigl(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
X_j^{\alpha^{(0)}}\bigr)[\Delta]+ G_{i'j'}^{\beta^-_q}[\Delta']
\bigl(\mathcal{D}^{{\alpha+}^{(0)}}_q\otimes
X_{j'}^{\beta^{(0)}}\bigr)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\biggr\}\\
&-g\bar{g}\biggl\{\text{see appendix~\ref{ggbar}}\biggr\}\\
{X}_{i'}^{\alpha}[\Delta']{X}_i^{\alpha}[\Delta]&=
{X}_{i'}^{\alpha^{(0)}}[\Delta']{X}_i^{\alpha^{(0)}}[\Delta]\\
&+g\biggl\{{X}_{i'}^{\alpha^{(0)}}[\Delta']
G_{ij}^{\alpha^+_q}[\Delta] \bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]
+G_{i'j'}^{\alpha^+_q}[\Delta'] \bigl(X_{j'}^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\biggr\}\\
&-\bar{g}\biggl\{{X}_{i'}^{\alpha^{(0)}}[\Delta']
G_{ij}^{\alpha^-_q}[\Delta]
\bigl(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
X_j^{\alpha^{(0)}}\bigr)[\Delta] +G_{i'j'}^{\alpha^-_q}[\Delta']
\bigl(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
X_{j'}^{\alpha^{(0)}}\bigr)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\biggr\}\\
&-g\bar{g}\biggl\{\text{see appendix~\ref{ggbar}}\biggr\}.
\end{aligned}
\end{equation}
Obviously, the power expansion of the expectation values can be derived
from the quantum average of the preceding equations, but not as easily
as it seems. Indeed, if one formally writes:
\begin{equation}
\left\{\begin{aligned}
X_{i'}^{\alpha'}[\Delta']{X}_i^{\alpha}[\Delta]&=\sum_{ab}O(a,b) g^a\bar{g}^b\\
\langle X_{i'}^{\alpha'}[\Delta']{X}_i^{\alpha}[\Delta]\rangle
&=\sum_{ab}C(a,b) g^a\bar{g}^b
\end{aligned}\right.,
\end{equation}
then $C(a,b)$ is not simply equal to
$\langle O(a,b)\rangle$. Actually, $C(a,b)$ depends on all
$\langle O(a',b')\rangle$ for $(a',b')\le(a,b)$, and this for two
reasons:
\begin{itemize}
\item[$\bullet$] for a given atom $\alpha$, the frequency correlation
functions $\langle F_p^{\alpha}[\Delta']F_q^{\alpha}[\Delta]\rangle$
are given by $2\pi\delta[\Delta'+\Delta]D_{pq}$, where $D_{pq}$
depends on the stationary values. But the latter are modified by the
second atom and, thus, must also be expanded in power of $g$ and
$\bar{g}$. This implies, for example, that the first term
${X}_{i'}^{\alpha^{(0)}}[\Delta']{X}_i^{\alpha^{(0)}}[\Delta]$ in
the expansion of ${X}_{i'}^{\alpha}[\Delta']{X}_i^{\alpha}[\Delta]$
(Eq.~\eqref{corrggb}) will contribute to all coefficients of
$\langle {X}_{i'}^{\alpha}[\Delta']{X}_i^{\alpha}[\Delta]\rangle$.
\item[$\bullet$] the Langevin forces acting on two different atoms are
correlated since they both originate from the vacuum quantum
field. More precisely, their frequency correlation
functions depend on their relative distance. This dependence
is analogous to the correlation function of a speckle pattern
(resulting from the random superposition of plane waves with
the same wavelength but arbitrary directions):
\begin{equation}
\begin{aligned}
\label{corrfafab} \langle
F_{i'}^{\beta}[\Delta']F_i^{\alpha}[\Delta]\rangle&=
2\pi\delta[\Delta'+\Delta]\frac{3}{2}\Gamma\frac{\sin{kR}}{kR}
T_{i'j'}^{q'+}\mathcal{P}^{\textbf{R}}_{q'q}T_{ij}^{q-}
\langle X_{j'}^{\beta}X_j^{\alpha}\rangle\\
&=-\frac{1}{2}\biggl(g+\bar{g}\biggr)2\pi\delta[\Delta'+\Delta]
T_{i'j'}^{q'+}\mathcal{P}^{\textbf{R}}_{q'q}T_{ij}^{q-}
\langle X_{j'}^{\beta}X_j^{\alpha}\rangle\\
&=-\frac{1}{2}\biggl(g+\bar{g}\biggr)2\pi\delta[\Delta'+\Delta]
D^{\beta\alpha}_{i'i}.
\end{aligned}
\end{equation}
Thus, terms like $ {X}_{i'}^{\beta^{(0)}}[\Delta']
\bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]$ appearing in
equation~\eqref{corrggb} will also contribute to higher-order
coefficients in the power expansion of
$\langle {X}_{i'}^{\beta}[\Delta']{X}_i^{\alpha}[\Delta]\rangle$. One must note
that, when $R\rightarrow 0$,
$\mathcal{P}^{\textbf{R}}_{q'q}\rightarrow\frac{2}{3}\delta_{q'q}$
and one recovers the single atom correlation functions given by
Eq.~\eqref{corr11}, which emphasizes the consistency of the
present approach.
\end{itemize}
Despite these subtleties, it is nevertheless possible to calculate
power expansions of the atomic correlation functions. More
precisely, in order to emphasize the validity of the present
approach, we will compare the results obtain from the OB equations
and from the Langevin approach. Indeed from the atomic correlation
functions, the stationary solutions can be calculated by inverse
Fourier transform as follows:
\begin{equation}
\langle X_{i'}^{\alpha}X_{i}^{\alpha'}\rangle=\frac{1}{(2\pi)^2}\iint
d\Delta'd\Delta\langle X_{i'}^{\alpha}[\Delta']X_{i}^{\alpha'}[\Delta]\rangle.
\end{equation}
As a specific example, the coefficient proportional to $g$ in the
perturbative expansion of
$\langle{X}_{i'}^{\beta}[\Delta']{X}_i^{\alpha}[\Delta]\rangle$ is
given by:
\begin{equation}
\begin{aligned}
\langle{X}_{i'}^{\beta}[\Delta']{X}_i^{\alpha}[\Delta]\rangle^{(g)}&=
\underline{\langle{X}_{i'}^{\beta^{(0)}}[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\rangle^{(g)}}
+\langle{X}_{i'}^{\beta^{(0)}}[\Delta']
G_{ij}^{\alpha^+_q}[\Delta] \bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]\rangle^{(0)}\\
&+\langle G_{i'j'}^{\beta^+_q}[\Delta']
\bigl(X_{j'}^{\beta^{(0)}}\otimes
\mathcal{D}^{{\alpha-}^{(0)}}_q\bigr)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\rangle^{(0)}\\
&=\underline{G^{\beta}_{i'j'}[\Delta']G^{\alpha}_{ij}[\Delta]
\langle
F_{j'}^{\beta}[\Delta']F_{j}^{\alpha}[\Delta]\rangle^{(g)}}+
G_{ij}^{\alpha^+_q}[\Delta]\langle X_j^{\alpha^{(0)}}\rangle
\langle{X}_{i'}^{\beta^{(0)}}[\Delta']
\mathcal{D}^{{\beta-}^{(0)}}_q[\Delta]\rangle^{(0)}\\
&+G_{i'j'}^{\beta^+_q}[\Delta'] \langle
X_{j'}^{\beta^{(0)}}\rangle \langle
\mathcal{D}^{{\alpha-}^{(0)}}_q[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\rangle^{(0)},
\end{aligned}
\end{equation}
where we have used the fact that terms like $\langle
X^{\alpha^{(0)}}X^{\beta^{(0)}}\rangle^{(0)}$ (\emph{i.e.} zeroth
order) actually factorize into $\langle X^{\alpha}\rangle \langle
X^{\beta}\rangle$ since
their fluctuating parts necessarily give rise to higher orders in $g$
and $\bar{g}$, see Eq.~\eqref{corrfafab}. The underlined terms
correspond to the non-vanishing correlations of the quantum vacuum
fluctuations evaluated at the two atom positions.
Finally, separating elastic and inelastic part, one gets:
\begin{equation}
\begin{aligned}
\langle{X}_{i'}^{\beta}[\Delta']{X}_i^{\alpha}[\Delta]\rangle^{(g)}&=
(2\pi)^2\delta[\Delta']\delta[\Delta]
\biggl(G_{ij}^{\alpha^+_q}[0]\langle X_j^{\alpha^{(0)}}\rangle
\langle{X}_{i'}^{\beta^{(0)}}\rangle
\langle\mathcal{D}^{{\beta-}^{(0)}}_q\rangle
+G_{i'j'}^{\beta^+_q}[0] \langle X_{j'}^{\beta^{(0)}}\rangle
\langle \mathcal{D}^{{\alpha-}^{(0)}}_q\rangle
\langle{X}_i^{\alpha^{(0)}}\rangle\biggr)\\
&+2\pi\delta[\Delta'+\Delta]\biggl(-\frac{1}{2}
\underline{G^{\beta}_{i'j'}[\Delta']G^{\alpha}_{ij}[\Delta]
D^{\beta\alpha^{(0)}}_{j'j}}
+G_{ij}^{\alpha^+_q}[\Delta]G_{i'j'}^{\beta}[\Delta']
G_{\mathcal{D}^-_qk'}^{\beta}[\Delta]
D^{\beta\beta^{(0)}}_{j'k'}\langle X_j^{\alpha^{(0)}}\rangle
\biggr.\\
&\phantom{+2\pi\delta[\Delta'+\Delta]\biggl.-\frac{1}{2}}
+G_{i'j'}^{\beta^+_q}[\Delta']G_{\mathcal{D}^-_qk}^{\alpha}[\Delta']
G_{ij}^{\alpha}[\Delta]D^{\alpha\alpha^{(0)}}_{kj}\langle
X_{j'}^{\beta^{(0)}}\rangle\biggr).
\end{aligned}
\end{equation}
The corresponding stationary solution then reads:
\begin{equation}
\begin{aligned}
\langle{X}_{i'}^{\beta}{X}_i^{\alpha}\rangle^{(g)}&=
G_{ij}^{\alpha^+_q}[0]\langle X_j^{\alpha^{(0)}}\rangle
\langle{X}_{i'}^{\beta^{(0)}}\rangle
\langle\mathcal{D}^{{\beta-}^{(0)}}_q\rangle
+G_{i'j'}^{\beta^+_q}[0] \langle X_{j'}^{\beta^{(0)}}\rangle
\langle \mathcal{D}^{{\alpha-}^{(0)}}_q\rangle
\langle{X}_i^{\alpha^{(0)}}\rangle\\
&+\frac{1}{2\pi}\int d\Delta\biggl(-\frac{1}{2}
\underline{G^{\beta}_{i'j'}[-\Delta]G^{\alpha}_{ij}[\Delta]
D^{\beta\alpha^{(0)}}_{j'j}}
+G_{ij}^{\alpha^+_q}[\Delta]G_{i'j'}^{\beta}[-\Delta]
G_{\mathcal{D}^-_qk'}^{\beta}[\Delta]
D^{\beta\beta^{(0)}}_{j'k'}\langle X_j^{\alpha^{(0)}}\rangle
\biggr.\\
&\phantom{+\frac{1}{2\pi}\int d\Delta\biggl.-\frac{1}{2}}
+G_{i'j'}^{\beta^+_q}[-\Delta]G_{\mathcal{D}^-_qk}^{\alpha}[-\Delta]
G_{ij}^{\alpha}[\Delta]D^{\alpha\alpha^{(0)}}_{kj}\langle
X_{j'}^{\beta^{(0)}}\rangle\biggr).
\end{aligned}
\end{equation}
All quantities above only depend on the stationary values without
coupling between the atoms and thus can be calculated from the
single atom solutions. Furthermore, the integration over $\Delta$
can be performed either numerically or analytically by the theorem
of residues once the poles of $G$ (\emph{i.e.} the complex
eigenvalues of $M$) are known. Because of causality, they
all lie in the lower-half of the complex plane. In
practice, we have checked that we effectively recover, from the
preceding expressions, the results obtained from the full OB
equations. In particular, the contribution of the correlations of
the quantum vacuum
fluctuations evaluated at the two atom positions (the underlined term) is
essential to get the correct results.
The same kind of expressions can be derived for $g\bar{g}$ terms,
but they are slightly more complicated, since they explicitly
involve three-body correlation functions, more precisely terms
like:
\begin{equation}
\left\{\begin{aligned}
&G_{ij}^{\alpha^+_q}[\Delta]\left\langle{X}_{i'}^{\beta^{(0)}}[\Delta']
\bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]\right\rangle^{(\bar{g})}\\
&G_{ij}^{\alpha^+_q}[\Delta]\left\langle {X}_{i'}^{\beta}[\Delta']
\left(G^{\alpha^-_p}_{jj'}
\left(\mathcal{D}^{{\beta+}^{(0)}}_p\otimes
X_{j'}^{\alpha^{(0)}}\right)\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\right)[\Delta]\right\rangle^{(0)}\\
\end{aligned}\right.,
\end{equation}
which require the calculation of three-points Langevin force
correlation functions like:
\begin{equation}
\left\{\begin{aligned}
&G_{ij}^{\alpha^+_q}[\Delta]G^{\beta}_{i'j'}[\Delta']
\frac{1}{2\pi}\iint
d\Delta_1d\Delta_2\delta[\Delta_1+\Delta_2-\Delta]
G_{jk}^{\alpha}[\Delta_1]G_{\mathcal{D}^-_qk'}^{\beta}[\Delta_2]
\left\langle F^{\beta}_{j'}[\Delta']F^{\alpha}_{k}[\Delta_1]
F^{\beta}_{k'}[\Delta_2]\right\rangle^{(\bar{g})}\\
&G_{ij}^{\alpha^+_q}[\Delta]G^{\beta}_{i'k'}[\Delta']
\frac{1}{2\pi}\iint
d\Delta_1d\Delta_2\delta[\Delta_1+\Delta_2-\Delta]
G^{\alpha^-_p}_{jj'}[\Delta_1]G_{\mathcal{D}^+_pk}^{\beta}[\Delta_1]
G_{\mathcal{D}^+_pk''}^{\beta}[\Delta_2] \left\langle
F^{\beta}_{k'}[\Delta']F^{\beta}_{k}[\Delta_1]
F^{\beta}_{k''}[\Delta_2]\right\rangle^{(0)}
\end{aligned}\right..
\end{equation}
These correlations functions are non-zero even if they involve an
odd number of Langevin forces, emphasizing that the statistical
properties of the vacuum field fluctuations are far from Gaussian.
Nevertheless, the explicit expressions of the above quantities can
be derived (see appendix~\ref{matdthreebody}). They lead to rather
complicated and tedious formulae for the atomic correlation
functions at order $g\bar{g}$. From that, we get the corresponding
stationary expectations values. Again, we have checked that we
indeed recover the OB results.
\subsection{Incorporation of an effective medium}
Finally, and in sharp contrast to optical Bloch equations,
it is very easy to adapt all the preceding results to the
case of propagation in a medium with a frequency-dependent
complex susceptibility. Indeed, propagation is controlled by the complex
amplitude $g$ so that the field radiated by an atom at a distance
$R$ and at frequency $\Delta$ will be given by:
\begin{equation}
\Omega^+_q[\Delta]= i \, g \, \mathcal{P}^{\textbf{R}}_{qq'}
\mathcal{D}^-_{q'}[\Delta]
\exp{\left(-\frac12\frac{R}{\ell^{+}[\Delta]}\right)},
\end{equation}
where $\ell^+[\Delta]$ is the (complex) scattering mean-free path
satisfying the dilute regime condition $k|\ell^+[\Delta]|\gg 1$.
The real part of $1/\ell^+[\Delta]$ describes the exponential
attenuation of the field during its propagation in the medium
while the imaginary part describes the additional dephasing
induced by the medium.
More complicated formulas, accounting for possible variations of
$\ell$ with position, birefringence effects, or even
nonlinearities in propagation, can be derived in the same spirit.
In all preceding equations, leading to the calculation of the
correlation functions, any occurrence of the dipole operators must
then simply be replaced by:
\begin{equation}
\mathcal{D}^{\mp} \quad \to \quad
\mathcal{D}^{\mp}\exp{\left(-\frac{R}{2\ell^{\pm}[\Delta]}\right)}
\end{equation}
while keeping the same "medium-free" coupling constant $g$. In
this way, the present approach can be easily extended to the
situation where the two atoms are embedded in a medium. In the
case of a nonlinear medium, this could lead to a self-consistent
set of nonlinear equations.
It is important to stress that accounting for the effective medium
is rather straightforward in this frequency-domain approach but is
a much more difficult task in the temporal-domain approach.
Indeed, one basic hypothesis for deducing OB equations from the
Langevin approach -- see section~\ref{equivalence_ob} -- is that
the light propagation time between the two atoms is much shorter
than any typical atomic timescale. When this condition is
fulfilled, it is possible to evaluate expectation values at equal
times for both atoms, producing the set of closed OB equations. In
the presence of a surrounding medium, propagation between the two
atoms is affected and this basic assumption may fail. If the
refraction index of the dilute medium is smoothly varying with
frequency, then the corresponding propagation term is also
smoothly varying with frequency and can be factored out. Thus,
except for the exponential attenuation, one may recover the OB
equations where equal times must be used for atoms 1 and 2. On the
contrary, if the propagation term has a complicated frequency
dependence, the problem cannot be simply reduced to OB equations.
It will rather involve operators evaluated at the other atom, but
\emph{at different times}, thus leading to a much more complicated
structure. This difficulty may even take place in a dilute medium
with refraction index close to unity. Indeed, the important
parameter is the time delay induced by the medium, itself related
to the \emph{derivative} of the index of refraction with respect
to frequency. If the medium is composed of atoms having sharp
resonances, the effective group velocity can be reduced by several
orders of magnitude, consequently increasing by the same amount
the propagation time between the two atoms. Around the atomic
resonance line, the typical propagation time delay induced by the
medium over one mean free path depends on the laser detuning but
is of the order of the atomic timescale for the internal dynamics,
namely $\Gamma^{-1}$~\cite{Labeyrie:radiation_trapping}. In this
case, only the full Langevin treatment developed in this paper can
properly account for the effect of the average atomic medium. Its
practical implementation calls for an investigation on its own and
is thus postponed to a future paper. We must also note that, if
the surrounding medium is composed of the same atoms than the
scatterers, it is not completely clear that propagation in the
medium can be described ``classically", \emph{i.e.} that the
correlation between the Langevin forces acting on the scatterers
and the Langevin forces acting on the medium can be safely
neglected.
For the rest of this paper, we will consider two isolated atoms in vacuum.
\section{Main results}
\subsection{Scattered field correlation functions in the CBS configuration}
In the case of a large number of atoms and for a given
configuration, the interference between all possible multiple
scattering paths gives rise to a speckle pattern. When averaging
the intensity scattered off the sample over all possible positions
of the atoms, one recovers the CBS phenomenon: the intensity
radiated in the direction opposite to the incident beam is up to
twice larger than the background intensity and gradually decreases
to the background value over an angular range $\Delta\theta$
scaling essentially as $(k\ell)^{-1}$, with $\ell$ the scattering
mean-free path. In the present case, the averaging procedure is
performed numerically by integrating over the relative positions
of the two atoms. As will be seen below, the far-field condition
$kR\gg 1$ allows for an \emph{a priori} selection of the dominant
terms contributing to the CBS signal.
The field radiated by the two atoms in the direction $\textbf{n}$
at a distance $r\gg R\gg\lambda$, in the polarization channel
$\boldsymbol{\epsilon}^{\mathrm{out}}$ orthogonal to $\textbf{n}$
($\boldsymbol{\epsilon}^{\mathrm{out}}\cdot \textbf{n} =0$), is
given by:
\begin{equation}
\Omega^{+}_{\mathrm{out}}[\textbf{n},\Delta]=-\frac{3}{2}\Gamma
\epsilon^{\mathrm{out}}_q
\left(\mathcal{D}^{1-}_{q}[\Delta]e^{-ik\textbf{n}\cdot\textbf{R}_1}
+\mathcal{D}^{2-}_{q}[\Delta]e^{-ik\textbf{n}\cdot\textbf{R}_2}\right)
\frac{e^{ikr}}{kr},
\end{equation}
so that the field correlation function in this channel reads:
\begin{multline}
\label{field1}
\langle\Omega^{-}_{\mathrm{out}}[\textbf{n},\Delta']
\Omega^{+}_{\mathrm{out}}[\textbf{n},\Delta]\rangle=
\big(\frac{3\Gamma}{2kr}\big)^2 \,
{\epsilon}^{\mathrm{out}}_q\epsilon^{\mathrm{out}}_{p} \biggl\{
\langle\mathcal{D}^{1+}_{p}[\Delta']\mathcal{D}^{1-}_{q}[\Delta]\rangle+
\langle\mathcal{D}^{2+}_{p}[\Delta']
\mathcal{D}^{2-}_{q}[\Delta]\rangle\biggr.\\
\biggl.+e^{ik\textbf{n}\cdot\textbf{R}}
\langle\mathcal{D}^{2+}_{p}[\Delta']\mathcal{D}^{1-}_{q}[\Delta]\rangle+
e^{-ik\textbf{n}\cdot\textbf{R}}
\langle\mathcal{D}^{1+}_{p}[\Delta']\mathcal{D}^{2-}_{q}[\Delta]\rangle
\biggr\}.
\end{multline}
The CBS effect occurs when the total phase in the interference
terms in the preceding expression becomes independent of the
positions of the atom. This phase accumulates during the
propagation of the incident laser beam to the atoms and during the
propagation of the radiated field between the two atoms. The phase
factor due to the incoming laser beam (a plane wave with wave
number $\textbf{k}_L = k \, \textbf{n}_L$) can be explicitly
factorized out of the atomic operators as follows:
\begin{equation}
\tilde{\mathcal{D}}^{\alpha\pm}_q =\mathcal{D}^{\alpha\pm}_q\,
e^{\pm i\textbf{k}_L\cdot\textbf{R}_{\alpha}}.
\end{equation}
The other components of $\tilde{X}$, cf. Eq.~(\ref{def_X}),
are populations and not affected by this phase factor.
In the single atom case, the expectation values of the hereby defined
operators
$\tilde{\mathcal{D}}^{\alpha\pm}_q$ are independent of the
positions of the atoms. Defining $\phi=\textbf{k}_L\cdot\textbf{R}$
and
\begin{equation}
g_1=ge^{i\phi}\qquad g_2=ge^{-i\phi} ,
\end{equation}
the Langevin equations~\eqref{langevin2} then become:
\begin{equation}
\label{langevin3}
\tilde{\mathbf{X}}^{\alpha}[\Delta]=\tilde{\mathbf{X}}^{\alpha^{(0)}}[\Delta]+
g_{\alpha}\tilde{G}^{\alpha_q^+}[\Delta]\left(\tilde{\mathbf{X}}^{\alpha}\otimes
\tilde{\mathcal{D}}^{\beta-}_{q}\right)[\Delta]+
\bar{g}_{\alpha}\tilde{G}^{\alpha_q^-}[\Delta]
\left(\tilde{\mathcal{D}}^{\beta+}_q
\otimes\tilde{\mathbf{X}}^{\alpha}\right)[\Delta],
\end{equation}
In the preceding equation, the Green's functions $\tilde{G}$ are
now independent of the position of the atoms, so that the phase
information due to the incident laser beam is entirely contained
in the coefficients $g_{\alpha}$.
Frequency correlation functions of the Langevin forces,
eq.~\eqref{corrfafab}, must also be modified accordingly:
\begin{equation}
\langle \tilde{F}_{i'}^{\beta}[\Delta']\tilde{F}_i^{\alpha}[\Delta]\rangle=
-\frac{1}{2}\biggl(g_{\beta}+\bar{g}_{\alpha}\biggr)
2\pi\delta[\Delta'+\Delta]\tilde{D}^{\beta\alpha}_{i'i}.
\end{equation}
Dropping for simplicity, the tilde notation, the field correlation
function~\eqref{field1}, in the backward direction
$\textbf{n}=-\textbf{n}_L$, becomes:
\begin{multline}
\label{field2}
\langle\Omega^{-}_{\mathrm{out}}[-\textbf{n}_L,\Delta']
\Omega^{+}_{\mathrm{out}}[-\textbf{n}_L,\Delta]\rangle=
\left(\frac{\Gamma}{kr}\right)^2 \,
{\epsilon}^{\mathrm{out}}_q\epsilon^{\mathrm{out}}_{p}\biggl\{
\langle\mathcal{D}^{1+}_{p}[\Delta']\mathcal{D}^{1-}_{q}[\Delta]\rangle+
\langle\mathcal{D}^{2+}_{p}[\Delta']\mathcal{D}^{2-}_{q}[\Delta]\rangle
\biggr.\\
\biggl.+e^{-2i\phi}
\langle\mathcal{D}^{2+}_{p}[\Delta']\mathcal{D}^{1-}_{q}[\Delta]\rangle+
e^{2i\phi}
\langle\mathcal{D}^{1+}_{p}[\Delta']\mathcal{D}^{2-}_{q}[\Delta]\rangle
\biggr\}.
\end{multline}
The configuration average is then performed in two steps. Since we
are working in the limit $kR\gg 1$, the first one is to keep only
terms with a total phase independent of $kR$. In the power
expansion with respect to the four parameters $g_1$, $g_2$,
$\bar{g}_1$ and $\bar{g}_2$, this simply amounts to keep terms
with even powers of $g_{\alpha}\bar{g}_{\alpha'}$. This obviously
cancels any $\phi$ dependence. More precisely, the
field correlation function in the backward direction, beside
the trivial zeroth order (in $g$) term, is given by:
\begin{equation}
\label{fieldmoy}
\begin{aligned}
\langle\Omega^{-}_{\mathrm{out}}[-\textbf{n}_L,\Delta']
\Omega^{+}_{\mathrm{out}}[-\textbf{n}_L,\Delta]\rangle^{(2)}&=
\left(\frac{\Gamma}{kr}\right)^2 \,
{\epsilon}^{\mathrm{out}}_q\epsilon^{\mathrm{out}}_{p}\biggl\{
\langle\mathcal{D}^{1+}_{p}[\Delta']\mathcal{D}^{1-}_{q}[\Delta]
\rangle^{(g_1\bar{g}_1)}+ \langle\mathcal{D}^{2+}_{p}[\Delta']
\mathcal{D}^{2-}_{q}[\Delta]\rangle^{(g_2\bar{g}_2)}\biggr.\\
&\phantom{\frac{1}{k^2R^2}\frac{9}{4}\Gamma^2
{\epsilon}^{\mathrm{out}}_q\epsilon^{\mathrm{out}}_{p}}\biggl.+
\langle\mathcal{D}^{2+}_{p}[\Delta']\mathcal{D}^{1-}_{q}[\Delta]
\rangle^{(g_1\bar{g}_2)}+
\langle\mathcal{D}^{1+}_{p}[\Delta']\mathcal{D}^{2-}_{q}[\Delta]
\rangle^{(g_2\bar{g}_1)}
\biggr\}\\
&= \left(\frac{\Gamma}{kr}\right)^2 \,
\big(L[\Delta',\Delta]+C[\Delta',\Delta]\big).
\end{aligned}
\end{equation}
The preceding field correlation function still depends on the
relative orientation of the atoms through the projector
$\mathcal{P}^{\textbf{R}}$, so that, in a second step, an
additional average over $\textbf{R}$ must be performed. In the
preceding equation, the first two terms correspond to the usual
``ladder'' terms $L[\Delta',\Delta]$ (they are actually
independent of the direction of observation), whereas the two
other terms correspond to the usual ``maximally crossed'' terms
$C[\Delta',\Delta]$:
\begin{equation}
\begin{aligned}
L[\Delta',\Delta] =\frac{9}{4}
{\epsilon}^{\mathrm{out}}_q\epsilon^{\mathrm{out}}_{p} \biggl\{
\langle\mathcal{D}^{1+}_{p}[\Delta']\mathcal{D}^{1-}_{q}[\Delta]
\rangle^{(g_1\bar{g}_1)}+ \langle\mathcal{D}^{2+}_{p}[\Delta']
\mathcal{D}^{2-}_{q}[\Delta]\rangle^{(g_2\bar{g}_2)}\biggl\} \\
C[\Delta',\Delta] =\frac{9}{4}
{\epsilon}^{\mathrm{out}}_q\epsilon^{\mathrm{out}}_{p}
\biggl\{\langle\mathcal{D}^{2+}_{p}[\Delta']\mathcal{D}^{1-}_{q}[\Delta]
\rangle^{(g_1\bar{g}_2)}+
\langle\mathcal{D}^{1+}_{p}[\Delta']\mathcal{D}^{2-}_{q}[\Delta]
\rangle^{(g_2\bar{g}_1)}\biggl\}
\end{aligned}
\end{equation}
\subsection{CBS enhancement factor}
In the case of linear scatterers, the CBS enhancement factor
achieves its maximal value 2 (recall that the CBS phenomenon is an
incoherent sum of two-wave interference patterns all starting with
a bright fringe at exact backscattering) if the single scattering
contribution can be removed from the total signal and provided
reciprocity holds. This is the case for scatterers with spherical
symmetry in the so-called polarization preserving channel $h
\parallel h$~\cite{BvTMaynard}.
In this polarization channel, we have calculated the relevant
quantities for an evaluation of the CBS enhancement factor
\emph{when no frequency filtering of the outgoing signal is made}.
We have thus derived the elastic and inelastic ladder terms and
the elastic and inelastic crossed terms, together with their
corresponding frequency spectra, for different values of the
on-resonance saturation parameter $s_0=2|\Omega_L|^2/\Gamma^2$.
This parameter measures the intensity strength of the incident
laser beam in units of the natural atomic transition line width
$\Gamma$, \emph{i.e.} its compares the on-resonance transition
rate induced by the laser to the atomic spontaneous emission rate.
For a detuned laser beam, the saturation parameter is $s(\delta)$
and is defined as:
\begin{equation}
s(\delta) = \frac{s_0}{1+(2\delta/\Gamma)^2}
\end{equation}
In the following, different values of the laser detuning have also
been considered:
\begin{equation*}
\begin{array}{ll}
(a)\quad \delta=0,\, s = s_0 = 0.02 &\quad
(b)\quad \delta=0,\, s = s_0 = 2.00 \\
(c)\quad \delta=5\Gamma, \, s_0=2.00, \, s=0.02
&\quad(d)\quad \delta=0,\, s = s_0 = 50.0
\qquad\qquad
\end{array}.
\end{equation*}
The ladder and crossed terms~\eqref{fieldmoy} are separated into
their elastic and inelastic parts according to:
\begin{equation}
\label{laddcrossplot}
\begin{aligned}
L[\Delta',\Delta]&=2\pi\delta(\Delta+\Delta')
\, \bigl\{2\pi\delta(\Delta) \, L_{\mathrm{el}}+L_{\mathrm{inel}}(\Delta)\bigr\}\\
C[\Delta',\Delta]&=2\pi\delta(\Delta+\Delta') \,
\bigl\{2\pi\delta(\Delta) \,
C_{\mathrm{el}}+C_{\mathrm{inel}}(\Delta)\bigr\}
\end{aligned}
\end{equation}
\begin{figure}[h]
\includegraphics[angle=-90,width=15cm]{figure1.eps}
\caption{\label{specfig} Backscattered light spectrum in the
helicity-preserving polarization channel $h\parallel h$. The solid lines
represent
the ladder term (average background intensity value) and the dotted lines represent the crossed
(interference) term. For both terms, the plotted values
corresponds to
$I_{\mathrm{inel}}(\Delta)/(C^{\mathrm{tot}}+L^{\mathrm{tot}})$,
see Eq.\eqref{laddcrossplot},
where $C^{\mathrm{tot}}+L^{\mathrm{tot}}$ is the total (elastic
\emph{plus} inelastic) intensity
scattered in the backward direction.
The
vertical dashed lines indicate the atomic
transition frequency. $\Delta$ corresponds to the scattered light angular frequency change
with respect to the initial laser angular frequency ($\Delta=0$ means thus that light is radiated at $\omega_L$).
Graph ($a$) corresponds to an on-resonance saturation parameter $s_0=0.02$ and
a laser detuning $\delta=0$ ; Graph ($b$) to $(s_0=2,\delta=0)$ ; Graph
($c$) to $(s_0=2,\delta=5\Gamma)$ and Graph ($d$) to $(s_0=50,\delta=0)$.
At low $s_0$, the inelastic contribution to the
total intensity is small and the ladder intensity is almost equal to
the crossed one. For a larger saturation parameter, firstly the inelastic contribution becomes
comparable to
the elastic one and secondly, the crossed term becomes smaller than
the
ladder one.
For a nonzero detuning, see graph ($c$), one clearly
observes an asymmetry in the
inelastic spectrum, which reflects the fact that the scattering
cross-section
of the atomic transition is maximal for resonant light: the
symmetric inelastic spectrum emitted by a single atom is filtered out
when scattered by the other one. At very large saturation (d),
the structure of the radiated spectrum becomes rather complicated.
}
\end{figure}
The corresponding inelastic spectra $L_{\mathrm{inel}}(\Delta)$
and $C_{\mathrm{inel}}(\Delta)$ are displayed in
figure~\ref{specfig}. For a sufficiently low saturation parameter
$s_0$, the inelastic contribution to the total intensity is small
and the ladder intensity is almost equal to the crossed one (see
graph~\ref{specfig}$a$). For larger saturation parameters (see
graphs~\ref{specfig}$b$ and \ref{specfig}$d$), there are two
effects : first, the inelastic contribution becomes comparable to
the elastic one and second, the crossed term is smaller than the
ladder one. For a nonzero detuning (see graph \ref{specfig}$c$),
one clearly observes an asymmetry in the inelastic spectrum, which
reflects that the scattering cross-section of the atomic
transition is maximal for resonant light (indicated by the
vertical dashed line): the symmetric inelastic spectrum emitted
by a single atom is filtered out when scattered by the other one.
We also observe that the crossed spectrum is much more reduced
than the ladder term, highlighting the non-linear effects in the
quantum correlations between the two atoms. Finally, for much
larger saturation parameters (see graph~\ref{specfig}$d$),
the scattered light almost entirely originates from the inelastic spectrum,
like for a single atom. However, contrary to the single atom case (for
which the scattered intensity reaches a constant value), the
total intensity scattered by the two atoms decreases when increasing
the incoming intensity. Indeed, since the atomic transitions become fully
saturated, the nonlinear scattering cross-section of each atom is
decreasing, resulting in a smaller total intensity scattered by
the two atoms compared to the one scattered by a single atom.
The CBS enhancement factor $\eta$ is defined as the peak to
background ratio. It thus reads:
\begin{equation}
\eta = 1 + \frac{C^{\mathrm{tot}}}{L^{\mathrm{tot}}}
\end{equation}
with:
\begin{equation}
\begin{aligned}
L^{\mathrm{tot}}&=L_{\mathrm{el}}+L_{\mathrm{inel}}^{\mathrm{tot}}=
L_{\mathrm{el}}+ \int \frac{d\Delta}{2\pi} \,L_{\mathrm{inel}}(\Delta)\\
C^{\mathrm{tot}}&=C_{\mathrm{el}}+C_{\mathrm{inel}}^{\mathrm{tot}}=
C_{\mathrm{el}}+ \int
\frac{d\Delta}{2\pi}\,C_{\mathrm{inel}}(\Delta)
\end{aligned}
\end{equation}
If the CBS phenomenon is reducible to a two-wave
interference, as it is the case here, then the enhancement factor $\eta$
is simply related to the degree of coherence $\gamma$ of the scattered light \cite{coherence}.
If the single scattering contribution can be removed from the
detected signal, and this is the case in the $h
\parallel h$ channel, one has simply $\eta=1+\gamma$ and consequently $\gamma =
C^{\mathrm{tot}}/L^{\mathrm{tot}}$. The maximal value for $\eta$
is 2, meaning that full coherence $\gamma=1$ is maintained for the
scattered field since then $C^{\mathrm{tot}}=L^{\mathrm{tot}}$.
If all interference effects disappear, meaning
$C^{\mathrm{tot}}=0$, $\eta$ reaches its minimal value 1 and
correspondingly coherence is fully lost $\gamma=0$. Furthermore,
one can show that in the $h \parallel h$ polarization channel,
$L_{\mathrm{el}} = C_{\mathrm{el}}$~\cite{prl94SMB}. Consequently,
as soon as $C_{\mathrm{inel}}^{\mathrm{tot}} <
L_{\mathrm{inel}}^{\mathrm{tot}}$ in this channel, the
coherence of the scattered light field is partially destroyed, since
then $\eta < 2$ and $\gamma <1$.
\begin{table}[ht]
\caption{\label{table1} Ladder (average background) and crossed
(interference) terms, see Eq.\eqref{laddcrossplot}, contributing
to the light scattered in the backward direction in the
helicity-preserving polarization channel $h\parallel h$. The given
values are relative to the incoming saturation parameter $s$. At
low $s_0$, the inelastic contributions are small and almost equal.
Thus $C^{\mathrm{tot}} \approx L^{\mathrm{tot}}$ and the maximum
enhancement factor 2 of the linear case is thus recovered, meaning
that full coherence $\gamma=1$ is maintained. At larger $s_0$,
elastic and inelastic terms become comparable. For very large
$s_0$, the contributions from the elastic terms vanish, like in
the single atom case. The inelastic contributions are also
decreasing, reflecting the fact that the probability for the light
to be scattered by a saturated atom becomes smaller with
increasing saturation. Furthermore, the inelastic crossed term is
\emph{always} smaller than the inelastic ladder one. This is a
signature of a coherence loss $\gamma <1$ induced by the quantum
vacuum fluctuations. However, the ratio
$C_{\mathrm{inel}}^{\mathrm{tot}}/L_{\mathrm{inel}}^{\mathrm{tot}}$
does not go to zero as $s_0 \to \infty$ but reaches the limit
value 0.096 (for $\delta=0$). Also, contrary to the single atom case, the
properties of the scattered light are not solely determined by
the saturation parameter $s$, but additionally depend on the
detuning $\delta,$ as exemplified by cases (a) and (c),
highlighting the role of the inelastic processes.}
\begin{ruledtabular}
\begin{tabular}{rd@{}d@{}d@{}d@{}}
& \multicolumn{1}{r}{$(a)\,s=s_0=0.02, \delta=0$}
& \multicolumn{1}{r}{$(b)\,s=s_0=2.00, \delta=0$}
& \multicolumn{1}{r}{$(c)\,s=0.02, s_0=2.00, \delta=5\Gamma$}
& \multicolumn{1}{r}{$(d)\,s=s_0=50.0, \delta=0$} \\
\multicolumn{1}{l}{$L_{\mathrm{el}}$} & 0.624 &0.833E-02 &
0.612 & 0.998E-07\\
\multicolumn{1}{l}{$L_{\mathrm{inel}}^{\mathrm{tot}}$} & 0.220E-01
&0.573E-01 & 0.328 & 0.487E-03 \\
\multicolumn{1}{l}{$L^{\mathrm{tot}}$} & 0.646 &0.656E-01 &
0.946 & 0.487E-03 \\
\multicolumn{1}{l}{$C_{\mathrm{el}}$} & 0.624 & 0.833E-02 &
0.612 & 0.998E-07\\
\multicolumn{1}{l}{$C_{\mathrm{inel}}^{\mathrm{tot}}$} & 0.188E-01 &
0.295E-01 & 0.157E-01 & 0.466E-04 \\
\multicolumn{1}{l}{$C^{\mathrm{tot}}$} & 0.642 & 0.378E-01 &
0.634 & 0.467E-04\\
\multicolumn{1}{l}{$\eta=1+\gamma$} & 1.994 & 1.576 &
1.670 & 1.096\\
\end{tabular}
\end{ruledtabular}
\end{table}
Our results are summarized in table~\ref{table1}. At low
saturation parameter $s_0$, $\eta$ reaches its maximal value 2 and
$\gamma=1$. This is so because the ladder and crossed inelastic
components are almost equal as evidenced in \ref{specfig}$a$.
Increasing $s_0$ reduces further
$C_{\mathrm{inel}}^{\mathrm{tot}}$ with respect to
$L_{\mathrm{inel}}^{\mathrm{tot}}$, thus decreasing $\eta$ and
$\gamma$. In the strongly saturated regime, one thus expects
$\gamma$ to decrease. However, there is no reason for the ratio
$C_{\mathrm{inel}}^{\mathrm{tot}}/
L_{\mathrm{inel}}^{\mathrm{tot}}$ to tend to zero as $s_0\to
\infty.$ It rather tends to a finite value, which depends on the
detuning, in agreement with the results published in
\cite{prl94SMB}. Furthermore, keeping $s_0$ fixed and decreasing
the saturation parameter $s$, situation $(c)$, $\eta$ increases,
as expected, but to a value which strongly depends on $s_0$. In
other words, contrary to the single atom case, the properties of
the scattered light, are not only determined by the saturation
parameter $s$~\cite{pra70WGDM}. Indeed, in both situations $(a)$
and $(c)$, $s$ has the same (small) value, but the enhancement
factor strongly differs, mainly because the inelastic ladder term
has increased. This highlights the crucial role of the inelastic
processes and of the rather complicated quantum correlations
between the two atoms.
This is not however the full story. Depending on the $s$ and
$\delta$ parameters, a rich variety of situations can be observed,
with various physical interpretations. These are beyond the scope
of this paper, which instead concentrates on the basic ingredients
of the quantum Langevin approach and will be published elsewhere.
\subsection{Linear response model}
Some insight on the relative behavior of
$C_{\mathrm{inel}}(\Delta)$ and $L_{\mathrm{inel}}(\Delta)$ can
be found by comparing the respective formulae from which these
quantities are extracted:
\begin{multline}
\label{corrcbscross} \left\langle{X}_{i'}^{\beta}[\Delta']
{X}_i^{\alpha}[\Delta]\right\rangle^{(\bar{g}_{\beta}g_{\alpha})}=
g_{\alpha}\left\langle{X}_{i'}^{\beta^{(0)}}[\Delta']
G_{ij}^{\alpha^+_q}[\Delta] \bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]
\right\rangle^{(\bar{g}_{\beta})} -\bar{g}_{\beta} \left\langle
G_{i'j'}^{\beta^-_q}[\Delta']
\bigl(\mathcal{D}^{{\alpha+}^{(0)}}_q\otimes
X_{j'}^{\beta^{(0)}}\bigr)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\right\rangle^{(g_{{\alpha}})}\\
-g_{\alpha}\bar{g}_{\beta}\biggl\{
\left\langle{X}_{i'}^{\beta}[\Delta']
G_{ij}^{\alpha^+_q}[\Delta]\left(X_j^{\alpha^{(0)}} \otimes
G^{\beta^-_p}_{\mathcal{D}_q^-j'}
\left(\mathcal{D}^{{\alpha+}^{(0)}}_p
\otimes X^{\beta^{(0)}}_{j'}\right)\right)[\Delta]\right\rangle^{(0)}\\
+\left\langle G_{i'j'}^{\beta^-_q}[\Delta']
\left(G^{\alpha^+_p}_{\mathcal{D}_q^-j}
\left(X_{j}^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_p\right)\otimes
X_{j'}^{\beta^{(0)}}\right)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\right\rangle^{(0)}\\
+\biggl.\left\langle\biggl[G_{i'j'}^{\beta^-_p}[\Delta']
\bigl(\mathcal{D}^{{\alpha+}^{(0)}}_p\otimes
X_{j'}^{\beta^{(0)}}\bigr)[\Delta']\biggr]
\biggl[G_{ij}^{\alpha^+_q}[\Delta] \bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]
\biggr]\right\rangle^{(0)}\biggr\}
\end{multline}
and
\begin{multline}
\label{corrcbsladder}
\left\langle{X}_{i'}^{\alpha}[\Delta']
{X}_i^{\alpha}[\Delta]\right\rangle^{(\bar{g}_{\alpha}g_{\alpha})}=
\left\langle{X}_{i'}^{\alpha^{(0)}}[\Delta']{X}_i^{\alpha^{(0)}}[\Delta]
\right\rangle^{(\bar{g}_{\alpha}g_{\alpha})}\\
+g_{\alpha}\biggl\{\left\langle{X}_{i'}^{\alpha^{(0)}}[\Delta']
G_{ij}^{\alpha^+_q}[\Delta] \bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]
\right\rangle^{(\bar{g}_{\alpha})} +\left\langle
G_{i'j'}^{\alpha^+_q}[\Delta'] \bigl(X_{j'}^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\right\rangle^{(\bar{g}_{\alpha})}
\biggr\}\\
-\bar{g}_{\alpha}\biggl\{\left\langle{X}_{i'}^{\alpha^{(0)}}[\Delta']
G_{ij}^{\alpha^-_q}[\Delta]
\bigl(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
X_j^{\alpha^{(0)}}\bigr)[\Delta]\right\rangle^{(g_{\alpha})}
+\left\langle G_{i'j'}^{\alpha^-_q}[\Delta']
\bigl(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
X_{j'}^{\alpha^{(0)}}\bigr)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\right\rangle^{(g_{\alpha})}
\biggr\}\\
-\bar{g}_{\alpha}g_{\alpha}\biggl\{\left\langle{X}_{i'}^{\alpha}[\Delta']
G_{ij}^{\alpha^+_q}[\Delta]\left(G^{\alpha^-_p}_{jj'}
\left(\mathcal{D}^{{\beta+}^{(0)}}_p\otimes
X_{j'}^{\alpha^{(0)}}\right)\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\right)[\Delta]\right\rangle^{(0)}
\biggr.\\
+\left\langle{X}_{i'}^{\alpha}[\Delta']G_{ij}^{\alpha^-_q}[\Delta]
\left(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
G^{\alpha^+_p}_{jj'}\left(X_{j'}^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_p\right)\right)[\Delta]
\right\rangle^{(0)}\\
+\left\langle
G_{i'j'}^{\alpha^+_q}[\Delta']\left(G^{\alpha^-_p}_{j'j}
\left(\mathcal{D}^{{\beta+}^{(0)}}_p\otimes
X_{j}^{\alpha^{(0)}}\right)\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\right)[\Delta']{
X}_i^{\alpha^{(0)}}[\Delta]\right\rangle^{(0)}\\
+\left\langle G_{i'j'}^{\alpha^-_q}[\Delta']
\left(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
G^{\alpha^+_p}_{j'j}\left(X_{j}^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_p\right)\right)[\Delta']
{X}_i^{\alpha^{(0)}}[\Delta]\right\rangle^{(0)} \\
+\left\langle\biggl[G_{i'j'}^{\alpha^+_p}[\Delta']
\bigl(X_{j'}^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_p\bigr)[\Delta']\biggr]
\biggl[G_{ij}^{\alpha^-_q}[\Delta]
\bigl(\mathcal{D}^{{\beta+}^{(0)}}_q\otimes
X_j^{\alpha^{(0)}}\bigr)[\Delta]\biggr]\right\rangle^{(0)}\\
+\biggl.\left\langle\biggl[G_{i'j'}^{\alpha^-_p}[\Delta']
\bigl(\mathcal{D}^{{\beta+}^{(0)}}_p\otimes
X_{j'}^{{\alpha}^{(0)}}\bigr)[\Delta']\biggr]
\biggl[G_{ij}^{\alpha^+_q}[\Delta] \bigl(X_j^{\alpha^{(0)}}\otimes
\mathcal{D}^{{\beta-}^{(0)}}_q\bigr)[\Delta]
\biggr]\right\rangle^{(0)}\biggr\}.
\end{multline}
There are twice as many terms contributing to the ladder
terms as to the crossed terms. A rather simple explanation of this
fact is
borrowed from the usual linear response theory. Indeed, each atom
is exposed to two fields : the incoming monochromatic field
(angular frequency $\omega_L$, wave vector $\textbf{k}_L$) and the
field scattered by the other atom (angular frequency $\omega_L +
\Delta$, wave vector $\textbf{k}_p$). In the far-field regime $R
\gg \lambda$, the incoming field is more intense than the
scattered field. It thus plays the role of a pump beam with
angular Rabi frequency $\Omega_L$, while the second weaker field
plays the role of a probe beam with angular Rabi frequency
$\Omega_p$. In this case, the response of each atom is simply
described by its nonlinear susceptibility~\cite{Cohenrouge,boyd}.
More precisely, forgetting about polarization effects, we have:
\begin{equation}
\begin{aligned}
\delta\mathcal{D}^+[\Delta]&=
e^{-i(2\textbf{k}_L-\textbf{k}_p)\cdot\textbf{R}_{\alpha}} \,
\chi_{\scriptscriptstyle ++}[\Delta] \, \Omega_p^+
+e^{-i\textbf{k}_p\cdot\textbf{R}_{\alpha}} \, \chi_{\scriptscriptstyle +-}[\Delta] \, \Omega_p^-\\
\delta\mathcal{D}^-[\Delta]&=
e^{i\textbf{k}_p\cdot\textbf{R}_{\alpha}} \, \chi_{\scriptscriptstyle
-+}[\Delta] \, \Omega_p^+
+e^{i(2\textbf{k}_L-\textbf{k}_p)\cdot\textbf{R}_{\alpha}} \,
\chi_{\scriptscriptstyle --}[\Delta] \, \Omega_p^-.
\end{aligned}
\end{equation}
where the phases due to the light fields have been explicitly
factorized.
As obviously seen, the two terms $\chi_{\scriptscriptstyle +-}$ and $\chi_{\scriptscriptstyle
-+}$ generate the forward propagation of the probe whereas the two
other terms $\chi_{\scriptscriptstyle ++}$ and $\chi_{\scriptscriptstyle --}$ can generate an
additional field in the direction $2\textbf{k}_L-\textbf{k}_p$
provided phase-matching conditions are fulfilled. This corresponds
to the usual forward four-wave mixing mechanism
(FFWM)~\cite{boyd,Cohenrouge}. If we now replace the probe field
by the field radiated by the other atom $\beta$, we get:
\begin{equation}
\begin{aligned}
\delta\mathcal{D}^+_{\beta\rightarrow\alpha}[\Delta]&=
\frac{1}{kR}\left\{ e^{-i(kR +
2\textbf{k}_L\cdot\textbf{R}_{\alpha} -
\textbf{k}_L\cdot\textbf{R}_{\beta})}\chi_{\scriptscriptstyle ++}[\Delta]
\,\mathcal{D}^-_{\beta}
+e^{i(kR-\textbf{k}_L\cdot\textbf{R}_{\beta})}\chi_{\scriptscriptstyle
+-}[\Delta]
\,\mathcal{D}^+_{\beta}\right\}\\
\delta\mathcal{D}^-_{\beta\rightarrow\alpha}[\Delta]&=
\frac{1}{kR}\left\{
e^{-i(kR-\textbf{k}_L\cdot\textbf{R}_{\beta})}\chi_{\scriptscriptstyle
-+}[\Delta] \,\mathcal{D}^-_{\beta}
+e^{i(2\textbf{k}_L\cdot\textbf{R}_{\alpha}+kR
-\textbf{k}_L\cdot\textbf{R}_{\beta})}\chi_{\scriptscriptstyle --}[\Delta]
\,\mathcal{D}^+_{\beta}\right\}.
\end{aligned}
\end{equation}
Hence the ladder and crossed contributions are given by (dropping
for sake of clarity any frequency dependence):
\begin{equation}
\begin{aligned}
C^{(2)}&\approx\delta\mathcal{D}^+_{\alpha\rightarrow\beta}
\delta\mathcal{D}^-_{\beta\rightarrow\alpha}
e^{i(-\textbf{k}_L\cdot\textbf{R}_{\beta}+\textbf{k}_L\cdot\textbf{R}_{\alpha})}\\
&\approx
e^{i(2\textbf{k}_L\cdot(\textbf{R}_{\alpha}-\textbf{R}_{\beta})-2kR)}
\chi_{\scriptscriptstyle ++}\chi_{\scriptscriptstyle
-+}\mathcal{D}^-_{\alpha}\mathcal{D}^-_{\beta}
+e^{4i\textbf{k}_L\cdot(\textbf{R}_{\alpha}-\textbf{R}_{\beta})}
\chi_{\scriptscriptstyle ++}\chi_{\scriptscriptstyle --}\mathcal{D}^-_{\alpha}\mathcal{D}^+_{\beta}\\
&\phantom{\approx} +\chi_{\scriptscriptstyle +-}\chi_{\scriptscriptstyle
-+}\mathcal{D}^+_{\alpha}\mathcal{D}^-_{\beta}
+e^{i(2\textbf{k}_L\cdot(\textbf{R}_{\alpha}-\textbf{R}_{\beta})+2kR)}
\chi_{\scriptscriptstyle +-}\chi_{\scriptscriptstyle --}\mathcal{D}^+_{\alpha}\mathcal{D}^+_{\beta}\\
\phantom{}\\
L^{(2)}&\approx\delta\mathcal{D}^+_{\beta\rightarrow\alpha}
\delta\mathcal{D}^-_{\beta\rightarrow\alpha}\\
&\approx
e^{i(2\textbf{k}_L\cdot(\textbf{R}_{\beta}-\textbf{R}_{\alpha})-2kR)}
\chi_{\scriptscriptstyle ++}\chi_{\scriptscriptstyle
-+}\mathcal{D}^-_{\beta}\mathcal{D}^-_{\beta}
+\chi_{\scriptscriptstyle ++}\chi_{\scriptscriptstyle --}\mathcal{D}^-_{\beta}\mathcal{D}^+_{\beta}\\
&\phantom{\approx} +\chi_{\scriptscriptstyle +-}\chi_{\scriptscriptstyle
-+}\mathcal{D}^+_{\beta}\mathcal{D}^-_{\beta}
+e^{i(2\textbf{k}_L\cdot(\textbf{R}_{\alpha}-\textbf{R}_{\beta})+2kR)}
\chi_{\scriptscriptstyle +-}\chi_{\scriptscriptstyle
--}\mathcal{D}^+_{\beta}\mathcal{D}^+_{\beta}.
\end{aligned}
\end{equation}
Averaging these expressions over the positions $\textbf{R}_\alpha$
and $\textbf{R}_\beta$ of the atoms while keeping $R\gg \lambda$
fixed, only terms with position-independent phases
survive, giving rise to:
\begin{equation}
\begin{aligned}
C^{(2)}&\approx\chi_{\scriptscriptstyle +-}\chi_{\scriptscriptstyle
-+}\,\mathcal{D}^+_{\alpha}
\mathcal{D}^-_{\beta}\\
L^{(2)}&\approx\chi_{\scriptscriptstyle ++}\chi_{\scriptscriptstyle --}\,\mathcal{D}^-_{\beta}
\mathcal{D}^+_{\beta}+ \chi_{\scriptscriptstyle +-}\chi_{\scriptscriptstyle
-+}\,\mathcal{D}^+_{\beta}\mathcal{D}^-_{\beta}.
\end{aligned}
\end{equation}
This simple model allows to understand clearly why there are twice
more terms in the ladder expression than in the crossed one.
Fields generated in the FFWM process \emph{always} interfere
constructively in the case of the ladder, since they originate
from the same atom. Of course, in the preceding explanation, we
have discarded polarization effects and inelastic processes in the
nonlinear susceptibilities. Nevertheless, even if in that case the
situation becomes more involved, the differences between the
ladder and crossed expressions still arise from this local four
wave-mixing process. For example, in the last line of
Eqs.~\eqref{corrcbscross} and \eqref{corrcbsladder}, we see that
the operator
$\big(G_{ij}^{\alpha^+_q}[\Delta]X_j^{\alpha^{(0)}}\otimes\big)$
plays the role of a generalized nonlinear susceptibility
(actually, the standard ones are recovered from the elastic part
of $X_j^{\alpha^{(0)}}$). Thus we recover the same structure as
previously depicted, which leads to similar conclusions.
Finally, as mentioned above, for large saturation parameters
$s_0$, even if in that case the total scattered intensities
(ladder and crossed) are dominated by the inelastic spectrum, we
numerically observe that the enhancement factor does not vanish
but rather goes to a finite limit $1.096$ (for $\delta=0$). Field
coherence is thus
not fully erased, which, at first glance, could be surprising
since the inelastic spectrum is a noise spectrum at the heart of
the temporal decoherence of the radiated field. But this only
means that both crossed and ladder become vanishingly small
relatively to the incident intensity. Nevertheless, even if it
would be hard to derive it analytically from
Eqs.~\eqref{corrcbscross} and \eqref{corrcbsladder}, they actually
decrease at the same rate, resulting in a finite (but small)
enhancement factor.
\section{Conclusion}
In the case of two atoms, even if the quantum Langevin approach
leads to calculations more tedious and involved than the direct
optical Bloch method, it nevertheless gives rise to an
understanding closer to the usual scattering approach developed in
the linear regime. In this way, one also gets direct information
about the inelastic spectrum of the radiated light. In particular,
it clearly outlines the crucial roles played by the inelastic
nonlinear susceptibilities and by the quantum correlations of the
vacuum fluctuations. Furthermore, since the framework of the
quantum Langevin approach is set in the frequency domain,
frequency-dependent propagation (\emph{i.e.} frequency-dependent
mean-free paths) between the atoms can be naturally included.
The next step would be to adapt the present approach to
"macroscopic" configurations (\emph{i.e.} at least many atoms),
allowing for a more direct comparison with existing
experiments~\cite{thierry}. This would provide a better
understanding of light transport properties in nonlinear atomic
media where vacuum fluctuations play a role. In particular, for
given values of the incident laser intensity and detuning, the
nonlinear mean-free path becomes negative in well-defined
frequency windows. This means that light \textit{amplification}
can be achieved in these frequency windows~\cite{pra5M,prl38WEDM}.
The atomic media would then constitute a very simple realization
of a coherent random laser.
\begin{acknowledgments}
We would like to thank Cord~M\"uller, Oliver~Sigwarth, Andreas
Buchleitner, Vyacheslav Shatokhin, Serge Reynaud and Jean-Michel
Courty for stimulating discussions. T.W. has been supported by the
DFG Emmy Noether program. Laboratoire Kastler Brossel is
laboratoire de l'Universit\'e Pierre et Marie Curie et de l'Ecole
Normale Sup\'erieure, UMR 8552 du CNRS.
\end{acknowledgments}
|
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| 461
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@extends('layouts.admin')
@section('contenido')
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<h3>Nuevo Puesto</h3>
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<li>{{$error}}</li>
@endforeach
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<div class="col-lg-6 col-md-6 col-sm-6 col-xs-12">
{!!Form::open(array('url'=>'admin/puesto','method'=>'POST','autocomplete'=>'off'))!!}
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|
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"redpajama_set_name": "RedPajamaGithub"
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| 6,587
|
\section{Introduction}
\label{Introduction}
In recent years, the development of GPU computation brought tremendous advances in deep learning. As a result, reinforcement learning could be applied to various fields. Reinforcement learning (RL) is a powerful technique that trains the agent to maximize the reward gain, so the agent can efficiently solve the problem. There are many applications that reinforcement learning is being used such as robotic manipulation tasks \cite{multiagentsystem2}, Atari games\cite{atarigame}, multiagent systems \cite{roboticmanupulation1}, and Alpha go\cite{alphago}. However, reinforcement learning's natural reward function is usually sparse since a reward is given to the agent only when the agent completes the task. Moreover, conventional reinforcement learning problems are formulated with the agent blind to the goal of the task. This property will help the agent to discover optimal policy without any guidance from the human, but it often takes a large amount of computational power and samples to learn.
In our work, we introduce novel reverse curriculum reinforcement learning, which uses trajectories of the agent in reverse order to train the agent. This enables our agent to start the training process by recognizing the goal of the task. As a result, agents will be trained in a sample efficient way by utilizing strong reward signals during "reverse learning." Our method does not need to modify any of the neural network structures and requires any other previous knowledge to train the agent. Moreover, reverse learning can be simply applied to any state-of-the-art algorithms such as PPO\cite{ppo3}, A3C\cite{a3c4}, and SAC\cite{sac5} within only two simple steps. First, collect the trajectory of the agent as usual, then flip the order of the trajectory of the episode to train our agent. Our reverse learning is using the concept of curriculum learning, which manually changes the order of the training process to train more efficiently. We empirically tested our reverse learning method using REINFORCE\cite{reinforce7} and A2C[\cite{a2c8} algorithms on CartPole-v1 and Lunar Lander-v2 environments in an Open AI gym\cite{openaigym10}, which both use discrete action spaces. Moreover, we further investigated the effects of the return normalization technique\cite{returnnorm9}, variation of learning rate, and structure of neural network on the performance of reverse learning.
\section{Related Works}
\label{Related Works}
There are some approaches published currently that have similarities with our reverse learning method. First, imitation learning\cite{imitationlearning11} requires the demonstration of experts who explains to the agent how to reach the goal of the task efficiently. This approach can use samples efficiently to train our agents. However, imitation learning limits agents to take advantage of exploration. Moreover, it requires lots of work since we need to find experts to perform completing the task, then record the process to train our agent. Instead, our reverse learning doesn't need extra work, but we only need to reverse the trajectory to train. In this case, we are using our backward trajectory as an expert in imitation learning.
Curriculum learning, which modifies the schedule of the learning process, has been applied to various machine learning tasks. The basic idea of curriculum learning is training on the easier example first, then constantly increasing the level of difficulty to solve the problem\cite{curriculum6}. In our reverse learning, we are training the agent using from the end of the episode to the beginning of the episode, so we are following the concept of curriculum learning since it is easier for the agent to complete the task from near the goal state rather than start state. Previously, curriculum learning has been applied to pre-specified tasks such as shooting the ball into a goal \cite{curriculumlearningshootingball}. Similar work was proposed by Barnes that solves difficult robotic problems\cite{difficultrobot}. However, their method requires partitioning the full task space, which limits application to various problems. Instead, our reverse learning can be applied to various machine learning tasks since this method doesn't require previous knowledge to train the agent.
\section{Preliminaries}
\label{Preliminaries}
In this work, we consider finite discrete time horizon Markov decision process (MDP) by a tuple \(M=(S,A,P,r,\gamma,\rho_{0},T\)). We define \(S\) as a state set, \(A\) as an action set, \(P: S\times A\times S \rightarrow R\) is a transition probability. \(r\) is a bounded reward function, \(\gamma\) is a discounted reward factor, \(\rho_{0}\) is the initial state distribution, and \(T\) is the trajectory of the moving agent. In the Markov decision process(MDP), at each time step, the agent takes an action, receives reward, and moves to the next state using transition probability \(P\). The goal of the reinforcement learning is to find the optimal policy \(\pi_{\theta}(a_{t}|s_{t})\) that maximizes reward gain.
In this work, we reverse the order of the agent's trajectory \(T\) and call it \(T_{b}\), and use it to train the agent from goal state \(s_{g}\) to initial state \(s_{0}\). Since our agent will start the training near the goal state \(s_{g}\), it will give the agent a strong reward signal which gives meaningful guidance to the goal.
\section{Reverse Curriculum Learning}
\subsection{Reverse Curriculum Learning in REINFORCE}
REINFORCE algorithm is one of basic policy gradient algorithms that gives insights to build state-of-art policy gradient algorithms. We first applied "reverse curriculum learning" on this method since it is a basic policy gradient algorithm, which is easy to implement and intuitive. The REINFORCE algorithm has a simple structure which uses cumulative discounted returns and log probability of choosing an action to compute its gradient. However, the original REINFORCE algorithm has a sparse reward function that lowers the performance on complicated tasks. Reverse curriculum learning can solve this problem by letting the agent start updating its gradient from the end of the episode. Therefore, our agent will recognize the goal in the beginning of the training process, which will replace the natural sparse reward function to a strong reward signal. Our detailed algorithm is explained on Algorithm 2. We first collect the sample trajectory using the policy network, then we simply flip the order of episode to compute the loss function in reverse order. The major difference between the original REINFORCE algorithm is that it reverses the order of the episode before starting taking its gradient.
\begin{algorithm}[bh]
\caption{REINFORCE}
\begin{algorithmic}
\STATE Collect the sample trajectories following $\pi_{\theta}(a_{t}|s_{t})$
\STATE $T=eqepisode \;length$
\FOR{$t=1$ {\bfseries to} $t=T-1$}
\STATE Compute Return $G$
\STATE $\theta \leftarrow \theta_{old} + G \nabla(\log\pi_{\theta}(a_{t}|s_{t}))$
\STATE Update $Optimizer$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[bh]
\caption{Backward Curriculum REINFORCE}
\begin{algorithmic}
\STATE Collect the sample trajectories following $\pi_{\theta}(a_{t}|s_{t})$
\STATE $T=eqepisode \;length$
\FOR{$t=T-1$ {\bfseries to} $t=1$}
\STATE Compute Return $G$
\STATE $\theta \leftarrow \theta_{old} + G \nabla(\log\pi_{\theta}(a_{t}|s_{t}))$
\STATE Update $Optimizer$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{Reverse Curriculum Learning in REINFORCE with Baseline}
The main problem with the REINFORCE algorithm is dealing with a high variance that may cause a divergence of policy network parameters. The common way to reduce the variance is to subtract the baseline \cite{reinforcebaseline}. In the REINFORCE algorithm, a value function is an appropriate baseline to subtract from return G. Agent will have both policy network and value network, and our loss function will be $\theta \leftarrow \theta_{old} + (G-V) \nabla(\log\pi_{\theta}(a_{t}|s_{t}))$. This baseline will decrease variance by reducing the step size of the gradient. In our reverse REINFORCE with the baseline algorithm, we first collect the trajectories using our policy, then flip the order of the episode to compute the loss function as shown on algorithm 4.
\begin{algorithm}[ht]
\caption{REINFORCE with Baseline}
\label{algorithm3}
\begin{algorithmic}
\STATE Collect the sample trajectories following $\pi_{\theta}(a_{t}|s_{t})$
\STATE $T=eqepisode \;length$
\FOR{$t=1$ {\bfseries to} $t=T-1$}
\STATE Compute Return $G$
\STATE $\theta \leftarrow \theta_{old} + (G-V) \nabla(\log\pi_{\theta}(a_{t}|s_{t}))$
\STATE Update $Optimizer$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[ht]
\caption{Backward REINFORCE with Baseline}
\label{algorithm4}
\begin{algorithmic}
\STATE Collect the sample trajectories following $\pi_{\theta}(a_{t}|s_{t})$
\STATE $T=eqepisode \;length$
\FOR{$t=T-1$ {\bfseries to} $t=1$}
\STATE Compute Return $G$
\STATE $\theta \leftarrow \theta_{old} + (G-V) \nabla(\log\pi_{\theta}(a_{t}|s_{t}))$
\STATE Update $Optimizer$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\section{Experimental Result}
We used a reinforcement learning environment from Open AI Gym \cite{openaigym10} to test the performance of our backward curriculum learning algorithm both on REINFORCE and REINFORCE with baseline algorithms. For experiments, we used Cart pole-v1 and Lunar Lander-v2 environments. The default structure of the network is a multi-layer perceptron network with 2 layers and 128 neurons as a baseline. Moreover, we analyzed the effect of return normalization as well as modifying the structure of networks.
\subsection{Cart Pole Environment}
In this section, we first tested our backward curriculum algorithm on the REINFORCE algorithm. The goal of the Cartpole-v1 algorithm is to balance the pendulum on the cart from falling by applying a discrete action domain with the force of +1 or -1 to the cart. The environment is considered solved when the agent reaches a score of 500 without falling down the pole from the cart.
The comparison of the backward curriculum algorithm and original REINFORCE is shown in figure 1. Until 1000 episodes, there is no conspicuous difference, but after that point, the backward curriculum algorithm clearly outperforms the original algorithm and solves the environment within 3000 episodes.
Next, we took experiments using REINFORCE with a baseline algorithm to analyze the effect of the backward curriculum learning algorithm. In figure 2, the backward curriculum algorithm solved the environment within 250 episodes, but the original REINFORCE with the baseline algorithm not only took longer episodes but also has a high variance that disturbs the agent from remaining in the goal state. We can further investigate that adding a baseline to the REINFORCE algorithm effectively solved a high variance problem that also improved the performance of the algorithm since it took about 1500 episodes to solve the environment using REINFORCE algorithm, but adding baseline results into solving the Cart pole environment less than 500 episodes as shown on figure 3.
\begin{figure}[t]
\centering
\includegraphics[width = 0.45\textwidth]{cartpolereinforceforwardbackward.png}
\caption{Comparison between backward and forward update method on REINFORCE algorithm on Cart Pole environment.}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{comparingREINFORCEbaselineforwardandbackward.png}
\caption{Comparison between backward and forward update method on REINFORCE with baseline algorithm on Cart Pole environment.}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{ComparingREINFORCEvsREINFORCEwithbaseline.png}
\caption{Comparison between REINFORCE and REINFORCE with baseline algorithm on Cart Pole environment.}
\end{figure}
\subsection{Lunar Lander Environment}
The next environment that we used to test our algorithms is the LunarLander-v2 environment from OpenAI gym. The goal of this environment is to make the agent safely land down on the goal region from the sky without crashing down the agent. The action space consists of 4 actions, which are resting, firing left, right, and main engines. When an agent reaches 200 scores without crashing, this environment is considered solved. In this section, we not only compared the performance of the backward curriculum algorithm and the original algorithm, but also we further experimented with the effect of return normalization.
As shown in figure 4, our reverse REINFORCE algorithm reaches the goal state with an average of 200 scores within 2000 episodes, but the REINFORCE solves the environment within 3000 episodes, which clearly shows that letting the agent know the goal of the environment at the beginning of the training process significantly helps the agent to reach the goal using fewer samples. Moreover, in figure 5, the REINFORCE with baseline using the reverse curriculum learning algorithm finishes the task even faster with 1000 episodes using a reverse method, however, the original algorithm took more than 2500 episodes. Finally, we compared the performance of reverse REINFORCE and reverse REINFORCE with the baseline in figure 6. As we can observe in figure 6, both methods successfully reach the goal score, but reverse curriculum learning on REINFORCE with baseline has less variance and uses fewer samples to solve the task.
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{comparingREINFORCEonLL2.png}
\caption{Comparison between backward and forward update method on REINFORCE algorithm on Lunar Lander environment.}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{ComparingREINFORCEwithbaselineonLL2forwardvsbackward.png}
\caption{Comparison between backward and forward update method on REINFORCE with baseline algorithm on Lunar Lander environment.}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{ComparingbackwardmethodwithREINFORCEvsREINFORCEbaselinewithbackward.png}
\caption{Comparison between REINFORCE and REINFORCE with baseline algorithm on Lunar Lander environment.}
\end{figure}
\subsection{Return Normalization}
\label{Return Normalization}
Even though reverse learning on the REINFORCE algorithm greatly optimized sample efficiency, it still has a problem with high variance. If we take a close look at the original REINFORCE loss function $\theta \leftarrow \theta_{old} + G \nabla(\log\pi_{\theta}(a_{t}|s_{t}))$, the step size of the gradient mainly depends on $G$ and log $\log\pi_{\theta}(a_{t}|s_{t})$. The log of action probability is usually acceptable, but the return has a high magnitude which will dynamically increase the step size of the gradient. The choice of step size is crucial in reinforcement learning since a small step size slows down the convergence rate, while a large step size may cause oscillations or divergence of policy networks due to overshooting\cite{overshooting_problem}. Therefore, we can apply return normalization to our method to stabilize the high variance. The basic idea of return normalization is making the mean of return to 0 and variance with 1, so we can scale down the magnitude of the return to avoid overshooting problems. We can do this by subtracting the average of return and dividing it by the standard deviation of return by following equation 1\cite{returnnorm9}. As a result, applying return normalization will make the reverse REINFORCE algorithm stable, which will increase the performance of the algorithm
We have applied return normalization both on the original algorithm and our backward curriculum algorithm simulated on the Lunar Lander environment.
As we can observe in figure 7, before applying return normalization, the learning curve has high variance and some of the learning rates result in divergence. Instead, applying return normalization decreased the variance significantly and an agent with a learning rate 1e-4 solved the environment within 3000 episodes as shown in figure 8. This result demonstrates that return normalization not only optimized the performance but also minimized the variance as shown in figure 9.
\begin{equation}
Return\:Norm = \frac{return - average\:return}{standard\:deviation \: of \: return }
\end{equation}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{REINFORCEwithoutreturnnormalizationLL-v2.png}
\caption{Without using return normalization in backward curriculum REINFORCE on Lunar Lander environment}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{REINFORCEreturnnormwithreturnnormalizationLL-v2.png}
\caption{Effect of return normalization in backward curriculum REINFORCE using Lunar Lander environment}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{ComparingbestlearningrateforreturnvswithoutreturnnormalizationLLv2REINFORCE.png}
\caption{Comparing backward curriculum REINFORCE using return normalization}
\end{figure}
\subsection{Comparison between Deep and Shallow Network}
The depth of the network is one of the important factors that affect the performance of the algorithm. In general, a deep neural network requires more computation power since increasing the number of layers will yield more parameters to compute the gradient. Instead, a shallow network has the benefit of quick computation since there are fewer parameters on fewer layers. Both shallow and deep networks have different benefits, but we discovered that for simple environments like Cartpole, the shallow network works better than the deep network. For a simple environment, the agent can solve the problem by applying simple heuristics without wasting computation efforts \cite{deep_shallow}. If we use deeper networks to solve the simple environment, it may lead to overthinking, which causes computational waste, and eventually leads to a misclassification \cite{deep_shallow}.
We took experiments to compare the performance in the Cartpole environment using both shallow and deep networks. Our baseline network structure is two layers with 128 neurons, and we compared our baseline network to a deeper netork, which has 3 layers with 256 neurons. We used backward curriculum REINFORCE with baseline and original REINFORCE with baseline method to observe the difference between them on deeper networks. The result was interesting since our backward curriculum method didn't improve the performance as we expected as shown in figure 10. We concluded that on a deeper network, the network already has an accurate approximation, so letting the agent start the training process while recognizing the goal of the task has little effect on the performance. Moreover, we compared the performance between shallow and deep networks in the Cartpole environment. As shown in figure 11, the agent with shallow networks could finish the task using under 500 episode samples, but the agent with deeper networks took more than 1500 episodes to reach the goal state. Therefore, we can conclude that backward curriculum learning works best in a shallow network to solve simple tasks.
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{comparingforwardandbackwardondeepernet.png}
\caption{Comparing backward curriculum learning with original REINFORCE baseline using deeper network}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45\textwidth]{comparingdeepandshallowreinforcebase.png}
\caption{Comparing backward curriculum REINFORCE with baseline using deep and shallow network}
\end{figure}
\section{Conclusion}
In this paper, we proposed a novel reverse curriculum reinforcement learning that addresses the natural sparse reward function problem. Our method simply reverses the order of the episode before starting the training process, so it will let the agent recognize the goal of the task in the beginning. Therefore, it replaces the natural sparse reward function with a strong reward signal, which will optimize the sample efficiency. Unlike the previously proposed method, reverse curriculum learning doesn't require many steps to modify the structure of the code, so we can directly apply our method to current state-of-art algorithms. We chose REINFORCE and REINFORCE with a baseline algorithm to test our method since it is the baseline of state-of-art algorithms. We empirically proved that reverse curriculum learning uses fewer samples to solve the given tasks. Moreover, we tested the effect of return normalization and depth of the network. We discovered that reverse curriculum learning best fits while solving simple tasks like Cartpole with a shallow network. In future work, we will apply our reverse curriculum learning on current state-of-art algorithms to test on various environments.
\section{Acknowledgement}
This work is supported by Georgia Tech's summer undergraduate in engineering program (SURE), which is part of the National Science Foundation (NSF).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,623
|
from typing import Dict, List, Optional, TYPE_CHECKING, Union
from ... import _serialization
if TYPE_CHECKING:
# pylint: disable=unused-import,ungrouped-imports
from .. import models as _models
class SubResource(_serialization.Model):
"""Reference to another subresource.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar id: Resource ID.
:vartype id: str
:ivar name: The name of the resource that is unique within a resource group. This name can be
used to access the resource.
:vartype name: str
:ivar type: Resource type.
:vartype type: str
"""
_validation = {
"id": {"readonly": True},
"name": {"readonly": True},
"type": {"readonly": True},
}
_attribute_map = {
"id": {"key": "id", "type": "str"},
"name": {"key": "name", "type": "str"},
"type": {"key": "type", "type": "str"},
}
def __init__(self, **kwargs):
""" """
super().__init__(**kwargs)
self.id = None
self.name = None
self.type = None
class AgentPool(SubResource): # pylint: disable=too-many-instance-attributes
"""Agent Pool.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar id: Resource ID.
:vartype id: str
:ivar name: The name of the resource that is unique within a resource group. This name can be
used to access the resource.
:vartype name: str
:ivar type: Resource type.
:vartype type: str
:ivar count: Number of agents (VMs) to host docker containers. Allowed values must be in the
range of 1 to 100 (inclusive). The default value is 1.
:vartype count: int
:ivar vm_size: Size of agent VMs. Known values are: "Standard_A1", "Standard_A10",
"Standard_A11", "Standard_A1_v2", "Standard_A2", "Standard_A2_v2", "Standard_A2m_v2",
"Standard_A3", "Standard_A4", "Standard_A4_v2", "Standard_A4m_v2", "Standard_A5",
"Standard_A6", "Standard_A7", "Standard_A8", "Standard_A8_v2", "Standard_A8m_v2",
"Standard_A9", "Standard_B2ms", "Standard_B2s", "Standard_B4ms", "Standard_B8ms",
"Standard_D1", "Standard_D11", "Standard_D11_v2", "Standard_D11_v2_Promo", "Standard_D12",
"Standard_D12_v2", "Standard_D12_v2_Promo", "Standard_D13", "Standard_D13_v2",
"Standard_D13_v2_Promo", "Standard_D14", "Standard_D14_v2", "Standard_D14_v2_Promo",
"Standard_D15_v2", "Standard_D16_v3", "Standard_D16s_v3", "Standard_D1_v2", "Standard_D2",
"Standard_D2_v2", "Standard_D2_v2_Promo", "Standard_D2_v3", "Standard_D2s_v3", "Standard_D3",
"Standard_D32_v3", "Standard_D32s_v3", "Standard_D3_v2", "Standard_D3_v2_Promo", "Standard_D4",
"Standard_D4_v2", "Standard_D4_v2_Promo", "Standard_D4_v3", "Standard_D4s_v3",
"Standard_D5_v2", "Standard_D5_v2_Promo", "Standard_D64_v3", "Standard_D64s_v3",
"Standard_D8_v3", "Standard_D8s_v3", "Standard_DS1", "Standard_DS11", "Standard_DS11_v2",
"Standard_DS11_v2_Promo", "Standard_DS12", "Standard_DS12_v2", "Standard_DS12_v2_Promo",
"Standard_DS13", "Standard_DS13-2_v2", "Standard_DS13-4_v2", "Standard_DS13_v2",
"Standard_DS13_v2_Promo", "Standard_DS14", "Standard_DS14-4_v2", "Standard_DS14-8_v2",
"Standard_DS14_v2", "Standard_DS14_v2_Promo", "Standard_DS15_v2", "Standard_DS1_v2",
"Standard_DS2", "Standard_DS2_v2", "Standard_DS2_v2_Promo", "Standard_DS3", "Standard_DS3_v2",
"Standard_DS3_v2_Promo", "Standard_DS4", "Standard_DS4_v2", "Standard_DS4_v2_Promo",
"Standard_DS5_v2", "Standard_DS5_v2_Promo", "Standard_E16_v3", "Standard_E16s_v3",
"Standard_E2_v3", "Standard_E2s_v3", "Standard_E32-16s_v3", "Standard_E32-8s_v3",
"Standard_E32_v3", "Standard_E32s_v3", "Standard_E4_v3", "Standard_E4s_v3",
"Standard_E64-16s_v3", "Standard_E64-32s_v3", "Standard_E64_v3", "Standard_E64s_v3",
"Standard_E8_v3", "Standard_E8s_v3", "Standard_F1", "Standard_F16", "Standard_F16s",
"Standard_F16s_v2", "Standard_F1s", "Standard_F2", "Standard_F2s", "Standard_F2s_v2",
"Standard_F32s_v2", "Standard_F4", "Standard_F4s", "Standard_F4s_v2", "Standard_F64s_v2",
"Standard_F72s_v2", "Standard_F8", "Standard_F8s", "Standard_F8s_v2", "Standard_G1",
"Standard_G2", "Standard_G3", "Standard_G4", "Standard_G5", "Standard_GS1", "Standard_GS2",
"Standard_GS3", "Standard_GS4", "Standard_GS4-4", "Standard_GS4-8", "Standard_GS5",
"Standard_GS5-16", "Standard_GS5-8", "Standard_H16", "Standard_H16m", "Standard_H16mr",
"Standard_H16r", "Standard_H8", "Standard_H8m", "Standard_L16s", "Standard_L32s",
"Standard_L4s", "Standard_L8s", "Standard_M128-32ms", "Standard_M128-64ms", "Standard_M128ms",
"Standard_M128s", "Standard_M64-16ms", "Standard_M64-32ms", "Standard_M64ms", "Standard_M64s",
"Standard_NC12", "Standard_NC12s_v2", "Standard_NC12s_v3", "Standard_NC24", "Standard_NC24r",
"Standard_NC24rs_v2", "Standard_NC24rs_v3", "Standard_NC24s_v2", "Standard_NC24s_v3",
"Standard_NC6", "Standard_NC6s_v2", "Standard_NC6s_v3", "Standard_ND12s", "Standard_ND24rs",
"Standard_ND24s", "Standard_ND6s", "Standard_NV12", "Standard_NV24", and "Standard_NV6".
:vartype vm_size: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMSizeTypes
:ivar os_disk_size_gb: OS Disk Size in GB to be used to specify the disk size for every machine
in this master/agent pool. If you specify 0, it will apply the default osDisk size according to
the vmSize specified.
:vartype os_disk_size_gb: int
:ivar vnet_subnet_id: VNet SubnetID specifies the VNet's subnet identifier.
:vartype vnet_subnet_id: str
:ivar max_pods: Maximum number of pods that can run on a node.
:vartype max_pods: int
:ivar os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default to
Linux. Known values are: "Linux" and "Windows".
:vartype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:ivar max_count: Maximum number of nodes for auto-scaling.
:vartype max_count: int
:ivar min_count: Minimum number of nodes for auto-scaling.
:vartype min_count: int
:ivar enable_auto_scaling: Whether to enable auto-scaler.
:vartype enable_auto_scaling: bool
:ivar type_properties_type: AgentPoolType represents types of an agent pool. Known values are:
"VirtualMachineScaleSets" and "AvailabilitySet".
:vartype type_properties_type: str or
~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolType
:ivar orchestrator_version: Version of orchestrator specified when creating the managed
cluster.
:vartype orchestrator_version: str
:ivar provisioning_state: The current deployment or provisioning state, which only appears in
the response.
:vartype provisioning_state: str
:ivar availability_zones: (PREVIEW) Availability zones for nodes. Must use
VirtualMachineScaleSets AgentPoolType.
:vartype availability_zones: list[str]
:ivar enable_node_public_ip: Enable public IP for nodes.
:vartype enable_node_public_ip: bool
:ivar scale_set_priority: ScaleSetPriority to be used to specify virtual machine scale set
priority. Default to regular. Known values are: "Low" and "Regular".
:vartype scale_set_priority: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetPriority
:ivar scale_set_eviction_policy: ScaleSetEvictionPolicy to be used to specify eviction policy
for low priority virtual machine scale set. Default to Delete. Known values are: "Delete" and
"Deallocate".
:vartype scale_set_eviction_policy: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetEvictionPolicy
:ivar node_taints: Taints added to new nodes during node pool create and scale. For example,
key=value:NoSchedule.
:vartype node_taints: list[str]
"""
_validation = {
"id": {"readonly": True},
"name": {"readonly": True},
"type": {"readonly": True},
"os_disk_size_gb": {"maximum": 1023, "minimum": 0},
"provisioning_state": {"readonly": True},
}
_attribute_map = {
"id": {"key": "id", "type": "str"},
"name": {"key": "name", "type": "str"},
"type": {"key": "type", "type": "str"},
"count": {"key": "properties.count", "type": "int"},
"vm_size": {"key": "properties.vmSize", "type": "str"},
"os_disk_size_gb": {"key": "properties.osDiskSizeGB", "type": "int"},
"vnet_subnet_id": {"key": "properties.vnetSubnetID", "type": "str"},
"max_pods": {"key": "properties.maxPods", "type": "int"},
"os_type": {"key": "properties.osType", "type": "str"},
"max_count": {"key": "properties.maxCount", "type": "int"},
"min_count": {"key": "properties.minCount", "type": "int"},
"enable_auto_scaling": {"key": "properties.enableAutoScaling", "type": "bool"},
"type_properties_type": {"key": "properties.type", "type": "str"},
"orchestrator_version": {"key": "properties.orchestratorVersion", "type": "str"},
"provisioning_state": {"key": "properties.provisioningState", "type": "str"},
"availability_zones": {"key": "properties.availabilityZones", "type": "[str]"},
"enable_node_public_ip": {"key": "properties.enableNodePublicIP", "type": "bool"},
"scale_set_priority": {"key": "properties.scaleSetPriority", "type": "str"},
"scale_set_eviction_policy": {"key": "properties.scaleSetEvictionPolicy", "type": "str"},
"node_taints": {"key": "properties.nodeTaints", "type": "[str]"},
}
def __init__(
self,
*,
count: Optional[int] = None,
vm_size: Optional[Union[str, "_models.ContainerServiceVMSizeTypes"]] = None,
os_disk_size_gb: Optional[int] = None,
vnet_subnet_id: Optional[str] = None,
max_pods: Optional[int] = None,
os_type: Union[str, "_models.OSType"] = "Linux",
max_count: Optional[int] = None,
min_count: Optional[int] = None,
enable_auto_scaling: Optional[bool] = None,
type_properties_type: Optional[Union[str, "_models.AgentPoolType"]] = None,
orchestrator_version: Optional[str] = None,
availability_zones: Optional[List[str]] = None,
enable_node_public_ip: Optional[bool] = None,
scale_set_priority: Union[str, "_models.ScaleSetPriority"] = "Regular",
scale_set_eviction_policy: Union[str, "_models.ScaleSetEvictionPolicy"] = "Delete",
node_taints: Optional[List[str]] = None,
**kwargs
):
"""
:keyword count: Number of agents (VMs) to host docker containers. Allowed values must be in the
range of 1 to 100 (inclusive). The default value is 1.
:paramtype count: int
:keyword vm_size: Size of agent VMs. Known values are: "Standard_A1", "Standard_A10",
"Standard_A11", "Standard_A1_v2", "Standard_A2", "Standard_A2_v2", "Standard_A2m_v2",
"Standard_A3", "Standard_A4", "Standard_A4_v2", "Standard_A4m_v2", "Standard_A5",
"Standard_A6", "Standard_A7", "Standard_A8", "Standard_A8_v2", "Standard_A8m_v2",
"Standard_A9", "Standard_B2ms", "Standard_B2s", "Standard_B4ms", "Standard_B8ms",
"Standard_D1", "Standard_D11", "Standard_D11_v2", "Standard_D11_v2_Promo", "Standard_D12",
"Standard_D12_v2", "Standard_D12_v2_Promo", "Standard_D13", "Standard_D13_v2",
"Standard_D13_v2_Promo", "Standard_D14", "Standard_D14_v2", "Standard_D14_v2_Promo",
"Standard_D15_v2", "Standard_D16_v3", "Standard_D16s_v3", "Standard_D1_v2", "Standard_D2",
"Standard_D2_v2", "Standard_D2_v2_Promo", "Standard_D2_v3", "Standard_D2s_v3", "Standard_D3",
"Standard_D32_v3", "Standard_D32s_v3", "Standard_D3_v2", "Standard_D3_v2_Promo", "Standard_D4",
"Standard_D4_v2", "Standard_D4_v2_Promo", "Standard_D4_v3", "Standard_D4s_v3",
"Standard_D5_v2", "Standard_D5_v2_Promo", "Standard_D64_v3", "Standard_D64s_v3",
"Standard_D8_v3", "Standard_D8s_v3", "Standard_DS1", "Standard_DS11", "Standard_DS11_v2",
"Standard_DS11_v2_Promo", "Standard_DS12", "Standard_DS12_v2", "Standard_DS12_v2_Promo",
"Standard_DS13", "Standard_DS13-2_v2", "Standard_DS13-4_v2", "Standard_DS13_v2",
"Standard_DS13_v2_Promo", "Standard_DS14", "Standard_DS14-4_v2", "Standard_DS14-8_v2",
"Standard_DS14_v2", "Standard_DS14_v2_Promo", "Standard_DS15_v2", "Standard_DS1_v2",
"Standard_DS2", "Standard_DS2_v2", "Standard_DS2_v2_Promo", "Standard_DS3", "Standard_DS3_v2",
"Standard_DS3_v2_Promo", "Standard_DS4", "Standard_DS4_v2", "Standard_DS4_v2_Promo",
"Standard_DS5_v2", "Standard_DS5_v2_Promo", "Standard_E16_v3", "Standard_E16s_v3",
"Standard_E2_v3", "Standard_E2s_v3", "Standard_E32-16s_v3", "Standard_E32-8s_v3",
"Standard_E32_v3", "Standard_E32s_v3", "Standard_E4_v3", "Standard_E4s_v3",
"Standard_E64-16s_v3", "Standard_E64-32s_v3", "Standard_E64_v3", "Standard_E64s_v3",
"Standard_E8_v3", "Standard_E8s_v3", "Standard_F1", "Standard_F16", "Standard_F16s",
"Standard_F16s_v2", "Standard_F1s", "Standard_F2", "Standard_F2s", "Standard_F2s_v2",
"Standard_F32s_v2", "Standard_F4", "Standard_F4s", "Standard_F4s_v2", "Standard_F64s_v2",
"Standard_F72s_v2", "Standard_F8", "Standard_F8s", "Standard_F8s_v2", "Standard_G1",
"Standard_G2", "Standard_G3", "Standard_G4", "Standard_G5", "Standard_GS1", "Standard_GS2",
"Standard_GS3", "Standard_GS4", "Standard_GS4-4", "Standard_GS4-8", "Standard_GS5",
"Standard_GS5-16", "Standard_GS5-8", "Standard_H16", "Standard_H16m", "Standard_H16mr",
"Standard_H16r", "Standard_H8", "Standard_H8m", "Standard_L16s", "Standard_L32s",
"Standard_L4s", "Standard_L8s", "Standard_M128-32ms", "Standard_M128-64ms", "Standard_M128ms",
"Standard_M128s", "Standard_M64-16ms", "Standard_M64-32ms", "Standard_M64ms", "Standard_M64s",
"Standard_NC12", "Standard_NC12s_v2", "Standard_NC12s_v3", "Standard_NC24", "Standard_NC24r",
"Standard_NC24rs_v2", "Standard_NC24rs_v3", "Standard_NC24s_v2", "Standard_NC24s_v3",
"Standard_NC6", "Standard_NC6s_v2", "Standard_NC6s_v3", "Standard_ND12s", "Standard_ND24rs",
"Standard_ND24s", "Standard_ND6s", "Standard_NV12", "Standard_NV24", and "Standard_NV6".
:paramtype vm_size: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMSizeTypes
:keyword os_disk_size_gb: OS Disk Size in GB to be used to specify the disk size for every
machine in this master/agent pool. If you specify 0, it will apply the default osDisk size
according to the vmSize specified.
:paramtype os_disk_size_gb: int
:keyword vnet_subnet_id: VNet SubnetID specifies the VNet's subnet identifier.
:paramtype vnet_subnet_id: str
:keyword max_pods: Maximum number of pods that can run on a node.
:paramtype max_pods: int
:keyword os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default
to Linux. Known values are: "Linux" and "Windows".
:paramtype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:keyword max_count: Maximum number of nodes for auto-scaling.
:paramtype max_count: int
:keyword min_count: Minimum number of nodes for auto-scaling.
:paramtype min_count: int
:keyword enable_auto_scaling: Whether to enable auto-scaler.
:paramtype enable_auto_scaling: bool
:keyword type_properties_type: AgentPoolType represents types of an agent pool. Known values
are: "VirtualMachineScaleSets" and "AvailabilitySet".
:paramtype type_properties_type: str or
~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolType
:keyword orchestrator_version: Version of orchestrator specified when creating the managed
cluster.
:paramtype orchestrator_version: str
:keyword availability_zones: (PREVIEW) Availability zones for nodes. Must use
VirtualMachineScaleSets AgentPoolType.
:paramtype availability_zones: list[str]
:keyword enable_node_public_ip: Enable public IP for nodes.
:paramtype enable_node_public_ip: bool
:keyword scale_set_priority: ScaleSetPriority to be used to specify virtual machine scale set
priority. Default to regular. Known values are: "Low" and "Regular".
:paramtype scale_set_priority: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetPriority
:keyword scale_set_eviction_policy: ScaleSetEvictionPolicy to be used to specify eviction
policy for low priority virtual machine scale set. Default to Delete. Known values are:
"Delete" and "Deallocate".
:paramtype scale_set_eviction_policy: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetEvictionPolicy
:keyword node_taints: Taints added to new nodes during node pool create and scale. For example,
key=value:NoSchedule.
:paramtype node_taints: list[str]
"""
super().__init__(**kwargs)
self.count = count
self.vm_size = vm_size
self.os_disk_size_gb = os_disk_size_gb
self.vnet_subnet_id = vnet_subnet_id
self.max_pods = max_pods
self.os_type = os_type
self.max_count = max_count
self.min_count = min_count
self.enable_auto_scaling = enable_auto_scaling
self.type_properties_type = type_properties_type
self.orchestrator_version = orchestrator_version
self.provisioning_state = None
self.availability_zones = availability_zones
self.enable_node_public_ip = enable_node_public_ip
self.scale_set_priority = scale_set_priority
self.scale_set_eviction_policy = scale_set_eviction_policy
self.node_taints = node_taints
class AgentPoolAvailableVersions(_serialization.Model):
"""The list of available versions for an agent pool.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar id: Id of the agent pool available versions.
:vartype id: str
:ivar name: Name of the agent pool available versions.
:vartype name: str
:ivar type: Type of the agent pool available versions.
:vartype type: str
:ivar agent_pool_versions: List of versions available for agent pool.
:vartype agent_pool_versions:
list[~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolAvailableVersionsPropertiesAgentPoolVersionsItem]
"""
_validation = {
"id": {"readonly": True},
"name": {"readonly": True},
"type": {"readonly": True},
}
_attribute_map = {
"id": {"key": "id", "type": "str"},
"name": {"key": "name", "type": "str"},
"type": {"key": "type", "type": "str"},
"agent_pool_versions": {
"key": "properties.agentPoolVersions",
"type": "[AgentPoolAvailableVersionsPropertiesAgentPoolVersionsItem]",
},
}
def __init__(
self,
*,
agent_pool_versions: Optional[List["_models.AgentPoolAvailableVersionsPropertiesAgentPoolVersionsItem"]] = None,
**kwargs
):
"""
:keyword agent_pool_versions: List of versions available for agent pool.
:paramtype agent_pool_versions:
list[~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolAvailableVersionsPropertiesAgentPoolVersionsItem]
"""
super().__init__(**kwargs)
self.id = None
self.name = None
self.type = None
self.agent_pool_versions = agent_pool_versions
class AgentPoolAvailableVersionsPropertiesAgentPoolVersionsItem(_serialization.Model):
"""AgentPoolAvailableVersionsPropertiesAgentPoolVersionsItem.
:ivar default: Whether this version is the default agent pool version.
:vartype default: bool
:ivar kubernetes_version: Kubernetes version (major, minor, patch).
:vartype kubernetes_version: str
:ivar is_preview: Whether Kubernetes version is currently in preview.
:vartype is_preview: bool
"""
_attribute_map = {
"default": {"key": "default", "type": "bool"},
"kubernetes_version": {"key": "kubernetesVersion", "type": "str"},
"is_preview": {"key": "isPreview", "type": "bool"},
}
def __init__(
self,
*,
default: Optional[bool] = None,
kubernetes_version: Optional[str] = None,
is_preview: Optional[bool] = None,
**kwargs
):
"""
:keyword default: Whether this version is the default agent pool version.
:paramtype default: bool
:keyword kubernetes_version: Kubernetes version (major, minor, patch).
:paramtype kubernetes_version: str
:keyword is_preview: Whether Kubernetes version is currently in preview.
:paramtype is_preview: bool
"""
super().__init__(**kwargs)
self.default = default
self.kubernetes_version = kubernetes_version
self.is_preview = is_preview
class AgentPoolListResult(_serialization.Model):
"""The response from the List Agent Pools operation.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar value: The list of agent pools.
:vartype value: list[~azure.mgmt.containerservice.v2019_08_01.models.AgentPool]
:ivar next_link: The URL to get the next set of agent pool results.
:vartype next_link: str
"""
_validation = {
"next_link": {"readonly": True},
}
_attribute_map = {
"value": {"key": "value", "type": "[AgentPool]"},
"next_link": {"key": "nextLink", "type": "str"},
}
def __init__(self, *, value: Optional[List["_models.AgentPool"]] = None, **kwargs):
"""
:keyword value: The list of agent pools.
:paramtype value: list[~azure.mgmt.containerservice.v2019_08_01.models.AgentPool]
"""
super().__init__(**kwargs)
self.value = value
self.next_link = None
class AgentPoolUpgradeProfile(_serialization.Model):
"""The list of available upgrades for an agent pool.
Variables are only populated by the server, and will be ignored when sending a request.
All required parameters must be populated in order to send to Azure.
:ivar id: Id of the agent pool upgrade profile.
:vartype id: str
:ivar name: Name of the agent pool upgrade profile.
:vartype name: str
:ivar type: Type of the agent pool upgrade profile.
:vartype type: str
:ivar kubernetes_version: Kubernetes version (major, minor, patch). Required.
:vartype kubernetes_version: str
:ivar os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default to
Linux. Known values are: "Linux" and "Windows".
:vartype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:ivar upgrades: List of orchestrator types and versions available for upgrade.
:vartype upgrades:
list[~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolUpgradeProfilePropertiesUpgradesItem]
"""
_validation = {
"id": {"readonly": True},
"name": {"readonly": True},
"type": {"readonly": True},
"kubernetes_version": {"required": True},
"os_type": {"required": True},
}
_attribute_map = {
"id": {"key": "id", "type": "str"},
"name": {"key": "name", "type": "str"},
"type": {"key": "type", "type": "str"},
"kubernetes_version": {"key": "properties.kubernetesVersion", "type": "str"},
"os_type": {"key": "properties.osType", "type": "str"},
"upgrades": {"key": "properties.upgrades", "type": "[AgentPoolUpgradeProfilePropertiesUpgradesItem]"},
}
def __init__(
self,
*,
kubernetes_version: str,
os_type: Union[str, "_models.OSType"] = "Linux",
upgrades: Optional[List["_models.AgentPoolUpgradeProfilePropertiesUpgradesItem"]] = None,
**kwargs
):
"""
:keyword kubernetes_version: Kubernetes version (major, minor, patch). Required.
:paramtype kubernetes_version: str
:keyword os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default
to Linux. Known values are: "Linux" and "Windows".
:paramtype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:keyword upgrades: List of orchestrator types and versions available for upgrade.
:paramtype upgrades:
list[~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolUpgradeProfilePropertiesUpgradesItem]
"""
super().__init__(**kwargs)
self.id = None
self.name = None
self.type = None
self.kubernetes_version = kubernetes_version
self.os_type = os_type
self.upgrades = upgrades
class AgentPoolUpgradeProfilePropertiesUpgradesItem(_serialization.Model):
"""AgentPoolUpgradeProfilePropertiesUpgradesItem.
:ivar kubernetes_version: Kubernetes version (major, minor, patch).
:vartype kubernetes_version: str
:ivar is_preview: Whether Kubernetes version is currently in preview.
:vartype is_preview: bool
"""
_attribute_map = {
"kubernetes_version": {"key": "kubernetesVersion", "type": "str"},
"is_preview": {"key": "isPreview", "type": "bool"},
}
def __init__(self, *, kubernetes_version: Optional[str] = None, is_preview: Optional[bool] = None, **kwargs):
"""
:keyword kubernetes_version: Kubernetes version (major, minor, patch).
:paramtype kubernetes_version: str
:keyword is_preview: Whether Kubernetes version is currently in preview.
:paramtype is_preview: bool
"""
super().__init__(**kwargs)
self.kubernetes_version = kubernetes_version
self.is_preview = is_preview
class CloudErrorBody(_serialization.Model):
"""An error response from the Container service.
:ivar code: An identifier for the error. Codes are invariant and are intended to be consumed
programmatically.
:vartype code: str
:ivar message: A message describing the error, intended to be suitable for display in a user
interface.
:vartype message: str
:ivar target: The target of the particular error. For example, the name of the property in
error.
:vartype target: str
:ivar details: A list of additional details about the error.
:vartype details: list[~azure.mgmt.containerservice.v2019_08_01.models.CloudErrorBody]
"""
_attribute_map = {
"code": {"key": "code", "type": "str"},
"message": {"key": "message", "type": "str"},
"target": {"key": "target", "type": "str"},
"details": {"key": "details", "type": "[CloudErrorBody]"},
}
def __init__(
self,
*,
code: Optional[str] = None,
message: Optional[str] = None,
target: Optional[str] = None,
details: Optional[List["_models.CloudErrorBody"]] = None,
**kwargs
):
"""
:keyword code: An identifier for the error. Codes are invariant and are intended to be consumed
programmatically.
:paramtype code: str
:keyword message: A message describing the error, intended to be suitable for display in a user
interface.
:paramtype message: str
:keyword target: The target of the particular error. For example, the name of the property in
error.
:paramtype target: str
:keyword details: A list of additional details about the error.
:paramtype details: list[~azure.mgmt.containerservice.v2019_08_01.models.CloudErrorBody]
"""
super().__init__(**kwargs)
self.code = code
self.message = message
self.target = target
self.details = details
class ContainerServiceDiagnosticsProfile(_serialization.Model):
"""Profile for diagnostics on the container service cluster.
All required parameters must be populated in order to send to Azure.
:ivar vm_diagnostics: Profile for diagnostics on the container service VMs. Required.
:vartype vm_diagnostics:
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMDiagnostics
"""
_validation = {
"vm_diagnostics": {"required": True},
}
_attribute_map = {
"vm_diagnostics": {"key": "vmDiagnostics", "type": "ContainerServiceVMDiagnostics"},
}
def __init__(self, *, vm_diagnostics: "_models.ContainerServiceVMDiagnostics", **kwargs):
"""
:keyword vm_diagnostics: Profile for diagnostics on the container service VMs. Required.
:paramtype vm_diagnostics:
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMDiagnostics
"""
super().__init__(**kwargs)
self.vm_diagnostics = vm_diagnostics
class ContainerServiceLinuxProfile(_serialization.Model):
"""Profile for Linux VMs in the container service cluster.
All required parameters must be populated in order to send to Azure.
:ivar admin_username: The administrator username to use for Linux VMs. Required.
:vartype admin_username: str
:ivar ssh: SSH configuration for Linux-based VMs running on Azure. Required.
:vartype ssh: ~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceSshConfiguration
"""
_validation = {
"admin_username": {"required": True, "pattern": r"^[A-Za-z][-A-Za-z0-9_]*$"},
"ssh": {"required": True},
}
_attribute_map = {
"admin_username": {"key": "adminUsername", "type": "str"},
"ssh": {"key": "ssh", "type": "ContainerServiceSshConfiguration"},
}
def __init__(self, *, admin_username: str, ssh: "_models.ContainerServiceSshConfiguration", **kwargs):
"""
:keyword admin_username: The administrator username to use for Linux VMs. Required.
:paramtype admin_username: str
:keyword ssh: SSH configuration for Linux-based VMs running on Azure. Required.
:paramtype ssh:
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceSshConfiguration
"""
super().__init__(**kwargs)
self.admin_username = admin_username
self.ssh = ssh
class ContainerServiceMasterProfile(_serialization.Model):
"""Profile for the container service master.
Variables are only populated by the server, and will be ignored when sending a request.
All required parameters must be populated in order to send to Azure.
:ivar count: Number of masters (VMs) in the container service cluster. Allowed values are 1, 3,
and 5. The default value is 1. Known values are: 1, 3, and 5.
:vartype count: int or ~azure.mgmt.containerservice.v2019_08_01.models.Count
:ivar dns_prefix: DNS prefix to be used to create the FQDN for the master pool. Required.
:vartype dns_prefix: str
:ivar vm_size: Size of agent VMs. Required. Known values are: "Standard_A1", "Standard_A10",
"Standard_A11", "Standard_A1_v2", "Standard_A2", "Standard_A2_v2", "Standard_A2m_v2",
"Standard_A3", "Standard_A4", "Standard_A4_v2", "Standard_A4m_v2", "Standard_A5",
"Standard_A6", "Standard_A7", "Standard_A8", "Standard_A8_v2", "Standard_A8m_v2",
"Standard_A9", "Standard_B2ms", "Standard_B2s", "Standard_B4ms", "Standard_B8ms",
"Standard_D1", "Standard_D11", "Standard_D11_v2", "Standard_D11_v2_Promo", "Standard_D12",
"Standard_D12_v2", "Standard_D12_v2_Promo", "Standard_D13", "Standard_D13_v2",
"Standard_D13_v2_Promo", "Standard_D14", "Standard_D14_v2", "Standard_D14_v2_Promo",
"Standard_D15_v2", "Standard_D16_v3", "Standard_D16s_v3", "Standard_D1_v2", "Standard_D2",
"Standard_D2_v2", "Standard_D2_v2_Promo", "Standard_D2_v3", "Standard_D2s_v3", "Standard_D3",
"Standard_D32_v3", "Standard_D32s_v3", "Standard_D3_v2", "Standard_D3_v2_Promo", "Standard_D4",
"Standard_D4_v2", "Standard_D4_v2_Promo", "Standard_D4_v3", "Standard_D4s_v3",
"Standard_D5_v2", "Standard_D5_v2_Promo", "Standard_D64_v3", "Standard_D64s_v3",
"Standard_D8_v3", "Standard_D8s_v3", "Standard_DS1", "Standard_DS11", "Standard_DS11_v2",
"Standard_DS11_v2_Promo", "Standard_DS12", "Standard_DS12_v2", "Standard_DS12_v2_Promo",
"Standard_DS13", "Standard_DS13-2_v2", "Standard_DS13-4_v2", "Standard_DS13_v2",
"Standard_DS13_v2_Promo", "Standard_DS14", "Standard_DS14-4_v2", "Standard_DS14-8_v2",
"Standard_DS14_v2", "Standard_DS14_v2_Promo", "Standard_DS15_v2", "Standard_DS1_v2",
"Standard_DS2", "Standard_DS2_v2", "Standard_DS2_v2_Promo", "Standard_DS3", "Standard_DS3_v2",
"Standard_DS3_v2_Promo", "Standard_DS4", "Standard_DS4_v2", "Standard_DS4_v2_Promo",
"Standard_DS5_v2", "Standard_DS5_v2_Promo", "Standard_E16_v3", "Standard_E16s_v3",
"Standard_E2_v3", "Standard_E2s_v3", "Standard_E32-16s_v3", "Standard_E32-8s_v3",
"Standard_E32_v3", "Standard_E32s_v3", "Standard_E4_v3", "Standard_E4s_v3",
"Standard_E64-16s_v3", "Standard_E64-32s_v3", "Standard_E64_v3", "Standard_E64s_v3",
"Standard_E8_v3", "Standard_E8s_v3", "Standard_F1", "Standard_F16", "Standard_F16s",
"Standard_F16s_v2", "Standard_F1s", "Standard_F2", "Standard_F2s", "Standard_F2s_v2",
"Standard_F32s_v2", "Standard_F4", "Standard_F4s", "Standard_F4s_v2", "Standard_F64s_v2",
"Standard_F72s_v2", "Standard_F8", "Standard_F8s", "Standard_F8s_v2", "Standard_G1",
"Standard_G2", "Standard_G3", "Standard_G4", "Standard_G5", "Standard_GS1", "Standard_GS2",
"Standard_GS3", "Standard_GS4", "Standard_GS4-4", "Standard_GS4-8", "Standard_GS5",
"Standard_GS5-16", "Standard_GS5-8", "Standard_H16", "Standard_H16m", "Standard_H16mr",
"Standard_H16r", "Standard_H8", "Standard_H8m", "Standard_L16s", "Standard_L32s",
"Standard_L4s", "Standard_L8s", "Standard_M128-32ms", "Standard_M128-64ms", "Standard_M128ms",
"Standard_M128s", "Standard_M64-16ms", "Standard_M64-32ms", "Standard_M64ms", "Standard_M64s",
"Standard_NC12", "Standard_NC12s_v2", "Standard_NC12s_v3", "Standard_NC24", "Standard_NC24r",
"Standard_NC24rs_v2", "Standard_NC24rs_v3", "Standard_NC24s_v2", "Standard_NC24s_v3",
"Standard_NC6", "Standard_NC6s_v2", "Standard_NC6s_v3", "Standard_ND12s", "Standard_ND24rs",
"Standard_ND24s", "Standard_ND6s", "Standard_NV12", "Standard_NV24", and "Standard_NV6".
:vartype vm_size: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMSizeTypes
:ivar os_disk_size_gb: OS Disk Size in GB to be used to specify the disk size for every machine
in this master/agent pool. If you specify 0, it will apply the default osDisk size according to
the vmSize specified.
:vartype os_disk_size_gb: int
:ivar vnet_subnet_id: VNet SubnetID specifies the VNet's subnet identifier.
:vartype vnet_subnet_id: str
:ivar first_consecutive_static_ip: FirstConsecutiveStaticIP used to specify the first static ip
of masters.
:vartype first_consecutive_static_ip: str
:ivar storage_profile: Storage profile specifies what kind of storage used. Choose from
StorageAccount and ManagedDisks. Leave it empty, we will choose for you based on the
orchestrator choice. Known values are: "StorageAccount" and "ManagedDisks".
:vartype storage_profile: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceStorageProfileTypes
:ivar fqdn: FQDN for the master pool.
:vartype fqdn: str
"""
_validation = {
"dns_prefix": {"required": True},
"vm_size": {"required": True},
"os_disk_size_gb": {"maximum": 1023, "minimum": 0},
"fqdn": {"readonly": True},
}
_attribute_map = {
"count": {"key": "count", "type": "int"},
"dns_prefix": {"key": "dnsPrefix", "type": "str"},
"vm_size": {"key": "vmSize", "type": "str"},
"os_disk_size_gb": {"key": "osDiskSizeGB", "type": "int"},
"vnet_subnet_id": {"key": "vnetSubnetID", "type": "str"},
"first_consecutive_static_ip": {"key": "firstConsecutiveStaticIP", "type": "str"},
"storage_profile": {"key": "storageProfile", "type": "str"},
"fqdn": {"key": "fqdn", "type": "str"},
}
def __init__(
self,
*,
dns_prefix: str,
vm_size: Union[str, "_models.ContainerServiceVMSizeTypes"],
count: Union[int, "_models.Count"] = 1,
os_disk_size_gb: Optional[int] = None,
vnet_subnet_id: Optional[str] = None,
first_consecutive_static_ip: str = "10.240.255.5",
storage_profile: Optional[Union[str, "_models.ContainerServiceStorageProfileTypes"]] = None,
**kwargs
):
"""
:keyword count: Number of masters (VMs) in the container service cluster. Allowed values are 1,
3, and 5. The default value is 1. Known values are: 1, 3, and 5.
:paramtype count: int or ~azure.mgmt.containerservice.v2019_08_01.models.Count
:keyword dns_prefix: DNS prefix to be used to create the FQDN for the master pool. Required.
:paramtype dns_prefix: str
:keyword vm_size: Size of agent VMs. Required. Known values are: "Standard_A1", "Standard_A10",
"Standard_A11", "Standard_A1_v2", "Standard_A2", "Standard_A2_v2", "Standard_A2m_v2",
"Standard_A3", "Standard_A4", "Standard_A4_v2", "Standard_A4m_v2", "Standard_A5",
"Standard_A6", "Standard_A7", "Standard_A8", "Standard_A8_v2", "Standard_A8m_v2",
"Standard_A9", "Standard_B2ms", "Standard_B2s", "Standard_B4ms", "Standard_B8ms",
"Standard_D1", "Standard_D11", "Standard_D11_v2", "Standard_D11_v2_Promo", "Standard_D12",
"Standard_D12_v2", "Standard_D12_v2_Promo", "Standard_D13", "Standard_D13_v2",
"Standard_D13_v2_Promo", "Standard_D14", "Standard_D14_v2", "Standard_D14_v2_Promo",
"Standard_D15_v2", "Standard_D16_v3", "Standard_D16s_v3", "Standard_D1_v2", "Standard_D2",
"Standard_D2_v2", "Standard_D2_v2_Promo", "Standard_D2_v3", "Standard_D2s_v3", "Standard_D3",
"Standard_D32_v3", "Standard_D32s_v3", "Standard_D3_v2", "Standard_D3_v2_Promo", "Standard_D4",
"Standard_D4_v2", "Standard_D4_v2_Promo", "Standard_D4_v3", "Standard_D4s_v3",
"Standard_D5_v2", "Standard_D5_v2_Promo", "Standard_D64_v3", "Standard_D64s_v3",
"Standard_D8_v3", "Standard_D8s_v3", "Standard_DS1", "Standard_DS11", "Standard_DS11_v2",
"Standard_DS11_v2_Promo", "Standard_DS12", "Standard_DS12_v2", "Standard_DS12_v2_Promo",
"Standard_DS13", "Standard_DS13-2_v2", "Standard_DS13-4_v2", "Standard_DS13_v2",
"Standard_DS13_v2_Promo", "Standard_DS14", "Standard_DS14-4_v2", "Standard_DS14-8_v2",
"Standard_DS14_v2", "Standard_DS14_v2_Promo", "Standard_DS15_v2", "Standard_DS1_v2",
"Standard_DS2", "Standard_DS2_v2", "Standard_DS2_v2_Promo", "Standard_DS3", "Standard_DS3_v2",
"Standard_DS3_v2_Promo", "Standard_DS4", "Standard_DS4_v2", "Standard_DS4_v2_Promo",
"Standard_DS5_v2", "Standard_DS5_v2_Promo", "Standard_E16_v3", "Standard_E16s_v3",
"Standard_E2_v3", "Standard_E2s_v3", "Standard_E32-16s_v3", "Standard_E32-8s_v3",
"Standard_E32_v3", "Standard_E32s_v3", "Standard_E4_v3", "Standard_E4s_v3",
"Standard_E64-16s_v3", "Standard_E64-32s_v3", "Standard_E64_v3", "Standard_E64s_v3",
"Standard_E8_v3", "Standard_E8s_v3", "Standard_F1", "Standard_F16", "Standard_F16s",
"Standard_F16s_v2", "Standard_F1s", "Standard_F2", "Standard_F2s", "Standard_F2s_v2",
"Standard_F32s_v2", "Standard_F4", "Standard_F4s", "Standard_F4s_v2", "Standard_F64s_v2",
"Standard_F72s_v2", "Standard_F8", "Standard_F8s", "Standard_F8s_v2", "Standard_G1",
"Standard_G2", "Standard_G3", "Standard_G4", "Standard_G5", "Standard_GS1", "Standard_GS2",
"Standard_GS3", "Standard_GS4", "Standard_GS4-4", "Standard_GS4-8", "Standard_GS5",
"Standard_GS5-16", "Standard_GS5-8", "Standard_H16", "Standard_H16m", "Standard_H16mr",
"Standard_H16r", "Standard_H8", "Standard_H8m", "Standard_L16s", "Standard_L32s",
"Standard_L4s", "Standard_L8s", "Standard_M128-32ms", "Standard_M128-64ms", "Standard_M128ms",
"Standard_M128s", "Standard_M64-16ms", "Standard_M64-32ms", "Standard_M64ms", "Standard_M64s",
"Standard_NC12", "Standard_NC12s_v2", "Standard_NC12s_v3", "Standard_NC24", "Standard_NC24r",
"Standard_NC24rs_v2", "Standard_NC24rs_v3", "Standard_NC24s_v2", "Standard_NC24s_v3",
"Standard_NC6", "Standard_NC6s_v2", "Standard_NC6s_v3", "Standard_ND12s", "Standard_ND24rs",
"Standard_ND24s", "Standard_ND6s", "Standard_NV12", "Standard_NV24", and "Standard_NV6".
:paramtype vm_size: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMSizeTypes
:keyword os_disk_size_gb: OS Disk Size in GB to be used to specify the disk size for every
machine in this master/agent pool. If you specify 0, it will apply the default osDisk size
according to the vmSize specified.
:paramtype os_disk_size_gb: int
:keyword vnet_subnet_id: VNet SubnetID specifies the VNet's subnet identifier.
:paramtype vnet_subnet_id: str
:keyword first_consecutive_static_ip: FirstConsecutiveStaticIP used to specify the first static
ip of masters.
:paramtype first_consecutive_static_ip: str
:keyword storage_profile: Storage profile specifies what kind of storage used. Choose from
StorageAccount and ManagedDisks. Leave it empty, we will choose for you based on the
orchestrator choice. Known values are: "StorageAccount" and "ManagedDisks".
:paramtype storage_profile: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceStorageProfileTypes
"""
super().__init__(**kwargs)
self.count = count
self.dns_prefix = dns_prefix
self.vm_size = vm_size
self.os_disk_size_gb = os_disk_size_gb
self.vnet_subnet_id = vnet_subnet_id
self.first_consecutive_static_ip = first_consecutive_static_ip
self.storage_profile = storage_profile
self.fqdn = None
class ContainerServiceNetworkProfile(_serialization.Model):
"""Profile of network configuration.
:ivar network_plugin: Network plugin used for building Kubernetes network. Known values are:
"azure" and "kubenet".
:vartype network_plugin: str or ~azure.mgmt.containerservice.v2019_08_01.models.NetworkPlugin
:ivar network_policy: Network policy used for building Kubernetes network. Known values are:
"calico" and "azure".
:vartype network_policy: str or ~azure.mgmt.containerservice.v2019_08_01.models.NetworkPolicy
:ivar pod_cidr: A CIDR notation IP range from which to assign pod IPs when kubenet is used.
:vartype pod_cidr: str
:ivar service_cidr: A CIDR notation IP range from which to assign service cluster IPs. It must
not overlap with any Subnet IP ranges.
:vartype service_cidr: str
:ivar dns_service_ip: An IP address assigned to the Kubernetes DNS service. It must be within
the Kubernetes service address range specified in serviceCidr.
:vartype dns_service_ip: str
:ivar docker_bridge_cidr: A CIDR notation IP range assigned to the Docker bridge network. It
must not overlap with any Subnet IP ranges or the Kubernetes service address range.
:vartype docker_bridge_cidr: str
:ivar load_balancer_sku: The load balancer sku for the managed cluster. Known values are:
"standard" and "basic".
:vartype load_balancer_sku: str or
~azure.mgmt.containerservice.v2019_08_01.models.LoadBalancerSku
:ivar load_balancer_profile: Profile of the cluster load balancer.
:vartype load_balancer_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterLoadBalancerProfile
"""
_validation = {
"pod_cidr": {"pattern": r"^([0-9]{1,3}\.){3}[0-9]{1,3}(\/([0-9]|[1-2][0-9]|3[0-2]))?$"},
"service_cidr": {"pattern": r"^([0-9]{1,3}\.){3}[0-9]{1,3}(\/([0-9]|[1-2][0-9]|3[0-2]))?$"},
"dns_service_ip": {
"pattern": r"^(?:(?:25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)\.){3}(?:25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)$"
},
"docker_bridge_cidr": {"pattern": r"^([0-9]{1,3}\.){3}[0-9]{1,3}(\/([0-9]|[1-2][0-9]|3[0-2]))?$"},
}
_attribute_map = {
"network_plugin": {"key": "networkPlugin", "type": "str"},
"network_policy": {"key": "networkPolicy", "type": "str"},
"pod_cidr": {"key": "podCidr", "type": "str"},
"service_cidr": {"key": "serviceCidr", "type": "str"},
"dns_service_ip": {"key": "dnsServiceIP", "type": "str"},
"docker_bridge_cidr": {"key": "dockerBridgeCidr", "type": "str"},
"load_balancer_sku": {"key": "loadBalancerSku", "type": "str"},
"load_balancer_profile": {"key": "loadBalancerProfile", "type": "ManagedClusterLoadBalancerProfile"},
}
def __init__(
self,
*,
network_plugin: Union[str, "_models.NetworkPlugin"] = "kubenet",
network_policy: Optional[Union[str, "_models.NetworkPolicy"]] = None,
pod_cidr: str = "10.244.0.0/16",
service_cidr: str = "10.0.0.0/16",
dns_service_ip: str = "10.0.0.10",
docker_bridge_cidr: str = "172.17.0.1/16",
load_balancer_sku: Optional[Union[str, "_models.LoadBalancerSku"]] = None,
load_balancer_profile: Optional["_models.ManagedClusterLoadBalancerProfile"] = None,
**kwargs
):
"""
:keyword network_plugin: Network plugin used for building Kubernetes network. Known values are:
"azure" and "kubenet".
:paramtype network_plugin: str or ~azure.mgmt.containerservice.v2019_08_01.models.NetworkPlugin
:keyword network_policy: Network policy used for building Kubernetes network. Known values are:
"calico" and "azure".
:paramtype network_policy: str or ~azure.mgmt.containerservice.v2019_08_01.models.NetworkPolicy
:keyword pod_cidr: A CIDR notation IP range from which to assign pod IPs when kubenet is used.
:paramtype pod_cidr: str
:keyword service_cidr: A CIDR notation IP range from which to assign service cluster IPs. It
must not overlap with any Subnet IP ranges.
:paramtype service_cidr: str
:keyword dns_service_ip: An IP address assigned to the Kubernetes DNS service. It must be
within the Kubernetes service address range specified in serviceCidr.
:paramtype dns_service_ip: str
:keyword docker_bridge_cidr: A CIDR notation IP range assigned to the Docker bridge network. It
must not overlap with any Subnet IP ranges or the Kubernetes service address range.
:paramtype docker_bridge_cidr: str
:keyword load_balancer_sku: The load balancer sku for the managed cluster. Known values are:
"standard" and "basic".
:paramtype load_balancer_sku: str or
~azure.mgmt.containerservice.v2019_08_01.models.LoadBalancerSku
:keyword load_balancer_profile: Profile of the cluster load balancer.
:paramtype load_balancer_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterLoadBalancerProfile
"""
super().__init__(**kwargs)
self.network_plugin = network_plugin
self.network_policy = network_policy
self.pod_cidr = pod_cidr
self.service_cidr = service_cidr
self.dns_service_ip = dns_service_ip
self.docker_bridge_cidr = docker_bridge_cidr
self.load_balancer_sku = load_balancer_sku
self.load_balancer_profile = load_balancer_profile
class ContainerServiceSshConfiguration(_serialization.Model):
"""SSH configuration for Linux-based VMs running on Azure.
All required parameters must be populated in order to send to Azure.
:ivar public_keys: The list of SSH public keys used to authenticate with Linux-based VMs. Only
expect one key specified. Required.
:vartype public_keys:
list[~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceSshPublicKey]
"""
_validation = {
"public_keys": {"required": True},
}
_attribute_map = {
"public_keys": {"key": "publicKeys", "type": "[ContainerServiceSshPublicKey]"},
}
def __init__(self, *, public_keys: List["_models.ContainerServiceSshPublicKey"], **kwargs):
"""
:keyword public_keys: The list of SSH public keys used to authenticate with Linux-based VMs.
Only expect one key specified. Required.
:paramtype public_keys:
list[~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceSshPublicKey]
"""
super().__init__(**kwargs)
self.public_keys = public_keys
class ContainerServiceSshPublicKey(_serialization.Model):
"""Contains information about SSH certificate public key data.
All required parameters must be populated in order to send to Azure.
:ivar key_data: Certificate public key used to authenticate with VMs through SSH. The
certificate must be in PEM format with or without headers. Required.
:vartype key_data: str
"""
_validation = {
"key_data": {"required": True},
}
_attribute_map = {
"key_data": {"key": "keyData", "type": "str"},
}
def __init__(self, *, key_data: str, **kwargs):
"""
:keyword key_data: Certificate public key used to authenticate with VMs through SSH. The
certificate must be in PEM format with or without headers. Required.
:paramtype key_data: str
"""
super().__init__(**kwargs)
self.key_data = key_data
class ContainerServiceVMDiagnostics(_serialization.Model):
"""Profile for diagnostics on the container service VMs.
Variables are only populated by the server, and will be ignored when sending a request.
All required parameters must be populated in order to send to Azure.
:ivar enabled: Whether the VM diagnostic agent is provisioned on the VM. Required.
:vartype enabled: bool
:ivar storage_uri: The URI of the storage account where diagnostics are stored.
:vartype storage_uri: str
"""
_validation = {
"enabled": {"required": True},
"storage_uri": {"readonly": True},
}
_attribute_map = {
"enabled": {"key": "enabled", "type": "bool"},
"storage_uri": {"key": "storageUri", "type": "str"},
}
def __init__(self, *, enabled: bool, **kwargs):
"""
:keyword enabled: Whether the VM diagnostic agent is provisioned on the VM. Required.
:paramtype enabled: bool
"""
super().__init__(**kwargs)
self.enabled = enabled
self.storage_uri = None
class CredentialResult(_serialization.Model):
"""The credential result response.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar name: The name of the credential.
:vartype name: str
:ivar value: Base64-encoded Kubernetes configuration file.
:vartype value: bytes
"""
_validation = {
"name": {"readonly": True},
"value": {"readonly": True},
}
_attribute_map = {
"name": {"key": "name", "type": "str"},
"value": {"key": "value", "type": "bytearray"},
}
def __init__(self, **kwargs):
""" """
super().__init__(**kwargs)
self.name = None
self.value = None
class CredentialResults(_serialization.Model):
"""The list of credential result response.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar kubeconfigs: Base64-encoded Kubernetes configuration file.
:vartype kubeconfigs: list[~azure.mgmt.containerservice.v2019_08_01.models.CredentialResult]
"""
_validation = {
"kubeconfigs": {"readonly": True},
}
_attribute_map = {
"kubeconfigs": {"key": "kubeconfigs", "type": "[CredentialResult]"},
}
def __init__(self, **kwargs):
""" """
super().__init__(**kwargs)
self.kubeconfigs = None
class Resource(_serialization.Model):
"""The Resource model definition.
Variables are only populated by the server, and will be ignored when sending a request.
All required parameters must be populated in order to send to Azure.
:ivar id: Resource Id.
:vartype id: str
:ivar name: Resource name.
:vartype name: str
:ivar type: Resource type.
:vartype type: str
:ivar location: Resource location. Required.
:vartype location: str
:ivar tags: Resource tags.
:vartype tags: dict[str, str]
"""
_validation = {
"id": {"readonly": True},
"name": {"readonly": True},
"type": {"readonly": True},
"location": {"required": True},
}
_attribute_map = {
"id": {"key": "id", "type": "str"},
"name": {"key": "name", "type": "str"},
"type": {"key": "type", "type": "str"},
"location": {"key": "location", "type": "str"},
"tags": {"key": "tags", "type": "{str}"},
}
def __init__(self, *, location: str, tags: Optional[Dict[str, str]] = None, **kwargs):
"""
:keyword location: Resource location. Required.
:paramtype location: str
:keyword tags: Resource tags.
:paramtype tags: dict[str, str]
"""
super().__init__(**kwargs)
self.id = None
self.name = None
self.type = None
self.location = location
self.tags = tags
class ManagedCluster(Resource): # pylint: disable=too-many-instance-attributes
"""Managed cluster.
Variables are only populated by the server, and will be ignored when sending a request.
All required parameters must be populated in order to send to Azure.
:ivar id: Resource Id.
:vartype id: str
:ivar name: Resource name.
:vartype name: str
:ivar type: Resource type.
:vartype type: str
:ivar location: Resource location. Required.
:vartype location: str
:ivar tags: Resource tags.
:vartype tags: dict[str, str]
:ivar identity: The identity of the managed cluster, if configured.
:vartype identity: ~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterIdentity
:ivar provisioning_state: The current deployment or provisioning state, which only appears in
the response.
:vartype provisioning_state: str
:ivar max_agent_pools: The max number of agent pools for the managed cluster.
:vartype max_agent_pools: int
:ivar kubernetes_version: Version of Kubernetes specified when creating the managed cluster.
:vartype kubernetes_version: str
:ivar dns_prefix: DNS prefix specified when creating the managed cluster.
:vartype dns_prefix: str
:ivar fqdn: FQDN for the master pool.
:vartype fqdn: str
:ivar agent_pool_profiles: Properties of the agent pool.
:vartype agent_pool_profiles:
list[~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterAgentPoolProfile]
:ivar linux_profile: Profile for Linux VMs in the container service cluster.
:vartype linux_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceLinuxProfile
:ivar windows_profile: Profile for Windows VMs in the container service cluster.
:vartype windows_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterWindowsProfile
:ivar service_principal_profile: Information about a service principal identity for the cluster
to use for manipulating Azure APIs.
:vartype service_principal_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterServicePrincipalProfile
:ivar addon_profiles: Profile of managed cluster add-on.
:vartype addon_profiles: dict[str,
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterAddonProfile]
:ivar node_resource_group: Name of the resource group containing agent pool nodes.
:vartype node_resource_group: str
:ivar enable_rbac: Whether to enable Kubernetes Role-Based Access Control.
:vartype enable_rbac: bool
:ivar enable_pod_security_policy: (DEPRECATING) Whether to enable Kubernetes pod security
policy (preview). This feature is set for removal on October 15th, 2020. Learn more at
aka.ms/aks/azpodpolicy.
:vartype enable_pod_security_policy: bool
:ivar network_profile: Profile of network configuration.
:vartype network_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceNetworkProfile
:ivar aad_profile: Profile of Azure Active Directory configuration.
:vartype aad_profile: ~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterAADProfile
:ivar api_server_access_profile: Access profile for managed cluster API server.
:vartype api_server_access_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterAPIServerAccessProfile
"""
_validation = {
"id": {"readonly": True},
"name": {"readonly": True},
"type": {"readonly": True},
"location": {"required": True},
"provisioning_state": {"readonly": True},
"max_agent_pools": {"readonly": True},
"fqdn": {"readonly": True},
}
_attribute_map = {
"id": {"key": "id", "type": "str"},
"name": {"key": "name", "type": "str"},
"type": {"key": "type", "type": "str"},
"location": {"key": "location", "type": "str"},
"tags": {"key": "tags", "type": "{str}"},
"identity": {"key": "identity", "type": "ManagedClusterIdentity"},
"provisioning_state": {"key": "properties.provisioningState", "type": "str"},
"max_agent_pools": {"key": "properties.maxAgentPools", "type": "int"},
"kubernetes_version": {"key": "properties.kubernetesVersion", "type": "str"},
"dns_prefix": {"key": "properties.dnsPrefix", "type": "str"},
"fqdn": {"key": "properties.fqdn", "type": "str"},
"agent_pool_profiles": {"key": "properties.agentPoolProfiles", "type": "[ManagedClusterAgentPoolProfile]"},
"linux_profile": {"key": "properties.linuxProfile", "type": "ContainerServiceLinuxProfile"},
"windows_profile": {"key": "properties.windowsProfile", "type": "ManagedClusterWindowsProfile"},
"service_principal_profile": {
"key": "properties.servicePrincipalProfile",
"type": "ManagedClusterServicePrincipalProfile",
},
"addon_profiles": {"key": "properties.addonProfiles", "type": "{ManagedClusterAddonProfile}"},
"node_resource_group": {"key": "properties.nodeResourceGroup", "type": "str"},
"enable_rbac": {"key": "properties.enableRBAC", "type": "bool"},
"enable_pod_security_policy": {"key": "properties.enablePodSecurityPolicy", "type": "bool"},
"network_profile": {"key": "properties.networkProfile", "type": "ContainerServiceNetworkProfile"},
"aad_profile": {"key": "properties.aadProfile", "type": "ManagedClusterAADProfile"},
"api_server_access_profile": {
"key": "properties.apiServerAccessProfile",
"type": "ManagedClusterAPIServerAccessProfile",
},
}
def __init__(
self,
*,
location: str,
tags: Optional[Dict[str, str]] = None,
identity: Optional["_models.ManagedClusterIdentity"] = None,
kubernetes_version: Optional[str] = None,
dns_prefix: Optional[str] = None,
agent_pool_profiles: Optional[List["_models.ManagedClusterAgentPoolProfile"]] = None,
linux_profile: Optional["_models.ContainerServiceLinuxProfile"] = None,
windows_profile: Optional["_models.ManagedClusterWindowsProfile"] = None,
service_principal_profile: Optional["_models.ManagedClusterServicePrincipalProfile"] = None,
addon_profiles: Optional[Dict[str, "_models.ManagedClusterAddonProfile"]] = None,
node_resource_group: Optional[str] = None,
enable_rbac: Optional[bool] = None,
enable_pod_security_policy: Optional[bool] = None,
network_profile: Optional["_models.ContainerServiceNetworkProfile"] = None,
aad_profile: Optional["_models.ManagedClusterAADProfile"] = None,
api_server_access_profile: Optional["_models.ManagedClusterAPIServerAccessProfile"] = None,
**kwargs
):
"""
:keyword location: Resource location. Required.
:paramtype location: str
:keyword tags: Resource tags.
:paramtype tags: dict[str, str]
:keyword identity: The identity of the managed cluster, if configured.
:paramtype identity: ~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterIdentity
:keyword kubernetes_version: Version of Kubernetes specified when creating the managed cluster.
:paramtype kubernetes_version: str
:keyword dns_prefix: DNS prefix specified when creating the managed cluster.
:paramtype dns_prefix: str
:keyword agent_pool_profiles: Properties of the agent pool.
:paramtype agent_pool_profiles:
list[~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterAgentPoolProfile]
:keyword linux_profile: Profile for Linux VMs in the container service cluster.
:paramtype linux_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceLinuxProfile
:keyword windows_profile: Profile for Windows VMs in the container service cluster.
:paramtype windows_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterWindowsProfile
:keyword service_principal_profile: Information about a service principal identity for the
cluster to use for manipulating Azure APIs.
:paramtype service_principal_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterServicePrincipalProfile
:keyword addon_profiles: Profile of managed cluster add-on.
:paramtype addon_profiles: dict[str,
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterAddonProfile]
:keyword node_resource_group: Name of the resource group containing agent pool nodes.
:paramtype node_resource_group: str
:keyword enable_rbac: Whether to enable Kubernetes Role-Based Access Control.
:paramtype enable_rbac: bool
:keyword enable_pod_security_policy: (DEPRECATING) Whether to enable Kubernetes pod security
policy (preview). This feature is set for removal on October 15th, 2020. Learn more at
aka.ms/aks/azpodpolicy.
:paramtype enable_pod_security_policy: bool
:keyword network_profile: Profile of network configuration.
:paramtype network_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceNetworkProfile
:keyword aad_profile: Profile of Azure Active Directory configuration.
:paramtype aad_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterAADProfile
:keyword api_server_access_profile: Access profile for managed cluster API server.
:paramtype api_server_access_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterAPIServerAccessProfile
"""
super().__init__(location=location, tags=tags, **kwargs)
self.identity = identity
self.provisioning_state = None
self.max_agent_pools = None
self.kubernetes_version = kubernetes_version
self.dns_prefix = dns_prefix
self.fqdn = None
self.agent_pool_profiles = agent_pool_profiles
self.linux_profile = linux_profile
self.windows_profile = windows_profile
self.service_principal_profile = service_principal_profile
self.addon_profiles = addon_profiles
self.node_resource_group = node_resource_group
self.enable_rbac = enable_rbac
self.enable_pod_security_policy = enable_pod_security_policy
self.network_profile = network_profile
self.aad_profile = aad_profile
self.api_server_access_profile = api_server_access_profile
class ManagedClusterAADProfile(_serialization.Model):
"""AADProfile specifies attributes for Azure Active Directory integration.
All required parameters must be populated in order to send to Azure.
:ivar client_app_id: The client AAD application ID. Required.
:vartype client_app_id: str
:ivar server_app_id: The server AAD application ID. Required.
:vartype server_app_id: str
:ivar server_app_secret: The server AAD application secret.
:vartype server_app_secret: str
:ivar tenant_id: The AAD tenant ID to use for authentication. If not specified, will use the
tenant of the deployment subscription.
:vartype tenant_id: str
"""
_validation = {
"client_app_id": {"required": True},
"server_app_id": {"required": True},
}
_attribute_map = {
"client_app_id": {"key": "clientAppID", "type": "str"},
"server_app_id": {"key": "serverAppID", "type": "str"},
"server_app_secret": {"key": "serverAppSecret", "type": "str"},
"tenant_id": {"key": "tenantID", "type": "str"},
}
def __init__(
self,
*,
client_app_id: str,
server_app_id: str,
server_app_secret: Optional[str] = None,
tenant_id: Optional[str] = None,
**kwargs
):
"""
:keyword client_app_id: The client AAD application ID. Required.
:paramtype client_app_id: str
:keyword server_app_id: The server AAD application ID. Required.
:paramtype server_app_id: str
:keyword server_app_secret: The server AAD application secret.
:paramtype server_app_secret: str
:keyword tenant_id: The AAD tenant ID to use for authentication. If not specified, will use the
tenant of the deployment subscription.
:paramtype tenant_id: str
"""
super().__init__(**kwargs)
self.client_app_id = client_app_id
self.server_app_id = server_app_id
self.server_app_secret = server_app_secret
self.tenant_id = tenant_id
class ManagedClusterAccessProfile(Resource):
"""Managed cluster Access Profile.
Variables are only populated by the server, and will be ignored when sending a request.
All required parameters must be populated in order to send to Azure.
:ivar id: Resource Id.
:vartype id: str
:ivar name: Resource name.
:vartype name: str
:ivar type: Resource type.
:vartype type: str
:ivar location: Resource location. Required.
:vartype location: str
:ivar tags: Resource tags.
:vartype tags: dict[str, str]
:ivar kube_config: Base64-encoded Kubernetes configuration file.
:vartype kube_config: bytes
"""
_validation = {
"id": {"readonly": True},
"name": {"readonly": True},
"type": {"readonly": True},
"location": {"required": True},
}
_attribute_map = {
"id": {"key": "id", "type": "str"},
"name": {"key": "name", "type": "str"},
"type": {"key": "type", "type": "str"},
"location": {"key": "location", "type": "str"},
"tags": {"key": "tags", "type": "{str}"},
"kube_config": {"key": "properties.kubeConfig", "type": "bytearray"},
}
def __init__(
self, *, location: str, tags: Optional[Dict[str, str]] = None, kube_config: Optional[bytes] = None, **kwargs
):
"""
:keyword location: Resource location. Required.
:paramtype location: str
:keyword tags: Resource tags.
:paramtype tags: dict[str, str]
:keyword kube_config: Base64-encoded Kubernetes configuration file.
:paramtype kube_config: bytes
"""
super().__init__(location=location, tags=tags, **kwargs)
self.kube_config = kube_config
class ManagedClusterAddonProfile(_serialization.Model):
"""A Kubernetes add-on profile for a managed cluster.
All required parameters must be populated in order to send to Azure.
:ivar enabled: Whether the add-on is enabled or not. Required.
:vartype enabled: bool
:ivar config: Key-value pairs for configuring an add-on.
:vartype config: dict[str, str]
"""
_validation = {
"enabled": {"required": True},
}
_attribute_map = {
"enabled": {"key": "enabled", "type": "bool"},
"config": {"key": "config", "type": "{str}"},
}
def __init__(self, *, enabled: bool, config: Optional[Dict[str, str]] = None, **kwargs):
"""
:keyword enabled: Whether the add-on is enabled or not. Required.
:paramtype enabled: bool
:keyword config: Key-value pairs for configuring an add-on.
:paramtype config: dict[str, str]
"""
super().__init__(**kwargs)
self.enabled = enabled
self.config = config
class ManagedClusterAgentPoolProfileProperties(_serialization.Model): # pylint: disable=too-many-instance-attributes
"""Properties for the container service agent pool profile.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar count: Number of agents (VMs) to host docker containers. Allowed values must be in the
range of 1 to 100 (inclusive). The default value is 1.
:vartype count: int
:ivar vm_size: Size of agent VMs. Known values are: "Standard_A1", "Standard_A10",
"Standard_A11", "Standard_A1_v2", "Standard_A2", "Standard_A2_v2", "Standard_A2m_v2",
"Standard_A3", "Standard_A4", "Standard_A4_v2", "Standard_A4m_v2", "Standard_A5",
"Standard_A6", "Standard_A7", "Standard_A8", "Standard_A8_v2", "Standard_A8m_v2",
"Standard_A9", "Standard_B2ms", "Standard_B2s", "Standard_B4ms", "Standard_B8ms",
"Standard_D1", "Standard_D11", "Standard_D11_v2", "Standard_D11_v2_Promo", "Standard_D12",
"Standard_D12_v2", "Standard_D12_v2_Promo", "Standard_D13", "Standard_D13_v2",
"Standard_D13_v2_Promo", "Standard_D14", "Standard_D14_v2", "Standard_D14_v2_Promo",
"Standard_D15_v2", "Standard_D16_v3", "Standard_D16s_v3", "Standard_D1_v2", "Standard_D2",
"Standard_D2_v2", "Standard_D2_v2_Promo", "Standard_D2_v3", "Standard_D2s_v3", "Standard_D3",
"Standard_D32_v3", "Standard_D32s_v3", "Standard_D3_v2", "Standard_D3_v2_Promo", "Standard_D4",
"Standard_D4_v2", "Standard_D4_v2_Promo", "Standard_D4_v3", "Standard_D4s_v3",
"Standard_D5_v2", "Standard_D5_v2_Promo", "Standard_D64_v3", "Standard_D64s_v3",
"Standard_D8_v3", "Standard_D8s_v3", "Standard_DS1", "Standard_DS11", "Standard_DS11_v2",
"Standard_DS11_v2_Promo", "Standard_DS12", "Standard_DS12_v2", "Standard_DS12_v2_Promo",
"Standard_DS13", "Standard_DS13-2_v2", "Standard_DS13-4_v2", "Standard_DS13_v2",
"Standard_DS13_v2_Promo", "Standard_DS14", "Standard_DS14-4_v2", "Standard_DS14-8_v2",
"Standard_DS14_v2", "Standard_DS14_v2_Promo", "Standard_DS15_v2", "Standard_DS1_v2",
"Standard_DS2", "Standard_DS2_v2", "Standard_DS2_v2_Promo", "Standard_DS3", "Standard_DS3_v2",
"Standard_DS3_v2_Promo", "Standard_DS4", "Standard_DS4_v2", "Standard_DS4_v2_Promo",
"Standard_DS5_v2", "Standard_DS5_v2_Promo", "Standard_E16_v3", "Standard_E16s_v3",
"Standard_E2_v3", "Standard_E2s_v3", "Standard_E32-16s_v3", "Standard_E32-8s_v3",
"Standard_E32_v3", "Standard_E32s_v3", "Standard_E4_v3", "Standard_E4s_v3",
"Standard_E64-16s_v3", "Standard_E64-32s_v3", "Standard_E64_v3", "Standard_E64s_v3",
"Standard_E8_v3", "Standard_E8s_v3", "Standard_F1", "Standard_F16", "Standard_F16s",
"Standard_F16s_v2", "Standard_F1s", "Standard_F2", "Standard_F2s", "Standard_F2s_v2",
"Standard_F32s_v2", "Standard_F4", "Standard_F4s", "Standard_F4s_v2", "Standard_F64s_v2",
"Standard_F72s_v2", "Standard_F8", "Standard_F8s", "Standard_F8s_v2", "Standard_G1",
"Standard_G2", "Standard_G3", "Standard_G4", "Standard_G5", "Standard_GS1", "Standard_GS2",
"Standard_GS3", "Standard_GS4", "Standard_GS4-4", "Standard_GS4-8", "Standard_GS5",
"Standard_GS5-16", "Standard_GS5-8", "Standard_H16", "Standard_H16m", "Standard_H16mr",
"Standard_H16r", "Standard_H8", "Standard_H8m", "Standard_L16s", "Standard_L32s",
"Standard_L4s", "Standard_L8s", "Standard_M128-32ms", "Standard_M128-64ms", "Standard_M128ms",
"Standard_M128s", "Standard_M64-16ms", "Standard_M64-32ms", "Standard_M64ms", "Standard_M64s",
"Standard_NC12", "Standard_NC12s_v2", "Standard_NC12s_v3", "Standard_NC24", "Standard_NC24r",
"Standard_NC24rs_v2", "Standard_NC24rs_v3", "Standard_NC24s_v2", "Standard_NC24s_v3",
"Standard_NC6", "Standard_NC6s_v2", "Standard_NC6s_v3", "Standard_ND12s", "Standard_ND24rs",
"Standard_ND24s", "Standard_ND6s", "Standard_NV12", "Standard_NV24", and "Standard_NV6".
:vartype vm_size: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMSizeTypes
:ivar os_disk_size_gb: OS Disk Size in GB to be used to specify the disk size for every machine
in this master/agent pool. If you specify 0, it will apply the default osDisk size according to
the vmSize specified.
:vartype os_disk_size_gb: int
:ivar vnet_subnet_id: VNet SubnetID specifies the VNet's subnet identifier.
:vartype vnet_subnet_id: str
:ivar max_pods: Maximum number of pods that can run on a node.
:vartype max_pods: int
:ivar os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default to
Linux. Known values are: "Linux" and "Windows".
:vartype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:ivar max_count: Maximum number of nodes for auto-scaling.
:vartype max_count: int
:ivar min_count: Minimum number of nodes for auto-scaling.
:vartype min_count: int
:ivar enable_auto_scaling: Whether to enable auto-scaler.
:vartype enable_auto_scaling: bool
:ivar type: AgentPoolType represents types of an agent pool. Known values are:
"VirtualMachineScaleSets" and "AvailabilitySet".
:vartype type: str or ~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolType
:ivar orchestrator_version: Version of orchestrator specified when creating the managed
cluster.
:vartype orchestrator_version: str
:ivar provisioning_state: The current deployment or provisioning state, which only appears in
the response.
:vartype provisioning_state: str
:ivar availability_zones: (PREVIEW) Availability zones for nodes. Must use
VirtualMachineScaleSets AgentPoolType.
:vartype availability_zones: list[str]
:ivar enable_node_public_ip: Enable public IP for nodes.
:vartype enable_node_public_ip: bool
:ivar scale_set_priority: ScaleSetPriority to be used to specify virtual machine scale set
priority. Default to regular. Known values are: "Low" and "Regular".
:vartype scale_set_priority: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetPriority
:ivar scale_set_eviction_policy: ScaleSetEvictionPolicy to be used to specify eviction policy
for low priority virtual machine scale set. Default to Delete. Known values are: "Delete" and
"Deallocate".
:vartype scale_set_eviction_policy: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetEvictionPolicy
:ivar node_taints: Taints added to new nodes during node pool create and scale. For example,
key=value:NoSchedule.
:vartype node_taints: list[str]
"""
_validation = {
"os_disk_size_gb": {"maximum": 1023, "minimum": 0},
"provisioning_state": {"readonly": True},
}
_attribute_map = {
"count": {"key": "count", "type": "int"},
"vm_size": {"key": "vmSize", "type": "str"},
"os_disk_size_gb": {"key": "osDiskSizeGB", "type": "int"},
"vnet_subnet_id": {"key": "vnetSubnetID", "type": "str"},
"max_pods": {"key": "maxPods", "type": "int"},
"os_type": {"key": "osType", "type": "str"},
"max_count": {"key": "maxCount", "type": "int"},
"min_count": {"key": "minCount", "type": "int"},
"enable_auto_scaling": {"key": "enableAutoScaling", "type": "bool"},
"type": {"key": "type", "type": "str"},
"orchestrator_version": {"key": "orchestratorVersion", "type": "str"},
"provisioning_state": {"key": "provisioningState", "type": "str"},
"availability_zones": {"key": "availabilityZones", "type": "[str]"},
"enable_node_public_ip": {"key": "enableNodePublicIP", "type": "bool"},
"scale_set_priority": {"key": "scaleSetPriority", "type": "str"},
"scale_set_eviction_policy": {"key": "scaleSetEvictionPolicy", "type": "str"},
"node_taints": {"key": "nodeTaints", "type": "[str]"},
}
def __init__(
self,
*,
count: Optional[int] = None,
vm_size: Optional[Union[str, "_models.ContainerServiceVMSizeTypes"]] = None,
os_disk_size_gb: Optional[int] = None,
vnet_subnet_id: Optional[str] = None,
max_pods: Optional[int] = None,
os_type: Union[str, "_models.OSType"] = "Linux",
max_count: Optional[int] = None,
min_count: Optional[int] = None,
enable_auto_scaling: Optional[bool] = None,
type: Optional[Union[str, "_models.AgentPoolType"]] = None,
orchestrator_version: Optional[str] = None,
availability_zones: Optional[List[str]] = None,
enable_node_public_ip: Optional[bool] = None,
scale_set_priority: Union[str, "_models.ScaleSetPriority"] = "Regular",
scale_set_eviction_policy: Union[str, "_models.ScaleSetEvictionPolicy"] = "Delete",
node_taints: Optional[List[str]] = None,
**kwargs
):
"""
:keyword count: Number of agents (VMs) to host docker containers. Allowed values must be in the
range of 1 to 100 (inclusive). The default value is 1.
:paramtype count: int
:keyword vm_size: Size of agent VMs. Known values are: "Standard_A1", "Standard_A10",
"Standard_A11", "Standard_A1_v2", "Standard_A2", "Standard_A2_v2", "Standard_A2m_v2",
"Standard_A3", "Standard_A4", "Standard_A4_v2", "Standard_A4m_v2", "Standard_A5",
"Standard_A6", "Standard_A7", "Standard_A8", "Standard_A8_v2", "Standard_A8m_v2",
"Standard_A9", "Standard_B2ms", "Standard_B2s", "Standard_B4ms", "Standard_B8ms",
"Standard_D1", "Standard_D11", "Standard_D11_v2", "Standard_D11_v2_Promo", "Standard_D12",
"Standard_D12_v2", "Standard_D12_v2_Promo", "Standard_D13", "Standard_D13_v2",
"Standard_D13_v2_Promo", "Standard_D14", "Standard_D14_v2", "Standard_D14_v2_Promo",
"Standard_D15_v2", "Standard_D16_v3", "Standard_D16s_v3", "Standard_D1_v2", "Standard_D2",
"Standard_D2_v2", "Standard_D2_v2_Promo", "Standard_D2_v3", "Standard_D2s_v3", "Standard_D3",
"Standard_D32_v3", "Standard_D32s_v3", "Standard_D3_v2", "Standard_D3_v2_Promo", "Standard_D4",
"Standard_D4_v2", "Standard_D4_v2_Promo", "Standard_D4_v3", "Standard_D4s_v3",
"Standard_D5_v2", "Standard_D5_v2_Promo", "Standard_D64_v3", "Standard_D64s_v3",
"Standard_D8_v3", "Standard_D8s_v3", "Standard_DS1", "Standard_DS11", "Standard_DS11_v2",
"Standard_DS11_v2_Promo", "Standard_DS12", "Standard_DS12_v2", "Standard_DS12_v2_Promo",
"Standard_DS13", "Standard_DS13-2_v2", "Standard_DS13-4_v2", "Standard_DS13_v2",
"Standard_DS13_v2_Promo", "Standard_DS14", "Standard_DS14-4_v2", "Standard_DS14-8_v2",
"Standard_DS14_v2", "Standard_DS14_v2_Promo", "Standard_DS15_v2", "Standard_DS1_v2",
"Standard_DS2", "Standard_DS2_v2", "Standard_DS2_v2_Promo", "Standard_DS3", "Standard_DS3_v2",
"Standard_DS3_v2_Promo", "Standard_DS4", "Standard_DS4_v2", "Standard_DS4_v2_Promo",
"Standard_DS5_v2", "Standard_DS5_v2_Promo", "Standard_E16_v3", "Standard_E16s_v3",
"Standard_E2_v3", "Standard_E2s_v3", "Standard_E32-16s_v3", "Standard_E32-8s_v3",
"Standard_E32_v3", "Standard_E32s_v3", "Standard_E4_v3", "Standard_E4s_v3",
"Standard_E64-16s_v3", "Standard_E64-32s_v3", "Standard_E64_v3", "Standard_E64s_v3",
"Standard_E8_v3", "Standard_E8s_v3", "Standard_F1", "Standard_F16", "Standard_F16s",
"Standard_F16s_v2", "Standard_F1s", "Standard_F2", "Standard_F2s", "Standard_F2s_v2",
"Standard_F32s_v2", "Standard_F4", "Standard_F4s", "Standard_F4s_v2", "Standard_F64s_v2",
"Standard_F72s_v2", "Standard_F8", "Standard_F8s", "Standard_F8s_v2", "Standard_G1",
"Standard_G2", "Standard_G3", "Standard_G4", "Standard_G5", "Standard_GS1", "Standard_GS2",
"Standard_GS3", "Standard_GS4", "Standard_GS4-4", "Standard_GS4-8", "Standard_GS5",
"Standard_GS5-16", "Standard_GS5-8", "Standard_H16", "Standard_H16m", "Standard_H16mr",
"Standard_H16r", "Standard_H8", "Standard_H8m", "Standard_L16s", "Standard_L32s",
"Standard_L4s", "Standard_L8s", "Standard_M128-32ms", "Standard_M128-64ms", "Standard_M128ms",
"Standard_M128s", "Standard_M64-16ms", "Standard_M64-32ms", "Standard_M64ms", "Standard_M64s",
"Standard_NC12", "Standard_NC12s_v2", "Standard_NC12s_v3", "Standard_NC24", "Standard_NC24r",
"Standard_NC24rs_v2", "Standard_NC24rs_v3", "Standard_NC24s_v2", "Standard_NC24s_v3",
"Standard_NC6", "Standard_NC6s_v2", "Standard_NC6s_v3", "Standard_ND12s", "Standard_ND24rs",
"Standard_ND24s", "Standard_ND6s", "Standard_NV12", "Standard_NV24", and "Standard_NV6".
:paramtype vm_size: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMSizeTypes
:keyword os_disk_size_gb: OS Disk Size in GB to be used to specify the disk size for every
machine in this master/agent pool. If you specify 0, it will apply the default osDisk size
according to the vmSize specified.
:paramtype os_disk_size_gb: int
:keyword vnet_subnet_id: VNet SubnetID specifies the VNet's subnet identifier.
:paramtype vnet_subnet_id: str
:keyword max_pods: Maximum number of pods that can run on a node.
:paramtype max_pods: int
:keyword os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default
to Linux. Known values are: "Linux" and "Windows".
:paramtype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:keyword max_count: Maximum number of nodes for auto-scaling.
:paramtype max_count: int
:keyword min_count: Minimum number of nodes for auto-scaling.
:paramtype min_count: int
:keyword enable_auto_scaling: Whether to enable auto-scaler.
:paramtype enable_auto_scaling: bool
:keyword type: AgentPoolType represents types of an agent pool. Known values are:
"VirtualMachineScaleSets" and "AvailabilitySet".
:paramtype type: str or ~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolType
:keyword orchestrator_version: Version of orchestrator specified when creating the managed
cluster.
:paramtype orchestrator_version: str
:keyword availability_zones: (PREVIEW) Availability zones for nodes. Must use
VirtualMachineScaleSets AgentPoolType.
:paramtype availability_zones: list[str]
:keyword enable_node_public_ip: Enable public IP for nodes.
:paramtype enable_node_public_ip: bool
:keyword scale_set_priority: ScaleSetPriority to be used to specify virtual machine scale set
priority. Default to regular. Known values are: "Low" and "Regular".
:paramtype scale_set_priority: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetPriority
:keyword scale_set_eviction_policy: ScaleSetEvictionPolicy to be used to specify eviction
policy for low priority virtual machine scale set. Default to Delete. Known values are:
"Delete" and "Deallocate".
:paramtype scale_set_eviction_policy: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetEvictionPolicy
:keyword node_taints: Taints added to new nodes during node pool create and scale. For example,
key=value:NoSchedule.
:paramtype node_taints: list[str]
"""
super().__init__(**kwargs)
self.count = count
self.vm_size = vm_size
self.os_disk_size_gb = os_disk_size_gb
self.vnet_subnet_id = vnet_subnet_id
self.max_pods = max_pods
self.os_type = os_type
self.max_count = max_count
self.min_count = min_count
self.enable_auto_scaling = enable_auto_scaling
self.type = type
self.orchestrator_version = orchestrator_version
self.provisioning_state = None
self.availability_zones = availability_zones
self.enable_node_public_ip = enable_node_public_ip
self.scale_set_priority = scale_set_priority
self.scale_set_eviction_policy = scale_set_eviction_policy
self.node_taints = node_taints
class ManagedClusterAgentPoolProfile(
ManagedClusterAgentPoolProfileProperties
): # pylint: disable=too-many-instance-attributes
"""Profile for the container service agent pool.
Variables are only populated by the server, and will be ignored when sending a request.
All required parameters must be populated in order to send to Azure.
:ivar count: Number of agents (VMs) to host docker containers. Allowed values must be in the
range of 1 to 100 (inclusive). The default value is 1.
:vartype count: int
:ivar vm_size: Size of agent VMs. Known values are: "Standard_A1", "Standard_A10",
"Standard_A11", "Standard_A1_v2", "Standard_A2", "Standard_A2_v2", "Standard_A2m_v2",
"Standard_A3", "Standard_A4", "Standard_A4_v2", "Standard_A4m_v2", "Standard_A5",
"Standard_A6", "Standard_A7", "Standard_A8", "Standard_A8_v2", "Standard_A8m_v2",
"Standard_A9", "Standard_B2ms", "Standard_B2s", "Standard_B4ms", "Standard_B8ms",
"Standard_D1", "Standard_D11", "Standard_D11_v2", "Standard_D11_v2_Promo", "Standard_D12",
"Standard_D12_v2", "Standard_D12_v2_Promo", "Standard_D13", "Standard_D13_v2",
"Standard_D13_v2_Promo", "Standard_D14", "Standard_D14_v2", "Standard_D14_v2_Promo",
"Standard_D15_v2", "Standard_D16_v3", "Standard_D16s_v3", "Standard_D1_v2", "Standard_D2",
"Standard_D2_v2", "Standard_D2_v2_Promo", "Standard_D2_v3", "Standard_D2s_v3", "Standard_D3",
"Standard_D32_v3", "Standard_D32s_v3", "Standard_D3_v2", "Standard_D3_v2_Promo", "Standard_D4",
"Standard_D4_v2", "Standard_D4_v2_Promo", "Standard_D4_v3", "Standard_D4s_v3",
"Standard_D5_v2", "Standard_D5_v2_Promo", "Standard_D64_v3", "Standard_D64s_v3",
"Standard_D8_v3", "Standard_D8s_v3", "Standard_DS1", "Standard_DS11", "Standard_DS11_v2",
"Standard_DS11_v2_Promo", "Standard_DS12", "Standard_DS12_v2", "Standard_DS12_v2_Promo",
"Standard_DS13", "Standard_DS13-2_v2", "Standard_DS13-4_v2", "Standard_DS13_v2",
"Standard_DS13_v2_Promo", "Standard_DS14", "Standard_DS14-4_v2", "Standard_DS14-8_v2",
"Standard_DS14_v2", "Standard_DS14_v2_Promo", "Standard_DS15_v2", "Standard_DS1_v2",
"Standard_DS2", "Standard_DS2_v2", "Standard_DS2_v2_Promo", "Standard_DS3", "Standard_DS3_v2",
"Standard_DS3_v2_Promo", "Standard_DS4", "Standard_DS4_v2", "Standard_DS4_v2_Promo",
"Standard_DS5_v2", "Standard_DS5_v2_Promo", "Standard_E16_v3", "Standard_E16s_v3",
"Standard_E2_v3", "Standard_E2s_v3", "Standard_E32-16s_v3", "Standard_E32-8s_v3",
"Standard_E32_v3", "Standard_E32s_v3", "Standard_E4_v3", "Standard_E4s_v3",
"Standard_E64-16s_v3", "Standard_E64-32s_v3", "Standard_E64_v3", "Standard_E64s_v3",
"Standard_E8_v3", "Standard_E8s_v3", "Standard_F1", "Standard_F16", "Standard_F16s",
"Standard_F16s_v2", "Standard_F1s", "Standard_F2", "Standard_F2s", "Standard_F2s_v2",
"Standard_F32s_v2", "Standard_F4", "Standard_F4s", "Standard_F4s_v2", "Standard_F64s_v2",
"Standard_F72s_v2", "Standard_F8", "Standard_F8s", "Standard_F8s_v2", "Standard_G1",
"Standard_G2", "Standard_G3", "Standard_G4", "Standard_G5", "Standard_GS1", "Standard_GS2",
"Standard_GS3", "Standard_GS4", "Standard_GS4-4", "Standard_GS4-8", "Standard_GS5",
"Standard_GS5-16", "Standard_GS5-8", "Standard_H16", "Standard_H16m", "Standard_H16mr",
"Standard_H16r", "Standard_H8", "Standard_H8m", "Standard_L16s", "Standard_L32s",
"Standard_L4s", "Standard_L8s", "Standard_M128-32ms", "Standard_M128-64ms", "Standard_M128ms",
"Standard_M128s", "Standard_M64-16ms", "Standard_M64-32ms", "Standard_M64ms", "Standard_M64s",
"Standard_NC12", "Standard_NC12s_v2", "Standard_NC12s_v3", "Standard_NC24", "Standard_NC24r",
"Standard_NC24rs_v2", "Standard_NC24rs_v3", "Standard_NC24s_v2", "Standard_NC24s_v3",
"Standard_NC6", "Standard_NC6s_v2", "Standard_NC6s_v3", "Standard_ND12s", "Standard_ND24rs",
"Standard_ND24s", "Standard_ND6s", "Standard_NV12", "Standard_NV24", and "Standard_NV6".
:vartype vm_size: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMSizeTypes
:ivar os_disk_size_gb: OS Disk Size in GB to be used to specify the disk size for every machine
in this master/agent pool. If you specify 0, it will apply the default osDisk size according to
the vmSize specified.
:vartype os_disk_size_gb: int
:ivar vnet_subnet_id: VNet SubnetID specifies the VNet's subnet identifier.
:vartype vnet_subnet_id: str
:ivar max_pods: Maximum number of pods that can run on a node.
:vartype max_pods: int
:ivar os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default to
Linux. Known values are: "Linux" and "Windows".
:vartype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:ivar max_count: Maximum number of nodes for auto-scaling.
:vartype max_count: int
:ivar min_count: Minimum number of nodes for auto-scaling.
:vartype min_count: int
:ivar enable_auto_scaling: Whether to enable auto-scaler.
:vartype enable_auto_scaling: bool
:ivar type: AgentPoolType represents types of an agent pool. Known values are:
"VirtualMachineScaleSets" and "AvailabilitySet".
:vartype type: str or ~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolType
:ivar orchestrator_version: Version of orchestrator specified when creating the managed
cluster.
:vartype orchestrator_version: str
:ivar provisioning_state: The current deployment or provisioning state, which only appears in
the response.
:vartype provisioning_state: str
:ivar availability_zones: (PREVIEW) Availability zones for nodes. Must use
VirtualMachineScaleSets AgentPoolType.
:vartype availability_zones: list[str]
:ivar enable_node_public_ip: Enable public IP for nodes.
:vartype enable_node_public_ip: bool
:ivar scale_set_priority: ScaleSetPriority to be used to specify virtual machine scale set
priority. Default to regular. Known values are: "Low" and "Regular".
:vartype scale_set_priority: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetPriority
:ivar scale_set_eviction_policy: ScaleSetEvictionPolicy to be used to specify eviction policy
for low priority virtual machine scale set. Default to Delete. Known values are: "Delete" and
"Deallocate".
:vartype scale_set_eviction_policy: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetEvictionPolicy
:ivar node_taints: Taints added to new nodes during node pool create and scale. For example,
key=value:NoSchedule.
:vartype node_taints: list[str]
:ivar name: Unique name of the agent pool profile in the context of the subscription and
resource group. Required.
:vartype name: str
"""
_validation = {
"os_disk_size_gb": {"maximum": 1023, "minimum": 0},
"provisioning_state": {"readonly": True},
"name": {"required": True, "pattern": r"^[a-z][a-z0-9]{0,11}$"},
}
_attribute_map = {
"count": {"key": "count", "type": "int"},
"vm_size": {"key": "vmSize", "type": "str"},
"os_disk_size_gb": {"key": "osDiskSizeGB", "type": "int"},
"vnet_subnet_id": {"key": "vnetSubnetID", "type": "str"},
"max_pods": {"key": "maxPods", "type": "int"},
"os_type": {"key": "osType", "type": "str"},
"max_count": {"key": "maxCount", "type": "int"},
"min_count": {"key": "minCount", "type": "int"},
"enable_auto_scaling": {"key": "enableAutoScaling", "type": "bool"},
"type": {"key": "type", "type": "str"},
"orchestrator_version": {"key": "orchestratorVersion", "type": "str"},
"provisioning_state": {"key": "provisioningState", "type": "str"},
"availability_zones": {"key": "availabilityZones", "type": "[str]"},
"enable_node_public_ip": {"key": "enableNodePublicIP", "type": "bool"},
"scale_set_priority": {"key": "scaleSetPriority", "type": "str"},
"scale_set_eviction_policy": {"key": "scaleSetEvictionPolicy", "type": "str"},
"node_taints": {"key": "nodeTaints", "type": "[str]"},
"name": {"key": "name", "type": "str"},
}
def __init__(
self,
*,
name: str,
count: Optional[int] = None,
vm_size: Optional[Union[str, "_models.ContainerServiceVMSizeTypes"]] = None,
os_disk_size_gb: Optional[int] = None,
vnet_subnet_id: Optional[str] = None,
max_pods: Optional[int] = None,
os_type: Union[str, "_models.OSType"] = "Linux",
max_count: Optional[int] = None,
min_count: Optional[int] = None,
enable_auto_scaling: Optional[bool] = None,
type: Optional[Union[str, "_models.AgentPoolType"]] = None,
orchestrator_version: Optional[str] = None,
availability_zones: Optional[List[str]] = None,
enable_node_public_ip: Optional[bool] = None,
scale_set_priority: Union[str, "_models.ScaleSetPriority"] = "Regular",
scale_set_eviction_policy: Union[str, "_models.ScaleSetEvictionPolicy"] = "Delete",
node_taints: Optional[List[str]] = None,
**kwargs
):
"""
:keyword count: Number of agents (VMs) to host docker containers. Allowed values must be in the
range of 1 to 100 (inclusive). The default value is 1.
:paramtype count: int
:keyword vm_size: Size of agent VMs. Known values are: "Standard_A1", "Standard_A10",
"Standard_A11", "Standard_A1_v2", "Standard_A2", "Standard_A2_v2", "Standard_A2m_v2",
"Standard_A3", "Standard_A4", "Standard_A4_v2", "Standard_A4m_v2", "Standard_A5",
"Standard_A6", "Standard_A7", "Standard_A8", "Standard_A8_v2", "Standard_A8m_v2",
"Standard_A9", "Standard_B2ms", "Standard_B2s", "Standard_B4ms", "Standard_B8ms",
"Standard_D1", "Standard_D11", "Standard_D11_v2", "Standard_D11_v2_Promo", "Standard_D12",
"Standard_D12_v2", "Standard_D12_v2_Promo", "Standard_D13", "Standard_D13_v2",
"Standard_D13_v2_Promo", "Standard_D14", "Standard_D14_v2", "Standard_D14_v2_Promo",
"Standard_D15_v2", "Standard_D16_v3", "Standard_D16s_v3", "Standard_D1_v2", "Standard_D2",
"Standard_D2_v2", "Standard_D2_v2_Promo", "Standard_D2_v3", "Standard_D2s_v3", "Standard_D3",
"Standard_D32_v3", "Standard_D32s_v3", "Standard_D3_v2", "Standard_D3_v2_Promo", "Standard_D4",
"Standard_D4_v2", "Standard_D4_v2_Promo", "Standard_D4_v3", "Standard_D4s_v3",
"Standard_D5_v2", "Standard_D5_v2_Promo", "Standard_D64_v3", "Standard_D64s_v3",
"Standard_D8_v3", "Standard_D8s_v3", "Standard_DS1", "Standard_DS11", "Standard_DS11_v2",
"Standard_DS11_v2_Promo", "Standard_DS12", "Standard_DS12_v2", "Standard_DS12_v2_Promo",
"Standard_DS13", "Standard_DS13-2_v2", "Standard_DS13-4_v2", "Standard_DS13_v2",
"Standard_DS13_v2_Promo", "Standard_DS14", "Standard_DS14-4_v2", "Standard_DS14-8_v2",
"Standard_DS14_v2", "Standard_DS14_v2_Promo", "Standard_DS15_v2", "Standard_DS1_v2",
"Standard_DS2", "Standard_DS2_v2", "Standard_DS2_v2_Promo", "Standard_DS3", "Standard_DS3_v2",
"Standard_DS3_v2_Promo", "Standard_DS4", "Standard_DS4_v2", "Standard_DS4_v2_Promo",
"Standard_DS5_v2", "Standard_DS5_v2_Promo", "Standard_E16_v3", "Standard_E16s_v3",
"Standard_E2_v3", "Standard_E2s_v3", "Standard_E32-16s_v3", "Standard_E32-8s_v3",
"Standard_E32_v3", "Standard_E32s_v3", "Standard_E4_v3", "Standard_E4s_v3",
"Standard_E64-16s_v3", "Standard_E64-32s_v3", "Standard_E64_v3", "Standard_E64s_v3",
"Standard_E8_v3", "Standard_E8s_v3", "Standard_F1", "Standard_F16", "Standard_F16s",
"Standard_F16s_v2", "Standard_F1s", "Standard_F2", "Standard_F2s", "Standard_F2s_v2",
"Standard_F32s_v2", "Standard_F4", "Standard_F4s", "Standard_F4s_v2", "Standard_F64s_v2",
"Standard_F72s_v2", "Standard_F8", "Standard_F8s", "Standard_F8s_v2", "Standard_G1",
"Standard_G2", "Standard_G3", "Standard_G4", "Standard_G5", "Standard_GS1", "Standard_GS2",
"Standard_GS3", "Standard_GS4", "Standard_GS4-4", "Standard_GS4-8", "Standard_GS5",
"Standard_GS5-16", "Standard_GS5-8", "Standard_H16", "Standard_H16m", "Standard_H16mr",
"Standard_H16r", "Standard_H8", "Standard_H8m", "Standard_L16s", "Standard_L32s",
"Standard_L4s", "Standard_L8s", "Standard_M128-32ms", "Standard_M128-64ms", "Standard_M128ms",
"Standard_M128s", "Standard_M64-16ms", "Standard_M64-32ms", "Standard_M64ms", "Standard_M64s",
"Standard_NC12", "Standard_NC12s_v2", "Standard_NC12s_v3", "Standard_NC24", "Standard_NC24r",
"Standard_NC24rs_v2", "Standard_NC24rs_v3", "Standard_NC24s_v2", "Standard_NC24s_v3",
"Standard_NC6", "Standard_NC6s_v2", "Standard_NC6s_v3", "Standard_ND12s", "Standard_ND24rs",
"Standard_ND24s", "Standard_ND6s", "Standard_NV12", "Standard_NV24", and "Standard_NV6".
:paramtype vm_size: str or
~azure.mgmt.containerservice.v2019_08_01.models.ContainerServiceVMSizeTypes
:keyword os_disk_size_gb: OS Disk Size in GB to be used to specify the disk size for every
machine in this master/agent pool. If you specify 0, it will apply the default osDisk size
according to the vmSize specified.
:paramtype os_disk_size_gb: int
:keyword vnet_subnet_id: VNet SubnetID specifies the VNet's subnet identifier.
:paramtype vnet_subnet_id: str
:keyword max_pods: Maximum number of pods that can run on a node.
:paramtype max_pods: int
:keyword os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default
to Linux. Known values are: "Linux" and "Windows".
:paramtype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:keyword max_count: Maximum number of nodes for auto-scaling.
:paramtype max_count: int
:keyword min_count: Minimum number of nodes for auto-scaling.
:paramtype min_count: int
:keyword enable_auto_scaling: Whether to enable auto-scaler.
:paramtype enable_auto_scaling: bool
:keyword type: AgentPoolType represents types of an agent pool. Known values are:
"VirtualMachineScaleSets" and "AvailabilitySet".
:paramtype type: str or ~azure.mgmt.containerservice.v2019_08_01.models.AgentPoolType
:keyword orchestrator_version: Version of orchestrator specified when creating the managed
cluster.
:paramtype orchestrator_version: str
:keyword availability_zones: (PREVIEW) Availability zones for nodes. Must use
VirtualMachineScaleSets AgentPoolType.
:paramtype availability_zones: list[str]
:keyword enable_node_public_ip: Enable public IP for nodes.
:paramtype enable_node_public_ip: bool
:keyword scale_set_priority: ScaleSetPriority to be used to specify virtual machine scale set
priority. Default to regular. Known values are: "Low" and "Regular".
:paramtype scale_set_priority: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetPriority
:keyword scale_set_eviction_policy: ScaleSetEvictionPolicy to be used to specify eviction
policy for low priority virtual machine scale set. Default to Delete. Known values are:
"Delete" and "Deallocate".
:paramtype scale_set_eviction_policy: str or
~azure.mgmt.containerservice.v2019_08_01.models.ScaleSetEvictionPolicy
:keyword node_taints: Taints added to new nodes during node pool create and scale. For example,
key=value:NoSchedule.
:paramtype node_taints: list[str]
:keyword name: Unique name of the agent pool profile in the context of the subscription and
resource group. Required.
:paramtype name: str
"""
super().__init__(
count=count,
vm_size=vm_size,
os_disk_size_gb=os_disk_size_gb,
vnet_subnet_id=vnet_subnet_id,
max_pods=max_pods,
os_type=os_type,
max_count=max_count,
min_count=min_count,
enable_auto_scaling=enable_auto_scaling,
type=type,
orchestrator_version=orchestrator_version,
availability_zones=availability_zones,
enable_node_public_ip=enable_node_public_ip,
scale_set_priority=scale_set_priority,
scale_set_eviction_policy=scale_set_eviction_policy,
node_taints=node_taints,
**kwargs
)
self.name = name
class ManagedClusterAPIServerAccessProfile(_serialization.Model):
"""Access profile for managed cluster API server.
:ivar authorized_ip_ranges: Authorized IP Ranges to kubernetes API server.
:vartype authorized_ip_ranges: list[str]
:ivar enable_private_cluster: Whether to create the cluster as a private cluster or not.
:vartype enable_private_cluster: bool
"""
_attribute_map = {
"authorized_ip_ranges": {"key": "authorizedIPRanges", "type": "[str]"},
"enable_private_cluster": {"key": "enablePrivateCluster", "type": "bool"},
}
def __init__(
self,
*,
authorized_ip_ranges: Optional[List[str]] = None,
enable_private_cluster: Optional[bool] = None,
**kwargs
):
"""
:keyword authorized_ip_ranges: Authorized IP Ranges to kubernetes API server.
:paramtype authorized_ip_ranges: list[str]
:keyword enable_private_cluster: Whether to create the cluster as a private cluster or not.
:paramtype enable_private_cluster: bool
"""
super().__init__(**kwargs)
self.authorized_ip_ranges = authorized_ip_ranges
self.enable_private_cluster = enable_private_cluster
class ManagedClusterIdentity(_serialization.Model):
"""Identity for the managed cluster.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar principal_id: The principal id of the system assigned identity which is used by master
components.
:vartype principal_id: str
:ivar tenant_id: The tenant id of the system assigned identity which is used by master
components.
:vartype tenant_id: str
:ivar type: The type of identity used for the managed cluster. Type 'SystemAssigned' will use
an implicitly created identity in master components and an auto-created user assigned identity
in MC_ resource group in agent nodes. Type 'None' will not use MSI for the managed cluster,
service principal will be used instead. Known values are: "SystemAssigned" and "None".
:vartype type: str or ~azure.mgmt.containerservice.v2019_08_01.models.ResourceIdentityType
"""
_validation = {
"principal_id": {"readonly": True},
"tenant_id": {"readonly": True},
}
_attribute_map = {
"principal_id": {"key": "principalId", "type": "str"},
"tenant_id": {"key": "tenantId", "type": "str"},
"type": {"key": "type", "type": "str"},
}
def __init__(self, *, type: Optional[Union[str, "_models.ResourceIdentityType"]] = None, **kwargs):
"""
:keyword type: The type of identity used for the managed cluster. Type 'SystemAssigned' will
use an implicitly created identity in master components and an auto-created user assigned
identity in MC_ resource group in agent nodes. Type 'None' will not use MSI for the managed
cluster, service principal will be used instead. Known values are: "SystemAssigned" and "None".
:paramtype type: str or ~azure.mgmt.containerservice.v2019_08_01.models.ResourceIdentityType
"""
super().__init__(**kwargs)
self.principal_id = None
self.tenant_id = None
self.type = type
class ManagedClusterListResult(_serialization.Model):
"""The response from the List Managed Clusters operation.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar value: The list of managed clusters.
:vartype value: list[~azure.mgmt.containerservice.v2019_08_01.models.ManagedCluster]
:ivar next_link: The URL to get the next set of managed cluster results.
:vartype next_link: str
"""
_validation = {
"next_link": {"readonly": True},
}
_attribute_map = {
"value": {"key": "value", "type": "[ManagedCluster]"},
"next_link": {"key": "nextLink", "type": "str"},
}
def __init__(self, *, value: Optional[List["_models.ManagedCluster"]] = None, **kwargs):
"""
:keyword value: The list of managed clusters.
:paramtype value: list[~azure.mgmt.containerservice.v2019_08_01.models.ManagedCluster]
"""
super().__init__(**kwargs)
self.value = value
self.next_link = None
class ManagedClusterLoadBalancerProfile(_serialization.Model):
"""Profile of the managed cluster load balancer.
:ivar managed_outbound_i_ps: Desired managed outbound IPs for the cluster load balancer.
:vartype managed_outbound_i_ps:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterLoadBalancerProfileManagedOutboundIPs
:ivar outbound_ip_prefixes: Desired outbound IP Prefix resources for the cluster load balancer.
:vartype outbound_ip_prefixes:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterLoadBalancerProfileOutboundIPPrefixes
:ivar outbound_i_ps: Desired outbound IP resources for the cluster load balancer.
:vartype outbound_i_ps:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterLoadBalancerProfileOutboundIPs
:ivar effective_outbound_i_ps: The effective outbound IP resources of the cluster load
balancer.
:vartype effective_outbound_i_ps:
list[~azure.mgmt.containerservice.v2019_08_01.models.ResourceReference]
"""
_attribute_map = {
"managed_outbound_i_ps": {
"key": "managedOutboundIPs",
"type": "ManagedClusterLoadBalancerProfileManagedOutboundIPs",
},
"outbound_ip_prefixes": {
"key": "outboundIPPrefixes",
"type": "ManagedClusterLoadBalancerProfileOutboundIPPrefixes",
},
"outbound_i_ps": {"key": "outboundIPs", "type": "ManagedClusterLoadBalancerProfileOutboundIPs"},
"effective_outbound_i_ps": {"key": "effectiveOutboundIPs", "type": "[ResourceReference]"},
}
def __init__(
self,
*,
managed_outbound_i_ps: Optional["_models.ManagedClusterLoadBalancerProfileManagedOutboundIPs"] = None,
outbound_ip_prefixes: Optional["_models.ManagedClusterLoadBalancerProfileOutboundIPPrefixes"] = None,
outbound_i_ps: Optional["_models.ManagedClusterLoadBalancerProfileOutboundIPs"] = None,
effective_outbound_i_ps: Optional[List["_models.ResourceReference"]] = None,
**kwargs
):
"""
:keyword managed_outbound_i_ps: Desired managed outbound IPs for the cluster load balancer.
:paramtype managed_outbound_i_ps:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterLoadBalancerProfileManagedOutboundIPs
:keyword outbound_ip_prefixes: Desired outbound IP Prefix resources for the cluster load
balancer.
:paramtype outbound_ip_prefixes:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterLoadBalancerProfileOutboundIPPrefixes
:keyword outbound_i_ps: Desired outbound IP resources for the cluster load balancer.
:paramtype outbound_i_ps:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterLoadBalancerProfileOutboundIPs
:keyword effective_outbound_i_ps: The effective outbound IP resources of the cluster load
balancer.
:paramtype effective_outbound_i_ps:
list[~azure.mgmt.containerservice.v2019_08_01.models.ResourceReference]
"""
super().__init__(**kwargs)
self.managed_outbound_i_ps = managed_outbound_i_ps
self.outbound_ip_prefixes = outbound_ip_prefixes
self.outbound_i_ps = outbound_i_ps
self.effective_outbound_i_ps = effective_outbound_i_ps
class ManagedClusterLoadBalancerProfileManagedOutboundIPs(_serialization.Model):
"""Desired managed outbound IPs for the cluster load balancer.
:ivar count: Desired number of outbound IP created/managed by Azure for the cluster load
balancer. Allowed values must be in the range of 1 to 100 (inclusive). The default value is 1.
:vartype count: int
"""
_validation = {
"count": {"maximum": 100, "minimum": 1},
}
_attribute_map = {
"count": {"key": "count", "type": "int"},
}
def __init__(self, *, count: int = 1, **kwargs):
"""
:keyword count: Desired number of outbound IP created/managed by Azure for the cluster load
balancer. Allowed values must be in the range of 1 to 100 (inclusive). The default value is 1.
:paramtype count: int
"""
super().__init__(**kwargs)
self.count = count
class ManagedClusterLoadBalancerProfileOutboundIPPrefixes(_serialization.Model):
"""Desired outbound IP Prefix resources for the cluster load balancer.
:ivar public_ip_prefixes: A list of public IP prefix resources.
:vartype public_ip_prefixes:
list[~azure.mgmt.containerservice.v2019_08_01.models.ResourceReference]
"""
_attribute_map = {
"public_ip_prefixes": {"key": "publicIPPrefixes", "type": "[ResourceReference]"},
}
def __init__(self, *, public_ip_prefixes: Optional[List["_models.ResourceReference"]] = None, **kwargs):
"""
:keyword public_ip_prefixes: A list of public IP prefix resources.
:paramtype public_ip_prefixes:
list[~azure.mgmt.containerservice.v2019_08_01.models.ResourceReference]
"""
super().__init__(**kwargs)
self.public_ip_prefixes = public_ip_prefixes
class ManagedClusterLoadBalancerProfileOutboundIPs(_serialization.Model):
"""Desired outbound IP resources for the cluster load balancer.
:ivar public_i_ps: A list of public IP resources.
:vartype public_i_ps: list[~azure.mgmt.containerservice.v2019_08_01.models.ResourceReference]
"""
_attribute_map = {
"public_i_ps": {"key": "publicIPs", "type": "[ResourceReference]"},
}
def __init__(self, *, public_i_ps: Optional[List["_models.ResourceReference"]] = None, **kwargs):
"""
:keyword public_i_ps: A list of public IP resources.
:paramtype public_i_ps: list[~azure.mgmt.containerservice.v2019_08_01.models.ResourceReference]
"""
super().__init__(**kwargs)
self.public_i_ps = public_i_ps
class ManagedClusterPoolUpgradeProfile(_serialization.Model):
"""The list of available upgrade versions.
All required parameters must be populated in order to send to Azure.
:ivar kubernetes_version: Kubernetes version (major, minor, patch). Required.
:vartype kubernetes_version: str
:ivar name: Pool name.
:vartype name: str
:ivar os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default to
Linux. Known values are: "Linux" and "Windows".
:vartype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:ivar upgrades: List of orchestrator types and versions available for upgrade.
:vartype upgrades:
list[~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterPoolUpgradeProfileUpgradesItem]
"""
_validation = {
"kubernetes_version": {"required": True},
"os_type": {"required": True},
}
_attribute_map = {
"kubernetes_version": {"key": "kubernetesVersion", "type": "str"},
"name": {"key": "name", "type": "str"},
"os_type": {"key": "osType", "type": "str"},
"upgrades": {"key": "upgrades", "type": "[ManagedClusterPoolUpgradeProfileUpgradesItem]"},
}
def __init__(
self,
*,
kubernetes_version: str,
os_type: Union[str, "_models.OSType"] = "Linux",
name: Optional[str] = None,
upgrades: Optional[List["_models.ManagedClusterPoolUpgradeProfileUpgradesItem"]] = None,
**kwargs
):
"""
:keyword kubernetes_version: Kubernetes version (major, minor, patch). Required.
:paramtype kubernetes_version: str
:keyword name: Pool name.
:paramtype name: str
:keyword os_type: OsType to be used to specify os type. Choose from Linux and Windows. Default
to Linux. Known values are: "Linux" and "Windows".
:paramtype os_type: str or ~azure.mgmt.containerservice.v2019_08_01.models.OSType
:keyword upgrades: List of orchestrator types and versions available for upgrade.
:paramtype upgrades:
list[~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterPoolUpgradeProfileUpgradesItem]
"""
super().__init__(**kwargs)
self.kubernetes_version = kubernetes_version
self.name = name
self.os_type = os_type
self.upgrades = upgrades
class ManagedClusterPoolUpgradeProfileUpgradesItem(_serialization.Model):
"""ManagedClusterPoolUpgradeProfileUpgradesItem.
:ivar kubernetes_version: Kubernetes version (major, minor, patch).
:vartype kubernetes_version: str
:ivar is_preview: Whether Kubernetes version is currently in preview.
:vartype is_preview: bool
"""
_attribute_map = {
"kubernetes_version": {"key": "kubernetesVersion", "type": "str"},
"is_preview": {"key": "isPreview", "type": "bool"},
}
def __init__(self, *, kubernetes_version: Optional[str] = None, is_preview: Optional[bool] = None, **kwargs):
"""
:keyword kubernetes_version: Kubernetes version (major, minor, patch).
:paramtype kubernetes_version: str
:keyword is_preview: Whether Kubernetes version is currently in preview.
:paramtype is_preview: bool
"""
super().__init__(**kwargs)
self.kubernetes_version = kubernetes_version
self.is_preview = is_preview
class ManagedClusterServicePrincipalProfile(_serialization.Model):
"""Information about a service principal identity for the cluster to use for manipulating Azure APIs.
All required parameters must be populated in order to send to Azure.
:ivar client_id: The ID for the service principal. Required.
:vartype client_id: str
:ivar secret: The secret password associated with the service principal in plain text.
:vartype secret: str
"""
_validation = {
"client_id": {"required": True},
}
_attribute_map = {
"client_id": {"key": "clientId", "type": "str"},
"secret": {"key": "secret", "type": "str"},
}
def __init__(self, *, client_id: str, secret: Optional[str] = None, **kwargs):
"""
:keyword client_id: The ID for the service principal. Required.
:paramtype client_id: str
:keyword secret: The secret password associated with the service principal in plain text.
:paramtype secret: str
"""
super().__init__(**kwargs)
self.client_id = client_id
self.secret = secret
class ManagedClusterUpgradeProfile(_serialization.Model):
"""The list of available upgrades for compute pools.
Variables are only populated by the server, and will be ignored when sending a request.
All required parameters must be populated in order to send to Azure.
:ivar id: Id of upgrade profile.
:vartype id: str
:ivar name: Name of upgrade profile.
:vartype name: str
:ivar type: Type of upgrade profile.
:vartype type: str
:ivar control_plane_profile: The list of available upgrade versions for the control plane.
Required.
:vartype control_plane_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterPoolUpgradeProfile
:ivar agent_pool_profiles: The list of available upgrade versions for agent pools. Required.
:vartype agent_pool_profiles:
list[~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterPoolUpgradeProfile]
"""
_validation = {
"id": {"readonly": True},
"name": {"readonly": True},
"type": {"readonly": True},
"control_plane_profile": {"required": True},
"agent_pool_profiles": {"required": True},
}
_attribute_map = {
"id": {"key": "id", "type": "str"},
"name": {"key": "name", "type": "str"},
"type": {"key": "type", "type": "str"},
"control_plane_profile": {"key": "properties.controlPlaneProfile", "type": "ManagedClusterPoolUpgradeProfile"},
"agent_pool_profiles": {"key": "properties.agentPoolProfiles", "type": "[ManagedClusterPoolUpgradeProfile]"},
}
def __init__(
self,
*,
control_plane_profile: "_models.ManagedClusterPoolUpgradeProfile",
agent_pool_profiles: List["_models.ManagedClusterPoolUpgradeProfile"],
**kwargs
):
"""
:keyword control_plane_profile: The list of available upgrade versions for the control plane.
Required.
:paramtype control_plane_profile:
~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterPoolUpgradeProfile
:keyword agent_pool_profiles: The list of available upgrade versions for agent pools. Required.
:paramtype agent_pool_profiles:
list[~azure.mgmt.containerservice.v2019_08_01.models.ManagedClusterPoolUpgradeProfile]
"""
super().__init__(**kwargs)
self.id = None
self.name = None
self.type = None
self.control_plane_profile = control_plane_profile
self.agent_pool_profiles = agent_pool_profiles
class ManagedClusterWindowsProfile(_serialization.Model):
"""Profile for Windows VMs in the container service cluster.
All required parameters must be populated in order to send to Azure.
:ivar admin_username: Specifies the name of the administrator account. :code:`<br>`:code:`<br>`
**restriction:** Cannot end in "." :code:`<br>`:code:`<br>` **Disallowed values:**
"administrator", "admin", "user", "user1", "test", "user2", "test1", "user3", "admin1", "1",
"123", "a", "actuser", "adm", "admin2", "aspnet", "backup", "console", "david", "guest",
"john", "owner", "root", "server", "sql", "support", "support_388945a0", "sys", "test2",
"test3", "user4", "user5". :code:`<br>`:code:`<br>` **Minimum-length:** 1 character
:code:`<br>`:code:`<br>` **Max-length:** 20 characters. Required.
:vartype admin_username: str
:ivar admin_password: Specifies the password of the administrator account.
:code:`<br>`:code:`<br>` **Minimum-length:** 8 characters :code:`<br>`:code:`<br>`
**Max-length:** 123 characters :code:`<br>`:code:`<br>` **Complexity requirements:** 3 out of 4
conditions below need to be fulfilled :code:`<br>` Has lower characters :code:`<br>`Has upper
characters :code:`<br>` Has a digit :code:`<br>` Has a special character (Regex match [\W_])
:code:`<br>`:code:`<br>` **Disallowed values:** "abc@123", "P@$$w0rd", "P@ssw0rd",
"P@ssword123", "Pa$$word", "pass@word1", "Password!", "Password1", "Password22", "iloveyou!".
:vartype admin_password: str
"""
_validation = {
"admin_username": {"required": True},
}
_attribute_map = {
"admin_username": {"key": "adminUsername", "type": "str"},
"admin_password": {"key": "adminPassword", "type": "str"},
}
def __init__(self, *, admin_username: str, admin_password: Optional[str] = None, **kwargs):
"""
:keyword admin_username: Specifies the name of the administrator account.
:code:`<br>`:code:`<br>` **restriction:** Cannot end in "." :code:`<br>`:code:`<br>`
**Disallowed values:** "administrator", "admin", "user", "user1", "test", "user2", "test1",
"user3", "admin1", "1", "123", "a", "actuser", "adm", "admin2", "aspnet", "backup", "console",
"david", "guest", "john", "owner", "root", "server", "sql", "support", "support_388945a0",
"sys", "test2", "test3", "user4", "user5". :code:`<br>`:code:`<br>` **Minimum-length:** 1
character :code:`<br>`:code:`<br>` **Max-length:** 20 characters. Required.
:paramtype admin_username: str
:keyword admin_password: Specifies the password of the administrator account.
:code:`<br>`:code:`<br>` **Minimum-length:** 8 characters :code:`<br>`:code:`<br>`
**Max-length:** 123 characters :code:`<br>`:code:`<br>` **Complexity requirements:** 3 out of 4
conditions below need to be fulfilled :code:`<br>` Has lower characters :code:`<br>`Has upper
characters :code:`<br>` Has a digit :code:`<br>` Has a special character (Regex match [\W_])
:code:`<br>`:code:`<br>` **Disallowed values:** "abc@123", "P@$$w0rd", "P@ssw0rd",
"P@ssword123", "Pa$$word", "pass@word1", "Password!", "Password1", "Password22", "iloveyou!".
:paramtype admin_password: str
"""
super().__init__(**kwargs)
self.admin_username = admin_username
self.admin_password = admin_password
class OperationListResult(_serialization.Model):
"""The List Compute Operation operation response.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar value: The list of compute operations.
:vartype value: list[~azure.mgmt.containerservice.v2019_08_01.models.OperationValue]
"""
_validation = {
"value": {"readonly": True},
}
_attribute_map = {
"value": {"key": "value", "type": "[OperationValue]"},
}
def __init__(self, **kwargs):
""" """
super().__init__(**kwargs)
self.value = None
class OperationValue(_serialization.Model):
"""Describes the properties of a Compute Operation value.
Variables are only populated by the server, and will be ignored when sending a request.
:ivar origin: The origin of the compute operation.
:vartype origin: str
:ivar name: The name of the compute operation.
:vartype name: str
:ivar operation: The display name of the compute operation.
:vartype operation: str
:ivar resource: The display name of the resource the operation applies to.
:vartype resource: str
:ivar description: The description of the operation.
:vartype description: str
:ivar provider: The resource provider for the operation.
:vartype provider: str
"""
_validation = {
"origin": {"readonly": True},
"name": {"readonly": True},
"operation": {"readonly": True},
"resource": {"readonly": True},
"description": {"readonly": True},
"provider": {"readonly": True},
}
_attribute_map = {
"origin": {"key": "origin", "type": "str"},
"name": {"key": "name", "type": "str"},
"operation": {"key": "display.operation", "type": "str"},
"resource": {"key": "display.resource", "type": "str"},
"description": {"key": "display.description", "type": "str"},
"provider": {"key": "display.provider", "type": "str"},
}
def __init__(self, **kwargs):
""" """
super().__init__(**kwargs)
self.origin = None
self.name = None
self.operation = None
self.resource = None
self.description = None
self.provider = None
class ResourceReference(_serialization.Model):
"""A reference to an Azure resource.
:ivar id: The fully qualified Azure resource id.
:vartype id: str
"""
_attribute_map = {
"id": {"key": "id", "type": "str"},
}
def __init__(self, *, id: Optional[str] = None, **kwargs): # pylint: disable=redefined-builtin
"""
:keyword id: The fully qualified Azure resource id.
:paramtype id: str
"""
super().__init__(**kwargs)
self.id = id
class TagsObject(_serialization.Model):
"""Tags object for patch operations.
:ivar tags: Resource tags.
:vartype tags: dict[str, str]
"""
_attribute_map = {
"tags": {"key": "tags", "type": "{str}"},
}
def __init__(self, *, tags: Optional[Dict[str, str]] = None, **kwargs):
"""
:keyword tags: Resource tags.
:paramtype tags: dict[str, str]
"""
super().__init__(**kwargs)
self.tags = tags
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 457
|
Люк Газдич 27 июля 1989, Торонто, Канада) — канадский хоккеист хорватского происхождения, нападающий. Выбран под общим 172-м номером на драфте НХЛ 2007 командой «Даллас Старз». На данный момент выступает в клубе Американской хоккейной лиги (АХЛ) «Сан-Диего Галлз».
Выполняет роль тафгая.
Семья
Его братья Бенджамин и Марк также хоккеисты. Его отец, Майк Газдич , был задрафтован на драфте НХЛ 1978 клубом Баффало Сейбрз
Карьера
В 2005 году был задрафтован клубом «Эри Оттерз». Выбран под общим номером 170.
В сезоне 2007/08 выступал за команду «Эри Оттерз», в составе которой провел 188 игр, набрал 72 (42+30) очков и 406 минут штрафа.
2010 году перешёл в клуб «Техас Старз», в составе которой провел 256 игр, набрал 54 (27+28) очков и 447 минут штрафа.
2013 году был переведён в клуб «Эдмонтон Ойлерз», в составе которой провел 90 игр, набрал 6 (3+3) очков и 158 минут штрафа.
Выступление
Последнее обновление: 13 мая 2015 года
Хоккеисты НХЛ
Задрафтованные ХК «Даллас Старз»
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,837
|
Holy Rewatch Batman!
Holy Rewatch Batman! "The Joker's Flying Saucer"
Keith R.A. DeCandido
Fri Mar 31, 2017 3:00pm 19 comments 1 Favorite [+]
"The Joker's Flying Saucer"
Written by Charles Hoffman
Directed by Sam Strangis
Production code 1720
Original air date: February 29, 1968
The Bat-signal: The citizenry of Gotham City is convinced that there will be an alien invasion, despite assurances by Gordon to the contrary. Professor Greenleaf is trying to convince Barbara (who's actually working in the library!) that humanity should submit to their new alien overlords. While Barbara doesn't buy Greenleaf's story, she does see a green-skinned and -haired man vandalizing the library.
Faced by near-harassment from the people of Gotham, Gordon does the same thing he always does when required to do his job: he calls Batman. The Dynamic Duo slide down the poles and drive to GCPD HQ.
Turns out the rumors were started by the Joker, who designed a flying saucer while in prison with the help of his pickpocket cellmate.
Batman, Robin, Gordon, and O'Hara question a Mrs. Green, who insists she saw a three-foot-tall Martian man in Gotham Central Park. And then Barbara arrives at Gordon's office completely a-quiver, telling our heroes about the little green man in the library. Said little green man, whose name is Verdigris, has also been in the Batmobile, where he's left a bomb to go off at midnight. The Dynamic Duo head back to the Batcave, not realizing the Batmobile is bombed, and eventually they figure out where they've seen Mrs. Green before: she was the front-woman for a bunko artist (which is what they called grifters in the 1960s).
Professor Greenleaf also turns out to be working for the Joker. The criminal clown's next move is to steal some beryllium for the flying saucer from the Wayne Foundation. Batman figures out that it's the Joker, so he has Alfred check on the Wayne Foundation security and also informs Gordon that the Joker is the likely culprit.
Barbara was in Gordon's office when Batman called, so she heads off to change to Batgirl. Batman and Robin get into the Batmobile at midnight, at which point the bomb goes off, half-destroying the Batcave.
Joker stole the beryllium, and also captured both Alfred and Batgirl (off-camera!) and bring them back to his hideout—an abandoned launching-pad factory—with Joker assuming that Alfred is a mad scientist.
Batman and Robin survived the bomb blast, though the Batcave is a disaster area, with all the phones and radios destroyed—including the Bat-phone.
Joker finishes the flying saucer and ties Batgirl to a rocket. His plan is to send her off into space while Joker will orbit the Earth a few times then launch his "invasion." Batman and Robin manage to rig up a radio, and Alfred finally gets through and reports to Batman.
Batgirl is able to keep from being shot into space, but Joker still takes her with him in his flying saucer, which goes into space, orbits the Earth a few times (allegedly getting too close to the sun at one point), then returns to earth. However, Alfred was able to put in some homing, er, uh, something in the beryllium that forces it to land back in Joker's hideout. Alfred surreptitiously lets Batman know this, and so the Dynamic Duo are waiting for Joker and his gang when they arrive back at the abandoned launching-pad factory. Fisticuffs ensue, and Batman, Robin, and Batgirl defeat the would-be invaders in time for Gordon and O'Hara to arrive to take them all to the hoosegow.
Batman and Robin put the Batcave back together, but then they're alerted by Gordon to a strange happening in Spiffany's…
Fetch the Bat-shark-repellant! We see the newest absurdly specific device in the Batcave: the Current Criminal Activity Bat-Disclosure Unit, which apparently provides the details from the script of the episode they're in. Also our heroes have taken to wearing Anti-Thermal Bat-T-Shirts under their costumes, which protect them from the bomb blast, er, somehow. With the Batmobile buried under rubble, our heroes get to use the Bat-cycle to drive to the Bat-copter. However, the Batmobile bomb-detector seems to be on the fritz, since it totally misses the bomb that was placed in the Batmobile…
Batgirl has a fuse extinguisher in her Batgirl utility belt.
Holy #@!%$, Batman! "Holy interplanetary yardstick" is Robin's clever rejoinder upon being told that Mrs. Green encountered a supposed Martian who was three feet tall. "Holy rock garden!" is his exclamation after the bomb has made a big mess of the Batcave. "Holy known unknown flying objects!" is Robin's bizarre response to Alfred's report on Joker's plan, which is so bizarre that Batman doesn't understand it and asks him to repeat it (it doesn't help).
Gotham City's finest. Apparently, everyone in the world feels that the logical thing to do when they see a flying saucer or hear about an alien invasion is call the police commissioner of Gotham City. Sure.
Special Guest Villain. This is Cesar Romero's swan song as the Joker, thus going out with a significant whimper.
Na-na na-na na-na na-na na.
"You suppose there's a working launching pad left in this abandoned launching-pad factory?"
"Yes, there's one in the launching-pad equipment locker, Joker."
–A delightful exchange between Joker and his henchman.
Trivial matters: This episode was discussed on The Batcave Podcast episode 66 by host John S. Drew with special guest chum, Jim Beard (editor of Gotham City 14 Miles).
The footage of the flying saucer in the sky is taken from the 1953 movie Invaders from Mars, while the footage of the Bat-copter is taken from the Batman feature film.
Joker assumes Alfred is a mad scientist, even though Alfred previously defeated the Joker singlehandedly in "Flop Goes the Joker." You'd think Joker would remember that.
Verdigris is played by Richard Bakalyan. It's never made clear who he really is or where he comes from. His name is a slightly more subtle take on the emerald theme of the episode, with the constant references to little green men from Mars and characters named Greenleaf, Emerald, Chartreuse, Shamrock, and Green.
Byron Keith makes his final appearance as Mayor Linseed. Fritz Feld returns as Greenleaf—he previously played Oliver Muzzy in "Pop Goes the Joker."
Pow! Biff! Zowie! "We'll return to Gotham City where I'll ultimate my ultimatum!" This episode isn't a total disaster, mostly by virtue of Richard Bakalyan, who cavorts beautifully with Cesar Romero, and also by virtue of Romero himself, who's never not fun.
But holy cow, what a misbegotten mess! This is perhaps the worst treatment of the Barbara Gordon/Batgirl character all season, as we start with Barbara screaming at the sight of Verdigris in her library, and continue to her being captured off-camera, and then being barely in evidence in the fight scene at the end. The one and only thing she accomplishes is to not be shot into space.
Not that Batman and Robin do much better. Aside from the fisticuffs at the end, they don't actually accomplish anything on their own, as the Batcave's computers tell them that it's the Joker, and it's Alfred who mostly saves the day. (Batman doesn't even notice the bomb in his car…)
And even by this show's standards, the plot's ridiculous—though in keeping with the Joker's previous plans. I mean, he's already managed time travel and robotics, why not space travel as well? And why not just use it for petty larceny? Sheesh.
Not the best episode for Romero to go out on, but the man himself is, as ever, having a grand old time cackling his way through the episode.
Bat-rating: 2
Keith R.A. DeCandido is running a Kickstarter for Mermaid Precinct, the long-awaited fifth novel in his series of fantasy police procedurals. Please consider supporting it! He will be a guest at the Central Pennsylvania Comic Con this weekend in York, Pennsylvania, where he'll have a table to sell and sign books.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,181
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{"url":"http:\/\/www.wikidoc.org\/index.php\/Psi_(letter)","text":"# Psi (letter)\n\nLook up \u03a8, \u03c8 in Wiktionary, the free dictionary.\nFor other uses, see Psi.\n\nPsi (uppercase \u03a8, lowercase \u03c8) is the 23rd letter of the Greek alphabet and has a numeric value of 700. In both Classical and Modern Greek, the letter indicates the combination \/ps\/ (like in English \"lapse\"). In Greek, this consonant cluster can occur in the syllable initial position, as in the Greek word \"\u03c8\u03ac\u03c1\u03b9\" [ps\u00e1ri (=fish)]. However, in some languages (including English) this combination is not possible at the beginning of a syllable. In Latin, Greek words beginning with psi are transcribed by ps-, but seem to have been pronounced simply as s-, with a quiescent p.[citation needed] This pronunciation has affected that of Greek loanwords beginning with the letter in several languages. In English, for example, psychology is pronounced with a silent p, and the name of the letter is often pronounced [sa\u026a] (\"sigh\"), although this is not always the case. Any person who has learned Greek is aware that the correct pronunciation is actually \"p'see\".[citation needed] The letter was adopted into the Old Italic alphabet, and its shape is continued into the Algiz rune of the Elder Futhark.\n\nThe letter may have originated from the practice of writing the sigma over the pi, eventually making the combination into a single letter.[citation needed] Psi was also adopted into the early Cyrillic alphabet. See psi (Cyrillic) (\u0470, \u0471). This may have also been since the letter, both in lower case and uppercase form, resembles the trident wielded by Poseidon, the Greek god of water\/ocean.\n\nThe letter psi is commonly used in physics for representing a wavefunction in quantum mechanics, particularly with the Schr\u00f6dinger equation and bra-ket notation: ${\\displaystyle \\langle \\phi |\\psi \\rangle }$. It is also used to represent the (generalized) positional states of a qubit in a quantum computer.\n\nPsi is also used as the symbol for the polygamma function, defined by\n\n${\\displaystyle \\psi _{n}(x)={\\frac {d^{(n)}}{dx^{(n)}}}{\\frac {\\Gamma '(x)}{\\Gamma (x)}}\\,\\!}$\n\nwhere ${\\displaystyle \\Gamma (x)}$ is the gamma function.\n\nThe letters \u03a8 or \u03c8 can also be a symbol for:","date":"2020-05-26 07:01:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 3, \"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\": 0.8479527235031128, \"perplexity\": 2012.0945990162593}, \"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-2020-24\/segments\/1590347390448.11\/warc\/CC-MAIN-20200526050333-20200526080333-00248.warc.gz\"}"}
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\begin{document}
\title{\huge A Vision of 6G Wireless Systems: Applications, Trends, Technologies, and Open Research Problems\vspace*{-0.4em}}
\author{{Walid Saad\IEEEauthorrefmark{1}}, Mehdi Bennis\IEEEauthorrefmark{2}, and Mingzhe Chen\IEEEauthorrefmark{3}$^,$\IEEEauthorrefmark{4}$^,$\IEEEauthorrefmark{1}
\vspace*{0em}\\
\IEEEauthorrefmark{1}\small Wireless@VT, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, USA, Email: \protect\url{walids@vt.edu}.\\
\IEEEauthorrefmark{2}\small CWC - Centre for Wireless Communications, University of Oulu, Finland, Email: \protect\url{mehdi.bennis@oulu.fi}.\\
\authorblockA{\small \IEEEauthorrefmark{3}The Future Network of Intelligence Institute, The Chinese University of Hong Kong, Shenzhen, China, Email: \protect\url{chenmingzhe@cuhk.edu.cn}. \\
\IEEEauthorrefmark{4}Department of Electrical Engineering, Princeton University, Princeton, NJ, USA.\\
\thanks{{This research was supported by the U.S. National Science Foundation under Grant CNS-1836802.}}
}\vspace*{-3em}
}
\maketitle
\vspace{0cm}
\begin{abstract}
The ongoing deployment of 5G cellular systems is continuously exposing the inherent limitations of this system, compared to its original premise as an enabler for Internet of Everything applications. These 5G drawbacks are spurring worldwide activities focused on defining the next-generation 6G wireless system that can truly integrate far-reaching applications ranging from autonomous systems to extended reality. Despite recent 6G initiatives\footnote{One example is the 6Genesis project in Finland (see https://www.oulu.fi/6gflagship/).}, the fundamental architectural and performance components of 6G remain largely undefined. In this paper, we present a holistic, forward-looking vision that defines the tenets of a 6G system. We opine that 6G will not be a mere exploration of more spectrum at high-frequency bands, but it will rather be a convergence of upcoming technological trends driven by exciting, underlying services. In this regard, we first identify the primary drivers of 6G systems, in terms of applications and accompanying technological trends. Then, we propose a new set of service classes and expose their target 6G performance requirements. We then identify the enabling technologies for the introduced 6G services and outline a comprehensive research agenda that leverages those technologies. We conclude by providing concrete recommendations for the roadmap toward 6G. Ultimately, the intent of this article is to serve as a basis for stimulating more out-of-the-box research around 6G.
\end{abstract}
\section{Introduction} \label{se:1}
To date, the wireless network evolution was primarily driven by a need for higher rates, which mandated a continuous 1000x increase in network capacity. While this demand for wireless capacity will continue to grow, the emergence of the Internet of Everything (IoE) system, connecting millions of people and billions of machines, is yielding a radical paradigm shift from the rate-centric enhanced mobile broadband (eMBB) services of yesteryears towards ultra-reliable, low latency communications (URLLC).
Although the fifth generation (5G) cellular system was marketed as the key IoE enabler, through concerted 5G standardization efforts that led to the first 5G new radio (5G NR) milestone and subsequent 3GPP releases, the initial premise of 5G -- as a true carrier of IoE services -- is yet to be realized. One can argue that the \emph{evolutionary} part of 5G (i.e., supporting rate-hungry eMBB services) has gained significant momentum, however, the promised \emph{revolutionary} outlook of 5G -- a system operating almost exclusively at high-frequency millimeter wave (mmWave) frequencies and enabling heterogeneous IoE services -- has thus far remained a mirage. Although the 5G systems that are currently being marketed will readily support basic IoE and URLLC services (e.g., factory automation), it is debatable whether they can deliver tomorrow's smart city IoE applications. Moreover, although 5G will eventually support fixed-access at mmWave frequencies, it is more likely that early 5G roll-outs will still use sub-6 GHz for supporting mobility.
Meanwhile, an unprecedented proliferation of new IoE services is ongoing. Examples range from extended reality (XR) services (encompassing augmented, mixed, and virtual reality (AR/MR/VR)) to telemedicine, haptics, flying vehicles, brain-computer interfaces, and connected autonomous systems. These applications will disrupt the original 5G goal of supporting short-packet, sensing-based URLLC services. To successfully operate IoE services such as XR and connected autonomous systems, a wireless system must simultaneously deliver high reliability, low latency, and high data rates, for heterogeneous devices, across uplink and downlink. Emerging IoE services will also require an \emph{end-to-end co-design of communication, control, and computing} functionalities, which to date has been largely overlooked. To cater for this new breed of services, unique challenges must be addressed ranging from characterizing the fundamental rate-reliability-latency tradeoffs governing their performance to exploiting frequencies beyond sub-6 GHz and transforming wireless systems into a self-sustaining, intelligent network fabric which flexibly provisions and orchestrates communication-computing-control-localization-sensing resources tailored to the requisite IoE scenario.
To overcome these challenges, a disruptive \emph{sixth generation (6G)} wireless system, whose design is inherently tailored to the performance requirements of IoE applications and their accompanying technological trends, is needed. The drivers of 6G will be a confluence of past trends (e.g., densification, higher rates, and massive antennas) and of emerging trends that include new services and the recent revolution in wireless devices (e.g., smart wearables, implants, XR devices, etc.), artificial intelligence (AI) \cite{Chen-AI}, computing, and sensing.
\begin{figure*}[!t]
\begin{center}
\vspace{0cm}
\includegraphics[width=15cm]{figure1.pdf}
\vspace{-0.2cm}
\caption{\label{figure1} 6G vision: Applications, trends, and technologies.}
\end{center}\vspace{-0.7cm}
\end{figure*}
The main contribution of this article is a bold, forward-looking vision of 6G systems (see Fig. \ref{figure1}) that identifies the applications, trends, performance metrics, and disruptive technologies, that will drive the 6G revolution. This vision will then delineate new 6G services and provide a concrete research roadmap and recommendations to facilitate the leap from current 5G systems towards 6G.
\section{6G Driving Applications, Metrics, and New Service Classes}\label{se:2}
Every new cellular generation is driven by innovative applications. 6G is no exception: It will be borne out of an unparalleled emergence of exciting new applications and technological trends that will shape its performance targets while radically redefining standard 5G services. Next, we first introduce the main applications that motivate 6G deployment and, then, discuss ensuing technological trends, target performance metrics, and new service requirements.
\subsection{\color{black}Driving Applications behind 6G and Their Requirements}\label{se:21}
While traditional applications, such as live multimedia streaming, will remain central to 6G, the key determinants of the system performance will be four new application domains:
\subsubsection{\bf Multisensory XR Applications} XR will yield many killer applications for 6G across the AR/MR/VR spectrum. Upcoming 5G systems still fall short of providing a full immersive XR experience capturing all sensory inputs due to their inability to deliver very low latencies for data-rate intensive XR applications. A truly immersive AR/MR/VR experience requires a joint design integrating not only engineering (wireless, computing, storage) requirements but also \emph{perceptual} requirements stemming from human senses, cognition, and physiology. Minimal and maximal perceptual requirements and limits must be factored into the engineering process (computing, processing, etc.). To do so, a new concept of {\emph{quality-of-physical-experience (QoPE)}} measure is needed to merge physical factors from the human user itself with classical QoS (e.g., latency and rate) and QoE (e.g., mean-opinion score) inputs. Some factors that affect QoPE include brain cognition, body physiology, and gestures. As an example, in \cite{kasgari2018human}, we have shown that the human brain may not be able to distinguish between different latency measures, within the URLLC regime. Meanwhile, in \cite{MehdiEdge}, we showed that visual and haptic perceptions are key for maximizing resource utilization. Concisely, the requirements of XR services are a blend of traditional URLLC and eMBB with incorporated perceptual factors that 6G must support.
\subsubsection{\bf Connected Robotics and Autonomous Systems (CRAS)} A primary driver behind 6G systems is the imminent deployment of CRAS including drone-delivery systems, autonomous cars, autonomous drone swarms, vehicle platoons, and autonomous robotics. The introduction of CRAS over the cellular domain is not a simple case of ``yet another short packet uplink IoE service''. Instead, CRAS mandate control system-driven latency requirements as well as the potential need for eMBB transmissions of high definition (HD) maps. The \emph{notion of QoPE applies once again for CRAS; however, the physical environment is now a control system}, potentially augmented with AI. CRAS are perhaps a prime use case that requires stringent requirements across the rate-reliability-latency spectrum; a balance that is not yet available in 5G.
\subsubsection{\bf Wireless Brain-Computer Interactions (BCI)} Beyond XR, tailoring wireless systems to their human user is mandatory to support services with direct BCI. Traditionally, BCI applications were limited to healthcare scenarios in which humans can control prosthetic limbs or neighboring computing devices using brain implants. However, the recent advent of wireless brain-computer interfaces and implants will revolutionize this field and introduce new use-case scenarios that require 6G connectivity. Such scenarios range from enabling brain-controlled movie input to fully-fledged multi-brain-controlled cinema \cite{Brain-Movies-2018}. Using wireless BCI technologies, instead of smartphones, people will interact with their environment and other people using discrete devices, some worn, some implanted, and some embedded in the world around them. This will allow individuals to control their environments through gestures and communicate with loved ones through haptic messages. Such \emph{empathic and haptic communications}, coupled with related ideas such as affective computing in which emotion-driven devices can match their functions to their user's mood, constitute important 6G use cases. Wireless BCI services require fundamentally different performance metrics compared to what 5G delivers. Similar to XR, wireless BCI services need high rates, ultra low latency, and high reliability. However, they are much more sensitive than XR to physical perceptions and necessitate QoPE guarantees.
\subsubsection{\bf Blockchain and Distributed Ledger Technologies (DLT)} Blockchains and DLT will be one of the most disruptive IoE technologies. Blockchain and DLT applications can be viewed as the next-generation of distributed sensing services whose need for connectivity will require a synergistic mix of URLLC and massive machine type communications (mMTC) to guarantee low-latency, reliable connectivity, and scalability.
\subsection{6G: Driving Trends and Performance Metrics}\label{se:22}
The applications of Section \ref{se:21} lead to new system-wide trends that will set the goals for 6G:
\begin{itemize}
\item {\bf Trend 1 -- More Bits, More spectrum, More Reliability:} Most of the driving applications of 6G require higher bit rates than 5G. To cater for applications such as XR and BCI, 6G must deliver yet another 1000x increase in data rates yielding a target of around 1 Terabit/second. This motivates a need for more spectrum resources, hence prompting further exploration of frequencies beyond sub-6 GHz. Meanwhile, the need for higher reliability will be pervasive across most 6G applications and will be more challenging to meet at high frequencies.
\item {\bf Trend 2 -- From Areal to Volumetric Spectral and Energy Efficiency:} 6G must deal with ground and aerial users, encompassing smartphones and XR/BCI devices along with flying vehicles. This 3D nature of 6G requires an evolution towards a volumetric rather than spatial {\color{black}{({areal})}} bandwidth definition. We envision that 6G systems must deliver high spectral and energy efficiency (SEE) requirements measured in bps/Hz/\textrm{m$^3$}/Joules. This is a natural evolution that started from 2G (bps) to 3G (bps/Hz), then 4G (bps/Hz/{m$^2$}) to 5G (bps/Hz/m$^2$/Joules).
\item {\bf Trend 3 -- Emergence of Smart Surfaces and Environments:} Current and past cellular systems used base stations (of different sizes and forms) for transmission. We are witnessing a revolution in electromagnetically active surfaces (e.g., using metamaterials) that include man-made structures such as walls, roads, and even entire buildings, as exemplified by the Berkeley ewallpaper project\footnote{See https://bwrc.eecs.berkeley.edu/projects/5605/ewallpaper.}. The use of such smart large intelligent surfaces and environments for wireless communications will drive the 6G architectural evolution.
\item {\bf Trend 4 -- Massive Availability of Small Data:} The data revolution will continue in the near future and shift from centralized, big data, towards massive, distributed ``small'' data. 6G systems must harness both big and small datasets across their infrastructure to enhance network functions and provide new services. This trend motivates new machine learning techniques that go beyond classical big data analytics.
\item {\bf Trend 5 -- From Self-Organizing Networks (SON) to Self-Sustaining Networks:} SON has only been scarcely integrated into 4G/5G networks due to a lack of real-world need. However, CRAS and DLT technologies motivate an immediate need for intelligent SON to manage network operations, resources, and optimization. 6G will require a paradigm shift from classical SON, whereby the network merely adapts its functions to specific environment states, into a \emph{self-sustaining network (SSN)} that can maintain its key performance indicators (KPIs), \emph{in perpetuity}, under highly dynamic and complex environments stemming from the rich 6G application domains. SSNs must be able to not only adapt their functions but to also sustain their resource usage and management (e.g., by harvesting energy and exploiting spectrum) to autonomously maintain high, long-term KPIs. SSN functions must leverage the recent revolution in AI technologies to create AI-powered 6G SSNs.
\item {\bf Trend 6 -- Convergence of Communications, Computing, Control, Localization, and Sensing (3CLS):} The past five generations of cellular systems had one exclusive function: wireless communications. {\color{black}However, 6G will disrupt this premise through a convergence (i.e., joint and simultaneous offering) of various functions that include communications, computing \cite{8265188}, control, localization, and sensing.} We envision 6G as a multi-purpose system that can deliver multiple 3CLS services which are particularly appealing and even necessary for applications such as XR, CRAS, and DLT where tracking, control, localization, and computing are an inherent feature. Moreover, sensing services will enable 6G systems to provide users with a \emph{3D mapping of the radio environment} across different frequencies. Hence, 6G systems must tightly integrate and manage 3CLS functions. {\color{black}Note that, the evolutions pertaining to previous trends will gradually enable 6G systems to readily provide 3CLS.}
\item {\bf Trend 7 -- End of the Smartphone Era:} Smartphones were central to 4G and 5G. However, recent years witnessed an increase in wearable devices whose functionalities are gradually replacing those of smartphones. This trend is further fueled by applications such as XR and BCI. The devices associated with those applications range from smart wearables to integrated headsets and smart body implants that can take direct sensory inputs from human senses; bringing an end to smartphones and potentially driving a majority of 6G use cases.
\end{itemize}
As shown in Table \ref{ta:1}, collectively, these trends impose new performance targets and requirements that will be met in two stages: a) A beyond 5G evolution and b) A revolutionary 6G step.
\begin{table*}
\centering
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2.4\end{tabular}}
\renewcommand\arraystretch{1.5}
\caption{
\vspace*{-0.05em}Requirements of 5G vs. Beyond 5G vs. 6G.}\label{ta:1}\vspace*{-0.5em}
\centering
\begin{tabular}{|c|l|l|l|
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{}}}&\multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{5G}}} & \multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{ Beyond 5G}}} & \multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{ 6G}}} \\
&&&\\
\hline
\multirow{5}{*}{{\bf Application Types }}& \multirow{1}{*}{{$\bullet$ eMBB.}} &\multirow{1}{*}{$\bullet$ Reliable eMBB.} &\multirow{1}{*}{New applications (see Section \ref{se:23}):} \\%& $\bullet$ UAV path planning $ \Rightarrow $ RNN-based RL algorithm. \\
&$\bullet$ URLLC.&$\bullet$ URLLC. &$\bullet$ MBRLLC.\\
&$\bullet$ mMTC.&$\bullet$ mMTC. & \multirow{1}{*}{$\bullet$ mURLLC.}\\
&&$\bullet$ Hybrid (URLLC + eMBB).&$\bullet$ HCS.\\
&&&$\bullet$ MPS. \\%& $\bullet$ Channel modeling for air-to-ground $ \Rightarrow $ SNN prediction algorithm.\\
\hline
\multirow{4}{*}{{\bf Device Types}}& \multirow{1}{*}{{$\bullet$ Smartphones.}} &\multirow{1}{*}{$\bullet$ Smartphones.} &\multirow{1}{*}{$\bullet$ Sensors and DLT devices.} \\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&$\bullet$ Sensors.&$\bullet$ Sensors.&$\bullet$ CRAS.\\
&$\bullet$ Drones.&$\bullet$ Drones. &$\bullet$ XR and BCI equipment. \\
&&\multirow{1}{*}{$\bullet$ XR equipment.} &$\bullet$ Smart implants. \\
\hline
\multirow{4}{3cm}{{\bf Spectral and Energy Efficiency Gains\footnotemark[3] with Respect to Today's Networks}}& \multirow{4}{*}{{10x in bps/Hz/m$^2$/Joules}} & \multirow{4}{*}{{100x in bps/Hz/m$^2$/Joules}}& \multirow{4}{*}{{1000x in bps/Hz/m$^3$/Joules (volumetric)}}\\%& $\bullet$ Analysis of content correlation. \\
&&&\\
&&&\\
&&&\\
\hline
\multirow{2}{3cm}{{\bf Rate Requirements}}& \multirow{2}{*}{{ 1 Gbps}} & \multirow{2}{*}{{100 Gbps}}& \multirow{2}{*}{{1 Tbps}}\\
&&&\\
\hline
\multirow{2}{3cm}{{\bf End-to-End Delay Requirements}}& \multirow{2}{*}{{ 5 ms}} & \multirow{2}{*}{{1 ms}}& \multirow{2}{*}{{< 1 ms}}\\
&&&\\
\hline
\multirow{2}{3cm}{{\bf Radio-Only Delay Requirements}}& \multirow{2}{*}{{ 100 ns}} & \multirow{2}{*}{{100 ns}}& \multirow{2}{*}{{10 ns}}\\
&&&\\
\hline
\multirow{2}{3cm}{{\bf Processing Delay}}& \multirow{2}{*}{{ 100 ns}} & \multirow{2}{*}{{50 ns}}& \multirow{2}{*}{{10 ns}}\\
&&&\\
\hline
\multirow{2}{3cm}{{\bf End-to-End Reliability Requirements}}& \multirow{2}{*}{{ \color{black}99.999\%}} & \multirow{2}{*}{{\color{black}99.9999\%}}& \multirow{2}{*}{{\color{black}99.99999\%}}\\
&&&\\
\hline
\multirow{3}{*}{{\bf Frequency Bands}}& \multirow{1}{*}{{$\bullet$ Sub-6 GHz.}} &\multirow{1}{*}{$\bullet$ Sub-6 GHz.} &\multirow{1}{*}{$\bullet$ Sub-6 GHz.} \\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&$\bullet$ MmWave for fixed access.&\multirow{2}{5cm}{$\bullet$ MmWave for fixed access at 26 GHz and 28GHz.}&$\bullet$ MmWave for mobile access.\\
&& &$\bullet$ Exploration of THz bands (above 300 GHz). \\
&& &$\bullet$ Non-RF (e.g., optical, VLC, etc.). \\
\hline
\multirow{6}{*}{{\bf Architecture}}& \multirow{3}{3cm}{{$\bullet$ Dense sub-6 GHz small base stations with umbrella macro base stations.}} &\multirow{2}{5cm}{$\bullet$ Denser sub-6 GHz small cells with umbrella macro base stations.} &\multirow{3}{4cm}{$\bullet$ Cell-free smart surfaces at high frequency supported by mmWave tiny cells for mobile and fixed access.} \\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&&& \\
&&$\bullet$ < 100 m tiny and dense mmWave cells.&\\
&\multirow{3}{3cm}{{$\bullet$ MmWave small cells of about 100 m (for fixed access).}} & &\multirow{2}{4cm}{$\bullet$ Temporary hotspots served by drone-carried base stations or tethered balloons.}\\
&&&\\
&&&$\bullet$ Trials of tiny THz cells.\\
\hline
\end{tabular}
\vspace{-0.2cm}
\end{table*}
\subsection{New 6G Service Classes}\label{se:23}
Beyond imposing new performance metrics, the new technological trends will redefine 5G application types by morphing classical URLLC, eMBB, and mMTC and introducing new services (summarized in Table \ref{ta:2}):
\subsubsection{\bf{Mobile Broadband Reliable Low Latency Communication}} {\color{black}As evident from Section \ref{se:22}, the distinction between eMBB and URLLC will no longer be sustainable to support applications such as XR, wireless BCI, or CRAS. This is because these applications require, not only high reliability and low latency but also high 5G-eMBB-level data rates.} Hence, we propose a new service class called \emph{mobile broadband reliable low latency communication (MBRLLC)} that allows 6G systems to deliver any required performance within the rate-reliability-latency space. As seen in Fig. \ref{figure2}, MBRLLC generalizes classical URLLC and eMBB services. Energy efficiency is central for MBRLLC, not only because of its impact on reliability and rate, but also because of the resource-limited nature of 6G devices.
\begin{figure*}[!t]
\begin{center}
\vspace{0cm}
\includegraphics[width=15cm]{figure2.pdf}
\vspace{-0.2cm}
\caption{\label{figure2}MBRLLC services and several special cases (including classical eMBB and URLLC) within the rate-reliability-latency space. Other involved, associated metrics that are not shown include energy and network scale.}
\end{center}\vspace{-0.7cm}
\end{figure*}
\subsubsection{\bf{Massive URLLC}} 5G URLLC meant meeting reliability and latency of very specific uplink IoE applications such as smart factories, for which prior work \cite{Popovski-100} provided the needed fundamentals. However, 6G must scale classical URLLC across the device dimension thereby leading to a new \emph{massive URLLC (mURLLC)} service that merges 5G URLLC with legacy mMTC. mURLLC brings forth a reliability-latency-scalability tradeoff which mandates a major departure from average-based network designs (e.g., average throughput/delay). Instead, a principled and scalable framework which accounts for delay, reliability, packet size, architecture, topology (across access, edge, and core) and decision-making under uncertainty is necessary \cite{MehdiMag}.
\subsubsection{\bf{Human-Centric Services}} We propose a new class of 6G services, dubbed \emph{human-centric services (HCS)}, that require QoPE targets (tightly coupled with their human users, as explained in Section \ref{se:21}) rather than raw rate-reliability-latency metrics. Wireless BCI are a prime example of HCS in which network performance is determined by the physiology of the human users and their actions. For such services, a whole new set of QoPE metrics must be defined and offered as function of raw QoS and QoE metrics.
\subsubsection{\bf{Multi-Purpose 3CLS and Energy Services}} 6G systems must jointly deliver 3CLS services and their derivatives. They can also potentially offer energy to small devices via wireless energy transfer. Such \emph{multi-purpose 3CLS and energy services (MPS)} will be particularly important for applications such as CRAS. MPS require joint uplink-downlink designs and must meet target performance for the control (e.g., stability), computing (e.g., computing latency), energy (e.g., target energy to transfer), localization (e.g., localization precision), as well as sensing and mapping functions (e.g., accuracy of a mapped radio environment).
\begin{table*}
\color{black}
\centering
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2.4\end{tabular}}
\renewcommand\arraystretch{1.5}
\caption{
\vspace*{-0.05em}Summary of 6G service classes, their performance indicators, and example applications. }\label{ta:2}\vspace*{-0.5em}
\centering
\begin{tabular}{|c|l|l|
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{Service}}}&\multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{Performance Indicators
}}} & \multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{ Example Applications}}} \\
&&\\
\hline
\multirow{4}{*}{{\bf MBRLLC}}& \multirow{1}{*}{{$\bullet$ Stringent rate-reliability-latency requirements.}} &\multirow{1}{*}{$\bullet$ \color{black}XR/AR/VR.}\\%& $\bullet$ UAV path planning $ \Rightarrow $ RNN-based RL algorithm. \\
&$\bullet$ Energy efficiency.&$\bullet$ \color{black}{Autonomous vehicular systems}. \\
&$\bullet$ Rate-reliability-latency in mobile environments.&$\bullet$ \color{black}{Autonomous drones}. \\
&&$\bullet$ \color{black}{Legacy eMBB and URLLC}.\\
\hline
\multirow{4}{*}{{\bf mURLLC}}& \multirow{1}{*}{{$\bullet$ Ultra high reliability.}} &\multirow{1}{*}{$\bullet$ Classical Internet of Things.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&$\bullet$ Massive connectivity.&$\bullet$ User tracking.\\
&$\bullet$ Massive reliability.&$\bullet$ Blockchain and DLT. \\
&$\bullet$ Scalable URLLC.
&\multirow{1}{*}{$\bullet$ Massive sensing.} \\
&&\multirow{1}{*}{$\bullet$ Autonomous robotics.} \\
\hline
\multirow{4}{*}{{\bf HCS}}& \multirow{2}{5cm}{{$\bullet$ QoPE capturing raw wireless metrics as well as human and physical factors.}} &\multirow{1}{*}{$\bullet$ BCI.} \\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&&$\bullet$ Haptics.\\
&&$\bullet$ Empathic communication. \\
&& $\bullet$ Affective communication. \\
\hline
\multirow{6}{*}{{\bf MPS}}& \multirow{1}{*}{{$\bullet$ Control stability.}} &\multirow{1}{4cm}{$\bullet$ CRAS.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
& $\bullet$ Computing latency. &$\bullet$ Telemedicine.\\
& $\bullet$ Localization accuracy.&$\bullet$ Environmental mapping and imaging.\\
&$\bullet$ Sensing and mapping accuracy. &$\bullet$ Some special cases of XR services. \\
&$\bullet$ Latency and reliability for communications. & \\
&$\bullet$ Energy. & \\
\hline
\end{tabular}
\vspace{-0.2cm}
\end{table*}
\footnotetext[3]{\color{black}{Here, spectral and energy efficiency gains are captured by the concept of area spectral and energy efficiency.}}
\section{6G: Enabling Technologies}\label{se:3}
To enable the aforementioned services and guarantee their performance, a cohort of new, disruptive technologies must be integrated into 6G.
\subsubsection{\bf{Above 6 GHz for 6G -- from Small Cells to Tiny Cells}} As per Trends 1 and 2, the need for higher data rates and SEE anywhere, anytime in 6G motivates exploring higher frequency bands beyond sub-6 GHz. As a first step, this includes further developing mmWave technologies to make \emph{mobile mmWave} a reality in early 6G systems. As 6G progresses, exploiting frequencies beyond mmWave, at the terahertz (THz) band, will become necessary \cite{Rappaport-6G}. To exploit higher mmWave and THz frequencies, the size of the 6G cells must shrink from small cells to ``tiny cells'' whose radius is only few tens meters. This motivates new architectural designs that need much denser deployments of tiny cells and new high-frequency mobility management techniques.
\subsubsection{\bf{Transceivers with Integrated Frequency Bands}} On their own, dense high-frequency tiny cells may not be able to provide the seamless connectivity required for mobile 6G services. Instead, an integrated system that can leverage multiple frequencies across the microwave/mmWave/THz spectra (e.g., using multi-mode base stations) is needed to provide seamless connectivity at both wide and local area levels.
\subsubsection{\bf{Communication with Large Intelligent Surfaces}} Massive MIMO will be integral to both 5G and 6G due to the need for better SEE, higher data rates, and higher frequencies (Trend 1). However, for 6G systems, as per Trend 3, we envision an initial leap from traditional massive MIMO towards large intelligent surfaces (LISs) and smart environments \cite{LIS-1} that can provide massive surfaces for wireless communications and for heterogeneous devices (Trend 7). LISs enable innovative ways for communication such as by using holographic radio frequency (RF) and holographic MIMO.
\subsubsection{\bf{Edge AI}} AI is witnessing an unprecedented interest from the wireless community \cite{Chen-AI} driven by recent breakthroughs in deep learning, the increase in available data (Trend 4), and the rise of smart devices (Trend 7). Imminent 6G use cases for AI (particularly for reinforcement learning) revolve around creating SSNs (Trend 5) that can autonomously sustain high KPIs and manage resources, functions, and network control. AI will also enable 6G to automatically provide MPS to its users and to send and create 3D radio environment maps (Trend 6). These short-term AI-enabled 6G functions will be complemented by a so-called ``collective network intelligence'' in which network intelligence is pushed at the edge, running AI and learning algorithms on edge devices (Trend 7) to provide distributed autonomy. This new edge AI leap will create a 6G system that can integrate the services of Section \ref{se:2}, realize 3CLS, and potentially replace classical frame structures.
\subsubsection{\bf{Integrated Terrestrial, Airborne, and Satellite Networks}} Beyond their inevitable role as 6G users, drones can be leveraged to complement terrestrial networks by providing connectivity to hotspots and to areas with scarce infrastructure. Meanwhile, both drones and terrestrial base stations may require satellite connectivity with low orbit satellites (LEO) and CubeSats to provide backhaul support and additional wide area coverage. Integrating terrestrial, airborne, and satellite networks \cite{Mohammad-2019} and \cite{Halim} into a single wireless system will be essential for 6G.
\subsubsection{\bf{Energy Transfer and Harvesting}} 6G could be the cellular system that can provide energy, along with 3CLS (Trend 6). As wireless energy transfer is maturing, we foresee 6G base stations providing basic power transfer for devices, particularly implants and sensors (Trend 7). Adjunct energy-centric ideas, such as energy harvesting and backscatter will also be a component of 6G.
\subsubsection{\bf{Beyond 6G}} A handful of technologies will mature along the same time of 6G and, hence, potentially play a role towards the end of the 6G standardization and research process. One prominent example is \emph{quantum computing and communications} that can provide security and long-distance networking. Currently, major research efforts are focused on the quantum realm and we expect them to intersect with 6G. Other similar beyond 6G technologies include integration of RF and non-RF links (including optical, neural, molecular, and other channels).
\section{6G: Research Agenda and Open Problems}\label{se:4}
\begin{figure*}[!t]
\begin{center}
\vspace{0cm}
\includegraphics[width=14cm]{figure3.pdf}
\vspace{-0.2cm}
\caption{\label{figure3} Necessary foundations and associated analytical tools for 6G.}
\end{center}\vspace{-0.7cm}
\end{figure*}
Building on the identified trends in Section \ref{se:2} and the enabling technologies in Section \ref{se:3}, we now put forward a research agenda for 6G (summarized in Table \ref{ta:3}).
\subsubsection{\bf{3D Rate-Reliability-Latency Fundamentals}} Fundamental 3D performance of 6G systems, in terms of rate-reliability-latency tradeoffs and SEE is needed. Such analysis must quantify the spectrum, energy, and communication requirements that 6G needs in order to support the identified driving applications. Recent works in \cite{MehdiMag} and \cite{Ali-ICC} provide a first step in this direction.
\subsubsection{\bf{Exploring Integrated, Heterogeneous High-Frequency Bands}} Exploiting mmWave and THz in 6G brings forth several new open problems. For mmWave, supporting high mobility at mmWave frequencies will be a central open problem. For THz, new transceiver architectures and propagation models are needed \cite{Rappaport-6G}. High power, high sensitivity, and low noise figure are key transceiver features needed to overcome the very high THz path loss. {\color{black}Once these physical layer aspects are well-understood, there is a need to develop new network and link-layer protocols to optimize the use of cross-frequency resources while taking into account the highly varying and uncertain nature of the mmWave and THz environments.} Another important direction is to study the co-existence of THz, mmWave, and microwave cells across all layers, building on early works such as \cite{Omid-2019}.
\subsubsection{\bf{3D Networking}} Due to the integration of ground and airborne networks, as outlined in Section \ref{se:3}, 6G must support communications in 3D space, including serving users in 3D and deploying 3D base stations (e.g., tethered balloons or temporary drones). This requires concerted research on various fronts. First, measurement and (data-driven) modeling of the 3D propagation environment is needed. Second, new approaches for 3D frequency and network planning (e.g., where to deploy base stations, tethered balloons, or even drone-base stations) must be developed. Our work in \cite{Mohammad-2019} showed that such 3D planning is substantially different from conventional 2D networks due to the new altitude dimension and the associated degrees of freedom. Finally, new network optimizations for mobility management, routing, and resource management in 3D are needed.
\subsubsection{\bf{Communication with LIS}} As per Trend 3, 6G will provide wireless connectivity via smart LIS environments that include active frequency selective surfaces, metallic passive reflectors, passive/active reflect arrays, as well as nonreconfigurable and reconfigurable metasurfaces. Open problems here range from the optimized deployment of passive reflectors and metasurfaces to AI-powered operation of LISs. Fundamental analysis to understand the performance of smart surfaces, in terms of rate, latency, reliability, and coverage is needed, building on the early work in \cite{LIS-1}. Another important direction is to investigate the potential of using LIS-based reflective surfaces to enhance the range and coverage of tiny cells and to dynamically modify the propagation environment.
\subsubsection{\bf{AI for Wireless}} AI brings forward many major research directions for 6G. Beyond the need for massive, small data analytics as well as using machine learning (ML) and AI-based SSNs (realized using reinforcement learning and game theory), there is also a need to operate ML algorithms reliably over 6G to deliver the applications of Section \ref{se:2}. To perform these critical application tasks, low-latency, high-reliability and scalable AI is needed, along with a reliable infrastructure \cite{Chen-AI} and \cite{MehdiEdge}. This joint design of ML and wireless networks is a key 6G research area.
\subsubsection{\bf{QoPE Metrics}} The design of QoPE metrics that integrate physical factors from human physiology (for HCS services) or from a control system (for CRAS) is an important 6G research area, especially in light of new, emerging devices (Trend 7). This requires both real-world psychophysics experiments as well as new, rigorous mathematical expressions for QoPE that combine QoS, QoE, and human perceptions. Theoretical development of QoPE can be achieved using techniques from other disciplines such as operations research (e.g., multi-attribute utility theory (see \cite{Chen-Saad-VR-2018})) and machine learning (see \cite{kasgari2018human}). 6G will be the first generation to enable a new breed of applications (wireless BCI) leveraging multiple human cognitive senses.
\subsubsection{\bf{ Joint Communication and Control}} 6G needs to pervasively support CRAS. The performance of CRAS is governed by real-world control systems whose operation requires data input from wireless 6G links. Therefore, operating CRAS over 6G systems requires a \emph{communication and control co-design}, whereby the performance of the 6G wireless links is optimized to cater for the stability of the control system and vice versa. Due to the traditional radio-centric focus (3GPP and IEEE fora), such a co-design has been overlooked in 5G. Meanwhile, prior works on networked control abstract the specifics of the wireless network and cannot apply to cellular communications. This makes the communication-control co-design a key research topic in 6G.
\subsubsection{\bf{3CLS}} The idea of joint communication and control must be extended to the joint design of the entire 3CLS functions. The interdependence between computing, communication, control, localization, sensing, energy, and mapping has not yet been fully explored in an end-to-end manner. Key questions range from how to jointly meet the performance of all 3CLS services to multi-modal sensor fusion for reconstructing 3D images and navigating in unknown environments for navigating robots, autonomous driving, etc. 3CLS is needed for various applications including CRAS, XR, and DLT.
\subsubsection{\bf{\color{black}6G Protocol Designs}}{\color{black}Owing to all trends discussed in Section \ref{se:22} and their challenges, compared to 5G, 6G will require radical new protocol designs. For instance, 6G must introduce new, AI-driven protocols for signaling, scheduling, and coordination that can replace conventional 5G protocols that rely on pre-determined network parameters and rigid frame structures. These new 6G protocols will, in contrast, continuously adapt to the current and projected state of the wireless environment. As 6G evolves, there will be a need for new, dynamic multiple access \cite{8663993} protocols that can dynamically change the type of multiple access (orthogonal or non-orthogonal, random or scheduled) used depending on the needs of the applications and the network state. Moreover, novel handover protocols must be designed to account for the 3D nature of the 6G system and the presence of different types of mobile devices. New protocols for authentication and identification will also be needed to handle the new breed of wireless devices that include drones, vehicles, as well as embedded and implanted devices. Finally, all 6G protocols must be distributed and able to leverage datasets distributed across the network edge.}
\subsubsection{\bf{RF and non-RF Link Integration}} 6G will witness a convergence of RF and non-RF links that encompass optical, visible light communication (VLC), molecular communication, and neuro-communication, among others. Design of such joint RF/non-RF systems is an open research area.
\subsubsection{\bf{Holographic Radio}} RF holography (including holographic MIMO) and spatial spectral holography can be made possible with 6G due to the use of LIS and similar structures. Holographic RF allows for control of the entire physical space and the full closed loop of the electromagnetic field through spatial spectral holography and spatial wave field synthesis. This greatly improves spectrum efficiency and network capacity, and helps the integration of imaging and wireless communication. How to realize holographic radio is a widely open area.
An overview on the necessary analytical tools and fundamentals related to these open problems is shown in Fig. \ref{figure3}.
\begin{table*}
\centering
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2.4\end{tabular}}
\renewcommand\arraystretch{1.2}
\caption{
\vspace*{-0.05em}Summary of Research Areas }\label{ta:3}\vspace*{-0.5em}
\centering
\begin{tabular}{|c|l|l|
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{Research Area}}}&\multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{Challenges
}}} & \multicolumn{1}{|c|}{\multirow{2}{*}{\textbf{ Open Problems}}} \\
&&\\
\hline
\multirow{4}{3.5cm}{{\bf3D Rate-Reliability-Latency Fundamentals}}& \multirow{1}{*}{{$\bullet$ Fundamental communication limits.}} &\multirow{1}{*}{$\bullet$ 3D performance analysis of rate-reliability-latency region.}\\%& $\bullet$ UAV path planning $ \Rightarrow $ RNN-based RL algorithm. \\
&$\bullet$ 3D nature of 6G systems.&$\bullet$ Characterization of achievable rate-reliability-latency targets. \\
&&$\bullet$ 3D SEE characterization. \\
&& \multirow{2}{6cm}{{$\bullet$ Characterization of energy and spectrum needs for rate-reliability-latency targets.}} \\
&&\\
\hline
\multirow{7}{3.5cm}{{\bf Exploring Integrated, Heterogeneous High-Frequency Bands}}& \multirow{1}{*}{{$\bullet$ Challenges of operation in highly mobile systems. }} &\multirow{1}{*}{$\bullet$ Effective mobility management for mmWave and THz systems.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&$\bullet$ Susceptibility to blockage.&$\bullet$ Cross-band physical, link, and network layer optimization. \\
&$\bullet$ Short range. &$\bullet$ Coverage and range improvement. \\
&$\bullet$ Lack of propagation models. &\multirow{1}{*}{$\bullet$ Design of mmWave and THz tiny cells.} \\
&$\bullet$ Need for high fidelity hardware. &\multirow{1}{*}{$\bullet$ Design of new high fidelity hardware for THz.} \\
&$\bullet$ Co-existence of frequency bands.&\multirow{2}{7cm}{$\bullet$ Propagation measurements and modeling across mmWave and THz bands.} \\
&&\\
\hline
\multirow{3}{*}{{\bf 3D Networking}}& \multirow{1}{*}{{$\bullet$ Presence of users and base stations in 3D.}} &\multirow{1}{*}{$\bullet$ 3D propagation modeling.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&$\bullet$ High mobility.&$\bullet$ 3D performance metrics.\\
&&$\bullet$ 3D mobility management and network optimization. \\
\hline
\multirow{7}{*}{{\bf Communication with LIS}}& \multirow{1}{*}{{$\bullet$ Complex nature of LIS surfaces.}} &\multirow{1}{*}{$\bullet$ Optimal deployment and location of LIS surfaces.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&$\bullet$ Lack of existing performance models.&$\bullet$ LIS reflectors vs. LIS base stations.\\
&$\bullet$ Lack of propagation models.&$\bullet$ LIS for energy transfer. \\
&$\bullet$ Heterogeneity of 6G devices and services.&\multirow{1}{*}{$\bullet$ AI-enabled LIS.} \\
& \multirow{2}{5cm}{{$\bullet$ Ability of LIS to provide different functions (reflectors, base stations, etc.).}} &\multirow{1}{*}{$\bullet$ LIS across 6G services.} \\
&& \multirow{2}{7cm}{{$\bullet$ Fundamental performance analysis of LIS transmitters and reflectors at various frequencies.}} \\
&&\\
\hline
\multirow{4}{*}{{\bf AI for Wireless}}& \multirow{1}{*}{{$\bullet$ Design of low-complexity AI solutions.}} &\multirow{1}{*}{$\bullet$ Reinforcement learning for SON.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
&$\bullet$ Massive, small data.&$\bullet$ Big and small data analytics.\\
&&$\bullet$ AI-powered network management. \\
& &\multirow{1}{*}{$\bullet$ Edge AI over wireless systems.} \\
\hline
\multirow{4}{*}{{\bf New QoPE Metrics}}& \multirow{1}{*}{{$\bullet$ Incorporate raw metrics with human perceptions.}} &\multirow{1}{*}{$\bullet$ Theoretical development of QoPE metrics.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
& \multirow{1}{5cm}{{$\bullet$ Accurate modeling of human perceptions and physiology.}}&$\bullet$ Empirical QoPE characterization.\\
&&$\bullet$ Real psychophysics experiments.\\
& &$\bullet$ Definition of realistic QoPE targets and measures. \\
\hline
\multirow{4}{*}{{\bf Joint Communication and Control}}& \multirow{1}{*}{{$\bullet$ Integration of control and communication metrics.}} &\multirow{1}{*}{$\bullet$ Communication and control systems co-design.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
& \multirow{1}{*}{{$\bullet$ Handling dynamics and multiple time scales.}}&$\bullet$ Control-enabled wireless metrics.\\
&&$\bullet$ Wireless-enabled control metrics.\\
& &$\bullet$ Joint optimization for CRAS. \\
\hline
\multirow{4}{*}{{\bf 3CLS }}& \multirow{1}{*}{{$\bullet$ Integration of multiple functions.}} &\multirow{1}{*}{$\bullet$ Design of 3CLS metrics.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
& \multirow{1}{*}{{$\bullet$ Lack of prior models.}}&$\bullet$ Joint 3CLS optimization.\\
&&$\bullet$ AI-enabled 3CLS.\\
& &$\bullet$ Energy efficient 3CLS. \\
\hline
\multirow{10}{*}{{\bf \color{black} 6G Protocol Designs }}& \multirow{2}{5.2cm}{{\color{black}$\bullet$ 6G protocols must operate in 3D space and across different propagation environments.}} &\multirow{3}{7cm}{{\color{black}$\bullet$ Design of scheduling, coordination, and signaling protocols that do not require pre-determined, rigid frame structures.}}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
& &\\
&\multirow{2}{5.2cm}{{\color{black}$\bullet$ Presence of heterogeneous devices with different capabilities and mobility patterns.}}&\\
& &\multirow{1}{7cm}{{\color{black}$\bullet$ Development of adaptive multiple access protocols.}} \\
&\multirow{2}{5.2cm}{{\color{black}$\bullet$ Need for protocols that can learn and adapt to the environment.}}&\multirow{2}{7cm}{{\color{black}$\bullet$ Design of proactive and dynamic handover mechanisms that can cope with different mobility patterns in 3D space.}} \\
& &\\
&&\multirow{2}{7cm}{{\color{black}$\bullet$ Introduction of new authentication and identification protocols tailored to 6G devices.}} \\
& &\\
&&\multirow{2}{7cm}{{\color{black}$\bullet$ Design of distributed, edge AI-inspired protocols for executing multiple 6G functions.} } \\
& &\\
\hline
\multirow{3}{*}{\bf RF and non-RF Link Integration}& \multirow{1}{*}{{$\bullet$ Different physical nature of RF/non-RF interfaces.}} &\multirow{1}{*}{$\bullet$ Design of joint RF/non-RF hardware.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
& &$\bullet$ System-level analysis of joint RF/non-RF systems.\\
&&$\bullet$ Use of RF/non-RF systems for various 6G services.\\
\hline
\multirow{4}{*}{{\bf Holographic Radio }}& \multirow{1}{*}{{$\bullet$ Lack of existing models.}} &\multirow{1}{*}{$\bullet$ Design of holographic MIMO using LIS.}\\%& $\bullet$ VR users' movement $ \Rightarrow $ RNNs prediction algorithm. \\
& \multirow{1}{*}{{$\bullet$ Hardware and physical layer challenges.}}&$\bullet$ Performance analysis of holographic RF.\\
&&$\bullet$ 3CLS over holographic radio.\\
& &$\bullet$ Network optimization with holographic radio. \\
\hline
\end{tabular}
\vspace{-0.2cm}
\end{table*}
\section{Conclusion and Recommendations}\label{se:5}
This article laid out a bold new vision for 6G systems that outlines the trends, challenges, and associated research. While many topics will come as a natural 5G evolution, new avenues of research such as LIS communication, 3CLS, holographic radio, and others will create an exciting research agenda for the next decade. We conclude with several recommendations:
\begin{itemize}
\item {\bf \underline {Recommendation 1}:} A first step towards 6G is to enable MBRLLC and mobility management at high-frequency mmWave bands and beyond (i.e., THz).
\item {\bf \underline {Recommendation 2}:} 6G requires a move from radio-centric system design (\`a-la-3GPP) towards an end-to-end 3CLS co-design under the orchestration of an AI-driven intelligence substrate.
\item {\bf \underline {Recommendation 3}:} The 6G vision will not be a simple case of exploring additional, high-frequency spectrum bands to provide more capacity. Instead, it will be driven by a diverse portfolio of applications, technologies, and techniques (see Figs. \ref{figure1} and \ref{figure3}).
\item {\bf \underline {Recommendation 4}:} 6G will transition from the smartphone-base station paradigm into a new era of smart surfaces communicating with human-embedded implants.
\item {\bf \underline {Recommendation 5}:} Performance analysis and optimization of 6G requires operating in 3D space and moving away from simple averaging towards fine-grained analysis that deals with tails, distributions, and QoPE.
\end{itemize}
\bibliographystyle{IEEEbib}
\def0.91{0.91}
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\section{Introduction}
A (Euclidean) lattice $L \subset \mathbb{R}^n$
in dimension $n$
is {\em unimodular} if
$L = L^{*}$, where
the dual lattice $L^{*}$ of $L$ is defined as
$\{ x \in {\mathbb{R}}^n \mid (x,y) \in \mathbb{Z} \text{ for all }
y \in L\}$ under the standard inner product $(x,y)$.
A unimodular lattice is called {\em even}
if the norm $(x,x)$ of every vector $x$ is even.
A unimodular lattice which is not even is called
{\em odd}.
An even unimodular lattice in dimension $n$ exists if and
only if $n \equiv 0 \pmod 8$, while
an odd unimodular lattice exists for every dimension.
Two lattices $L$ and $L'$ are {\em neighbors} if
both lattices contain a sublattice of index $2$
in common.
Rains and Sloane~\cite{RS-bound} showed that
the minimum norm $\min(L)$ of a unimodular
lattice $L$ in dimension $n$
is bounded by
$\min(L) \le 2 \lfloor n/24 \rfloor+2$
unless $n=23$ when $\min(L) \le 3$.
We say that a unimodular lattice meeting the upper
bound is {\em extremal}.
Gaulter~\cite{Gaulter} showed that
any unimodular lattice in dimension $24k$
meeting the upper
bound has to be even, which was conjectured by
Rains and Sloane.
Hence, an odd unimodular lattice $L$
in dimension $24k$
satisfies $\min(L) \le 2k+1$.
We say that an odd unimodular lattice $L$ in dimension $24k$
with $\min(L)=2k+1$ is {\em optimal}.
Shadows of odd unimodular lattices appeared
in~\cite{CS90-AMS} and \cite{CS-odd},
and shadows play an important role in the study of odd unimodular lattices.
For example, shadows are the main tool in \cite{RS-bound}.
Let $L$ be an odd unimodular lattice and let $L_0$ be the
subset of vectors of even norm.
Then $L_0$ is a sublattice of $L$ of index 2. The {\em shadow} of $L$ is
defined as
$S(L) = L_0^ * \setminus L$.
We define the {\em shadow minimum} of $L$
as $\sm(L) = \min \{ (x,x) \mid x\in S(L) \} $.
The aim of this note is to show the following:
\begin{thm}\label{thm:main}
If there is an extremal even unimodular lattice $\Lambda$ in dimension
$72$, then there is an optimal odd unimodular lattice $L$
in dimension $72$ with $\sm(L) = 2$, which is a
neighbor of $\Lambda$.
\end{thm}
Recently Nebe~\cite{Nebe72} has found an
extremal even unimodular lattice in dimension $72$.
It was a long-standing question to determine the
existence of such a lattice.
As a consequence of Theorem~\ref{thm:main},
we have the following:
\begin{cor}
There is an optimal odd unimodular lattice $L$
in dimension $72$ with $\sm(L)=2$.
\end{cor}
\section{An optimal odd unimodular lattice in dimension 72}
The theta series $\theta_{L}(q)$ of a lattice $L$ is
the formal power series
$\theta_{L}(q) = \sum_{x \in L} q^{(x, x)}$.
Conway and Sloane~\cite{CS90-AMS, CS-odd} showed that
when the theta series of an odd unimodular lattice $L$
in dimension $n$ is written as
\begin{equation}\label{eq:CS1}
\theta_L(q)=
\sum_{j =0}^{\lfloor n/8\rfloor} a_j\theta_3(q)^{n-8j}\Delta_8(q)^j,
\end{equation}
the theta series of the shadow $S(L)$ is written as
\begin{equation}\label{eq:CS2}
\theta_S(q)= \sum_{j=0}^{\lfloor n/8\rfloor}
\frac{(-1)^j}{16^j} a_j\theta_2(q)^{n-8j}\theta_4(q^2)^{8j}
= \sum_i B_i q^i \text{ (say),}
\end{equation}
where
$\Delta_8(q) = q \prod_{m=1}^{\infty} (1 - q^{2m-1})^8(1-q^{4m})^8$
and $\theta_2(q), \theta_3(q)$ and $\theta_4(q)$ are the Jacobi
theta series~\cite{SPLAG}.
As the additional conditions, it follows
from~\cite{CS90-AMS} and~\cite{CS-odd} that
\begin{equation}
\label{eq:C}
\begin{cases}
\text{$B_r =0$ unless $r \not\equiv n/4 \pmod 2$},\\
\text{there is at most one nonzero $B_r$ for $r < (\min(L)+2)/2$},\\
\text{$B_r=0$ for $r < \min(L)/4$,}\\
\text{$B_r \le 2$ for $r < \min(L)/2$.}
\end{cases}
\end{equation}
\begin{lem}\label{lem:T}
Let $L$ be an optimal odd unimodular lattice in dimension $72$
with $\sm(L)=2$.
Then the theta series of $L$ and $S(L)$ are uniquely
determined as
\begin{align}
\label{eq:T1}
\theta_{L}(q) =&
1 + 27918336 q^7 + 3165770864 q^8 + \cdots, \\
\label{eq:T2}
\theta_S(q) =&
2 q^2 + 127800 q^6 + \cdots,
\end{align}
respectively.
\end{lem}
\begin{proof}
In (\ref{eq:CS1}) and (\ref{eq:CS2}),
it follows from $\min(L)=7$ that
\begin{multline*}
a_0=1,
a_1=-144,
a_2=7056,
a_3=-136704,
\\
a_4=928656,
a_5=-1518336,
a_6=136704.
\end{multline*}
Since $S(L)$ does not have $\mathbf{0}$,
$a_9=0$.
Hence, we have the following possible theta series:
\begin{align}
\label{eq:T3}
\theta_{L}(q) =&
1
+ (28901376 + a_7)q^7
+ (3108623472 + a_8 - 24a_7 )q^8 + \cdots,\\
\label{eq:T4}
\theta_S(q )=&
\frac{a_8}{2^{24}} q^2
+\Big(\frac{-15 a_8}{2^{21}} - \frac{a_7}{2^{12}}\Big) q^4
+ \Big(136704 + \frac{1767 a_8}{2^{22}} + \frac{3a_7}{2^7}\Big)q^6
+ \cdots,
\end{align}
If $x \in S(L)$ with $(x,x)=2$ then $-x \in S(L)$.
It follows from (\ref{eq:C}) that $B_2=2$ and $B_4=0$.
Hence, we have that
\[
a_7= -15 \cdot 2^{16}, a_8=2^{25}.
\]
Therefore, the theta series of $L$ and $S(L)$ are uniquely
determined.
\end{proof}
Now we start on the proof of Theorem~\ref{thm:main}.
Let $\Lambda$ be an extremal even unimodular lattice in dimension
$72$. Since $\Lambda$ has minimum norm $8$,
there exists a vector $x\in\Lambda$ with $(x,x)=8$.
Fix such a vector $x$.
Put
\[
\Lambda_x^{+}=\{v\in\Lambda\mid(x,v)\equiv0\pmod2\}.
\]
If $(x,y)$ is even for all vectors $y \in \Lambda$
then $\frac{1}{2}x \in \Lambda^*=\Lambda$ and
$(\frac{1}{2}x,\frac{1}{2}x)=2<\min(\Lambda)$, which is a contradiction.
Thus, $\Lambda_x^{+}$ is a sublattice of $\Lambda$ of index $2$,
and there exists a vector $y\in\Lambda$ such
that $(x,y)$ is odd.
Fix such a vector $y$.
Define the lattice
\[
\Gamma_{x,y}=
\Lambda_x^+ \cup \Big(\frac{1}{2}x+y\Big)+\Lambda_x^+.
\]
It is easy to see that $\Gamma_{x,y}$ is an odd unimodular
lattice, which is a neighbor of $\Lambda$.
We show that $\Gamma_{x,y}$ has minimum norm $7$.
Since $\min (\Lambda_x^+) \geq 8$,
it suffices to show that
$(u,u) \geq 7$
for all vectors $u \in (\frac{1}{2}x+y)+\Lambda_x^{+}$.
Let $u=\frac{1}{2}x+y+\alpha$ ($\alpha \in \Lambda_x^{+}$).
Then we have
\begin{equation}\label{eq:1}
\Big(u,\frac{1}{2}x\Big)=\Big(\frac{1}{2}x,\frac{1}{2}x\Big)
+\Big(y,\frac{1}{2}x\Big)+\Big(\alpha,\frac{1}{2}x\Big)
\in \frac{1}{2}+\mathbb{Z}.
\end{equation}
Here, we may assume without loss of generality that
$(u, \frac{1}{2}x) \leq -\frac{1}{2}$.
Then
\[
\Big(u+\frac{1}{2}x, u+\frac{1}{2}x\Big)
=(u,u)+2+2\Big(u,\frac{1}{2}x\Big)
\leq(u,u)+1.
\]
If $u+\frac{1}{2}x$ is the zero vector $\mathbf{0}$ then
$(u,\frac{1}{2}x)=-(\frac{1}{2}x,\frac{1}{2}x)=-2$,
which contradicts (\ref{eq:1}).
Hence, $u+\frac{1}{2}x$ is a nonzero vector in $\Lambda$.
Then we obtain $8 \leq (u,u)+1$.
Therefore, $\Gamma_{x,y}$ is an odd unimodular lattice
with minimum norm $7$, which is a neighbor of $\Lambda$.
It follows that $(\Gamma_{x,y})_0=\Lambda_x^{+}$.
For any vector $\alpha \in \Lambda_x^{+}$,
$(\frac{1}{2}x,\alpha)=\frac{1}{2}(x,\alpha) \in \mathbb{Z}$.
Hence, $\frac{1}{2}x$ is a vector of norm $2$ in $S(\Gamma_{x,y})$.
Therefore, we have Theorem~\ref{thm:main}.
\begin{rem}
A similar argument can be found in~\cite{Venkov}
for dimension $48$.
\end{rem}
By Lemma~\ref{lem:T}, the theta series of
$\Gamma_{x,y}$ and $S(\Gamma_{x,y})$ are uniquely
determined as
(\ref{eq:T1}) and (\ref{eq:T2}), respectively.
\begin{rem}\label{rem:Z8}
The extremal even unimodular lattice in~\cite{Nebe72},
which we denote by $N_{72}$,
contains a sublattice
$\{(x,\mathbf{0},\mathbf{0}),(\mathbf{0},y,\mathbf{0}),(\mathbf{0},\mathbf{0},z) \mid x,y,z \in L_{24}\}$,
where $L_{24}$ is isomorphic to $\sqrt{2}\Lambda_{24}$
and $\Lambda_{24}$ is the Leech lattice.
Since $\Lambda_{24}$ contains many $4$-frames,
$N_{72}$ contains many $8$-frames
(see e.g.\ \cite{BDHO,HMV} for undefined terms in this remark).
Take one of the vectors of an $8$-frame $\mathcal{F}$ as $x$ in the construction of
$\Gamma_{x,y}$.
It follows that $\Gamma_{x,y} \supset \Lambda_x^+ \supset \mathcal{F}$.
Therefore, there is a
self-dual $\mathbb{Z}_8$-code $C_{72}$ of
length $72$ and minimum Euclidean weight $56$
such that $\Gamma_{x,y}$ is isomorphic to
the lattice obtained from $C_{72}$ by Construction A.
A generator matrix of $C_{72}$ can be obtained electronically from
\vspace{15pt}
http://sci.kj.yamagata-u.ac.jp/\~{}mharada/Paper/z8-72-I.txt
\end{rem}
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Wiwi Jury
Wiwi Jury: Denmark's Emmelie de Forest with "Only Teardrops"
This afternoon the Wiwi Jury—our in-house panel of musical unprofessionals—caught a bus to Copenhagen to photograph the Little Mermaid statue and to review Emmelie de Forest's "Only Teardrops". Did she leave us in tears with her beauty and poise? Or did we tell her to dry her tears and write a better song next time? Read on to find out..
Bogdan: If many songs don't sound like Eurovision songs, this one sounds too much like a Eurovision cliché to me, because it is very similar to past contenders. It has "Let Me Try" written all over it if you ask me (the flute, the drums… It'll sound very familiar to Romanian ears). Also, the "We Are the World" lyrics are sort of dated and lame. That being said, Emmelie delivers a good performance and I see it doing well, especially because it's so popular. But not in my book.
Vebooboo: Denmark has turned itself into a Eurovision powerhouse as of late. Sweden may be the most intensely fanatic nation when it comes to Eurovision, but Denmark in the past 5 years has delivered a string of solid performances that have ranked in the top 5. Along comes Emelie, and Denmark finally has itself a winner (or so I hope at least). This act has it all – a simple song that is easy to remember, a folksy flair that will appeal to eastern European peasants, a pretty girl, and sensational staging. And can I just point out the obvious — Emelie is Helena Bonham Carter's bastard child! Not only do they look exactly alike, but they both have the same eery, cooky style. And both are utterly fantabulous. Bottom line, unless Emelie gets a horrible lot when it comes to her Final starting position, I think she will be shedding a lot of teardrops when she wins. So start booking tickets to CPH, peeps. Funny, since half of the Eurovision community are already there this year in light of Malmö's atrocious reputation and utter lack of quality accommodation.
Score: 10/10
Deban: The Danx Melodi Grand Prix was interesting for many reasons. This year had the most submissions in it's five-decade history. Oddly enough, the 10 finalists that were selected were by no means indicative of high quality. In their defence, many of them brought something special to the table, but nobody had the overall package in my view. Mohammed had the best song, Simone had the best vocals, Brinck is a big name, Kate had the most sophisticated staging and Daze rocked a killer chorus. Unfortunately for all the other participants, it was Emelie the unknown starlet who bagged the trophy with Only Teardrops. Although, not a standout contestant or song by any means, she was the only one who successfully ticked all boxes in all categories without compromising. Will this do well in Malmö? It's hard to say, but my prediction is that it will qualify, but stall like her predecessor, Soluna Samay. Perhaps her Royal ties may win a few votes. Personally, I can't help but echo Soluna and think, 'Shouldn't Denmark have known better'?
HK Dick: This has it all and is surely the song to beat. It's original, has obligatory Eurovision drums, great vocals, a memorable chorus and even a bit of Irish flute to complete the package. Maybe I should just leave my suitcase in Copenhagen for 2014…
Alexander: Denmark's Eurovision entries in recent years have usually been either generic or knock-offs. Just as a few examples, Denmark 2006 was a knock-off of Chubby Checker's "The Twist", Denmark 2009 was a generic country song, Denmark 2010 was an ABBA knock-off, Denmark 2011 was accused of being a rip-off of other compositions, and last year's Danish entry was typical American country-pop. While I have enjoyed many of these entries, I have to sincerely applaud Denmark for going with something more unique and original this year. "Only Teardrops" is an excellent entry performed wonderfully by Emmelie. The song has a powerful and addictive chorus, the drumming is great, and the flute is absolutely beautiful. The lyrics also have a nice meaning as well. This should definitely receive a high placing in Malmö!
Mr Häggkvist: Just clean, pure, magical, different and not a commercial pop song… I can write a lot of things about this song, but I think it's all about Emmelie's way of performing and the feeling that she translates. It's just beautiful. At the very least this should make the top 10.
Wiwi: Of all the songs at Eurovision this year, "Only Teardrops" is the one that will play well in the most markets. It has East-West appeal, is radio-friendly and actually has a story to tell. Themes of division and unity apply to love, politics, and the environment—and cut across all cultures. The tribal drumming taps into our most basic instincts and would have stirred a reaction in the first homo sapiens, the mandarins, and Enlightenment thinkers, just as it will in today's bright young things, super models, and fans of pop. This song will age well as it's timeless and has no borders. Mature lyrics with a spiritual delivery. A musical orgasm. A perfect ten.
Wiwi Jury Verdict: 8.07/10
Wiwi Jury Final Ranking: #3 out of 39 contestants
You can see the latest reviews and standings on the Wiwi Jury page. You can also listen to all 39 entries on our contestants page.
Eurovision 2013 ratings
Only Teardrops
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WATCH: Emmelie de Forest says Danish "is not pretty to sing in" as she reflects on Eurovision and "Typical Love Song"
byLauren
What are the notes of "Only Teardrops" that are played on the tin whistle?
What are the notes that are played on the tin whistle for "Only Teardrops" ?
Cheers 😉
Mark Dowd
I can see this winning as easily as Alexander R for Norway in 2009. Douze points!!
Like it, like it, like it. Her breathy inability to hit a big note without her voice breaking – in that way that Dolores O'Riordan of the Cranberries has – is rather endearing and she overcomes it towards the end before it has the chance to irritate. Like the tin whistle, too. Very engaging.
The beautiful blonde for the win? possibly. Just teach her how to breathe while singing first. She's panting worse than an exasperated stallion… Staging is lovely, being barefoot on the floor with the neutral white flowing dress and the "just woke up" hairdo (took a page from 'euphoria' didn't they?). Get her a clean dress for the ESC since I reckon she wiped the danish stage floor sufficiently with the current model. photographs dirty for some reason… The song is actually refreshing, a sort of combination between traditional pop and folk (yes I am a sucker for the flute and… Read more »
NeoGabox
12 points from Venezuela :3
song is very beautiful and girl too
Emmlily
Kind of reminds me of Horehronie…we all know how that turned out.
A good entry with lots of ticks in the right places. It moves away from the clubbing mob and could draw voters who want to listen and appreciate the finer nuances of a song.
MrHäggkvist
Henrique like i always said, it's a matter of tastes, i don't see this entry as Poor, and Norway is good, but it wont connect with people like this!!!
You're absolutely right Dan, this song is simply quite poor, it has just been staged in a very grand, clever way to make people believe it's good. So many have been caught in the trap.
There's not much in this year's lot to be honest, so you'll probably find you have been wasting your time anyway, but I'd advise you to check on Norway. It's miles from all the rest.
Dan, maybe this song doesn't fit your type of style/genre. There are other good songs in different genres 🙂
I haven't listened until now to any of this year's songs, except for this one (being the hot favourite and all), and if this is the best 2013 has to offer, I won't bother and waste my time listening to any more entries…
Certainly deserves to be one of the favorites. A terrific song.
I was rather over this song after the first listen. I don't think it's anything special, it sounds very much a generic eurovision song that I have heard before.
I can apricate the singer and the performance and ill be eager to see it on the night, but interms of a song to listen too, a song to share, I find it dull and slightly cliched of eurovision, but not in a good way, in a dreary way.
I think a lot of the odds in her favour have to do with the fact that Denmark is the one scandinavian country that hasn't won in quite a while than with the real chances this song has. As far as I am concerned this sound would have been very succesful in the '90s (think Norway '95 or Ireland '96 for example) but now it sounds terribly middle of the road, adult-orientated radio kind of song.
I do believe this will prove as overrated as Denmark's song last year and will struggle to get in the top-10, let alone win.
Harriet Krohn
Hmm. The song is nice and all, but I don't quite get why so many people see this as the winner. It's way too harmless for that in my personal opinion, but who knows, maybe Europe will go for middle of the road, by the numbers again this year – there could be worse outcomes. But the flute is totally getting on my nerves by now.
After 2010, 2011 and 2012 being so close to victory I think 2013 could be the year of Denmark. Denmark was robbed in 2012 with Soluna placing a dismal 23rd. Let's bring 2014 Eurovision to Copenhagen!
Interesting fact: Denmark's odds are actually lower than Loreen's last year! Could Denmark peak over 400 points in the final? I hope so!
Yawn. This song is formulaic Eurovision.
*singing* "How many times will you do this, dude?"
After years and years of pure boredom and overrating, Denmark finally sends a decent, interesting song.
This may be in my top 10, but not really quite there. It's greatly arranged and great lyrics, but it may be getting more credit that it actually deserves. Still, very nice of you Denmark for not sending overrated rubbish like lots of years.
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Widerad (died in 1075) was abbot of Fulda Abbey (in the Kingdom of Germany). His dispute over precedence with Bishop Hezilo of Hildesheim contributed to the loss of significant parts of the estates of the abbey.
References
Sources
Year of birth unknown
1075 deaths
Abbots of Fulda
|
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"redpajama_set_name": "RedPajamaWikipedia"
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"redpajama_set_name": "RedPajamaC4"
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| 8,098
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{"url":"https:\/\/hal-ensta.archives-ouvertes.fr\/hal-00968742","text":"# Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions\n\nAbstract : We consider a linear system of PDEs of the form $$\\begin{array}{c} \\begin{array}{rcl} u_{tt} - c\\Delta u_t - \\Delta u = 0 & \\text{in} & \\Omega \\times (0,T)\\\\ u_{tt} + \\partial_n (u+cu_t) - \\Delta_\\Gamma (c \\alpha u_t + u) = 0 & \\text{on} & \\Gamma_1 \\times (0,T)\\\\ u = 0 & \\text{on} & \\Gamma_0 \\times (0,T) \\end{array}\\\\ (u(0),u_t(0),u|_{\\Gamma_1}(0),u_t|_{\\Gamma_1}(0)) \\in \\s{H} \\end{array}$$ on a bounded domain $\\Omega$ with boundary $\\Gamma = \\Gamma_1 \\cup \\Gamma_0$. We show that the system generates a strongly continuous semigroup $T(t)$ which is analytic for $\\alpha > 0$ and of Gevrey class for $\\alpha = 0$. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form $\\|(d\/dt)T(t)\\| \\lesssim |t|^{-1}$ for $\\alpha > 0$, $\\|(d\/dt)T(t)\\| \\lesssim |t|^{-2}$ for $\\alpha = 0$. Moreover, when $\\alpha = 0$ we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to $1\/2$ power of the generator.\nType de document :\nArticle dans une revue\nSemigroup Forum, Springer Verlag, 2013, \u300810.1007\/s00233-013-9534-3\u3009\nListe compl\u00e8te des m\u00e9tadonn\u00e9es\n\nhttps:\/\/hal-ensta.archives-ouvertes.fr\/hal-00968742\nContributeur : Aur\u00e9lien Arnoux <>\nSoumis le : mardi 1 avril 2014 - 14:46:30\nDerni\u00e8re modification le : mercredi 6 d\u00e9cembre 2017 - 16:46:01\n\n### Citation\n\nPhilip Jameson Graber, Irena Lasiecka. Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions. Semigroup Forum, Springer Verlag, 2013, \u300810.1007\/s00233-013-9534-3\u3009. \u3008hal-00968742\u3009\n\n### M\u00e9triques\n\nConsultations de la notice","date":"2019-02-21 10:51:00","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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.569444477558136, \"perplexity\": 1132.1765638303482}, \"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-2019-09\/segments\/1550247503844.68\/warc\/CC-MAIN-20190221091728-20190221113728-00201.warc.gz\"}"}
| null | null |
Dr David Fletcher has a PhD in fish physiology and is an honorary research associate at the University of Liverpool. He has consulted on commercial aquaculture programmes in Southeast Asia, South America and Africa. He is currently working in South Korea on the development of recirculation aquaculture systems (RAS).
For further information contact djfletcher4@aol.com or on 00447970 705495.
A new candidate for commercial aquaculture?
|
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| 7,733
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Drive up to this ideal Family Home or excellent Business Office. Come into the foyer and enjoy the spacious 2975 sq ft sunny upper level. There are 4 bedrooms and 2.5 bathrooms with 1.45 acres on the Shiawassee River. Do you need an elevator? It is wonderful to move things upstairs or downstairs. Do you want a 7 car garage or a heated garage? There are all new appliances, new kitchen cabinets with granite countertops and new washer and dryer. Upstairs is freshly painted neutral with built-ins, skylights and new laminate flooring throughout. The lower level is walk-out/ drive-out with additional rooms and the half bathroom. This 1950's home has updated plumbing, electrical, heating, AC and driveway. It is located close to schools, stores and freeway access. Agent related to seller.
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{
"redpajama_set_name": "RedPajamaC4"
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Q: Hide link title attribute based on screen width – won't unhide I've made an attempt here based on some things I have gleaned, but this is just plain removing it altogether; the title doesn't return back at <= 768px
<script>
if( $(window).width() > 767) {
$('[title]').each( function() {
var $this = $(this);
$this.data('title',$this.attr('title'));
$this.removeAttr('title');
});
}
</script>
See http://jsfiddle.net/2nHxV/
A: So put it back?
var $titles = [];
if( $(window).width()> 767) {
$('[title]').each( function() {
var $this = $(this);
$this.data('title',$this.attr('title'));
$this.removeAttr('title');
$titles.push($this);
});
} else {
$.each($titles, function(index, $this) {
$this.attr('title',$this.data('title'));
});
}
Working demo: http://jsfiddle.net/z3rr9d04/
You also might want to put this logic inside a $(window).on('resize', ...); handler since it'll only be executed once on page load as it stands currently.
A: if ($(window).width() > 767) {
$('[title]').each( function() {
var $this = $(this);
$this.data('title',$this.attr('title'));
$this.removeAttr('title');
});
} else {
// as in above `title` attribute removed and `data-title` added, so now you've
// to loop with data-title
$('[data-title]').each( function() {
var $this = $(this);
$this.data('title',$this.data('title'));
$this.removeAttr('data-title');
});
}
|
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
Probabilistic model checking is an important technique in the analysis of stochastic systems. Given a formal
description of the system in an appropriate modeling formalism and a requirement specification in an
appropriate system of formal logic, the problem is to decide whether the system
satisfies the requirement specification or not. Popular model checking techniques for such systems include numerical
model checking which is expensive but accurate, and statistical model checking wherein accuracy can be traded off for
speed~\cite{courcoubetis88,RV10a,NumVsStat}.
Modeling formalisms for stochastic systems are usually variants of Markov Chains like Discrete and Continuous Time
Markov Chains (DTMC and CTMC respectively)~\cite{CTMC},
Constrained Markov Chains~\cite{CMC} and Probabilistic
Automata~\cite{APA}.
Specification requirement queries are typically formulated in logics like
Probabilistic Computation Tree Logic (PCTL)~\cite{HanssonJ94} and Continuous Stochastic
Language (CSL)~\cite{ModelCheck}.
However, for more complex systems, it is convenient to use more powerful modeling techniques like
Discrete Event Simulation (DES) and agent based simulation, and statistical model checking for analysis~\cite{Younes}.
Indeed, statistical model checkers that can be coupled with discrete event simulators have been designed.
Tools like PLASMA~\cite{PLASMA1,PLASMA2} and
MultiVesta~\cite{MultiVesta},
which builds on the statistical model checker Vesta~\cite{Vesta} and its parallel variant PVesta~\cite{PVesta} are recent
popular examples.
While substantial
work has been done in the model checking domain,
important practical problems can arise due to the quality of the simulation tool itself.
For example, there could be stubs for unwritten modules in the simulation tool, or modules whose correctness is not yet established. It is not
clear how good such a simulator is for the purpose of model checking. Is it, for example, impossible to
verify the satisfaction of a given query on such an implementation? Or is it the case that in spite of lacunas
in the implementation, some model checking queries can still be answered?
In this paper, we demonstrate a simple algorithm towards answering this question.
The central idea originates from the observation that at an abstract level, the problem boils down
to the inability of assigning truth values to atomic propositions in a state of the model.
We demonstrate the approach using appropriately modified DTMC and PCTL. The proposed modifications are as
follows: In the state of a DTMC, an atomic proposition can take the value $Unknown$
(abbreviated ``$?$") in addition to the usual $True$ ($T$) or $False$ ($F$). The syntax and semantics of PCTL are modified so
that a PCTL formula can also take the value ``$?$".
Intuitively, the question that we ask is: Are there a sufficient number of paths in the DTMC that do not evaluate
to ``$?$"? If so, does the modified PCTL query evaluate to $True$ or to $False$ on this DTMC?
Our algorithm answers these questions by invoking the model checking tool twice (PRISM~\cite{prism} in our case) as
a subroutine. This is a crucial advantage, as it means that the model checker itself need not be changed to account for three valued logic.
We illustrate applications of the algorithm with examples of varying complexity. In particular, we demonstrate the
usefulness of our approach with an example program that has a module of unknown correctness.
The paper is arranged as follows. The next section briefly discusses some preliminary notations and definitions, and
relevant previous work done on model checking using three valued logic. Section 3 discusses our modifications
in the definitions of DTMC and PCTL and the modified model checking algorithm. Section 4 discusses implementation details, the examples,
and the results. Section 5 concludes the paper with a brief discussion on future directions.
\section{Preliminaries and Related Work}
This section briefly discusses some basic definitions and terminology that will be used subsequently in the paper.
For details, see~\cite{ModelCheck}.
\subsection{Discrete Time Markov Chains (DTMC)}
A Discrete Time Markov Chain (DTMC) is one in which transition from one state to another occurs in
discrete time steps.
\begin{definition} A DTMC is a tuple $M = (S, \mathbb{P}, s_{init} , AP, L) $ where $S$ is a nonempty
set of states, $\mathbb{P}$ : $S \times S \rightarrow [0, 1] $ is the transition probability function such that for all states
$s\ \in\ S$ :\[\sum_{s'\ \in \ S}\ \mathbb{P}(s,s') = 1\] $s_{init} \in S$ is the initial state, $AP$ is a set of atomic
propositions, and $L : S \rightarrow 2^{AP}$ is a labeling function, which assigns to each state a subset of $AP$
that are true in that state.
\end{definition}
\begin{definition}
A \textbf{path} $\pi$ in a DTMC $M$ is a sequence of states $s_{0} ,s_{1}, s_{2}...$ such that for all $i=0,\ 1,\ 2,...$ ,
$\mathbb{P}(s_{i},s_{i+1})\ >0$. The $(i+1)^{th}$ state in a path $\pi$ is written as $\pi[i]$. $Path(s)$ denotes the
set of all infinite paths which start from a state $s$ in the model, $M$. $Paths_{fin}(s)$ is the set of all
finite paths starting from state $s$.
\end{definition}
\begin{definition}
A \textbf{cylinder set}, $C(\omega)$ is the set of infinite paths that have a common finite prefix $\omega$ of length $n$. Let $\Sigma_{Path(s)}$
be the smallest $\sigma$-algebra generated by $\{C(\omega)\ |\ \omega \in Paths_{fin}(s)\}$. Then, we can define $\mu$ on the
measurable space $(Path(s)$,$\Sigma_{Path(s)})$ as the unique probability measure such that:
\[\mu(C(\omega)) = \prod_{i=0}^{n-1} \mathbb{P}(s_{i},s_{i+1})\]
\end{definition}
\subsection{Probabilistic Computation Tree Logic (PCTL)}
Probabilistic Computation Tree Logic (PCTL), an extension of Computation Tree Logic (CTL), was introduced by Hansson and
Johnson \cite{HanssonJ94} for analyzing discrete time probabilistic systems.\\
\\
\textbf{Syntax of PCTL:}
\[ \Phi\ ::=\ T\ |\ a\ |\ \Phi_{1} \land \Phi_{2} \ |\ \lnot\Phi \ |\ \mathbb{P}_{\bowtie \theta} [\psi]\]
\[ \psi\ ::=\ X \Phi \ | \ \Phi_{1} U\ \Phi_{2} \ | \ \Phi_{1} U^{\leq k}\ \Phi_{2}\]
where $\Phi$, $\Phi_{1}$, and $\Phi_{2}$ are state formulas, $\psi$ is a path formula, \textit{a} is an atomic proposition,
$\theta \in [0,1]$ is the probability constraint, $\bowtie\ \in \{ <, \ >,\ \leq,\ \geq \}$ represents the set of operators,
and $k\ \in\ \mathbb{N}\ $ is the time bound. The $ X$, $U$, and $U^{\leq k} $ operators are called \textit{Next}, \textit{Until}
and \textit{Bounded Until} respectively. \\
\\
\textbf{Semantics of PCTL:}
Let $M : (S,\mathbb{P}, s_{init},AP,L) $ be a Discrete Time Markov Chain. Let
$s \ \in\ S$, $a\ \in\ AP$, and $\Phi,\ \Phi_{1},\ \Phi_{2}$ be PCTL state formulas, and $\psi$ be a PCTL path formula.
Then, $\Phi$ is said to be satisfied in state $s$ i.e. $(s,\Phi)=T$ if:
\begin{center}
\begin{tabular}{ccc}
$ (s,T)\ =\ T $, \\
$ (s,a)\ =\ T $ & iff & $a\ \in L(s)$ , \\
$ (s,\lnot\Phi)\ =\ T $ & iff & $(s,\Phi) = F $ , \\
$ (s,\Phi_{1}\land\Phi_{2})\ =\ T $ & iff & $ (s,\Phi_{1})=T\ \land (s,\Phi_{2})=T $ ,\\
$ (s,\mathbb{P}_{\bowtie\ \theta} (\psi))\ =\ T $ & iff & $\ \ \mu(\{\pi \in Path(s)\ |\ (\pi,\psi)\ =\ T \})\ \bowtie \theta$
\end{tabular}
\end{center}
If $\pi\ = s_{0}\ s_{1}\ s_{2}...\ $ is a path in $Path(s_{0})$ then \(\Pr( (s,\psi) =T)\ =\ \mu\{\ \pi \in Path(s)\ |\ (\pi,\psi)=T\}\)
i.e. the probability of the set of paths starting from $s$ which satisfy the path formula $\psi$. The last satisfaction relation
for a state formula thus states that the probability that $\psi$ is true on paths starting at $s$ satisfies
$\bowtie \theta$. A path formula $\psi$ is said to be satisfied for path $\pi$ i.e. $(\pi,\psi)=T$ if:
\begin{center}
\begin{tabular}{ccc}
$ (\pi, X \Phi)=T$ & iff & $(\pi[1],\Phi)=T $ , \\
$ (\pi, (\Phi_{1}\ U\ \Phi_{2}))=T$ & iff & [$\exists i \geq 0 \ |\ (\pi[i],\Phi_{2})=T ]\ \land [\ \forall j < i, (\pi[j],\Phi_{1})=T\ ]$ , \\
$ (\pi,(\Phi_{1}\ U^{\leq k}\ \Phi_{2} ))=T $ & iff & $\ [\exists\ i \leq k \ |\ (\pi[i],\Phi_{2})=T\ ]\ \land [\ \forall j < i, (\pi[j],\Phi_{1})=T\ ]. $
\end{tabular}
\end{center}
\noindent \textbf{Problem Statement for PCTL model checking:} Given a DTMC $M$, decide whether a PCTL formula $\Phi$
evaluates to $T$ or $F$ on $M$.
\subsection{Three-valued logic and model checking}
Multi-valued logics have been comprehensively investigated in the past few decades. In addition to having a rich
theory, they have also found practical applications. Depending on the problem, classical binary language can be extended
to include additional truth values.
For example, an additional truth value can be used to represent inconsistent and incomplete information.
An application might also demand that we use two different values to denote inconsistent and incomplete information separately.
In this work, we will use three valued logic. We expand the logic associated with atomic propositions in the state of a DTMC to
include $Unknown$, denoted by the question mark symbol ``$?$". In what follows, we will use $Unknown$ and $?$ interchangeably. A number
of different truth tables have been
designed for three valued logics
\cite{mal93many,putnam57three,manyval}. The three valued logic used in this work has all the properties of a Quasi-Boolean
lattice and the
truth tables for logic operations are described in Tables \ref{and}, \ref{or} and \ref{not}.
\begin{table}[h]
\centering
\begin{minipage}{0.3\textwidth}
\centering
\begin{tabular}{c| c c c}
$\land$ & T & ? & F \\
\hline
T & T & ? & F \\
? & ? & ? & F \\
F & F & F & F
\end{tabular}
\caption{AND operator }
\label{and}
\end{minipage}%
\hfill
\begin{minipage}{0.3\textwidth}
\centering
\begin{tabular}{c| c c c}
$\lor$ & T & ? & F \\
\hline
T & T & T & T \\
? & T & ? & ? \\
F & T & ? & F
\end{tabular}
\caption{OR operator }
\label{or}
\end{minipage}%
\hfill
\begin{minipage}{0.3\textwidth}
\centering
\begin{tabular}{c| c}
$\lnot$ & \\
\hline
T & F \\
? & ? \\
F & T
\end{tabular}
\caption{NOT operator}
\label{not}
\end{minipage}
\end{table}
Indeed, three valued logic has been used in the past for model checking in non-probabilistic settings--for example,
LTL~\cite{bruns99model,bruns00gen,genLTL} and CTL~\cite{chechik00,chechik01model}. Chechik et al.~\cite{chechik00,chechik01model}
have used three valued
logic for atomic propositions as well as for the transition functions. In case of transition functions, the \textit{True}
and \textit{False} values denote the presence or absence of a transition between two states respectively.
The third truth value represents the lack of information about the transition.
Three valued logic has also been used with numerical model checking of probabilistic systems, but with a different
motivation and solution. To overcome
the problem of state-space explosion in numerical model checking, two or more states of a model are combined, yielding
an \emph{abstract} Markov chain.
However, over-abstraction often leads to a significant loss of information. Three valued logics have been
associated with abstract probabilistic systems wherein an $Unknown$ value represents loss of information, indicating
that the level of abstraction should be decreased. Model checking of an abstract Markov chain
is often done by reducing
it to a Markov decision process and then using model checking techniques
for Markov decision processes. For more details, please see~\cite{dontknow,huththree,klinkthree}.
\section{Problem Statement and Solution}
As mentioned earlier, the aim of this work is to study the effect of an information being unknown,
in asserting whether a given property is satisfied in the model
or not. To perform model checking on such three valued systems, both DTMC and PCTL
need to be modified. While in case of DTMC the labeling function $L$ is modified,
the semantics are altered for PCTL.
Intuitively, in our approach, the model checker aims to identify if there are too many paths
in a model wherein it is not known whether a property will be satisfied or not. Thus,
the model checker first evaluates whether the property is satisfied in the model and
if not, it examines the reason behind the lack of satisfaction.
In the coming subsections, we discuss the modifications in DTMC and PCTL, and the problem statement.
\subsection{DTMC with Question Marks}
A Discrete Time Markov Chain with question marks (qDTMC) is a tuple $M : (S,\mathbb{P},s_{init},AP,L)$ with a finite
non-empty set of states $S$, a transition probability function $\mathbb{P}:S\times S \rightarrow [0,1]$ such
that for all states $s\ \in\ S$ : \(\sum_{s'\ \in \ S}\ \mathbb{P}(s,s') = 1\), the initial state $s_{init} \in S$,
a set of atomic propositions $AP$ and labeling function $L:S\times AP\rightarrow \{T,F,\textrm{{\bf ?}}\}$.
\subsection{PCTL with Question Marks}
The syntax of PCTL in the context of three valued logic (hereafter referred to as qPCTL for convenience) remains
the same. The operators ($\land$, $\lor$, $\lnot$) and operands ($T,F,?$) however, are as defined in
Tables 1, 2 and 3 for three valued logic. Therefore, the structure of the queries remains unchanged.
However, the semantics need to be modified:
\noindent \textbf{Semantics:}
Let $M : (S,\mathbb{P},s_{init},AP,L) $ be a qDTMC model.
Let $s \ \in\ S$, $a\ \in\ AP$, $\Phi$, $\Phi_{1}$, $\Phi_{2}$ be qPCTL state formulas, and $\psi$ be a qPCTL
path formula. Then, semantics for $\Phi$ are as stated below:
\begin{center}
$(s,T)\ =\ T$ , \\
$(s,F)\ =\ F$ , \\
$(s,?)\ =\ ?$ .
\end{center}
\begin{equation*}
(s, a) = \left\{
\begin{array}{ccc}
T & \text{iff} & L(s,a)=T ,\\
F & \text{iff} & L(s,a)=F , \\
? & \text{iff} & L(s,a)=? .
\end{array} \right.
\end{equation*}
\begin{equation*}
(s, \lnot\Phi) = \left\{
\begin{array}{ccc}
T & \text{iff} & (s,\Phi)=F,\\
F & \text{iff} & (s,\Phi)=T,\\
? & \text{iff} & (s,\Phi)=? .
\end{array} \right.
\end{equation*}
\begin{equation*}
(s, \Phi_{1}\land\Phi_{2}) = \left\{
\begin{array}{ccc}
T & \text{iff} & (s,\Phi_{1})=T\ \land (s,\Phi_{2})=T ,\\
F & \text{iff} & (s,\Phi_{1})=F\ \lor (s,\Phi_{2})= F ,\\
? & & otherwise.
\end{array} \right.
\end{equation*}
The intuition behind the above definitions follows directly from three valued logic.
The semantics of the probabilistic state formula are defined as follows:
\begin{equation*}
(s, Pr_{\geq\theta}(\psi)) = \left\{
\begin{array}{rl}
T & \ \ \text{if } \mu\{\pi\in Path(s):(\pi,\psi)=T\} \geq \theta,\\
F & \ \ \text{if } \mu\{\pi\in Path(s):(\pi,\psi)=F\} \geq 1-\theta,\\
? & \ \ \text{if } (\mu\{\pi\in Path(s):(\pi,\psi)=T\} < \theta) \\
& \ \ \land (\mu\{\pi\in Path(s):(\pi,\psi)=F\} < 1-\theta) .
\end{array} \right.
\end{equation*}
We first note that every formula must evaluate to one of $T,F,$ or $?$.
The above definition follows from the intuition that $(s, Pr_{\geq\theta}(\psi))$ evaluates to $T$ if
at least $\theta$ fraction of the paths evaluate $\psi$ to $T$. Also, if $1-\theta$ (or more) fraction
of the paths evaluate to $F$, then there are sufficient number of paths to evaluate $(s, Pr_{\geq\theta}(\psi))$ to $F$.
However, if there does not exist enough paths to decisively tell whether or not the property $\psi$ holds, then
$(s, Pr_{\geq\theta}(\psi))$ evaluates to $?$. We now turn to the semantics of the path formulas.
\begin{equation*}
(\pi, X\Phi) = \left\{
\begin{array}{rl}
T & \text{if } (\pi[1],\Phi)=T ,\\
F & \text{if } (\pi[1],\Phi)=F ,\\
? & \text{if } (\pi[1],\Phi)=? .
\end{array} \right.
\end{equation*}
\begin{equation*}
(\pi, \Phi_1 U^{\leq k} \Phi_2) = \left\{
\begin{array}{rl}
T & \text{if } \exists i\leq k: (\pi[i],\Phi_2)=T \wedge \forall i'< i: (\pi[i'],\Phi_1)=T\\
F & \text{if } (\forall i\leq k:( \pi[i],\Phi_2)=F) \vee (\exists i\leq k : (\pi[i],\Phi_2)\neq F \wedge
\\ &\qquad \qquad\qquad\qquad\qquad\qquad\ \ \ \exists i'< i :(\pi[i'],\Phi_1)=F), \\
? & \text{otherwise}.
\end{array} \right.
\end{equation*}
\begin{equation*}
(\pi, \Phi_1 U \Phi_2) = \left\{
\begin{array}{rl}
T & \text{if } \exists i: (\pi[i],\Phi_2)=T \wedge \forall i'< i: (\pi[i'],\Phi_1)=T\\
F & \text{if } (\forall i:( \pi[i],\Phi_2)=F) \vee (\exists i : (\pi[i],\Phi_2)\neq F \wedge
\\ &\qquad \qquad\qquad\qquad\qquad \exists i'< i :(\pi[i'],\Phi_1)=F), \\
? & \text{otherwise}.
\end{array} \right.
\end{equation*}
First, we note that $(\pi,X\Phi)$ evaluates to $T,F$ or $?$ depending on whether $\Phi$ is $T,F$ or $?$ in
$\pi[1]$ that is, in the next state.
The \textit{bounded until} formulas in qPCTL are simple extension of
the corresponding formulas in the standard PTCL. For example, a \textit{bounded until} formula
$\Phi_1U^{\leq k}\Phi_2$ evaluates to $?$ if one of the following happens:
\begin{itemize}
\item $\Phi_2$ is $?$ in all the states up to $k$.
\item $\Phi_2$ is $?$ in at least one state and is never $T$ in any of the states along the path up to $k$,
but $\Phi_1$ is never $F$.
\item $\Phi_2$ is $T$ for some $i\leq k$ and $\Phi_1$ is $?$ in at least one state but never $F$ in any of
the states upto $i$.
\end{itemize}
\noindent \textit{Unbounded until} is also extended similarly for qPCTL. We are now in a
position to formally define the problem. \\
\noindent \textbf{Problem Statement:} Given a qDTMC $M$, and a qPCTL formula $\Phi$, decide whether $(s_{init},\Phi)$
evaluates to $T$, $F$ or $?$.
\subsection{The Algorithm}
As mentioned earlier, our algorithm for model checking qDTMC using qPCTL uses binary model
checkers as a subroutine. The algorithm involves modifying the input qDTMC suitably
before subjecting it to binary model checking queries. The central idea behind the algorithm is to use
these modifications to filter the three truth values successively, using only binary valued model checkers.
In what follows, for ease of exposition, we denote the outcomes of the binary model checker by $T'$ and
$F'$. Algorithm \ref{qMC} describes our approach.
\begin{algorithm}
\caption{\textbf{qMC}}
\label{qMC}
\begin{algorithmic}
\vspace{2mm}
\STATE INPUT: A qDTMC $M$ and a qPCTL formula $\Phi$.
\IF[\emph{Conditional 1}]{$\Phi$ contains an AP of form $\lnot a$}
\STATE Add new AP $a'= \lnot a$ in $M$.
\STATE Replace all instances of $\lnot a$ with $a'$ in $\Phi$.
\ENDIF
\STATE Set $?$ to $F$ in qDTMC $M$ to obtain binary DTMC $M^{(1)}$.
\IF[\emph{Conditional 2}]{BINARY\_MC$(M^{(1)}, \Phi) =T'$}
\RETURN $T$
\ELSE
\STATE Set $?$ to $T$ in qDTMC $M$ to obtain binary DTMC $M^{(2)}$.
\IF[\emph{Conditional 3}] {BINARY\_MC$(M^{(2)},\Phi) = F'$}
\RETURN $F$
\ELSE
\RETURN $?$
\ENDIF
\ENDIF
\end{algorithmic}
\end{algorithm}
Intuitively, the algorithm proceeds as follows.
In each phase of the algorithm, $?$ truth values in the qDTMC $M$ are identified with either $T$ or $F$.
Let $\Phi$ be a qPCTL formula consisting of an atomic proposition $a$. Then, all instances of the atomic
proposition $a$ with truth value $?$ are set to $F$ in the first phase of the algorithm. On the other hand,
the $?$ truth values are set to $T$ in the second phase of algorithm. While the first phase determines if
sufficient paths with $a=T$ exist in the model to verify $\Phi$, the second phase checks for the paths
with $a=F$ to disprove $\Phi$.
For atomic propositions of the form $\lnot a$,
the algorithm should search for paths with $a=F$ and $a=T$ in first and second phases respectively. Thus,
the sequence of mapping needs to be reversed for atomic propositions of the form $\lnot a$. To maintain
uniformity in the algorithm, for each atomic proposition of the form $\lnot a$ in $\Phi$, a new atomic
proposition $a' = \lnot a$ is added in $M$. Similarly, each instance of $\lnot a$ is replaced with $a'$
in $\Phi$. Note that new atomic propositions need to be added only
when a negated atomic proposition exists in $\Phi$.
Figures \ref{example_old} and \ref{example_new} show a qDTMC $M_{1}$
in which a new atomic proposition $t = \lnot p$ is added. The modified models in the two phases of the Algorithm
\ref{qMC} are given in figures \ref{small_step1} and \ref{small_step2}.
\begin{figure}
\centering
\begin{subfigure}[Example qDTMC $M_1$]{%
\label{example_old}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.05cm, thin, state/.style={circle,draw,font=\sffamily\tiny,initial text=,minimum size=0.75cm, inner sep =0mm}]
\node[initial,state,label=270 : {\tiny $s_{init}$}] (0) {$\lnot p q?$};
\node[state] (2) [right of=0] {$\ p\ q $};
\node[state] (1) [above of=2] {$p? \lnot q$};
\node[state] (3) [below of=2] {$\lnot p q?$};
\node[state] (4) [right of=2] {$p?q?$};
\node[state] (5) [right of=3] {$\ p \lnot q$};
\node[state] (6) [right of=1] {$\ p\ q $};
\path [every node/.style={font=\sffamily\tiny}]
(0) edge [bend left] node {$0.3$} (1)
edge node {$0.2$} (2)
edge node {$0.5$} (3)
(1) edge node {$0.1$} (0)
edge [bend left] node {$0.35$} (2)
edge node {$0.4$} (4)
edge node {$0.25$} (6)
(2) edge node {$0.1$} (1)
edge [bend left] node {$0.1$} (3)
edge node {$0.8$} (4)
(3) edge node {$0.5$} (2)
edge [bend right] node {$0.5$} (4)
(4) edge node {$0.33$} (6)
edge node {$0.67$} (5)
(5) edge [loop below] node {$0.9$} (5)
edge node {$0.1$} (3)
(6) edge [loop above] node { $1$ } (6);
\end{tikzpicture}
}
\end{subfigure}
\begin{subfigure}[Example qDTMC $M_1$ with new atomic proposition $t$, after Conditional 1 in Algorithm qMC]{
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.05cm, thin, state/.style={circle,draw,font=\sffamily\tiny,initial text=,minimum size=0.75cm, inner sep =0mm}]
\label{example_new}
\node[initial,state,label=270 : {\tiny $s_{init}$}] (0) {$\lnot p q?t$};
\node[state] (2) [right of=0] {$\ p\ q \lnot t $};
\node[state] (1) [above of=2] {$p? \lnot qt?$};
\node[state] (3) [below of=2] {$\lnot p q?t$};
\node[state] (4) [right of=2] {$p?q?t?$};
\node[state] (5) [right of=3] {$\ p \lnot q\lnot t$};
\node[state] (6) [right of=1] {$\ p\ q \lnot t\ $};
\path [every node/.style={font=\sffamily\tiny}]
(0) edge [bend left] node {$0.3$} (1)
edge node {$0.2$} (2)
edge node {$0.5$} (3)
(1) edge node {$0.1$} (0)
edge [bend left] node {$0.35$} (2)
edge node {$0.4$} (4)
edge node {$0.25$} (6)
(2) edge node {$0.1$} (1)
edge [bend left] node {$0.1$} (3)
edge node {$0.8$} (4)
(3) edge node {$0.5$} (2)
edge [bend right] node {$0.5$} (4)
(4) edge node {$0.33$} (6)
edge node {$0.67$} (5)
(5) edge [loop below] node {$0.9$} (5)
edge node {$0.1$} (3)
(6) edge [loop above] node { $1$ } (6);
\end{tikzpicture}
}\end{subfigure}
\caption{First step of Algorithm qMC} \label{example}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[Modified model $M_1^{(1)}$ in step 1.]{%
\label{small_step1}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.05cm, thin, state/.style={circle,draw,font=\sffamily\tiny,initial text=,minimum size=0.75cm, inner sep =0mm}]
\node[initial,state,label=270 : {\tiny $s_{init}$}] (0) {$\lnot p \lnot qt$};
\node[state] (2) [right of=0] {$\ p\ q \lnot t $};
\node[state] (1) [above of=2] {$\lnot p \lnot q\lnot t$};
\node[state] (3) [below of=2] {$\lnot p \lnot q\ t$};
\node[state] (4) [right of=2] {$\lnot p \lnot q \lnot t$};
\node[state] (5) [right of=3] {$\ p \lnot q\lnot t$};
\node[state] (6) [right of=1] {$\ p\ q \lnot t\ $};
\path [every node/.style={font=\sffamily\tiny}]
(0) edge [bend left] node {$0.3$} (1)
edge node {$0.2$} (2)
edge node {$0.5$} (3)
(1) edge node {$0.1$} (0)
edge [bend left] node {$0.35$} (2)
edge node {$0.4$} (4)
edge node {$0.25$} (6)
(2) edge node {$0.1$} (1)
edge [bend left] node {$0.1$} (3)
edge node {$0.8$} (4)
(3) edge node {$0.5$} (2)
edge [bend right] node {$0.5$} (4)
(4) edge node {$0.33$} (6)
edge node {$0.67$} (5)
(5) edge [loop below] node {$0.9$} (5)
edge node {$0.1$} (3)
(6) edge [loop above] node { $1$ } (6);
\end{tikzpicture}
}
\end{subfigure}
\begin{subfigure}[Modified model $M_1^{(2)}$ in step 2.]{%
\label{small_step2}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.05cm, thin, state/.style={circle,draw,font=\sffamily\tiny,initial text=,minimum size=0.75cm, inner sep =0mm}]
\node[initial,state,label=270 : {\tiny $s_{init}$}] (0) {$\lnot p q\ t$};
\node[state] (2) [right of=0] {$\ p\ q \lnot t $};
\node[state] (1) [above of=2] {$p\ \lnot qt$};
\node[state] (3) [below of=2] {$\lnot p q\ t$};
\node[state] (4) [right of=2] {$p\ q\ t$};
\node[state] (5) [right of=3] {$\ p \lnot q\lnot t$};
\node[state] (6) [right of=1] {$\ p\ q \lnot t\ $};
\path [every node/.style={font=\sffamily\tiny}]
(0) edge [bend left] node {$0.3$} (1)
edge node {$0.2$} (2)
edge node {$0.5$} (3)
(1) edge node {$0.1$} (0)
edge [bend left] node {$0.35$} (2)
edge node {$0.4$} (4)
edge node {$0.25$} (6)
(2) edge node {$0.1$} (1)
edge [bend left] node {$0.1$} (3)
edge node {$0.8$} (4)
(3) edge node {$0.5$} (2)
edge [bend right] node {$0.5$} (4)
(4) edge node {$0.33$} (6)
edge node {$0.67$} (5)
(5) edge [loop below] node {$0.9$} (5)
edge node {$0.1$} (3)
(6) edge [loop above] node { $1$ } (6);
\end{tikzpicture}
}\end{subfigure}
\caption{Two step modification of qDTMC $M_1$ in Fig. \ref{example_new}.}
\end{figure}
\begin{theorem}
The algorithm qMC solves the model checking problem for qPCTL: for a qDTMC $M$ and a qPCTL formula $\Phi$,
\begin{itemize}
\item qMC($M,\Phi$)=T (alt., F or ?) iff $(s_{init},\Phi)$=T (resp., F or ?)
\end{itemize}
\end{theorem}
\begin{proof}
The algorithm qMC solves the model checking problem for qPCTL, if it matches the semantics of qPCTL for all
state formulas. Recall that there are two types of state formulas: non-probabilistic and probabilistic.
The proof that the algorithm works for non-probabilistic state formulas is straightforward and omitted here.
If
$\psi$ is a path formula in a probabilistic state formula then:
\[
qMC(M,Pr_{\geq\theta}(\psi))=\textrm{T (alt., F or ?) iff } (s_{init},Pr_{\geq\theta}(\psi))\textrm{=T (resp., F or ?)}
\]
Recall that there are three path formulas : $X\Phi$, $\Phi_1U^{\leq k} \Phi_2$ and $\Phi_1U\Phi_2$. The state formulas
in these path formulas could in turn also be either non-probabilistic or probabilistic. This allows the algorithm
to verify properties with both nested and non-nested path formulas. We now prove that the algorithm qMC matches the semantics
of probabilistic state formulas.
\begin{itemize}
\item $Pr_{\geq\theta}(X\Phi)$ :
The correctness of \textit{next} operator can be proved through the following claims:
\begin{claim}
In the second conditional of Algorithm $qMC$, the formula $Pr_{\geq\theta}(X\Phi)$ evaluates to $T$ in the qDTMC $M$
if and only if it evaluates to $T'$ in the binary DTMC $M^{(1)}$.
\end{claim}
\begin{proof}
The state formula $Pr_{\geq\theta}(X\Phi)$ evaluates to $T$ in the qDTMC $M$, if there are at least $\theta$ fraction of
paths that evaluate $X\Phi$ to $T$ in $M$. The mapping of the truth values while constructing the binary DTMC $M^{(1)}$
in the second conditional does not disturb $T$. So, there continues to be at least $\theta$ fraction of the paths evaluating $ X\Phi$ to
$T$ in $M^{(1)}$. Therefore, $Pr_{\geq\theta}(X\Phi)$ evaluates to $T$ in the qDTMC $M$, if and only if it evaluates to $T'$
in the binary DTMC $M^{(1)}$.
\end{proof}
\begin{claim}
In the third conditional of Algorithm $qMC$, the formula $Pr_{\geq\theta}(X\Phi)$ evaluates to $F$ in the qDTMC $M$
if and only if it evaluates to $F'$ in the binary DTMC $M^{(2)}$.
\end{claim}
\begin{proof}
The state formula $Pr_{\geq\theta}(X\Phi)$ should evaluate to $F$ in the qDTMC $M$, if there are at least $1-\theta$
fraction of paths that evaluate $X\Phi$ to $F$ in $M$. The construction of the binary
DTMC $M^{(2)}$ does not disturb $F$ in the third conditional. If the binary model checker returns $F'$ for
$Pr_{\geq\theta}(X\Phi)$ in spite of $?$ being identified with $T$, it implies that more than $1-\theta$ fraction of
the paths evaluate to $F$ in both $M$ and $M^{(2)}$. Thus, $Pr_{\geq\theta}(X\Phi)$ evaluates to $F$ in the qDTMC $M$,
if and only if it evaluates to $F'$ in the binary DTMC $M^{(2)}$.
\end{proof}
\begin{claim}
In the third conditional of Algorithm $qMC$, the formula $Pr_{\geq\theta}(X\Phi)$ evaluates to $?$ in the qDTMC $M$
if and only if it evaluates to $T'$ in the binary DTMC $M^{(2)}$.
\end{claim}
\begin{proof}
In the third conditional, truth values $T$ and $?$ in the qDTMC $M$ are mapped to $T$ in the binary DTMC $M^{(2)}$. If
the formula $Pr_{\geq\theta}(X\Phi)$ evaluates to $T'$ in $M^{(2)}$, then it means that at least $\theta$ fraction of paths
evaluate $X\Phi$ to $T'$ in $M^{(2)}$. This fraction is the sum of fractions of the paths in which
$X\Phi$ is either $T$ or $?$ in $M$.
Let the fractions of paths that evaluate $X\Phi$ to $T$ and $F$ in $M$ be $p$ and $q$ respectively.
Then reaching the third conditional implies that $p < \theta$.
Since the formula $Pr_{\geq\theta}(X\Phi)$ evaluated to $T'$ in $M^{(2)}$, it is clear that $q < 1-\theta$. Therefore,
there exists a fraction of paths in $M$ for which $X \Phi$ evaluates to $?$, such that the formula $Pr_{\geq\theta}(X\Phi)$
can neither be $T$ nor $F$ in the qDTMC $M$. Hence, the formula is evaluated to $?$ in the qDTMC $M$. Thus, an output
of $T'$ by the binary model checker is correctly interpreted as $?$ in the qDTMC $M$.
\end{proof}
\begin{example}
Given a qDTMC $M_{1}$ in figure \ref{example_new}, a property $\phi=\Pr_{\geq\theta}(X\ p)$
can be verified for different values of $\theta$. For instance, if $\theta = 0.1$, then binary DTMC $M_1^{(1)}$,
given in figure \ref{small_step1}, has atleast $\theta$ fraction of the paths that evaluate
to $T'$. Thus, the property evaluates to $T$ in qDTMC $M_1$.
However, if $\theta = 0.8$, then $M_1^{(1)}$ does not have sufficient fraction of paths evaluating to $T$. The algorithm
now modifies $M_1$ to $M_{1}^{(2)}$, given in figure \ref{small_step2}, to check if sufficient number of paths that
disprove $\phi$ exist. Since more than $1-\theta$ fraction of paths in $M_{1}^{(2)}$ evaluate to $F'$,
for $\theta = 0.8$, the property is evaluated to $F$ in qDTMC $M_1$.
Similarly, for $\theta = 0.4$, $M_1^{(1)}$ does not have sufficient fraction of paths evaluating to $T$. But, $M_{1}^{(2)}$
also does not have sufficient paths evaluating to $F'$. In such a case, the
property $\Phi$ is evaluated to $?$ in qDTMC $M_1$ due to lack of
sufficient conclusive paths.
\end{example}
\item $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ :
The correctness of \textit{bounded until} operator is similarly proved using following claims:
\begin{claim}
In the second conditional, $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ evaluates to $T$ in the qDTMC $M$ if and only if
it evaluates to $T'$ in the binary DTMC $M^{(1)}$.
\end{claim}
\begin{proof}
The formula $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ is evaluated to $T$ in the qDTMC $M$, if there are at least
$\theta$ fraction of paths that evaluate $\Phi_1U^{\leq k} \Phi_2$ to $T$ in $M$. Also, the truth value mapping during the
construction of the binary DTMC $M^{(1)}$ does not alter $T$. Therefore, if $\Phi_1$ holds on a path until $\Phi_2$ becomes
true in the qDTMC $M$, it will continue to remain that way in the binary DTMC $M^{(1)}$.
Hence, $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ is evaluated to $T$ in the qDTMC $M$ if and only if it evaluated to $T'$ in the
binary DTMC $M^{(1)}$.
\end{proof}
\begin{claim}
In the third conditional, $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ evaluates to $F$ in the qDTMC $M$ if and only if
it evaluates to $F'$ in the binary DTMC $M^{(2)}$.
\end{claim}
\begin{proof}
In the qDTMC $M$, a path can evaluate $\Phi_1U^{\leq k}\Phi_2 $ to $F$ if one of the following happens:
\begin{itemize}
\item $\Phi_2$ is $F$ all along the path up to the $k^{th}$ state.
\item $\Phi_2$ is not $F$ for some $i\leq k$, but $\Phi_1$ is $F$ for some $j<i$.
\end{itemize}
Recall that the state formula $Pr_{\geq\theta}(\Phi_1U^{\leq k}\Phi_2 )$ evaluates to $F$ in the qDTMC $M$, if there are
at least $1-\theta$ fraction of paths that evaluate $\Phi_1U^{\leq k}\Phi_2$ to $F$ in $M$. The mapping of truth values
does not change $F$ in third conditional. So the paths that evaluated the formula $\Phi_1U^{\leq k}\Phi_2$ to $F$ in
the qDTMC $M$ would continue to do so in binary DTMC $M^{(2)}$. So, there are at least $1-\theta$ fraction of the
paths that evaluate $\Phi_1U^{\leq k} \Phi_2$ to $F$ in the qDTMC, if and only if at least $1-\theta$ fraction of the
paths evaluate the formula to $F'$ in the binary DTMC $M^{(2)}$. Thus, $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$
evaluates to $F$ in the qDTMC $M$ if and only if it evaluates to $F'$ in the binary DTMC $M^{(2)}$.
\end{proof}
\begin{claim}
In the third conditional, the formula $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ evaluates to $?$ in the qDTMC $M$ if
and only if it evaluates to $T'$ in the binary DTMC $M^{(2)}$.
\end{claim}
\begin{proof}
Recall that the formula $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ evaluates to $?$ in qDTMC $M$, if one of
the following occurs :
\begin{itemize}
\item $\Phi_2$ is $?$ in all the states up to $k$.
\item $\Phi_2$ is $?$ in at least one state and is never $T$ in any of the states along the path up to $k$,
but $\Phi_1$ is never $F$.
\item $\Phi_2$ is $T$ for some $i\leq k$ and $\Phi_1$ is $?$ in at least one state but never $F$ in any of
the states up to $i$.
\end{itemize}
For the third conditional, the truth values $T$ and $?$ in the qDTMC $M$ are mapped to $T$ in the binary DTMC $M^{(2)}$.
So, in all the above cases, the binary model checker outputs $T'$, because $?$ is mapped to $T$ in binary DTMC $M^{(2)}$.
If the binary model checker returns $T'$ at the third conditional, then there are at least $\theta$ fraction of paths in
the qDTMC $M$ that evaluate to either $T$ or $?$. If the fraction of paths that evaluate $\Phi_1U^{\leq k} \Phi_2$ to
$T$ in $M$ is $p$, then from the second conditional, $p < \theta$. Further, let the fraction of paths that evaluate $X\Phi$
to $F$ in $M$ be $q$. If the formula $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ evaluates to $T'$ in $M^{(2)}$, it is
clear that $q < 1-\theta$. It can then easily be concluded that there do not exist sufficient number of conclusive paths
(either $T$ or $F$)
in the qDTMC $M$, and the formula $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ evaluates to $?$.
\end{proof}
\item $Pr_{\geq\theta}(\Phi_1U^{\leq k} \Phi_2)$ : The proof for \textit{unbounded until} operator is a simple extension
of the \textit{bounded until} operator.
\end{itemize}
\qed
\end{proof}
\begin{remark}
If the state formula occurs in negated form as $\lnot\Phi$, then we use $\Phi'=\lnot\Phi$ in the $qMC$
algorithm, proceed as usual and negate the final answer as per the semantics of three valued logic.
\end{remark}
\begin{remark}
If the probabilistic query is of the type $Pr_{<\theta}(\psi)$, we use the identity
$Pr_{<\theta}(\psi)=\lnot Pr_{\geq \theta} (\psi)$ and proceed as usual.
\end{remark}
\begin{remark}
The Algorithm \ref{qMC} discussed here is symmetric in the sense that the result would not change if the order
of truth value mapping is swapped in the algorithm.
\end{remark}
\section{Implementation and Results}
We use PRISM~\cite{prism} for the binary model checker subroutine in the implementation of the $qMC$
algorithm. The algorithm works for both numerical and statistical model checking. The inputs to the model
checker are the three valued probabilistic model and the property specification. The model checker then
verifies the input property in the given model. If the input property contains nested probabilistic
operators, then each inner probabilistic formula is considered as a separate property and verified first.
The results of these sub-formulas are then replaced in the input property to remove nesting.
However, the current version of PRISM does not support statistical model checking of nested properties.
\subsection{Results with qDTMC}
We illustrate our approach with the qDTMCs $M_1$ (Fig~\ref{example}), $M_2$ (Fig~\ref{small_more}), $M_3$ (Fig~\ref{input_less}) and $M_4$ (Fig~\ref{input_more}).
Note that $M_1$ and $M_2$
(and $M_3$ and $M_4$) have the same state space and differ only in the number of \emph{unknowns}.
These models are checked against different properties to observe the effect of unknown information on the behaviour of the models.
The first set of verification tests was done on two small qDTMCs, $M_1$ and $M_2$, given in Fig~\ref{example}
and Fig~\ref{small_more}. In these models, there are two atomic propositions $p$ and $q$, each of which can have
a truth value from the set $\{T,F, ?\}$. These models are verified for two properties:
\(\Phi_1\ = \Pr_{\geq \theta}(\lnot p\ U\ r)\) and \(\Phi_2\ = \Pr_{\geq \theta}(X q)\). An additional atomic
proposition $t \equiv \lnot p $ is added in the models to handle the negation in $\Phi_1$. Thus, $\Phi_1$ can
now be written as $\Pr_{\geq \theta}(t\ U\ r)$. The results corresponding to various values of $\theta$ in
$\Phi_1$ and $\Phi_2$, for qDTMCs $M_1$ and $M_2$, are in tables \ref{small_prop1} and \ref{small_prop2}, respectively.
\begin{figure}
\centering
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.05cm, thin,
state/.style={circle,draw,font=\sffamily\tiny,initial text=,minimum size=0.75cm, inner sep =0mm}]
\node[initial,state,label=270 : {\tiny $s_{init}$}] (0) {$ \lnot p q? t?$};
\node[state] (2) [right of=0] {$\ p\ q \lnot t $};
\node[state] (1) [above of=2] {$p? q?t?$};
\node[state] (3) [below of=2] {$ p? q?t?$};
\node[state] (4) [right of=2] {$p?q?t?$};
\node[state] (5) [right of=3] {$\ p \lnot q\lnot t$};
\node[state] (6) [right of=1] {$\ p\ q\ \lnot t$};
\path [every node/.style={font=\sffamily\tiny}]
(0) edge [bend left] node {$0.3$} (1)
edge node {$0.2$} (2)
edge node {$0.5$} (3)
(1) edge node {$0.1$} (0)
edge [bend left] node {$0.35$} (2)
edge node {$0.4$} (4)
edge node {$0.25$} (6)
(2) edge node {$0.1$} (1)
edge [bend left] node {$0.1$} (3)
edge node {$0.8$} (4)
(3) edge node {$0.5$} (2)
edge [bend right] node {$0.5$} (4)
(4) edge node {$0.33$} (6)
edge node {$0.67$} (5)
(5) edge [loop below] node {$0.9$} (5)
edge node {$0.1$} (3)
(6) edge [loop above] node { $1$ } (6);
\end{tikzpicture}
\caption{Another example qDTMC with small state-space. This qDTMC has same state space as $M_1$, but more number of unknowns.}
\label{small_more}
\end{figure}
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&&&&&&&&& \\
$\theta$ & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\[6pt]
\hline &&&&&&&&& \\
$M_1$ & T & T & T & T & ? & ? & ? & ? & ? \\
\hline &&&&&&&&& \\
$M_2$ & T & T & ? & ? & ? & ? & ? & ? & ? \\
\hline
\end{tabular}
\caption{Results for various values of $\theta$ for the property \(\Phi_1\ = \Pr_{\geq \theta}(\lnot p\ U\ r)\)}
\label{small_prop1}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&&&&&&&&& \\
$\theta$ & 0.1\ & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\[6pt]
\hline &&&&&&&&& \\
$M_1$ & T & T & ? & ? & ? & ? & ? & F & F \\
\hline &&&&&&&&& \\
$M_2$ & T & T & ? & ? & ? & ? & ? & ? & ? \\
\hline
\end{tabular}
\caption{Results for various values of $\theta$ for the property $\Phi_2 = \Pr_{\geq \theta}(Xq)$}
\label{small_prop2}
\end{table}
It is evident from the results that when a model checker does not have enough paths to either accept or reject a proposition,
it generates $?$ as result. The $?$ truth value indicates to the model designer that the model needs more details, so as to be
certain of its required behavior. On the other hand, a $T$ or an $F$ result concludes that in spite of missing information,
the property or the behavior of the model can be easily verified. The difference in results for models $M_1$ and $M_2$ also
shows that with the increase in absence or uncertainty of the information in the models, uncertainty in the behavior
of the model increases as well.
\begin{figure}
\centering
\begin{subfigure}[qDTMC $M_3$ with less unknown values]{%
\label{input_less}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.75cm, thin, state/.style={circle,draw,font=\sffamily\tiny,initial text=,minimum size=0.85cm, inner sep =0mm}]
\node[initial,state,label=225 : {\tiny $s_{init}$}] (0) {$ p \lnot q\ r?$ };
\node[state] (1) [above of=0] {$p \lnot q r?$};
\node[state] (2) [right of=0] {$\lnot p\ q \ r$};
\node[state] (3) [below of=2] {$ p \lnot q \lnot r$};
\node[state] (4) [below of=0] {$p?q?r?$};
\node[state] (6) [right of=1] {$\ p \lnot q r?$};
\node[state] (5) [above of=6] {$\ p \ qr?$};
\node[state] (12) [right of=2] {$p \lnot q \lnot r$};
\node[state] (7) [below of= 12] {$p \ q\ r$};
\node[state] (8) [below right of=3] {$\lnot p \lnot q r\ $};
\node[state] (9) [below right of=4] {$\lnot p \lnot q r?$};
\node[state] (10) [right of=5] {$\lnot p \ q \lnot r\ $};
\node[state] (11) [right of=6] {$ p \ q \ r $};
\node[state] (13) [below right of=8] {$\lnot pqr\ $};
\node[state] (14) [below left of=8] {$\lnot p\lnot q\lnot r$};
\node[state] (15) [below of=4] {$p \ q \lnot r$};
\path [every node/.style={font=\sffamily\tiny}]
(0) edge node {$0.25$} (1)
edge node {$0.25$} (2)
edge node {$0.25$} (3)
edge node {$0.25$} (4)
(1) edge node {$0.5$} (5)
edge node {$0.17$} (2)
edge node {$0.33$} (6)
(2) edge node {$1$} (12)
(3) edge node {$0.5$} (7)
edge node {$0.5$} (8)
(4) edge node {$0.5$} (15)
edge node {$0.5$} (9)
(5) edge node {$0.5$} (10)
edge node {$0.25$} (11)
edge node {$0.25$} (6)
(6) edge node { $1$ } (12)
(7) edge [loop below] node { $1$ } (7)
(8) edge node {$0.75$} (13)
edge node {$0.25$} (14)
(9) edge [loop below] node { $1$ } (9)
(10) edge [loop above] node { $1$ } (10)
(11) edge [loop above] node { $1$ } (11)
(12) edge node {$0.5$} (11)
edge node {$0.5$} (7)
(13) edge [loop below] node { $1$ } (13)
(14) edge [loop below] node { $1$ } (14)
(15) edge [loop below] node { $1$ } (15);
\end{tikzpicture}
}
\end{subfigure}
\begin{subfigure}[qDTMC $M_4$ with more unknown values]{%
\label{input_more}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.75cm, thin, state/.style={circle,draw,font=\sffamily\tiny,initial text=,minimum size=0.85cm, inner sep =0mm}]
\node[initial,state,label=225 : {\tiny $s_{init}$}] (0) {$ p q? r?$};
\node[state] (1) [above of=0] {$p q? r?$};
\node[state] (2) [right of=0] {$\lnot p q? r?$};
\node[state] (3) [below of=2] {$ p q r?$};
\node[state] (4) [below of=0] {$p?q?r?$};
\node[state] (6) [right of=1] {$p \lnot q r?$};
\node[state] (5) [above of=6] {$ p \ qr?$};
\node[state] (12) [right of=2] {$\ p \lnot q r? $};
\node[state] (7) [below of= 12] {$p? q\ r$};
\node[state] (8) [below right of=3] {$p? q\ r? $};
\node[state] (9) [below right of=4] {$\lnot p \lnot q r?$};
\node[state] (10) [right of=5] {$\lnot p q \lnot r $};
\node[state] (11) [right of=6] {$ p \ q \ r$};
\node[state] (13) [below right of=8] {$\lnot pqr\ $};
\node[state] (14) [below left of=8] {$\lnot p\lnot q\lnot r$};
\node[state] (15) [below of=4] {$p \ q r? $};
\path [every node/.style={font=\sffamily\tiny}]
(0) edge node {$0.25$} (1)
edge node {$0.25$} (2)
edge node {$0.25$} (3)
edge node {$0.25$} (4)
(1) edge node {$0.5$} (5)
edge node {$0.17$} (2)
edge node {$0.33$} (6)
(2) edge node {$1$} (12)
(3) edge node {$0.5$} (7)
edge node {$0.5$} (8)
(4) edge node {$0.5$} (15)
edge node {$0.5$} (9)
(5) edge node {$0.5$} (10)
edge node {$0.25$} (11)
edge node {$0.25$} (6)
(6) edge node { $1$ } (12)
(7) edge [loop below] node { $1$ } (7)
(8) edge node {$0.75$} (13)
edge node {$0.25$} (14)
(9) edge [loop below] node { $1$ } (9)
(10) edge [loop above] node { $1$ } (10)
(11) edge [loop above] node { $1$ } (11)
(12) edge node {$0.5$} (11)
edge node {$0.5$} (7)
(13) edge [loop below] node { $1$ } (13)
(14) edge [loop below] node { $1$ } (14)
(15) edge [loop below] node { $1$ } (15);
\end{tikzpicture} }
\end{subfigure}
\caption{Example qDTMCs with large state-space}
\end{figure}
The qDTMCs $M_3$ and $M_4$ have a larger state-space as can be see in Figures \ref{input_less} and \ref{input_more}. Similar to the
previous case, these models also differ only in labeling functions but contain three atomic propositions
$p$, $q$ and $r$. These models are verified for both non-nested and nested properties. The behaviour of qDTMCs
$M_3$ and $M_4$ are verified using properties $\Phi_2 = \Pr_{\geq \theta}(Xq)$, $\Phi_3 = \Pr_{\geq \theta}(p\ U\ r)$,
$\Phi_4 = \Pr_{\geq \theta}(p\ U\ \Pr_{\geq 0.8}(Xr))$ and $\Phi_5 = \Pr_{\geq \theta}(\Pr_{\geq 0.2}(p\ U\ r)\ U\ q)$.
The results for these properties, for various values of $\theta$ can be found in Tables \ref{large_prop1},
\ref{large_prop2}, \ref{large_prop3} and \ref{large_prop4}. The results for these models concur with the ones
for models $M_1$ and $M_2$, and same conclusions can be made in this case as well.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&&&&&&&&& \\
$\theta$ & 0.1\ & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\[6pt]
\hline &&&&&&&&& \\
$M_3$ & T & T & ? & ? & ? & F & F & F & F \\
\hline &&&&&&&&& \\
$M_4$ & T & T & ? & ? & ? & ? & ? & ? & ? \\
\hline
\end{tabular}
\caption{Results for various values of $\theta$ for the property $\Phi_2 = \Pr_{\geq \theta}(Xq)$},
\label{large_prop1}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&&&&&&&&& \\
$\theta$ & 0.1\ & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\[6pt]
\hline &&&&&&&&& \\
$M_3$ & T & T & T & T & T & T & ? & ? & ? \\
\hline &&&&&&&&& \\
$M_4$ & T & T & ? & ? & ? & ? & ? & ? & ? \\
\hline
\end{tabular}
\caption{Results for various values of $\theta$ for the property $\Phi_3 = \Pr_{\geq \theta}(p\ U\ r)$}
\label{large_prop2}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&&&&&&&&& \\
$\theta$ & 0.1\ & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\[6pt]
\hline &&&&&&&&& \\
$M_3$ & T & T & T & ? & ? & ? & F & F & F \\
\hline &&&&&&&&& \\
$M_4$ & T & T & ? & ? & ? & ? & ? & ? & ? \\
\hline
\end{tabular}
\caption{Results for various values of $\theta$ for the property $\Phi_4 = \Pr_{\geq \theta}(p\ U\ \Pr_{\geq 0.8}(Xr))$}
\label{large_prop3}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&&&&&&&&& \\
$\theta$ & 0.1\ & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\[6pt]
\hline &&&&&&&&& \\
$M_3$ & T & T & T & T & T & T & T & ? & ? \\
\hline &&&&&&&&& \\
$M_4$ & T & T & T & T & ? & ? & ? & ? & ? \\
\hline
\end{tabular}
\caption{Results for various values of $\theta$ for the property $\Phi_5 = \Pr_{\geq \theta}(\Pr_{\geq 0.2}(p\ U\ r)\ U\ q)$}
\label{large_prop4}
\end{table}
\subsection{\emph{Unknowns} in Code}
Finally, we present an example of a code listing that has a module whose implementation details (and hence correctness properties etc) are
\emph{unknown}.
This could be due to several reasons: for instance, if a module in the system is not implemented yet and exists only in ``stub" form
or if an implementation of the module exists, but whose correctness is not established. Therefore, it would be
incorrect to assume any truth value for certain atomic propositions in the states that represent such a module.
Listing \ref{snippet} shows a code snippet of a system wherein $func$ is a function call whose internal
working is not known to the system designer. The value thus returned by this function is not known.
The system contains three atomic propositions, $p : var1=10$, $q : var2=z$ and $r : var3 \geq 0$, which
are true if and only if their respective conditions hold true in the system.
The truth values of these atomic propositions change with each set of assignment statements in the code. The module can now be
modeled as a $qDTMC$ using the above atomic propositions, as shown in Fig. \ref{code}.
Each state in the qDTMC represents the possible truth values of atomic propositions during a code execution. For instance, when the variables
$var1$, $var2$ and $var3$ are initialized to -1, then all three atomic propositions are false in the initial state of the qDTMC.
Algorithm qMC can now evaluate various properties for this module. The results of an example qPCTL query
$\Pr_{\geq \theta}(\lnot q \ U \ p)$ are shown in Table~\ref{code_DTMC}.
\begin{lstlisting}[caption={Code snippet},frame=single,label=snippet,captionpos=b,language= C,basicstyle=\small]
int x= randint(1,5);
int y= randint(1,10);
\\randint(int a,int b):returns a random int between a and b
int z=10;
int var1=-1, var2=-1, var3=-1;
var1= x+y;
if (var
var2=z;
else
var2=func(z);
if (var
var1=5;
var2=7;
var3=0; }
else {
var1=10;
var3=func(z); }
\end{lstlisting}
\begin{figure}[!ht]{}
\centering
\begin{tikzpicture}[framed,->,>=stealth',shorten >=1pt,auto,node distance=2cm, thin, state/.style={circle,draw,font=\sffamily\tiny,initial text=,minimum size=0.8cm, inner sep =0mm}]
\node[initial,state,label=270 : {\tiny $s_{init}$}] (0) {$\lnot p \lnot q \lnot r$};
\node[state] (2) [below right of=0] {$\lnot p \lnot q \lnot r$};
\node[state] (1) [above right of=0] {$\ p\ \lnot q \lnot r$};
\node[state] (6) [right of=2] {$\lnot p q? \lnot r$};
\node[state] (3) [right of=1] {$\ p\ q \lnot r$};
\node[state] (4) [right of=3] {$\ p \ q\ r?$};
\node[state] (5) [above right of=2] {$\lnot p\ q \lnot r $};
\node[state] (7) [right of=5] {$\lnot p \lnot q \ r$};
\node[state] (10)[right of=6] {$p\ q? r?$};
\path [every node/.style={font=\sffamily\tiny}]
(0) edge node {$0.1$} (1)
edge node {$0.9$} (2)
(1) edge node {$1.0$} (3)
(2) edge node {$0.4$} (5)
edge node {$0.6$} (6)
(3) edge node {$1.0$} (4)
(4) edge [loop right] node {$1.0$} (4)
(5) edge node {$0.18$} (7)
edge node {$0.82$} (4)
(6) edge node { $0.16$ } (7)
edge node { $0.84$ } (10)
(7) edge [loop right] node {$1.0$} (7)
(10) edge [loop right] node {$1.0$} (10) ;
\end{tikzpicture}
\caption{qDTMC $M_5$ for code snippet in Listing \ref{snippet}}
\label{code}
\end{figure}
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
&&&&&&&&& \\
$\theta$ & 0.1\ & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\[6pt]
\hline &&&&&&&&& \\
$M_5$ & T & ? & ? & ? & ? & F & F & F & F \\
\hline
\end{tabular}
\caption{Results for $ \Pr_{\geq \theta} (\lnot q U p)$}
\label{code_DTMC}
\end{table}
\section{Conclusions and Future Directions}
In this paper we presented a technique to determine the feasibility of model checking in the
presence of uncertainty in the implementation of a model of a stochastic system.
We have presented a small example of a code with an incompletely determined module.
However, it involved a manual conversion of the code to a qDTMC for the purpose of model checking.
It remains to be seen how well this conversion and our technique scales to a full-fledged
code of, say, a discrete event simulator.
It would also be interesting to see how this approach can be applied to other modeling-formalism/query-logic
pairs.
\bibliographystyle{splncs03}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 108
|
Resultados das 500 Milhas de Indianápolis de 1933, no circuito de Indianapolis na terça-feira, 30 de Maio de 1933.
1933
Indianapolis 500
Desporto nos Estados Unidos em 1933
|
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"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,694
|
Trew Knowledge Helps Olympic.ca Achieve Gold for User Experience in PyeongChang 2018
For the past five years, Trew Knowledge, in collaboration with Toronto-based brand agency Zync, have been overseeing the programming and web development of the official Team Canada website. In preparation for the 2018 PyeongChang Winter Olympics, it was critical that Olympic.ca & Olympique.ca deliver an Olympic worthy user experience for Canadian fans and global supporters during this year's games.
Building on the momentum of previous record-setting achievements from Rio 2016 & Sochi 2014, this Winter Olympics the Canadian Olympic Committee's goals were focused on optimizing the site for speed and performance, enhancing mobile user experience, and amplifying the Canadian Olympic Club fan experience.
The Trew Knowledge team enjoying the opening ceremonies of the Winter Olympic Games in PyeongChang 2018.
To improve speed and performance, Trew Knowledge worked with their partner WordPress VIP to focus on key areas of the site where the highest volumes of traffic would be impacted. This included athlete profiles, real-time results, schedules, and medal counts all being served from third-party APIs. It was critical that an intelligent caching mechanism was in place to avoid hitting rate-limits but also pull in statistics at regular intervals to ensure the results being displayed were up to date.
During peak traffic periods, the site was able to handle massive amounts of traffic without any outages or drops in performance, all while seeing load times being reduced to approximately 1-1.5 seconds.
Many different nationalities on our team, but it's @TeamCanada we support from a technical perspective! Great work by our partners @trewknowledge ahead of #PyeongChang #Olympics https://t.co/0MhPEmvPlE
— WordPress.com VIP (@WordPressVIP) February 9, 2018
To engage fans for PyeongChang 2018, the COC introduced several new additions to the Canadian Olympic Club. Fans can now add their favourite athletes and sports to their profile which will be used to tailor specific content based on their preferences.
Canadian Olympic Club members were also given access to exclusive "Be Olympic" content such as behind the scenes videos and photos of the new brand platform that highlights the Canadian ideals of virtue and victory.
A new Fan Fun section was introduced giving members the ability to complete quizzes, answer trivia questions, play interactive games, and watch their rank sore on the leaderboard.
New challenges were introduced for club members to unlock and earn points as well. During the Olympic games, the Canadian Olympic Committee offered fans the ability to enter contests to win prizes such as signed Team Canada gear from medal-winning athletes, Molson Canadian mini fridges, Samsung Galaxy View tablets, and much more.
With these new features and rewards, 100,000 new fans joined the Canadian Olympic Club via social login during PyeongChang 2018.
Not only was this a record-setting performance for Team Canada, earning 29 total medals (11 gold, 8 silver and 10 bronze), and finishing third overall, but off the field of play, olympic.ca posted its highest number page views, exceeding projections by more than 200% and doubling what was generated in Rio 2016 with more than 17 million page views.
In addition to the inspirational number of podium finishes, olympic.ca won a few medals of its own including a Gold W3 award for mobile user experience, Silver W3 award for best sports website and another Silver W2 award for structure and navigation.
Overall, the 2018 PyeongChang Winter Olympics meant a lot of late nights and early mornings for the Trew Knowledge team, but just like the Canadian Olympic Athletes, the team at Trew Knowledge was dedicated to making Canada proud with a user experience worthy of the podium.
Are you ready to level-up your user experience? Contact us today.
Posted in Press Release, WordPress
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 474
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Today, the Ohio Senate unanimously passed Senate Bill 9; legislation that would require a health plan issuer to release each group policyholder's monthly claims data.
Just last week, OSBC member, Victoria McCoy, of Association Insurance Group, testified in support of SB 9. The cost of health care is one of, if not the top expense for a business. We believe that this bill will help small businesses by allowing those with 50 or more full-time employees to access more data as they make their decision about which health plan they will choose.
However, we would like to see the threshold go below 50 full-time employees in order to capture and assist more small businesses. Nevertheless, Senate Bill 9 does serve as a positive step forward to aid small businesses in their effort to control health insurance costs.
The Ohio Small Business Council as a whole sent a letter of support of Senate Bill 9 and it is our hope that this bill will move through the House quickly and be implemented into law for the benefit of Ohio's business climate.
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{
"redpajama_set_name": "RedPajamaC4"
}
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require 'spec_helper'
include Helpers
describe 'TICK' do
it 'runs a standard tick' do
run('TICK').should == ''
end
it 'runs a verbose tick' do
run('TICKV').should =~ /^=== Tick 0 ===$/
end
it 'runs a quiet tick' do
run('TICKQ').should == ''
end
it 'runs multiple ticks' do
run('TICK 4', 'STATUS').should =~ /^ticks: 4$/
end
it 'errors for negative ticks' do
run('TICK -2').should == 'Tick count must be greater than zero'
end
it 'errors for non numerical ticks' do
run('TICK abc').should == 'Tick count must be numeric'
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,355
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Sir George Sitwell 4th Baronet
antiquary, writer, 4th Baronet Sitwell of Renishaw (from 1862), Member of Parliament (1885-1886), and Member of Parliament (1892-1895)
Sir George Reresby Sitwell, 4th Baronet (27 January 1860 – 9 July 1943) was a British antiquarian writer and Conservative politician who sat in the House of Commons between 1885 and 1895.
Sir Sitwell Reresby Sitwell 3rd Baronet mother?
Sir George Sitwell 4th Baronet Ida Sitwell
Dame Edith Sitwell DBE
Sir Sacheverell Sitwell CH 6th Baronet
Sir Osbert Sitwell CH 5th Baronet
Chris Hall on Geograph
Scarborough, United Kingdom
Wood End Here lived Sir George & Lady Ida Sitwell and their children, Edith, Osbert & Sacheverell
Woodend Creative Workspace, The Crescent, Scarborough, United Kingdom where they lived
Wood End former home of the Sitwell family
Gate pillar to Wood End, The Crescent, Scarborough, United Kingdom where they lived
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,085
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Q: Difference between two dates in minute, hours javascript I want to find difference between two dates. I have tried this code but it gives me wrong values. I want get total minutes between two dates, so I am converting hours to minutes and adding to minutes.
var hourDiff = timeEnd - timeStart;
var diffHrs = Math.round((hourDiff % 86400000) / 3600000);
var diffMins = Math.round(((hourDiff % 86400000) % 3600000) / 60000);
diffMins = diffMins + (diffHrs * 60);
Here timeEnd is Mon Jan 01 2007 11:30:00 GMT+0530 (India Standard Time),
and timeStart is Mon Jan 01 2007 11:00:00 GMT+0530 (India Standard Time).
Here if hours difference I am getting 1, it should be 0 and minutes I am getting 30 that is right. But hours should be 0. Am I doing something wrong here?
A: Try:
var diffHrs = Math.floor((hourDiff % 86400000) / 3600000);
Math.round rounded the 0.5 hour difference up to 1. You only want to get the "full" hours in your hours variable, do you remove all the minutes from the variable with the Math.floor()
A: Try this:
var startDate = new Date('Jan 01 2007 11:00:00');
var endDate = new Date('Jan 01 2007 11:30:00');
var starthour = parseInt(startDate.getHours());
var endhour = parseInt(endDate.getHours());
if(starthour>endhour){
alert('Hours diff:' + parseInt(starthour-endhour));
}
else{
alert('Hours diff:' + parseInt(endhour-starthour));
}
And here is the working fiddle.
A: Try this code (uses ms as initial units)
var timeStart = new Date("Mon Jan 01 2007 11:00:00 GMT+0530").getTime();
var timeEnd = new Date("Mon Jan 01 2007 11:30:00 GMT+0530").getTime();
var hourDiff = timeEnd - timeStart; //in ms
var secDiff = hourDiff / 1000; //in s
var minDiff = hourDiff / 60 / 1000; //in minutes
var hDiff = hourDiff / 3600 / 1000; //in hours
var humanReadable = {};
humanReadable.hours = Math.floor(hDiff);
humanReadable.minutes = minDiff - 60 * humanReadable.hours;
console.log(humanReadable); //{hours: 0, minutes: 30}
JSFiddle: http://jsfiddle.net/n2WgW/
A: If you are confident that the difference will be less that 24 hours, the following works.
var timeStart= new Date('2015-01-01 03:45:45.890');
var timeEnd = new Date('2015-01-01 05:12:34.567');
var timeDiff = new Date(timeEnd.getTime() - timeStart.getTime());
var humanTime = timeDiff.toISOString().substring(11, 23);
var diffHours = timeDiff.toISOString().substring(11, 12);
humanTime is 01:26:48.677, diffHours is 01
A: You can Try This:-
function diff_hours(dt2, dt1)
{
var diff =(dt2.getTime() - dt1.getTime()) / 1000;
diff /= (60 * 60); // For Hours
//diff /= (60); For Minutes
return Math.abs(Math.round(diff));
}
dt1 = new Date(2014,10,2);
dt2 = new Date(2014,10,3);
console.log(diff_hours(dt1, dt2)); // you will get '24' hours
dt1 = new Date("October 13, 2014 08:11:00");
dt2 = new Date("October 13, 2014 11:13:00");
console.log(diff_hours(dt1, dt2)); // you will get '3' hours
A: var timeDiff = function (date1, date2) {
var a = new Date(date1).getTime(),
b = new Date(date2).getTime(),
diff = {};
diff.milliseconds = a > b ? a % b : b % a;
diff.seconds = diff.milliseconds / 1000;
diff.minutes = diff.seconds / 60;
diff.hours = diff.minutes / 60;
diff.days = diff.hours / 24;
diff.weeks = diff.days / 7;
return diff;
}
A: You can get Time difference in hours minutes and secons like countdown by using following:
var diff = EndedTime - StartedTime;
var hours = Math.floor(diff / 3.6e6);
var minutes = Math.floor((diff % 3.6e6) / 6e4);
var seconds = Math.floor((diff % 6e4) / 1000);
var duration = hours+":"+minutes+":"+seconds;
Hope This Help. Thanks
A: You can get Time Difference in hours and minutes format like below
const startTime = new Date(event.startTime);
const endTime = new Date(event.endTime);
const diff = endTime.getTime() - startTime.getTime();
const hrDiff = diff / 3600 / 1000; // 1.555555
const totalHours = parseFloat((hrDiff).toFixed(2)); // 1.5
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,947
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\section{Introduction}\label{introductionsec}
A significant part of stars are located in double or multiple stellar systems. However, reviews of pulsating stars bound in binaries \citep[e.g.][]{szatmary1990,zhou2010} clearly show the lack of stellar pairs with an RR Lyrae component. The number of currently confirmed binaries comprising an RR Lyrae type pulsator can be counted on one hand.
The binarity of an object can be revealed in many different ways. For example, detection of eclipses, periodic radial velocity (RV) changes, or regular astrometric shifts in a visual binary can serve as a direct proof of binarity. A companion of a periodic variable star can also be detected indirectly through changes in timings of light extrema, the so-called light time effect (hereafter LiTE). RR Lyrae stars are generally located at larger distance from Earth, hence astrometric detection of binarity is highly unlikely. Since the spectra of RR Lyrae stars are influenced by pulsations, discovering the binary nature of stars through changes in the position of spectral lines is also difficult \citep[e.g.][]{fernley1997,solano1997}. Thus the most promising methods are detection of eclipses and LiTE.
In the Large Magellanic Cloud (LMC) three candidates for RR Lyraes in eclipsing binaries were detected \citep{soszynski2003}. However, these objects were identified as optical blends consisting of two objects, RR Lyrae star and eclipsing system \citep{soszynski2003,prsa2008}. A very interesting object was identified by \citet{soszynski2011} and subsequently studied by \citet{pietrzynski2012} and \citet{smolec2013}. This peculiar eclipsing system with an orbital period of 15.24\,d contains a component that mimics an RR Lyrae pulsator. The detailed study showed that this object, \object{OGLE-BLG-RRLYR-02792}, has a very low mass of only 0.26\,$\mathfrak{M}_{\odot,}$ which is too low for classical RR Lyrae stars. Other physical characteristics also indicate that this binary component is a very special object as a result of an evolution in a close binary followed by a life similar to a classical RR Lyrae star. \citet{pietrzynski2012} included the object in a new class of pulsating stars called binary evolution pulsators (BEP). \object{OGLE-LMC-RRLYR-03541} is another candidate for RR Lyr in eclipsing binary \citep{soszynski2009}. Nonetheless, the similarity in the shape of the light curve and the short orbital period of 16.229\,d might mean that it belongs to the class of BEP \citep{hajdu2015b}. In addition, it is not yet excluded that it could be actually a blend of two close stars.
\citet{szatmary1990} published a list of various types of pulsating stars bound in binary systems that were visually identified on the basis of the LiTE~in their \textit{O-C}~diagrams. However, detailed information and references are missing. The catalogue of various binary systems with pulsating components from \citet[][version December 2014]{zhou2010} contains in the RR Lyrae class \object{TU UMa}, \object{OGLE-BLG-RRLYR-02792}, and several tens of candidates without any closer information. Therefore, including some of these objects as binary systems (in both catalogues) is at least doubtful.
Several examples of RR Lyrae stars in binaries were very recently
identified through the analysis of their \textit{O-C}~diagrams. \citet{li2014} found that \object{FN~Lyr} and \object{V894~Cyg} are probably in pairs with brown dwarves. \citet{hajdu2015} found 20 additional candidates among OGLE bulge RRab variables. \citet{liska2015} presented an analysis of 11 new binary candidates located in the Galactic field. Nevertheless, all these candidates need to be confirmed spectroscopically or using another independent method.
In this paper, we present a new analysis of the LiTE~in \object{TU~UMa} \citep[23-year long cycle, detected by][]{szeidl1986} that is based on a much wider sample of \textit{O-C}~than was used for the LiTE~in \object{TU~UMa} before \citep{wade1999}. Highly accurate photometric observations that cover two thirds of the proposed orbital period, are newly available. In Sect.~\ref{historysec} we briefly discuss the history of observation of \object{TU~UMa} with an emphasis on its binary nature. In Sect.~\ref{datasec} we summarise the characteristics of our data sample. Except for data from various sources, we used the original measurements gathered in 2013--2014. We apply our LiTE~procedure (described in Sect.~\ref{modelsec} and Appendix~\ref{liteappendix1}) and model the \object{TU~UMa} data in Sect.~\ref{resultstuumasubsec}. Other proofs for binarity are discussed in Sect.~\ref{proofsec}, and all results are summarised in Sect.~\ref{summarysec}.
\section{History of \object{TU UMa} observations}\label{historysec}
TU Ursae Majoris = \object{AN 1.1929} = \object{BD+30 2162} = \object{HIP 56088} ($\alpha$ = 11$^{\rm h}$29$^{\rm m}$48\fs49, $\delta$ = +30\degr04\arcmin02\farcs4, J2000.0) is a pulsating RR Lyrae star of Bailey ab type. According to the Variable Star Index\footnote{http://www.aavso.org/vsx/} \citep{watson2006}, its brightness in $V$ band varies in the range of 9.26--10.24\,mag (spectral type A8--F8) with a period of about 0.558\,d. No signs of the Bla\v{z}ko effect have been reported for \object{TU UMa} up to now.
The variability of \object{TU UMa} was discovered by \citet{guthnick1929} on Babelsberg plates. Many authors thereafter studied the star using photoelectric photometry and spectroscopy. A detailed description of the history of \object{TU UMa} research was performed by \citet{szeidl1986}. Only the most important information about the LiTE~is briefly mentioned below, since about 180 articles with the keyword \object{TU~UMa} are currently retrievable at the NASA ADS portal.
\citet{payne-gaposchkin1939} was the first who noted cyclic variations in maxima timings and proposed a 12400-day (34-year) long cycle. Important results were obtained by \citet{szeidl1986}, who mentioned a secular period decrease that causes the parabolic trend in
the \textit{O-C}~diagram, and a probable 23-year (8400\,d) variation that
is possibly caused by the binarity of the star. \citet{saha1990a} detected systematic shifts in RVs that indicate binarity, but they used very few RV measurements. They modelled the LiTE~for the first time and determined orbital period of 7374.5\,d (20.19\,yr). Their analysis showed that the proposed stellar pair has an extremely eccentric orbit with $e\!=\!0.970$. They considered only a constant pulsation period in their model. The effect of neglecting the secular changes of $e$, $a\,\sin i$, and $\mathfrak{M}_{2}\,\sin^{3} i$ was tested by \citet{wade1992}.
The LiTE~with secular variation in the pulsation period was firstly solved by \citet{kiss1995}, who determined more accurate orbital elements and derived an orbital period of 8800\,d (24.1\,yr). \citet{wade1999} collected all available maxima timings and also RVs and successfully verified results from \citet{saha1990a}. They obtained five different groups of models of the LiTE~with respect to different subsets of maxima timings (all maxima without visual values, only photoelectric and CCD values, etc.). They derived orbital periods in the range from 20.26\,yr to 24.13\,yr depending on the particular data set and the number of fitting parameters (with or without parabolic trend). Subsequently, they used nine derived sets of orbital elements to reconstruct the orbital RV curve of the pulsating component and to compare it with shifts in observed RVs. Since then, \object{TU UMa} has not been
studied with regard to its LiTE for about 15 years. However, improvements of quadratic ephemeris were performed by \citet{arellanoferro2013},
for instance. Many high-accurate maxima timings (mainly CCD measurements) were published during this 15-year interval. The currently available data exceed five proposed orbital cycles.
\section{Data sources}\label{datasec}
\subsection{GEOS RR Lyrae database}\label{geossubsec}
Since the GEOS RR Lyrae database\footnote{http://rr-lyr.irap.omp.eu/dbrr/} \citep[{\it{Groupe Europ\'{e}en d\textquoteright Observations Stellaires}}, ][]{boninsegna2002,leborgne2007} is the most extended archive of times of RR Lyrae star maxima, we used this as the main data source for our analysis. The version of the database of November 2014 contains corrected values for three maxima timings from \citet{kiss1995} that originally contained incorrect heliocentric corrections \citep{wade1999}.
We discarded all visually determined maxima timings because they
were inaccurate and omitted two photographic maxima from \citet{robinson1940} and \citet{payne-gaposchkin1954}, which were replaced by our maxima from the DASCH project (see Sect.~\ref{othersourcessubsec}). We used also the latest maxima timings from \citet{hubscher2014,hubscher2015} that were not included in the version of the GEOS list we used.
We paid special attention to maxima timings based on data from sky surveys such as Hipparcos \citep{esa1997} or ROTSE = NSVS \citep{wozniak2004} for the GEOS values\footnote{GEOS data marked \textquoteleft
pr. com.\textquoteright~were not used.}. These timings were acquired
with a special method because in most cases they were determined statistically based on the combination of many points that are often spread out over a few years. \textit{O-C}~values determined from such data sets very often did not follow the general trend of \textit{O-C}~dependence\footnote{the original value of the maximum timing 2448500.0710 HJD from the Hipparcos satellite \citep{maintz2005},
e.g., has a residual value \textit{O-C}$_{\rm res} = 0.0096$\,d based on model 2 in Sect.~\ref{resultstuumasubsec}, but the standard deviation of CCD measurements determined from the model is only 0.0017\,d.}. They were therefore omitted. However, we re-analysed the original data of these surveys (Sect.~\ref{othersourcessubsec}). All used maxima timings are available at the CDS portal.
\subsection{Our observations}\label{observsubsec}
As an extension of the GEOS data we also used ten new maxima timings gathered by J.~Li\v{s}ka in 19 nights between December 2013 and June 2014. CCD photometric measurements were performed using two telescopes -- three nights with a 24-inch Newtonian telescope ($vby$ Str\"{o}mgren filters) at Masaryk University Observatory in Brno, and 16 nights with a small 1-inch refractor \citep[{\it green} filter with similar throughput as the Johnson $V$ filter,][]{liska2014a} at a private observatory in Brno. For the small-aperture telescope, five frames with exposure times of 30\,s each were combined to a single image to achieve a better signal-to-noise ratio. The time resolution of such a combined frame is about 170\,s. The comparison star \object{BD+30~2165} was the same for both instruments, but the control stars were \object{BD+30~2164} (for the 24-inch telescope) and \object{HD~99593} (for the 1-inch telescope). Maxima timings were determined via polynomial fitting\footnote{Used for observations with 24-inch telescope where a full light curve was not available.} or the template fitting described in Sect.~\ref{othersourcessubsec}. Monitoring with the small telescope resulted in the well-covered phase light curve shown in Fig.~\ref{Fig:LightCurve}. Except for maxima timings determination, the observations were also
important for detecting possible eclipse (Sect.~\ref{eclipsesec}). Our measurements are available at the CDS portal.
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_LC}
\caption{Differential magnitudes of \object{TU UMa} obtained with the 1-inch telescope folded with the pulsation period (black dots) plotted together with the model of $V$-band observations from \citep{liakos2011a}.}
\label{Fig:LightCurve}
\end{figure}
\subsection{Other sources}\label{othersourcessubsec}
We used high-cadence measurements from the SuperWASP project \citep{pollacco2006,butters2010} and Pi of the Sky project \citep[e.g.][]{burd2004,siudek2011} to determine maximum timings from the individual well-covered nights.
In addition, to extend the \textit{O-C}~dataset as far as possible, we also analysed data from other large sky surveys (Hipparcos, NSVS) and from the project DASCH \citep[photometry from scanned Harvard plates, e.g.][]{grindlay2009}. For these very sparse but very extended data, maxima timings were estimated using template fitting. The same process was applied to the data from the other sources.
Firstly, we chose the dataset with the best-quality data and with the best phase-coverage. Since the data from all surveys were of insufficient quality to construct the template light curve (e.g. in Hipparcos and NSVS data the phase around maximum brightness was not observed), $V$-band measurements from \citet{liakos2011a} were used. We then modelled the shape of the light curve $m(t)$ in Matlab with a non-linear least-squares method (hereafter LSM) with an $n$-order harmonic polynomial
\begin{equation}\label{eqfouriersum}
m(t) = A_{0} + \sum_{j=1}^{n} A_{j} \cos\left( 2\,\pi\,j\,\frac{t - M_{0}}{P_{\rm puls}} + \phi_{j}\ \right),
\end{equation}
where $A_{0}$ is the zeroth level of brightness in mag, $A_{j}$ are amplitudes of the components in mag, $t$ is the time of observation in Heliocentric Julian Date (HJD), $M_{0}$ is the zero epoch of maximal brightness in HJD, $P_{\rm puls}$ is the pulsation period in days, and $\phi_{j}$ represents the phase shifts in radians. After some experiments, we found that a polynomial with $n=15$ is sufficient for a good template model.
Nevertheless, using one template curve for various surveys brings some particularities. Firstly, each dataset has its own magnitude zero level, and the model has to be scaled because different surveys use different filters. Therefore we used the whole light curves for our fit. Individual amplitudes $A_{j}$ and phase shifts $\phi_{j}$ known from the template curve remained fixed, but the total amplitude was rescaled using the ratio of total amplitudes for the analysed and template datasets. The factor was typically
close to 1. In addition, pulsation period and zero epoch had to be slightly refined for each dataset. Subsequently, outliers were iteratively removed.
Individual observational instruments have different spectral sensitivities, therefore it was necessary to verify the usability of the same $V$-band template for different datasets. We used photometric measurements in \textit{UBVRI} filters from \citet{liakos2011a} and compared all curves. The main difference is in amplitude, which is highly dependent on the mean wavelength (the amplitude decreases from 1.3\,mag in $U$ to 0.6\,mag in $I$). When the amplitudes are normalised, the differences between shapes of $U$, $B$, $V$, and $R$ curves are almost negligible (comparable with the noise level). The same applies for the times of maxima. Only $I$-band light changes differ, apparently. Surveys typically observed in broad band filters and mean wavelengths are not known, or the values were only roughly estimated. We can fortunately
estimate them using comparison between amplitudes of survey data and \textit{UBVRI}-amplitudes. We found that effective wavelengths for all surveys lie between the wavelengths of the $B$ and $R$ filters; this means that our approach ($V$-band template) is applicable.
The good consistence of our template and the observed curves was controlled visually (see Fig.~\ref{Fig:LCSurveys}). For SuperWASP data, which were obtained using six different CCD cameras, a very careful selection of the best nights compared to the template was performed to avoid significant trends that are present in the data.
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_LC_surveys}
\caption{Light curves of \object{TU UMa} from different datasets: DA -- DASCH A, DB -- DASCH B, HIPP -- Hipparcos, LIAKOS -- \citet[][$V$-filter]{liakos2011a}, OUR -- this paper (1-inch, $green$-filter), WASP -- SuperWASP (CCD 144), PI -- Pi of the Sky, NSVS -- NSVS, MODEL -- used template curves (DASCH templates are described in Sect.~\ref{DASCHsubsec}). Light curves are vertically shifted to better display the light variability.}
\label{Fig:LCSurveys}
\end{figure}
When the data were sparse without well-defined maxima, it was necessary to divide the whole dataset into smaller subsamples containing typically about 30 points, with a time span from several days to hundreds of days. The data in these subsamples were then compared with the refined template light curve, and the time of maximum was determined. With this subsampling we were of course
only able to estimate the mean time of maximum for the time interval of particular subset. However, in comparison with the total time span of the \textit{O-C}~values, which cover several decades, this method is fully appropriate. The numbers of all used maxima timings from particular surveys are listed in Table~\ref{NumberMaximaTable}.
\begin{center}
\begin{table}[t]
\centering
\caption{Numbers of new maxima timings of \object{TU UMa} determined from individual projects and from our observations.}\label{NumberMaximaTable}
{\tiny
\def1.5{1.5}
\tabcolsep=3.0pt
\begin{tabular}{ccccccc}
\hline\hline
Hipparcos & NSVS & Pi of the Sky & DASCH & SuperWASP & Our\\
\hline
3 & 4 & 5 & 29 & 21 & 10 \\
\hline
\end{tabular}}
\end{table}
\end{center}
In addition, we determined five maxima times from the data that were omitted or only poorly determined from the analysis in \citet{boenigk1958}, \citet{liakos2011a,liakos2011b}, \citet{liu1989}, and \citet{preston1961}.
The uncertainty of each time of maximum determined by the template fitting method is influenced mainly by the choice of the used model (selection of polynomial degree), the time determined for
the template maximum, and by the quality of the fitted datasets. The first two components were estimated statistically from the template light curve, and their combination was set to 0.0008\,d. The influence of the third source of uncertainties was individually
estimated for each light curve directly from the LSM.
\subsection{Photographic measurements -- DASCH maxima timings}\label{DASCHsubsec}
Photographic measurements that were obtained with exposure times of an hour and longer are special cases.
Light curves resulting from these observations differ from real curves in amplitude and also in time of minima and maxima (for example the models of $B$-band measurements of \object{TU UMa} from \citet{liakos2011a} in Fig.~\ref{Fig:LightCurveExp}). We performed a detailed analysis of this problem and will publish the results elsewhere. Therefore we discuss only the most important findings. The extrema timings and amplitude change almost linearly with the duration of the exposure time. Maxima are delayed, minima occurred sooner, and
the amplitude is lower than for shorter exposures.
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_LC_B_convolution_exp}
\caption{Change of the light curve shape caused by long exposure time. The model of the real light curve based on $B$-band observations from \citet{liakos2011a} (red line) is plotted together with models of observed curves with exposures of 63\,min (black crosses) and 200\,min (blue dots).}
\label{Fig:LightCurveExp}
\end{figure}
The photographic plates that were digitalised in the DASCH project
have various integration times, mostly from 1 to 200\,min (Fig.~\ref{Fig:DaschExp}), therefore it is very difficult to correctly analyse the data (different models of the observing curve have to be used). We solved this problem by selecting the measurements with similar exposures (they can be approximated with the same template curve; the time difference is no more than 0.0005\,d).
At first, we selected the measurements according to the exposure -- group A (58--68\,min, older measurements) and group B (30--40\,min, newer measurements). Then we calculated template curves for the mean exposures 63\,min and 35\,min (deformed by the exposure) based on measurements of \citet{liakos2011a}. These two templates were used for the DASCH A and B dataset.
Our tests showed that for $\sim$\,60\,min exposure, for instance, the maximum brightness based on exposure-deformed light curve is delayed of about +0.004\,d in comparison with real changes. This difference will be apparent, for example, in maxima timings determined using polynomial fitting. Therefore, this shift has to be included in the final maxima times. When the template fitting method is applied with an improved time-corrected model (template from the deformed curve), the exposure-length discrepancy is fully reduced to zero.
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_time_distribution_exposure_DASCH}
\caption{Exposure times of the photographic plates of \object{TU UMa} in the DASCH project. The data were divided into two groups according to their integration times -- group A with 58--68\,min (red crosses) and B with 30--40\,min (green stars).}
\label{Fig:DaschExp}
\end{figure}
\section{Modelling the LiTE}\label{modelsec}
\subsection{Light time effect}\label{modellitesubsec}
The LiTE~was suggested at the end of the ninteenth century in
the \object{Algol} system by \citet{chandler1888}, while the first detailed theoretical analysis of the problem was performed by \citet{woltjer1922}. \citet{irwin1952a} solved important equations for a part of the direct solution of the LiTE~and described a graphical way to determine
the orbital elements.
Currently, the LiTE~is usually solved very accurately by applying equations of motion in a two-body system (the equations from \citet{irwin1952a} included a direct solution of Kepler's equation) or using numerical calculations of a perturbed orbit in a multiple system. Nevertheless, the inverse part of the calculations, in which the best solution is determined (minimisation), still remains a problem to be discussed -- various authors use various methods, for example, damped differential corrections \citep{pribulla2000}, the LSM \citep{panchatsaram1981}, the simplex (Levenberg-Marquart) method \citep{wade1999,lee2010}, or a combination of LSM and simplex method \citep{zasche2008}. For example, the simplex method has several versions that differ in the setting of the initial parameters, sorting conditions, or in the size of corrections. Thus, obtaining the same results, that is, repeat
the process, is very difficult, even impossible. To avoid this ambiguity, the inverse part of our code was constructed on the basis of the non-linear LSM described in \citep[e.g.,][]{mikulasek2013} applied to modelling the LiTE~(for details see Sect.~\ref{modelfittingsubsec} and Appendix~\ref{liteappendix1}). This way has previously been applied to analyse the LiTE~in the \object{AR~Aur} system \citep{mikulasek2011b,chrastina2013} and is similar to the one used by \citet{vanhamme2007} and \citet{wilson2014}.
\subsection{Fitting procedure}\label{modelfittingsubsec}
The code that we used is written in Matlab. It consists of several modules: loading measurements and setting initial input parameters, direct solution of the LiTE~including an optional parabolic trend, inverse minimisation method, and, finally, selecting the best solution and calculating uncertainties of individual parameters through bootstrap resampling (Sect.~\ref{modelbootstrapsubsec}).
A prediction of maxima timings $T_{\rm cal}$ calculated according to the relation
\begin{equation}
T_{\rm cal} = M_{0} + P_{\rm puls}\times N + \mathnormal{\Delta}
\label{eq:liteEphLin}
\end{equation}
contains linear ephemeris, as well as correction $\mathnormal{\Delta}$ for the LiTE. The parameter $M_{0}$ is the zero epoch of pulsations in HJD, $P_{\rm puls}$ is the pulsation period in days, and $N$ is the number of pulsation cycle from $M_{0}$.
The correction $\mathnormal{\Delta}$ for the LiTE~includes calculating the orbit of a pulsating star around the centre of mass of a binary in relative units and can be expressed by the equation adopted from \citet{irwin1952a}
\begin{equation}
\mathnormal{\Delta} = A \,\left[ (1-e^{2})\,\frac{\sin(\nu+\omega)}{1+e\,\cos\nu} + e\,\sin\omega\right],
\end{equation}\label{eq:liteDelta}
where $e$ is the numerical eccentricity, $\omega$ is the argument of periastron (usually in degrees), $A \doteq a_{1}\,\sin i/173.145$ is the projection of the semi-major axis of primary (pulsating) component $a_{1}$ in light days\footnote{the semi-amplitude of the
LiTE~changes in \textit{O-C}~diagram is then $A_{\rm LiTE} = A\,\sqrt{1-e^{2}\,\cos^{2}\omega}$.} according to the inclination of the orbit $i$, and $\nu$ is the true anomaly. The eccentric anomaly $E$, which is necessary to determine the true anomaly $\nu$, is solved with Kepler's equation by iteratively using Newton's method with a given precision higher than $1\times 10^{-9}\,$arcsec. Kepler's equation requires a mean anomaly $M$, which is determined from the orbital period $P_{\rm orbit}$ in days and the time of periastron passage $T_{0}$ in HJD.
Another more complex model that includes a parabolic trend in \textit{O-C}~\citep[e.g.][]{zhu2012}, uses the modified Eq.~(\ref{eq:liteEphLin}) in the form of
\begin{equation}
T_{\rm cal} = M_{0} + P_{\rm puls}\times N+ \frac{1}{2}\,P_{\rm puls}\,\dot{P}_{\rm puls}\times N^2 +\mathnormal{\Delta},
\label{eq:liteEphPar}
\end{equation}
where $P_{\rm puls}$ is the instantaneous pulsation period at the moment $M_{0}$ and the parameter $\dot{P}_{\rm puls} = {\rm d}P_{\rm puls}/{\rm d}t$ is the rate of changes in the pulsation period in [d\,d$^{-1}$]. For easy comparison with other RR Lyrae stars, we used prescriptions from \citet{leborgne2007}. Their parameter $a_{3}=1/2\,P_{\rm puls}\,\dot{P}_{\rm puls}$ is the rate of period changes per one cycle in [d\,cycle$^{-1}$], and the rate of period changes $\beta$ in [ms\,d$^{-1}$] is $\beta = 6.31152\times10^{10}\,a_{3}/P_{\rm puls}$ or $\beta = 0.07305\times10^{10}\,a_{3}/P_{\rm puls}$ in [d\,Myr$^{-1}$] \footnote{\citet{leborgne2007} probably used a year with 366\,d, therefore their constant $0.0732\times10^{10}$ is slightly different.}. The instantaneous pulsation period at arbitrary epoch $N$ is $P_{\rm puls}(N) = P_{\rm puls}(M_{0}) + \dot{P}_{\rm puls}\times N$.
The first step of the non-linear LSM is linearising the non-linear model function (Eqs.~\ref{eq:liteEphLin}~or~\ref{eq:liteEphPar}) by Taylor decomposition of the first order \citep[see][]{mikulasek2006,mikulasek2011a}
\begin{equation}
T_{\rm cal} \cong T_{\rm cal}(T,\textbf{b}_{0}) + \sum_{j=1}^{g} \Delta b_{j}\frac{\partial T_{\rm cal}(T,\textbf{b})}{\partial b_{j}},
\label{eq:liteEph}
\end{equation}
where $b_{j}$ are individual free parameters in vector $\textbf{b}$. Vector $\textbf{b}_{0}$ contains initial estimates of parameters, $\Delta b_{j}$ are their corrections, $g$ is the number of free parameters (the length of matrix $\textbf{b}$).
After linearisation, the problem can be solved in the same way as in the linear LSM, but with several necessary iterations to obtain a precise solution. Our code generates the
initial parameters quasi-randomly many times from large intervals with limits that are defined by user. The derivatives are solved analytically. For more details see Appendix~\ref{liteappendix1}.
The parameter $\chi^{2}(\textbf{b}_{k})$ or its normalised value $\chi_{\rm R}^{2}(\textbf{b}_{k}) = \chi^{2}(\textbf{b}_{k})/(n-g)$, where $n$ is a number of measurements, was used as an indicator of the quality of the $k$-fit.
Since many maxima timings from the GEOS database are given without errors or are often questionable, we used an alternative approach to
determine the weights. The dataset was divided into several groups according to the type of observations (photographic, photoelectric, CCD, DSLR), and weights were assigned to each of the groups with respect to the dispersion of points around the model. The weights were improved iteratively. Groups with fewer than five points (DSLR) were merged with another group with similar data quality to avoid unrealistic weight assignment (CCD+DSLR). During the fitting process, outliers differing by more than $5\,\sigma$ from the model were rejected. All steps of the analysis were supervised visually.
The LiTE~fitting process does not allow estimating the masses of the two stars, but only a mass function $f(\mathfrak{M})$
\begin{equation}
f(\mathfrak{M}) = \frac{( \mathfrak{M}_{2}\,\sin i )^{3}}{(\mathfrak{M}_{1} + \mathfrak{M}_{2} )^2} = \frac{4\,\pi^{2}}{G}\,\frac{(a_{1}\,\sin i)^{3}}{P_{\rm orbit}^2},
\label{eq:massFunction}
\end{equation}
where $\mathfrak{M}_{1}$, $\mathfrak{M}_{2}$ are masses of the components, $i$ is the inclination angle of the orbit, $P_{\rm orbit}$ is the orbital period, and $a_{1}$ is the semi-major axis of the primary component. Based on the studies of \citet{fernley1993} and \citet{skarka2014}, we adopted as the value for the mass of the RR Lyrae component $\mathfrak{M}_{1} = 0.55$\,$\mathfrak{M}_{\odot}$ and set the inclination angle to $i = 90\,^{\circ}$ ($\sin i = 1$). This allows computing the lowest mass of the second component by solving the cubic equation
\begin{equation}
\mathfrak{M}_{2}^3 - f(\mathfrak{M})\,\mathfrak{M}_{2}^2 - 2\,f(\mathfrak{M})\,\mathfrak{M}_{1}\,\mathfrak{M}_{2} - f(\mathfrak{M})\,\mathfrak{M}_{1}^2 = 0.
\label{eq:cubicEquation}
\end{equation}
We expect that the variation in the \textit{O-C}~diagram of \object{TU~UMa} can be well described using Eq.~\ref{eq:liteEphPar} and that other possible secular variation of the pulsation period can be neglected (it has a low amplitude or appears to be on a longer timescale).
\subsection{Bootstrap-resampling}\label{modelbootstrapsubsec}
The (non-linear) LSM itself gives an estimate of the uncertainty of all fitted parameters, but the method is extremely sensitive to data characteristics. A slight change of the dataset by adding one single measurement can cause a significant difference in the new parameters from the previous solution. We therefore decided to use a statistical approach represented by bootstrap-resampling to estimate the errors.
The parameters from the best solution were used as initial values to fit a new dataset, whose points were randomly selected from the original dataset. This procedure was repeated 5000 times. From the scattering of individual parameters of these five thousand solutions, we estimated their uncertainties. The errors in Tables~\ref{CLAurtable}
and~\ref{TuUMatable} correspond to 1\,$\sigma$.
\subsection{Test object CL Aurigae: an eclipsing binary with probable LiTE~and mass transfer}\label{CLAursubsec}
The code was, among others, tested on the well-known detached binary system \object{CL Aur}. This eclipsing binary was chosen because it shows
the LiTE~and a secular period change. In addition, \object{CL Aur} was studied in a similar way three times during the past 15 years \citep{wolf1999,wolf2007,lee2010}.
The times of minima of \object{CL Aur} taken from the \textit{O-C}~gateway database\footnote{http://var.astro.cz/ocgate/, all used minima timings are available at the CDS portal.} \citep{paschke2006} were used to construct \textit{O-C}~diagram and to determine parameters through the methods described above\footnote{The model was calculated for the half-value of the period. Subsequently, results were corrected for this effect.}. Our best model (Fig.~\ref{Fig:CLAur}) describes the \textit{O-C}~variations very well in the most recent part (precise CCD observations). The old part of the \textit{O-C}~diagram with visual and photographic measurements is highly scattered, but these measurements were also taken into account during the fitting process by assigning them a lower weight (model weights for different observation methods were found in the ratio pg:vis:ccd 1:11.5:403)\footnote{\citet{wolf2007} visually distinguished the data quality by weights in each category 0, 1, 2 for pg, 0, 1, 2 for visual, and 5, 10, 20 for CCD observations, respectively.}. We conclude that our results are comparable with previous results (Table~\ref{CLAurtable}). This example, as well as additional testing, showed that our code works well and is suitable for analysing RR Lyraes with suspected LiTE. The code was successfully used to model the LiTE~in \object{V2294 Cyg} \citep{liska2014b}.
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{CL_Aur_OC}
\caption{\textit{O-C}~diagram of the testing eclipsing binary CL Aurigae (double star without an RR Lyrae component) with the LiTE~and parabolic trend (black circles). The model of changes (red line) is based on our parameters from Table~\ref{CLAurtable}.}
\label{Fig:CLAur}
\end{figure}
\begin{center}
\begin{table}[t]
\caption{Our determined parameters from the testing object \object{CL~Aur} (right) together with results from previous studies (left).}\label{CLAurtable}
{\tiny
\begin{center}
\def1.5{1.5}
\tabcolsep=1.1pt
\begin{tabular}{lcc|c}
\hline\hline
Study & \citet{wolf2007} & \citet{lee2010} & Our model\\
\hline
$P(M_{0})$ [d] & 1.24437505(18) & 1.24437498(17) & 1.24437488$^{+16}_{-12}$\\
$M_{0}$ [HJD] & 2450097.2712(5) & 2450097.27082(46) & 2450097.2716$^{+6}_{-7}$\\
$10^{-10}$$\dot{P}$ & \multirow{2}*{$4.05(6)^{*}$} & \multirow{2}*{$3.92(55)^{*}$} & \multirow{2}*{3.76$^{+30}_{-25}$}\\
$$[d\,d$^{-1}$] & & &\\
$10^{-10}\,$$a_{3}$ & \multirow{2}*{$2.52(4)$} & \multirow{2}*{$2.44(34)$} & \multirow{2}*{2.34$^{+17}_{-14}$}\\
$$[d\,cycle$^{-1}$] & & &\\
$\beta$ [ms\,yr$^{-1}$] & $12.8(2)^{*}$ & $12.4(1.7)^{*}$ & 11.9$^{+9}_{-7}$\\
$\beta$ [d\,Myr$^{-1}$] & $0.148(2)$ & $0.143(20)$ & 0.137$^{+10}_{-8}$\\
$P_{3}$ [yr] & 21.7(2) & 21.63(14) & 21.61$^{+19}_{-18}$\\
$T_{0}$ [HJD] & 2443880(80) & 2444072(56) & 2444020$^{+140}_{-190}$\\
$e$ & 0.32(2) & 0.337(53) & 0.27$^{+5}_{-3}$\\
$\omega$ $[^{\circ}]$ & 209.2(1.2) & 218.9(2.7) & 218$^{+6}_{-9}$\\
$A$ [light day] & 0.0144(12)$^{*}$ & 0.01378(72)$^{*}$ & 0.01388$^{+35}_{-25}$\\
$a_{12}\sin i$ [au] & 2.49(22)$^{*}$ & 2.38(12) & 2.40$^{+6}_{-4}$\\
$f(\mathfrak{M}_{3})$ [$\mathfrak{M}_{\odot}$] & 0.034 & 0.0290(15) & 0.0297$^{+19}_{-14}$\\
$K_{12}$ [km\,s$^{-1}$] & $-$ & $-$ & 3.44$^{+10}_{-6}$\\
$\chi_{\rm R}^2$ & $-$ & $-$ & 1.04(10)\\
$N_{\rm min}$ & 144 & 198 & 203\\
\hline
\end{tabular}
\end{center}
{\bf Notes.} $^{(*)}$ Parameter calculated using values from the original study.}
\end{table}
\end{center}
\section{LiTE in \object{TU UMa} and analysis of the $\textrm{\textit{O-C}}$~diagram}\label{resultstuumasubsec}
Cyclic changes in \textit{O-C}~diagram of \object{TU UMa} have been well known for a long time and were analysed in detail by \citet{saha1990a}\footnote{Several parameters from \citet{saha1990a} e.g. $A$, $a\,\sin i$ were corrected in their erratum \citep{saha1990b}.}, \citet{kiss1995}, and \citet{wade1999}. The parameters determined by these authors are given in Table~\ref{TuUMatable}.
Our dataset described in Sect.~\ref{datasec} is much denser and more extended than in previous studies (in contrast to \citet{wade1999}, who used only 83, we used 253
maxima timings). Our \textit{O-C}~values span 113 years, which is mainly due to measurements recorded on the Harvard plates (provided by the project DASCH) in the first half of the twentieth century. Because the complete dataset is not homogeneous (it has incomparably better coverage over the last two decades with CCD measurements), we analysed LiTE~in \object{TU UMa} in two ways. Model 1 is based on the whole dataset\footnote{Weights for model 1 were found to be of the ratio pg\,:\,pe\,:\,CCD+DSLR 1.0\,:\,7.6\,:\,14.8, uncertainties are 0.0066\,d, 0.0024\,d, and 0.0017\,d.}, while model 2 describes only photoelectric, CCD, and DSLR observations\footnote{Weights for model 2 were of the ratio pe\,:\,CCD+DSLR 1.00\,:\,1.38, uncertainties are 0.0020\,d and 0.0017\,d.}. Since \object{TU UMa} experiences secular period decrease (in Fig.~\ref{Fig:TUUMa} represented by the parabolic dot-dashed curve) with the rate $1/2\,P_{\rm puls}\,\dot{P}_{\rm puls}$ of about $-2.9\times10^{-11}$ days per cycle \citep{wade1999}, we used the complex form of the model (Eq. \ref{eq:liteEphPar}).
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_OC_all_out_vis}
\includegraphics[width=1.00\hsize]{TU_UMa_OC_pe_ccd_model_pe_ccd}
\caption{\textit{O-C}~diagram of \object{TU UMa}. Black circles and blue stars display the maxima adopted from the GEOS database
and new maxima determined in this work. The period decrease manifested by the parabolic trend (dot--dashed line) is obvious. Cyclic changes due to an orbital motion are also clearly visible. Our model of LiTE is represented by the solid red line. The top panel shows model 1 with all available data, while the plot in the bottom panel shows the situation with only photoelectric, CCD, and DSLR observations (model 2).}
\label{Fig:TUUMa}
\end{figure}
Logically, model 1, which is based on the whole data set (covering five orbital periods) gives more precise, but slightly different results than model 2, which spans only two of the 23-year orbital cycles. Nevertheless, high-accurate photoelectric and CCD measurements cover the whole orbital cycle very well (see Fig.~\ref{Fig:TUUMaPhase}). Because of the shorter time base, our second model has a lower value of the secular period evolution, for example. This happens at the expense of an increasing eccentricity (0.66 and 0.69 for models 1 and 2, respectively) and change in the argument of periastron (181 and 184$^{\circ}$).
\begin{center}
\begin{table*}[t]
\caption{Parameters determined previously (left) and our results (right) for the system \object{TU UMa}. The mass limit of the second body was estimated from the mass function $f(\mathfrak{M})$, the orbit
inclination $(i=90^{\circ}),$ and the mass of the RR Lyrae star $\mathfrak{M}_{1} = 0.55$\,$\mathfrak{M}_{\odot}$ adopted from \citet{fernley1993} and \citet{skarka2014}. Our parameters (right part of the table) were calculated from all maxima timings for model~1 and only from photoelectric, CCD, and DSLR measurements for model~2. }\label{TuUMatable}
{\tiny
\begin{center}
\def1.5{1.3}
\begin{tabular}{lccc|cc}
\hline\hline
Study & \citet{saha1990a} & \citet{kiss1995} & \citet{wade1999}$^{C}$ & model 1 & model 2\\
\hline
$P_{\rm puls}(M_{0})$ [d] & 0.5576581097 & 0.5576581097 & 0.55765817(29)$^{D}$ & 0.557657598$^{+19}_{-20}$ & 0.557657477$^{+20}_{-25}$\\
$M_{0}$ [HJD] & 2425760.4364 & 2425760.4364 & 2425760.464(5)$^{D}$ & 2442831.4869$^{+4}_{-5}$ & 2442831.4864$^{+4}_{-4}$\\
$\dot{P}_{\rm puls}$ & \multirow{2}*{$\times$} & \multirow{2}*{$-31.48^{*}$} & \multirow{2}*{$-10.4(7)^{*}$} & \multirow{2}*{$-6.99^{+30}_{-25}$}& \multirow{2}*{$-4.55\,^{+50}_{-40}$}\\
$10^{-11}$[d\,d$^{-1}$] & & & &\\
$a_{3}\!=\!1/2\,P_{\rm puls}\dot{P}_{\rm puls}$ & \multirow{2}*{$\times$} & \multirow{2}*{$-8.78^{*}$} & \multirow{2}*{$-2.9(2)$} & \multirow{2}*{$-1.95^{+8}_{-7}$}& \multirow{2}*{$-1.27\,^{+13}_{-10}$}\\
$10^{-11}\,$[d\,cycle$^{-1}$] & & & &\\
$\beta\!=\!\dot{P}_{\rm puls}$ [ms\,yr$^{-1}$] & $\times$ & $-9.934^{*}$ & $-3.3(2)^{*}$ & $-2.21^{+9}_{-7}$ & $-1.44\,^{+15}_{-11}$\\
$\beta\!=\!\dot{P}_{\rm puls}$ [d\,Myr$^{-1}$] & $\times$ & $-0.11498^{*}$ & $-0.038(3)^{*}$ & $-0.0255^{+10}_{-8}$ & $-0.0166\,^{+17}_{-13}$\\
$P_{\rm orbit}$ [yr] & 20.19$^{*}$ & 24.1(3)$^{*}$ & 23.27(24)$^{*}$ & 23.30$^{+6}_{-6}$ & 23.27$^{+6}_{-8}$\\
$T_{0}$ [HJD] & 2425000$^{*}$ & 2447200(50) & 2421585(207) & 2447092$^{+40}_{-40}$ & 2447124$^{+45}_{-40}$\\
$e$ & 0.970 & 0.90(5) & 0.74(10) & 0.663$^{+30}_{-35}$ & 0.686$^{+25}_{-25}$\\
$\omega$ $[^{\circ}]$ & 196.1$^{*}$ & 178(3) & 183(5)$^{*}$ & 181.3$^{+2.5}_{-2.0}$ & 184.1$^{+2.0}_{-2.0}$\\
$A$ [light day] & 0.056 & 0.023(5)$^{*}$ & 0.0203(35)$^{*}$ & 0.0168$^{+5}_{-6}$ & 0.0172$^{+6}_{-5}$\\
$a_{1}\sin i$ [au] & 10 & 4.0(7)$^{*}$ & 3.52(61) & 2.91$^{+9}_{-10}$ & 2.99$^{+11}_{-9}$\\
$f(\mathfrak{M})$ [$\mathfrak{M}_{\odot}$] & $-$ & 0.11(1) & 0.080 & 0.046$^{+5}_{-4}$ & 0.049$^{+6}_{-4}$\\
$\mathfrak{M}_{\rm 2, min}$\,$^{*}$ [$\mathfrak{M}_{\odot}$] & $-$ & 0.17 & $-$ & 0.327$^{+14}_{-14}$ & 0.339$^{+16}_{-14}$\\
$K_{1}$ [km\,s$^{-1}$] & 60.7$^{*}$ & 11.4(5) & 6.6 & 4.97$^{+35}_{-35}$ & 5.25$^{+40}_{-30}$\\
$\chi_{\rm R}^2$ & $-$ & $-$ & $-$ & 1.05(9) & 1.03(10)\\
$N_{\rm max}$ & $\sim$\,43 & $\sim$\,42 & 67 & 253 & 220\\[1mm]
\hline
\end{tabular}
\end{center}
{\bf Notes.} $^{(*)}$ Parameter calculated using values from the original study, $^{(C)}$ their approach C was selected, $^{(D)}$ pulsation elements are known only from their approach D.}
\end{table*}
\end{center}
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_OC_pe_ccd_parabolic_model_pe_ccd_phased}
\caption{\textit{O-C}~diagram of \object{TU UMa} constructed only from photoelectric, CCD, and DSLR measurements after subtracting the parabolic trend and phased with the orbital period based on model~2.}
\label{Fig:TUUMaPhase}
\end{figure}
In comparison with orbital elements from previous studies given in Table~\ref{TuUMatable}, our results have a higher confidence level that is due to the larger and better dataset. Our values differ mainly in eccentricity and distance between components and in the mass function. All these values were found to be significantly lower than those from \citet{saha1990a}, \citet{kiss1995}, or \citet{wade1999}. Values from \citet{saha1990a} differ more because
they neglected the period decrease.
The eccentricity is not as extreme as proposed by \citet{saha1990a}. Its value is comparable with other systems from \citet{hajdu2015,li2014}.
Based on our results, it seems that \object{TU UMa} is very likely a member of a well-detached system with a dwarf component with a minimal mass of only 0.33\,$\mathfrak{M}_{\odot}$. Since no signs of the companion are observed in the light of \object{TU UMa}, it is probably a late-type main-sequence dwarf star. However, we cannot exclude the possibility that it is a white dwarf or a neutron star, since we do not know the inclination.
Except for the LiTE, which is the most remarkable feature of the \textit{O-C}~diagram, changes represented by the parabolic trend are also
apparent. The progression of the dependence suggests secular shortening of the pulsation period of \object{TU UMa}, which is almost certainly an evolutionary effect because the mass transfer, which is responsible for period changes in close binaries, can be excluded because of the very wide orbit of \object{TU UMa}. In addition, value $\beta = \dot{P}_{\rm puls} \!\sim\! -2.2$\,ms\,yr$^{-1}\!=\!-0.026$\,d\,Myr$^{-1}$ can correspond to a blueward evolution of the RR Lyrae component, but \citet{leborgne2007}, for instance, reported a higher median value $\beta = -0.20$\,d\,Myr$^{-1}$ for their sample of 21 stars with a significant period decrease.
After subtracting our model 1 from the whole dataset~(Fig.~\ref{Fig:ocResidual}), several photographic measurements (in the range of between JD 2425000 and 2429000) are deviate more than other values, with a systematic shift of about 15 minutes (0.01\,d). This might indicate that \object{TU UMa} undergoes more complex period changes than the LiTE~and parabolic trend alone. Some possible explanations such as cubic trend or an additional LiTE~could describe this variation. However, even though the influence of longer exposure of photographic measurements was taken into account, some other instrumental artefact might play a role.
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_OC_all_out_vis_residual}
\includegraphics[width=1.00\hsize]{TU_UMa_OC_pe_ccd_model_pe_ccd_residual}
\caption{Residual \textit{O-C}~diagram of \object{TU UMa} after subtracting the first LiTE model~(top panel) and second model -- only photoelectric, CCD, and DSLR observations (bottom). Black circles and blue stars display maxima adopted from the GEOS database
and new maxima determined in this work. The jump in the general trend of \textit{O-C}~in the range from JD 2425000 to 2429000 (top panel) might be an indication of a more complex period change.}
\label{Fig:ocResidual}
\end{figure}
\section{Other proofs for binarity}\label{proofsec}
Since the LiTE~is only an indirect manifestation of binarity, it is necessary to prove it in a different way. The analysis of the mean RVs can be considered as the most valuable test. In addition to this method, other possible approaches with which
the binarity of \object{TU UMa} can be confirmed are discussed in the next sections.
\subsection{Radial velocities}\label{rvsec}
The known orbital parameters from the analysis of the LiTE~allow us to predict, but also reconstruct, the RV curve from the past. The binarity can then be proved by comparing the model for the orbital RV curve and the spectroscopically determined centre-of-mass RV. This analysis was first performed for \object{TU UMa} by \citet{saha1990a} and a few years later by \citet{wade1999}. They noted systematic shifts in the RV determined at different times. Their predictions correlate fairly well with measurements.
We scanned the literature for RV measurements and found nine sources (Table~\ref{RVPublicationtable}). Unfortunately, the last available dataset with RVs was published in 1997. \citet{saha1990a} used only three RV sources, \citet{wade1999} did not give their values. In addition, both authors ignored the RV measurements from \citet{preston1964}. Other authors have reported slightly different values determined from the same dataset (Table~\ref{Tab:RVvalues}, Col. 2) without listing the mean time of observation. Therefore we decided to re-analyse all available RV measurements.
Determining the centre of mass of the RV for a binary with a pulsating star is more complicated than for a binary with non-variable stars (pulsations are often the dominant source of RV changes). Other inconveniences are connected with an RV based on different types of spectral lines (e.g. Balmer or metallic lines). They are formed at different depths, and therefore RV curves from different lines have different shapes, amplitudes, and zero points \citep[shown e.g. by][]{sanford1949,oke1962}. Since available RV measurements are based on various lines, it was necessary to unify them. This was done using the highly accurate normalised template curves from \citet{sesar2012}. Firstly we modelled these template curves with an $n$-order harmonic polynomial. The observed RV curve for the particular spectral line was then compared with the polynomial image of the template, and the amplitude and the central value of RV curve was simultaneously determined by the LSM for all datasets. Before this step, measurements were time-corrected for binary orbit and period shortening (based on model 1), and several of the datasets were divided into smaller groups to obtain a time resolution of about one year (see Table \ref{Tab:RVvalues} with the determined mean RVs for given epochs corresponding to the mean value of observation time). We did not find original RV measurements for two studies \citep{fernley1997,solano1997} and therefore adopted their mean RV values and estimated the
mean time of observation from the information provided in their papers. Finally, the mean centre of mass of the RV values were compared with the RV model resulting from our LiTE~analysis (Fig.~\ref{Fig:TUUMaRVmodel}). The points roughly follow the model RV curve.
\begin{center}
\begin{table}[t]
\centering
\caption{Sources of RV measurements for \object{TU UMa}, $S$ is a source number, $N_{\rm RV}$ is a number or RV measurements.}\label{RVPublicationtable}
{\tiny
\def1.5{1.5}
\tabcolsep=1.3pt
\begin{tabular}{llcc}
\hline\hline
$S$ & Publication & $N_{\rm RV}$ & Lines\\
\hline
{\bf 1} & \citet{abt1970} & 1 & Unknown\\
{\bf 2} & \citet{barnes1988} & 74 & Metallic\\
{\bf 3} & \citet{fernley1997} & 3? & OI triplet, (H$_{\alpha}$)\\
{\bf 4} & \citet{layden1993,layden1994} & 5 & Hydrogen, CaII K\\
{\bf 5} & \citet{liu1989} & 60 & Metallic\\
{\bf 6a} & \multirow{2}*{\citet{preston1961}} & 4 & Metallic\\
{\bf 6b} & & 21 & H$_{\gamma}$, H$_{\delta}$, H$_{8-11}$, Ca II K\\
{\bf 7a} & \multirow{2}*{\citet{preston1964}} & 12 & Metallic\\
{\bf 7b} & & 8+7 & Hydrogen\\
{\bf 8} & \citet{saha1990a} & 32 & Metallic\\
{\bf 9} & \citet{solano1997} & 3? & Metallic, (H$_{\gamma}$)\\
\hline
\end{tabular}}
\end{table}
\end{center}
\begin{center}
\begin{table}[t]
\begin{center}
\caption{Determined values of centre-of-mass velocities for \object{TU UMa} based on different RV measurements and templates from \citet{sesar2012}. The mean values published in different publications are present for comparison, $S$ is the source number of the original RV measurements from Table~\ref{RVPublicationtable}.}\label{Tab:RVvalues}
{\tiny
\def1.5{1.5}
\tabcolsep=1.3pt
\begin{tabular}{lc|ccc}
\hline\hline
$S$ & RV$_{\rm pub}$ & $T_{\rm mid}$ & RV$_{\rm our}$ & errRV$_{\rm our}$ \\
& [km\,s$^{-1}$] & [HJD] & [km\,s$^{-1}$] & [km\,s$^{-1}$] \\
\hline
{\bf 1} & 104(35)$^{La}$ & 2426076 & 104$^{La}$ & 35$^{La}$ \\
\hline
{\bf 2} & 90(2)$^{F}$, 90(2)$^{So}$ & 2443563 & 95 & 3 \\
& & 2443941 & 98 & 3 \\
& & 2444218 & 90 & 1 \\
& & 2444948 & 88 & 4 \\
\hline
{\bf 3} & 101(3)$^{F}$, 101(5)$^{So}$ & 2449520 & 101$^{F}$ & 3$^{F}$ \\
\hline
{\bf 4} & 75(17)$^{La}$, 75(17)$^{F}$ & 2447975 & 75$^{La}$ & 17$^{La}$ \\
\hline
{\bf 5} & 84.2$^{Li}$, 84$^{Sa}$, 84(1)$^{La}$, & 2446843 & 84 & 1 \\
& 84(1)$^{F}$, 84(2)$^{So}$ & 2447130 & 85 & 3 \\
\hline
{\bf 6a} & -- & 2436979 & 93 & 1 \\
\hline
{\bf 6b} & 92(1)$^{P}$, 87$^{H}$, & 2436648 & 93 & 2 \\
& 93(3)$^{Sa}$, 92(1)$^{F}$ & 2436979 & 94 & 4 \\
\hline
{\bf 7a} & -- & 2438039 & 94 & 2 \\
\hline
{\bf 7b} & -- & 2438039 & 94 & 3 \\
\hline
{\bf 8} & 77$^{Sa}$, 77(2)$^{F}$, 77(2)$^{So}$ & 2446894 & 76 & 1 \\
\hline
{\bf 9} & 96(3)$^{So}$ & 2449600 & 96$^{So}$ & 3$^{So}$ \\
\hline
\end{tabular}}
\end{center}
{\bf Notes.} Values adopted from $^{(F)}$\citet{fernley1997}, $^{(H)}$\citet{hemenway1975}, $^{(La)}$\citet{layden1994}, $^{(Li)}$\citet{liu1990},
$^{(P)}$\citet{preston1961}, $^{(Sa)}$\citet{saha1990a}, $^{(So)}$\citet{solano1997}.
\end{table}
\end{center}
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_RV_all_phase_value}
\caption{Models of variations in RV caused by orbit of pulsating component around mass-centre of the binary system (red and blue lines) and center-of-mass velocities determined for each dataset of RV measurements using template fitting or adopted from literature. The visually estimated correction $-91$\,km\,s$^{-1}$ for systematic velocity mass-centre of the system from Sun ($\gamma$-velocity) was applied.}
\label{Fig:TUUMaRVmodel}
\end{figure}
An alternative test for binarity using RV curves can be performed by comparing the observed RV curves (the top panel of Fig.~\ref{Fig:TUUMaRVphase}) and those in which the orbital RV curve from the model is subtracted (the bottom panel of Fig.~\ref{Fig:TUUMaRVphase}). In~this figure RV measurements are phased according to the pulsation period\footnote{The stitching in phase was possible only by taking the LiTE~and secular period change into account. Without this, the curve was scattered horizontally.}. Apparently, the phased RV curve with corrected velocities is significantly less vertically scattered than without the correction. The residual scatter in the bottom panel results from the different metallic lines that the RVs were based on.
Both tests clearly show that \object{TU UMa} is very likely bound in a binary system.
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_RV_all_obs_notcorr}
\includegraphics[width=1.00\hsize]{TU_UMa_RV_all_obs_corr}
\caption{Radial velocity curves from the metallic lines of \object{TU UMa} from different publications phased with the pulsation period corrected for the LiTE~and secular period changes. Uncorrected observed RVs (top) and corrected values after subtracting the
changes in RV caused by orbital motion based on our model 1 (bottom). RV values corrected for binary orbit are evidently less scattered than uncorrected RVs.}
\label{Fig:TUUMaRVphase}
\end{figure}
\subsection{Eclipses}\label{eclipsesec}
A detection of eclipses in the light curve (in the appropriate phase of the orbit) would be a strong proof for binarity of \object{TU UMa}. Several third components of eclipsing binary stars that
were known only from the LiTE\ were confirmed by detecting additional eclipses \citep[e.g. in the {\it Kepler} project,][]{slawson2011}. The probability of catching an eclipse in \object{TU~UMa} is very low because the expected orbital period of the binary system is very long and radius and luminosity of the secondary component are probably much smaller and lower than for the pulsating star. In addition, the inclination angle of the orbit is unknown. Our two preliminary LiTE models allow us to estimate the time of a possible eclipse, where the RR Lyrae component should transit the secondary one (January -- February 2014 or May -- June 2014), but the difference between the
two predictions is too large. However, we attempted to detect the proposed eclipse.
Observations with the small telescope described in Sect.~\ref{observsubsec} were dedicated for this purpose. Unfortunately, our measurements were insufficient for a reliable decision about eclipses. Weather conditions, limited object visibility and other influences allowed
us to observe in only 19 nights, which is hardly sufficient considering the imprecise eclipse prediction. At least we can conclude that no sign of an eclipse with an amplitude higher than 0.07\,mag was detected in our data (see Fig.~\ref{Fig:LightCurveResidual}).
\begin{figure}[t]
\centering
\includegraphics[width=1.00\hsize]{TU_UMa_LC_residual}
\caption{Residuum of the light curve of \object{TU UMa} after subtracting
the harmonic polynomial model. No signs of an eclipse with an
amplitude higher than 0.07\,mag was detected in the \textit{green} band.}
\label{Fig:LightCurveResidual}
\end{figure}
\section{Summary and conclusions}\label{summarysec}
We presented a new analysis of a probable LiTE~in \object{TU~UMa}. We used published maxima timings from the GEOS database (168 values) and added maxima values from our photometric observations and from the SuperWASP and Pi of the Sky surveys. We applied the template fitting method to determine maxima from these measurements and also from sparse data from the projects Hipparcos, NSVS and DASCH. Altogether, we analysed 253 maxima timing measurements, which is about three times more than were used in the dataset in the last study of \object{TU UMa} by \citet{wade1999}. This large and well covered dataset allowed us to determine a quadratic ephemeris of the pulsations and orbital elements of the binary system with much better accuracy than in previous studies (Table~\ref{TuUMatable}). All analyses were performed with a new code written in Matlab that uses a bootstrap method to estimate the errors. We calculated two models: model~1, which describes the whole dataset (without visual maxima timings), and model~2, which describes only high-accurate photoelectric, CCD and DSLR maxima. The second model is based on data with a significantly shorter time span than for model 1.
The second model gives a lower value of the period-decrease rate ($\beta=\dot{P}_{\rm puls} \!\sim\! -1.4$\,ms\,yr$^{-1}$), which causes the eccentricity to become higher ($e \!\sim\! 0.69$) than in the first model ($\beta \!\sim\! -2.2$\,ms\,yr$^{-1}$, $e \!\sim\! 0.66$). For comparison, \citet{arellanoferro2013} give $\beta \!=\! -1.3$\,ms\,yr$^{-1}$ without fitting the LiTE. Nevertheless, both our models have a lower eccentricity value, semi-major axis of pulsating component $a_{1}\,\sin i$ (2.9\,au or 3.0\,au), and semi-amplitude of RV variations of the pulsating star $K_{1}$ (5.0\,km\,s$^{-1}$ or 5.3\,km\,s$^{-1}$) than in previous works. Our values of the orbital period (identically 23.3\,yr) and argument of periastron $\omega$ (181$^{\circ}$ or 184$^{\circ}$) are comparable with values determined by previous authors. In addition, the lowest mass limit of the secondary component (0.33\,$\mathfrak{M}_{\odot}$ or 0.34\,$\mathfrak{M}_{\odot}$) was determined with an assumption for the mass of the RR Lyrae component of 0.55\,$\mathfrak{M}_{\odot}$.
The binary nature was tested in several ways. Firstly, our models of the orbit gave predictions of possible eclipses. Although the prediction was highly inaccurate and an eclipse is highly unlikely (wide orbit, unknown inclination, and other important parameters) we attempted to detect them, but were unfortunately not successful.
Binarity is expected manifest itself in cyclic changes of the mean RV. We adopted RV measurements from nine independent sources and corrected their values according to our model by subtracting the LiTE~and secular changes. When an observed RV curve was phased with the pulsation period, we obtained the typical RV curve for RR Lyrae, which was scattered. The scatter significantly decreased when our model was applied.
We also determined central RV values for each RV dataset using pulsation templates for different spectral lines from \citet{sesar2012}. We compared these values with our model of orbital RV variations based on orbital parameters known from the LiTE. The correlation is evident (Fig.~\ref{Fig:TUUMaRVmodel}).
The two successful proofs are important for confirming the binarity of \object{TU UMa}. However, only long-term spectroscopic measurements covering the whole orbital cycle can unambiguously confirm that \object{TU~UMa} is indeed a member of a binary system.
\begin{acknowledgements}
The DASCH project at Harvard is grateful for partial support from NSF grants AST-0407380, AST-0909073, and AST-1313370.
This paper makes use of data from the DR1 of the WASP data \citep{butters2010} as provided by the WASP consortium, and the computing and storage facilities at the CERIT Scientific Cloud, reg. no. CZ.1.05/3.2.00/08.0144, which is operated by Masaryk University, Czech Republic. This research has made use of NASA's Astrophysics Data System. Work on the paper has been supported by LH14300. MS acknowledges the support of the postdoctoral fellowship programme of the Hungarian Academy of Sciences at the Konkoly Observatory as a host institution. We are very grateful to the anonymous referee, who significantly helped to improve this paper.
\end{acknowledgements}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: How do I open a directory under the cursor in vim? A gf-like for directories For instance, if I had something like:
path = '/mnt/data/files/'
and my cursor was over '/mnt/data/files/', I'd hit a gf-like command and it would open vim :Explorer in that folder.
I've searched through StackOverflow, but I only found answers on how to open the directory of the current open file.
A: Pressing <C-r><C-f> in the command-line inserts the path under the cursor. This means that you can do the following to achieve your goal:
:Ex <C-r><C-f><CR>
From there, you could simply create a custom mapping:
nnoremap <key> :Explore <C-r><C-f><CR>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Weekly Market Recap: 9/21/2018
Market Recap: Week Ending 9/21/2018
In the markets:
U.S. Markets: U.S. stock indexes finished the week mixed with the Dow and the S&P 500 at or near record highs, while the NASDAQ continued to see pressure amid weakness in tech stocks. The Dow Jones Industrial Average rose 588 points, or 2.3%, to end the week at 26,744. The Nasdaq Composite gave up 23 points to close at 7,987, a loss of 0.3%. By market cap, large caps powered ahead 0.9%, while smaller cap indexes finished in the red. The mid cap S&P 400 finished the week down -0.3%, along with the small cap Russell 2000 which fell -0.6%.
International Markets: Canada's TSX rebounded 1.3% after last week's loss, while the United Kingdom's FTSE rose for a second week by adding 2.6%. On Europe's mainland, major markets finished well into the green. France's CAC 40 surged 2.7%, Germany's DAX added 2.5%, and Italy's Milan FTSE followed last week's 2.1% gain with an additional 3.1% gain this week. In Asia, China's Shanghai Composite shot higher by 4.3%, and Japan's Nikkei gained 3.4%. As grouped by Morgan Stanley Capital International, developed markets gained 2.8% while emerging markets added 3%.
Commodities: Energy had its second week of gains, with West Texas Intermediate crude oil rising 2.6%, or $1.79, to $70.78 per barrel. Precious metals finished the week in the green. Gold rose just 0.02% to $1201.30 an ounce, while Silver gained 1.5%, ending the week at $14.36. The industrial metal copper rebounded a big 8% after being down five of the last seven weeks. Copper is sometimes viewed as an indicator of global economic health due to its variety of uses.
U.S. Economic News: Initial claims for new unemployment benefits fell by 3,000 last week to their lowest level since November of 1969. The Labor Department reported Initial Jobless Claims fell to 201,000 in the seven days ended September 15. The reading was below forecasts of an increase of 6,000. The four-week average of new claims, smoothed to lower the volatility, also declined down 2,250 to 205,750—its lowest level since December 1969. Continuing claims, which counts the number of people already receiving benefits, declined by 55,000 to 1.65 million. That number is at its lowest level since 1973. These numbers continue to reflect an economy with very tight labor market conditions.
The number of new homes under construction rebounded 9.2% last month to their highest level since January. The Commerce Department reported housing starts ran at a seasonally-adjusted 1.282 million annual rate in August, beating expectations for a 5.3% increase and up 9.4% from the same time last year. However, the details of the report painted a slightly less robust picture. Single-family home starts, which most economists consider the most critical component by which to judge builder activity, edged up only slightly – just 1.9%. But multi-family starts (apartments and condominiums) surged 27.3%–the most since December 2016. In addition, building permits, an indicator of future building activity, fell -5.7%–the most since February of 2017. The decline suggests that there is less construction in the pipeline which could translate into slower housing starts in the coming months.
Sales of existing homes remained unchanged at 5.34 million last month, matching its lowest level since February of 2016. However, it was the first time in five months that sales did not decline, perhaps indicating stabilization. The reading missed forecasts of a 0.7% gain. Year-over-year, existing home sales were down 1.5%. By region, sales surged 7.6% in the Northeast, rose 2.4% in the Midwest, but decreased 0.4% in the South and slumped 5.9% in the West. Inventories, expressed as months of available supply, remained unchanged at 4.3 indicating continued housing shortages. More than half of the properties sold in August were on the market for less than a month.
Confidence among the nation's home builders held steady this month according to the National Association of Home Builders/Wells Fargo Housing Market Index (HMI). The HMI remained unchanged at 67 in September, at its lowest level in a year. The consensus was for a one point decline to 66. Although lumber prices had declined since earlier in the summer, homebuilders remained concerned about rising material costs amid growing trade tensions. Builders had a slightly more favorable view of current and expected single-family home sales, although buyer traffic remained unchanged. By region, confidence shot up in the Northeast, but either declined or remained unchanged in the other three regions.
The latest reading from the Conference Board's Leading Economic Index (LEI) suggests an economy poised for 3% growth for the rest of this year and broadly positive growth going into 2019. The LEI rose 0.4% last month, slightly below the consensus of 0.5%, but the previous month was revised up 0.1% to 0.7%. Seven of the ten LEI components made positive contributions, led by stronger new orders and better credit conditions. In addition, the year-over-year momentum held steady at 6.4%, near its fastest pace since July of 2014 and nearly three times the gain per annum historically. If the economy keeps it up, the U.S. might have its first full calendar year of 3% growth since 2005. Ataman Ozyildirim, economist at the board stated, "The leading indicators are consistent with a solid growth scenario in the second half of 2018 and at this stage of a maturing business cycle in the U.S., it doesn't get much better than this."
In the New York region, factories churned out goods at a slower but still flourishing pace this month according to a survey of executives. The New York Federal Reserve's Empire State Manufacturing index fell 6.6 points in September to 19.0—a five-month low. Economists had expected just a 3.6 point decline to 22. In the details, shipments and new orders growth eased, while employment and average hours worked pick up. Firms tempered their expectations about growth in the near term as the Future Business Conditions Index retreated 4.5 points—its second decline in the last three months. With regard to inflation pressures, executives reported prices paid rose at a quicker rate this month than prices received.
The Philadelphia Fed's measure of manufacturing activity in the mid-Atlantic area picked up by rebounding 11 points this month. The Philly Fed's General Business Activity Index came in with a reading of 22.9 this month, far exceeding forecasts of a 3.1 point gain to 15.0. Most of the individual indexes increased, led by the new orders index which jumped 11.5 points to 21.4. The outlook for growth in the next six months remained broadly positive, although the Future Activity Index slipped 2.5 points to 36.3. Overall, the future activity indicators pointed to continued expansion, despite a moderation in expected rates of growth over the course of this year.
International Economic News: The Conference Board of Canada now expects the Canadian economy to grow 2.0% in 2018, up 0.2% from its previous forecast. However, weakening employment growth and high levels of household debt are expected to slow the advance in the near future. Matthew Stewart, director of the national forecast stated, "The Canadian economy performed well in the second quarter of 2018, with consumers increasing their pace of spending, businesses raising investment in their capital stock and a double-digit increase in exports. However, significant challenges remain, which will slow growth over the remainder of the year and into 2019." Furthermore, if the US imposes significant tariffs and Canada retaliates, the Conference Board of Canada expects there would be about a 1.3% hit to Canada's economic growth over a two-year period.
The International Monetary Fund said there would be a "substantial cost" to the U.K. economy if a "no-deal" Brexit were to occur. "No-deal" is the term used to describe a scenario where the United Kingdom leaves the European Union without first agreeing to transitional arrangements. The U.K. is due to leave the European Union on March 29, 2019 but London and Brussels have yet to strike a deal over the terms of the UK's exit from the bloc. The IMF stated a "No-deal" Brexit will cause a "shock to the supply" of goods, shrink the UK's economy and reduce the value of its currency. IMF chief Christine Lagarde said the British economy would weaken under any outcome that involves leaving the bloc, but warned that a no-deal Brexit would cause an even greater and immediate slump.
On Europe's mainland, research firm IHS Markit's composite index of manufacturing and services for France fell 1.3 points this month to 53.6, missing economists' estimates of a 54.6 reading. The reading brings French economic growth to a 21-month low in a broad-based slowdown. Much of the slowdown appears to be due to a reduction in demand from the automotive sector, and the French economy has struggled to keep up the brisk pace it set in 2017.
German Chancellor Angela Merkel is facing a potential large scale exodus of German companies to the now "tax-friendlier" United States, according to the latest survey from the Munich-based Institute for Economic Research (IFO). The IFO study, which surveyed 1,250 companies, revealed that a shockingly high number – more than one in four – of the companies that already had representation in the United States wanted to expand their capacities as a result of the tax reform in the U.S. And some 10% of the company not already partially based across the Atlantic said they want to expand, naming the new tax reform as one of the main reasons for expanding.
Chinese Premier Li Keqiang stated this week that China was facing "greater difficulties" in keeping its economy stable. During an address at a World Economic Forum conference in Tianjin, China, Li acknowledged "China is confronted with a host of difficulties and challenges in economic development." Given China's deep integration into the world economy, Li stated, "The Chinese economy is inevitably affected by changes in the global economic and trade context. Indeed, we're facing greater difficulties in keeping stable performance of the Chinese economy."
The Bank of Japan kept its monetary policy steady this week and maintained its optimistic view of the economy even as growing global trade tensions clouded the outlook. In a move that was widely expected, the BOJ maintained its key short-term interest rate target at -0.1% and pledged to guide its 10-year government bond yields to near zero percent. The decision was an overwhelming 7-2 in favor. In its statement the BOJ noted, "Japan's economy is expanding moderately." The central bank also left unchanged a new forward guidance, adopted in July, which pledges to keep interest rates extremely low for an extended period.
Finally: The number of new jobless claims just reached its lowest level since November of 1969. That alone is impressive, but to truly get an understanding of the tightness of today's labor market it has to be compared to the size of the U.S. labor force. Indeed, the U.S. labor force has more than doubled over the last 49 years from 81,106,000 people to 161,776,000 last month. Jobless claims adjusted for the size of the U.S. labor force, are now at the lowest level since the Department of Labor started reporting monthly jobless claims in 1967.
(sources: all index return data from Yahoo Finance; Reuters, Barron's, Wall St Journal, Bloomberg.com, ft.com, guggenheimpartners.com, zerohedge.com, ritholtz.com, markit.com, financialpost.com, Eurostat, Statistics Canada, Yahoo! Finance, stocksandnews.com, marketwatch.com, wantchinatimes.com, BBC, 361capital.com, pensionpartners.com, cnbc.com, FactSet; Figs 1-5 source W E Sherman & Co, LLC)
Community Family Office
Team Member Spotlight
Weekly Market Recap: 12/28/2018
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
The study of the weak decays of quarks, and the measurement of the
corresponding CKM matrix elements, are consistently hampered by the presence
of the long distance QCD effects that are responsible for the binding of the
quarks into hadrons. These effects are hard to evaluate in a model independent
way, and so tend to bring large uncertainties to the theoretical predictions
for the weak decay amplitudes. They appear in the calculation of the matrix
elements of the weak Hamiltonian operators, between the initial and final
hadronic states. For practical purposes, they are included in standard sets of
independent, Lorentz invariant, form factors, that parametrize those hadronic
matrix elements in a convenient way, but are otherwise poorly known. Relations
between different form factors, that will hold under certain conditions or
approximations, can then be very useful: they will reduce the number of
uncertain quantities, and improve the accuracy of the theoretical predictions.
Moreover, they may help us understand better the general features of the
underlying long distance QCD effects.
Here, I am interested in the weak hadronic matrix elements for a pseudoscalar
$B$ to a vector meson $V$ transition (for definiteness, I use a $B$-meson in
my notation, but the results are quite general; they apply equally well to
$D$-mesons, or even to lighter pseudoscalars, if they were to decay weakly to
vector mesons). These matrix elements are parametrized by the form factors
$V$, $A_{0,1,2}$ and $F_{1,2,3}$, that are defined as follows:
\beqa
\lefteqn{\langle V(\vec{p^\prime},\vec{\varepsilon})|
\overline{q} \gamma^\mu b |B(\vec{p}) \rangle =
\frac{-1}{m_B + m_V} 2 i \epsilon^{\mu\alpha\beta\gamma}
\varepsilon_\alpha^\ast p_\beta^\prime p_\gamma V(k^2) \ ,}
\label{0}\\
\nonumber\\
\lefteqn{\langle V(\vec{p^\prime},\vec{\varepsilon})|
\overline{q} \gamma^\mu \gamma_5 b |B(\vec{p}) \rangle} \nonumber\\
& & = (m_B + m_V) \left( \varepsilon^{\ast\mu} -
\frac{\varepsilon^\ast.k}{k^2} k^\mu \right) A_1(k^2) \nonumber\\
& & - \frac{\varepsilon^\ast.k}{m_B + m_V}
\left( (p+p^\prime)^\mu - \frac{m_B^2 - m_V^2}{k^2} k^\mu \right) A_2(k^2)
\nonumber\\
& & + 2 m_V \frac{\varepsilon^\ast.k}{k^2} k^\mu A_0(k^2) \ ,
\label{1}\\
\nonumber\\
\lefteqn{\langle V(\vec{p^\prime},\vec{\varepsilon})|
\overline{q} i \sigma^{\mu\nu} k_\nu b |B(\vec{p}) \rangle
= i \epsilon^{\mu\alpha\beta\gamma}
\varepsilon_\alpha^\ast p_\beta^\prime p_\gamma F_1(k^2) \ ,}
\label{2}\\
\nonumber\\
\lefteqn{\langle V(\vec{p^\prime},\vec{\varepsilon})|
\overline{q} i \sigma^{\mu\nu} \gamma_5 k_\nu b |B(\vec{p})\rangle
\frac{}{}}
\nonumber\\
& & = \left( (m_B^2 - m^2_V) \varepsilon^{\ast\mu}
- \varepsilon^\ast.k (p+p^\prime)^\mu \right) F_2(k^2)
\nonumber\\
& & + \varepsilon^\ast.k \left( k^\mu - \frac{k^2}{m_B^2 - m^2_V}
(p+p^\prime)^\mu \right) F_3(k^2) \ ,
\label{3}
\eeqa
with $2 m_V A_0(0) = (m_B + m_V) A_1(0) - (m_B - m_V) A_2(0)$, $F_1(0) =
2 F_2(0)$ and $k \equiv p - p^\prime$.
In Ref.\ \cite{JMS}, a Quark Model description of the $B \to V$ hadronic
transition led to the following relations between the form factors $F_{1,2,3}$
and $A_{0,1,2}$:
\beqa
F_1(k^2) &=& 2 A_0(k^2) \ ,
\label{4}\\
\nonumber\\
m_V (m_B - m_V) F_2(k^2) &=& (m_B E^\prime - m_V^2) A_1(k^2)
\nonumber\\
& & - \frac{2 m_B^2 |\vec{p^\prime}|^2}{(m_B + m_V)^2} A_2(k^2) \ ,
\nonumber\\
\label{5}\\
(m_B E^\prime + m_V^2) F_2(k^2)
&-& \frac{2 m_B^2 |\vec{p^\prime}|^2}{m_B^2 - m_V^2} F_3(k^2)
\nonumber \\
&=& m_V (m_B + m_V) A_1(k^2) \ .
\label{6}
\eeqa
Note that these relations are valid in any reference frame; the energy
$E^\prime$ and momentum $|\vec{p^\prime}|$ of the vector meson $V$, in the
rest frame of the $B$-meson, are used as an abbreviation for the more
cumbersome invariant functions of $k^2$:
\beqa
E^\prime &=& \frac{m_B^2 + m_V^2 - k^2}{2 m_B} \ , \\
\label{7}
|\vec{p^\prime}| &=& \frac{\left[ (m_B^2 + m_V^2 - k^2)^2
- 4 m_B^2 m_V^2 \right]^{1/2}}{2 m_B} \ .
\label{8}
\eeqa
The fact that these relations were derived, independently of the details of
the momentum wavefunctions for the two mesons, suggests that they are a very
general Quark Model result. However, the proof in Ref.\ \cite{JMS} was not
entirely satisfactory, in that the origin of these form factor relations was
not clear, and the outcome seemed rather accidental. In what follows, I will
give an alternate and more general proof of the same relations. It will
clarify their origin, and the Quark Model assumptions in which they rely. The
end result will confirm that, in the Quark Model, the form factors $F_{1,2,3}$
for pseudoscalar to vector meson transitions are not independent form factors.
\section{Form factor relations}
The first step in deriving the form factor relations is to solve
eqs.~\ref{0}--\ref{3}, in order to write each form factor as a combination of
hadronic matrix elements. This is most easily done in the $B$ rest frame, with
the $z$-axis chosen along the momentum $\vec{p^\prime}$ of the vector meson
$V$. In this frame,
\beq
p = m_B (1, 0, 0, 0) \;\;\;\;\;\;\;
p^\prime = (E^\prime, 0, 0, |\vec{p^\prime}|) \ ,
\label{9}
\eeq
and the polarization vectors for the helicity states $\lambda = 0$, $\pm 1$
of the vector meson are
\beq
\varepsilon_0 = \frac{1}{m_V} (|\vec{p^\prime}|, 0, 0, E^\prime) \;\;\;\;\;\;
\varepsilon_\pm = \mp \frac{1}{\sqrt{2}} (0, 1, \pm i, 0) \ ;
\label{10}
\eeq
$k \equiv p - p^\prime = (m_B - E^\prime, 0, 0, -|\vec{p^\prime}|)$, and it is
convenient to define a 4-vector $r = N (|\vec{p^\prime}|, 0, 0, -m_B +
E^\prime)$, with arbitrary normalization, such that $r.k = 0$ and $\vec{r}$ is
parallel to $\vec{k}$, in the particular frame that we have chosen. The
expressions that are obtained for each of the form factors can then be
combined as follows:
\beqa
\lefteqn{F_1(k^2) = 2 A_0(k^2) \;\;
\frac{k_\mu \langle V(\lambda = \pm 1) | \overline{q} \frac{i}{\sqrt{2}}
( \sigma^{1 \mu} \pm i \sigma^{2 \mu} ) b | B \rangle}
{k_\mu \langle V(\lambda = 0) | \overline{q} \gamma^\mu \gamma_5 b
| B \rangle} \ ,}
\label{12}\\
\nonumber\\
\lefteqn{m_V (m_B - m_V) F_2(k^2)} \nonumber\\
& & = \left[ (m_B E^\prime - m_V^2) A_1(k^2)
- \frac{2 m_B^2 |\vec{p^\prime}|^2}{(m_B + m_V)^2} A_2(k^2) \right]
\nonumber \\
& & \times \;\; \frac{r_\mu \langle V(\lambda = \pm 1) | \overline{q}
\frac{i}{\sqrt{2}} ( \sigma^{1 \mu} \pm i \sigma^{2 \mu} ) b | B \rangle}
{r_\mu \langle V(\lambda = 0) | \overline{q} \gamma^\mu \gamma_5 b | B
\rangle} \ ,
\label{13}\\
\nonumber\\
\lefteqn{(m_B E^\prime + m_V^2) F_2(k^2)
- \frac{2 m_B^2 |\vec{p^\prime}|^2}{m_B^2 - m_V^2} F_3(k^2)}
\nonumber \\
& & = m_V (m_B + m_V) A_1(k^2) \; \nonumber\\
& & \times \;\; \frac{\varepsilon^\pm_\mu \langle V(\lambda = 0)
| \overline{q} \frac{i}{\sqrt{2}} ( \sigma^{1 \mu} \mp i \sigma^{2 \mu} )
b | B \rangle}
{\varepsilon^\pm_\mu \langle V(\lambda = \pm 1) | \overline{q}
\gamma^\mu \gamma_5 b | B \rangle} \ .
\label{14}
\eeqa
The remainder of the proof consists in showing that the ratios of hadronic
matrix elements, that appear on the right-hand-side (RHS) of
eqs.~\ref{12}--\ref{14}, are precisely equal to $1$, in the Quark Model.
In order to do so, we must relate the $\lambda = 0$ and $\lambda = \pm 1$
helicity states of the vector meson. If one is to rely on the Quark Model
angular momentum wavefunction for the vector meson, this can be done by
flipping the spin of the constituent quark $q$:
\beqa
\langle V(\lambda = \pm 1) | \overline{q} \Gamma b | B \rangle &=&
\sqrt{2} \langle V(\lambda = 0) | \; S_\pm^\dag \; \overline{q} \Gamma b
| B \rangle \ ,
\label{15}
\eeqa
where $S_\pm$ are the raising and lowering operators for the spin of the quark
$q$, along the direction of the vector meson momentum; $\Gamma$ is an
arbitrary combination of Dirac $\gamma$-matrices. The spin operator, for the
$q$-quark field, is \cite{I&Z}
\beqa
S_\sigma &=& \frac{1}{2 m} \epsilon_{\sigma\mu\nu\rho} J^{\mu\nu}
P^\rho \ ,
\label{16}
\eeqa
where
\beqa
J^{\mu\nu} &=& \int d^3x : q^\dag(x) \left[ i x^\mu D^\nu - i x^\nu D^\mu
+ \frac{1}{2} \sigma^{\mu\nu} \right] q(x) :
\label{17}
\eeqa
and
\beqa
P^\mu &=& \int d^3x : q^\dag(x) i D^\mu q(x) :
\label{18}
\eeqa
are the conserved angular and linear momentum operators; they satisfy the
commutation relations
\beqa
[J^{\mu\nu}, q(x)] &=& - ( i x^\mu D^\nu - i x^\nu D^\mu
+ \frac{1}{2} \sigma^{\mu\nu} ) q(x)
\label{19}
\eeqa
and
\beqa
[P^\mu, q(x)] &=& - i D^\mu q(x) \ .
\label{20}
\eeqa
The covariant derivative $D^\mu$ accounts for the strong interactions of the
quark field; it also appears in the equation of motion: $(i \not\!\!D - m)
q(x) = 0$. The spin projection operator along a direction $n$, with $P.n=0$
and $n^2=-1$, is $S.n$, and it is then easy to arrive at the central relation
in our proof,
\beqa
\langle V | (S.n)^\dag \overline{q} \Gamma b | B \rangle &=&
\frac{1}{2} \langle V | \overline{q} \gamma_5 \gamma_\sigma \Gamma b
| B \rangle n^{\sigma\ast} \ .
\label{21}
\eeqa
In the derivation, we assumed, as in the Quark Model, that the $B$-meson state
does not contain the quark $q$; then, $J^{\mu\nu}| B \rangle = P^\mu | B
\rangle = 0$.
In order to apply this result to the matrix element on the RHS of
eq.~\ref{15}, one must proceed with care. The spin projection operator along
the z-axis, $S_z \equiv S.n^{(3)}$, and the spin raising and lowering
operators, $S_\pm \equiv S.(n^{(1)} \pm i n^{(2)})$, take a simple form in the
$q$-quark rest frame, where
\beq
n^{(1)} = (0,1,0,0) \ , \;\;\;
n^{(2)} = (0,0,1,0) \;\;\; {\rm and} \;\;\;
n^{(3)} = (0,0,0,1) \ ;
\label{22}
\eeq
then, $S_z = S_3$ and $S_\pm = S_1 \pm i S_2$. In the frame that we have
chosen, with the B meson at rest and the vector meson momentum along the
z-axis, the direction vectors $n^{(k)}$ depend on the $q$-quark momentum in
that frame, and so the spin projection operators do not have, in general, such
a simple form. If, however, we can neglect the transverse momentum of the
$q$-quark inside the vector meson $V$, then $n^{(1)}$ and $n^{(2)}$ are
as before. Under that assumption, eqs.~\ref{15} and \ref{21} give
\beqa
\langle V(\lambda = \pm 1) | \overline{q} \Gamma b | B \rangle &=&
\frac{1}{\sqrt{2}} \langle V(\lambda = 0) | \overline{q} \gamma_5
( \gamma_1 \mp i \gamma_2 ) \Gamma b | B \rangle \ .
\label{23}
\eeqa
This relation is now applied to the hadronic matrix elements on the RHS of
eqs.~\ref{12}--\ref{14}, where $\Gamma = i/\sqrt{2} (\sigma^{1 \mu} \pm i
\sigma^{2 \mu}) k_\mu$, $i/\sqrt{2} (\sigma^{1 \mu} \pm i \sigma^{2 \mu})
r_\mu$ or $\varepsilon^\pm_\mu \gamma^\mu \gamma_5$. It is straightforward
to check that the ratio of hadronic matrix elements, in each one of the
equations, is exactly $1$, and so we recover the form factor relations of
eqs.~\ref{4}--\ref{6} and Ref.~\cite{JMS}.
\section{Measuring $|V_{ub}|$ in the Quark Model}
One possible application of the form factor relations derived in here is to
the extraction of the CKM parameter $|V_{ub}|$, from the $B \to \rho l^-
\overline{\nu}_l$ decay rate. Since this exclusive decay rate depends on the
form factors $V$ and $A_{0,1,2}$, that parametrize the matrix element $\langle
\rho | \overline{u} \gamma^\mu (1 - \gamma_5) b | B \rangle$, a clean
measurement of $|V_{ub}|$ is problematic \cite{IW}. However, we have seen
that, in the Quark Model, these form factors are not independent from the form
factors $F_{1,2,3}$; they are the form factors that parametrize the matrix
element $\langle \rho | \overline{d} i \sigma^{\mu\nu} k_\nu (1 + \gamma_5) b
| B \rangle$, that appears, for example, in the amplitude for the radiative
decay $B \to \rho \gamma$. A judicious comparison between the semi-leptonic
and radiative decays can then yield $|V_{ub}|$, free from any form factor
dependence \cite{BD}.
From the ratio of the exclusive $B \to \rho \gamma$ and the inclusive $B \to
\gamma + X$ decay rates,
\beqa
R_\rho &\equiv& \frac{\Gamma(B \to \rho \gamma)}{\Gamma(B \to \gamma + X)}
\nonumber\\
&=& \left| \frac{V_{td}}{V_{ts}} \right|^2
\left( 1 - \frac{m_\rho}{m_B}\right)^3
|\frac{1}{2} F_1(0)|^2 \ ,
\label{24}
\eeqa
we can obtain a measurement of the form factor $F_1(k^2)$ at $k^2 = p_\gamma^2
= 0$. On the other hand, the $B \to \rho l^- \overline{\nu}_l$ differential
decay rate, at the $k^2 = (p_{l^-} + p_{\overline{\nu}_l})^2 = 0$ boundary of
the Dalitz plot, is
\beqa
\lim_{k^2 \to 0} \frac{d\Gamma(B \to \rho l^- \overline{\nu}_l)}
{d(k^2/m_B^2)} &=&
\frac{G_F^2}{192 \pi^3}
\left( 1 - \frac{m_\rho}{m_B}\right)^2 m_B^5
|V_{ub}|^2 |A_0(0)|^2 \ .
\label{25}
\eeqa
Using the $F_1(k^2) = 2 A_0(k^2)$ relation of eq.~\ref{4}, the dependence on
the form factors can be eliminated between eqs.~\ref{24} and \ref{25}:
\beqa
\frac{1}{R_\rho} \; \lim_{k^2 \to 0}
\frac{d\Gamma(B \to \rho l^- \overline{\nu}_l)}{d(k^2/m_B^2)} &=&
\frac{G_F^2}{192 \pi^3} m_B^5
\left| \frac{V_{ts}}{V_{td}} \right|^2 |V_{ub}|^2 \ .
\label{26}
\eeqa
A measurement of the CKM parameters can then be obtained that is free of the
hadronic form factor contributions. One must stress, however, that such a
measurement is dependent on the Quark Model assumptions that led to the form
factor relations of the previous section. Nevertheless, within that model, it
is a very general result, as it does not depend on the particular choice for
the momentum wavefunctions of the mesons involved.
For the inclusive radiative decay rate in eq.~\ref{24}, I have taken $\Gamma(B
\to \gamma + X) \simeq \Gamma(b \to s \gamma)$, $m_b \simeq m_B$ and $m_s
\simeq 0$; for the present purpose, these are sufficiently good
approximations. In the exclusive $B \to \rho \gamma$ amplitude, corrections
proportional to $|V_{ub} V^\ast_{ud}|$, due to the difference between $u$ and
$c$ quark loops in the weak vertex, have been ignored. These corrections
have been estimated to be small, when compared to the dominant $t$ quark loop
contribution \cite{JMSb}. Alternatively, one can consider the decay $B \to
K^\ast \gamma$, where such corrections are irrelevant. The analogue of
eq.~\ref{26}, in that case, will not have the extraneous factor
$|V_{ts}/V_{td}|^2$, but it will be valid only up to $SU(3)_{flavor}$
symmetry-breaking corrections to the form factors. The semi-leptonic decay
rate has been observed by CLEO, with $BR(B^0 \to \rho^- l^+ \nu_l) = (2.5 \pm
0.4 ^{+0.5}_{-0.7} \pm 0.5) \times 10^{-4}$ \cite{CLEOa}; more data will be
necessary, in order to determine the differential decay rate in eq.~\ref{25}.
The inclusive radiative decay has also been measured, with $BR(B \to \gamma +
X) = (2.32 \pm 0.57 \pm 0.35) \times 10^{-4}$ \cite{CLEOb}. As for the
exclusive radiative decay, only the Cabibbo favored $B \to K^\ast \gamma$ has
been seen, with $BR(B \to K^\ast \gamma) = (4.2 \pm 0.8 \pm 0.6) \times
10^{-5}$ \cite{CLEOc,CLEOd}. The branching ratio for $B \to \rho \gamma$ is
expected to be about $20$ times smaller, whereas the present limit is $BR(B
\to (\rho,\omega) \gamma)/ BR(B \to K^\ast \gamma) < 0.19$ at $90\%$ C.L.
\cite{CLEOd}.
\section{Conclusion}
The Quark Model relations between the form factors $F_{1,2,3}$ and
$A_{0,1,2}$, that parametrize the weak hadronic transitions from pseudoscalar
to vector me\-sons, were obtained in here from a different, and more general,
argument than that in the original derivation of Ref.\ \cite{JMS}. This
alternate proof confirms that, in the Quark Model, $F_{1,2,3}$ are no longer
independent form factors. Moreover, it reveals how these form factor relations
originate from the spin wavefunctions for the mesons, but do not depend (with
one caveat) on the particular choice for the internal momentum wavefunctions.
In that respect, they are a very general result of the Quark Model. One should
also stress that these relations are valid throughout the entire $k^2$ range
of the hadronic transition, and for any value of the pseudoscalar and vector
meson masses. As for the caveat regarding the momentum wavefunctions, it
arises from the need to assume that the momentum of the quark produced at the
weak vertex is nearly parallel to the momentum of the vector meson into which
it hadronizes. This is a good approximation in at least two situations that,
together, nearly exhaust all scenarios that may occur in practice. One is the
case of heavy-to-heavy transitions, where the momentum of the heavy quark is
roughly that of the corresponding heavy meson. The other is the case of
heavy-to-light transitions, at sufficiently large recoil: the light quark
carries a fraction of the large vector meson momentum that dominates over its
small transverse momentum. As an example where the latter situation occurs,
the form factor relations were applied to the extraction of the CKM parameter
$|V_{ub}|$, from a comparison of the exclusive $B \to \rho l^-
\overline{\nu}_l$ and $B \to \rho \gamma$ decays, along similar lines to the
method that was suggested in Ref.~\cite{BD}.
The question that one would now like to answer is whether the form factor
relations obtained in here remain valid, to some degree, beyond the Quark
Model. In the case of heavy-to-heavy transitions, it is easy to check that the
model independent form factor relations, that follow from Heavy Quark Symmetry
(HQS) \cite{HQS}, do lead to the relations derived in here. This means that
the Quark Model result is indeed a true QCD result, in that limit; however, it
also means that it adds little to the well known, and more general, form
factor relations of HQS. On the other hand, for heavy-to-light transitions,
the Quark Model result provides a new set of form factor relations \cite{h2l}.
A comparison with lattice QCD may shed some light into their validity beyond
the Quark Model. Hopefully, future lattice calculations will address the
question and test these form factor relations, in the more general setting of
their approximation. Ultimately, when sufficient data is accumulated for
heavy-to-light decays, a comparison with the experimental results will decide
whether the Quark Model prediction for the form factors provides a reasonable
picture of QCD.
\section*{Acknowledgments} This research was funded in part by a grant from
the National Science Foundation. I wish to thank Jean-Bernard Zuber, John
Donoghue, Eugene Golowich and Lincoln Wolfenstein for helpful discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 971
|
package cf.paradoxie.dizzypassword.activity;
import android.os.Bundle;
import android.support.v7.widget.Toolbar;
import android.view.View;
import android.widget.TextView;
import cf.paradoxie.dizzypassword.MyApplication;
import cf.paradoxie.dizzypassword.R;
import cf.paradoxie.dizzypassword.utils.SPUtils;
import cf.paradoxie.dizzypassword.utils.ThemeUtils;
import cn.pedant.SweetAlert.SweetAlertDialog;
/**
* Created by xiehehe on 2017/10/31.
*/
public class AboutActivity extends BaseActivity {
private TextView version_info, note;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_about);
Toolbar toolbar = (Toolbar) findViewById(R.id.toolbar);
toolbar.setTitle("关于");
setSupportActionBar(toolbar);
toolbar.setNavigationIcon(R.drawable.back);
getSupportActionBar().setDisplayHomeAsUpEnabled(true);
toolbar.setNavigationOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
finish();
}
});
if (SPUtils.get("first_in", "") + "" == "") {
showNote();
}
note = (TextView) findViewById(R.id.note);
note.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
showNote();
}
});
version_info = (TextView) findViewById(R.id.version_info);
version_info.setText("V" + MyApplication.GetVersionName());
ThemeUtils.initStatusBarColor(AboutActivity.this, ThemeUtils.getPrimaryDarkColor(AboutActivity.this));
}
private void showNote() {
new SweetAlertDialog(AboutActivity.this, SweetAlertDialog.WARNING_TYPE)
.setTitleText("免责声明")
.setContentText(
"1.本app为非盈利开源项目,任何拷贝复制的相同项目与本app无关" +
"\n2.后台数据储存由Bmob云服务提供,作者承诺不会对后台用户数据进行任何操作" +
"\n3.任何加密技术都有被破解的可能性,由此造成的损失与本app及作者无关" +
"\n4.作为作者,建议用户自己申请Bmob云服务进行私人信息储存,相关教程请点击设置页面【修改签名】")
.setConfirmText("好的我知道啦")
.setConfirmClickListener(new SweetAlertDialog.OnSweetClickListener() {
@Override
public void onClick(SweetAlertDialog sDialog) {
SPUtils.put("first_in", "第一次进入app");
sDialog.cancel();
}
})
.show();
}
@Override
public void onBackPressed() {
finish();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,918
|
Q: The completeness assumption in Prokhorov's theorem Originally, I encountered this question on Terence Tao's blog, where the following exercise is presented:
Exercise 23 (Implications and equivalences) Let $X_n, X$ be random variables taking values in a $\sigma$compact metric space $R$.
[...]
(ii) Show that if $X_n$ converges in distribution to $X$, then $X_n$ has a tight sequence of distributions.
(iii) Show that if $X_n$ converges in probability to $X$, then $X_n$ converges in distribution to $X$. (Hint: first show tightness, then use the fact that on compact sets, continuous functions are uniformly continuous.)
Whats struck me as odd was that $R$ is merely assumed to be $\sigma$-compact, i.e. no completeness-assumption on $R$ is made (as the example $\Bbb{Q}$ shows, there are $\sigma$-compact spaces that are neither locally compact, nor complete).
This makes it rather hard to construct new compact subsets (from old ones). Indeed, the proofs of the above statement (ii) that I found (see e.g. https://www.math.leidenuniv.nl/~vangaans/jancol1.pdf Theorem 5.2) use the fact that
$$
K := \bigcap_j \bigcup_{i=1}^{k_j} \overline{B}(a_i, 1/j)
$$
is a compact set if $R$ is complete, because it is closed and totally bounded.
Nevertheless, Wikipedia (http://en.wikipedia.org/wiki/Prokhorov%27s_theorem) does also not assume that the metric space in question is complete, there the only requirement is separability.
Note that we need to produce compact subsets of $R$, because for tightness (see below), we have to show that for $\varepsilon > 0$ there is $K \subset R$ compact such that $\Bbb{P}(X_n \in K) \geq 1-\varepsilon$ holds for all $n$ sufficiently large.
In summary, my question is if the two statements as in the exercise above are correct even without further completeness assumptions on $R$. Hints/proofs/counterexamples would be highly appreciated.
For convenience of the reader, I repeat the necessary definitions below:
Definition 10 (Modes of convergence) Let $R = (R,d)$ be a $\sigma$-compact metric space (with the Borel $\sigma$-algebra), and let $X_n$ be a sequence of random variables taking values in $R$. Let $X$ be another random variable taking values in $R$.
[...]
*
*$X_n$ converges in probability to $X$ if, for every $\epsilon > 0$, one has
$$\liminf_{n \rightarrow \infty} {\bf P}( d(X_n,X) \leq \epsilon ) = 1$$
[...]
*$X_n$ converges in distribution to $X$ if, for every bounded continuous function $F: R \rightarrow {\bf R}$, one has $$\lim_{n \rightarrow\infty} \mathop{\bf E} F(X_n) = \mathop{\bf E} F(X)$$
*$X_n$ has a tight sequence of distributions if, for every $\epsilon > 0$, there exists a compact subset $K$ of $R$ such that $\mathop{\bf P}( X_n \in K ) \geq 1 - \epsilon$ for all sufficiently large $n$.
A: The assumption of $\sigma$-compactness is used to guarantee that each probability measure is tight. In this case, there is no need of completeness or to use the fact that a probability Borel measure on a Polish space is tight because if $X=\bigcup_n K_n$ where each $K_n$ is compact and $K_n\subset K_{n+1}$, then $\mu(K_n)\uparrow 1$.
In order to prove (ii), fix $\varepsilon$ and $n$ such that $\mathbb P(X\in K_n)\gt 1-\varepsilon$. Then consider a continuous function $F$ such that $F(x)=1$ if $x\in K_n$, $F(x)=0$ if $x\in K_{n+1}^c$ and $0\leqslant F\leqslant 1$. Using the definition of convergence in distribution, we obtain the wanted result.
Statement (iii) is also true and it can be shown using portmanteau theorem.
A: As it turns out, the claim is indeed true.
The $\sigma$-compactness of the space $R$ ensures that $R$ is still $\sigma$-compact (and thus a Borel-set) in the completion of $R$. Also (which turns out to be equivalent (for separable metric spaces)) it ensures that every finite measure on $R$ is tight.
These two (for separable metric spaces equivalent) conditions are also called universal measurability of $R$, cf. Dudley, Real Analysis and Probability, Theorem 11.5.1 and the definition before that theorem.
Then a result of Le Cam (cf. Dudley, Theorem 11.5.3) implies that if $X_n \rightarrow X$ in distribution, then the sequence of laws $\mathcal{L}(X_n)$ is uniformly tight, which is exactly what I wanted to know.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,859
|
\section{Introduction}\label{introd}
Polymer knots are abundant in nature and in artificial polymer
materials \cite{sauvage,arsuaga,ramirez,amabilino,tezuka}. They can be created
in the laboratory \cite{amabilino,tezuka,leigh} and have attracted a
considerable attention both from experimentalists and theoreticians of
several different disciplines including
chemistry\cite{chemistryarticles1,StauchDreuw,Schaufelberger},
engineering\cite{engineeringarticles1},
mathematics \cite{mathematicalarticles2,mathematicalarticles1} and
physics\cite{physicalarticles1,physicalarticles2,physicalarticles3,physicalarticles4,janke3}. In this work
we consider the static properties of knots
made by copolymers. We study in particular diblock copolymers
consisting of polymers with two different kinds of monomers $A$ and
$B$.
Part of the motivations
for this work come from biology. In fact, DNA and other biomolecules
are characterized by regions that have different
properties and can thus be regarded as copolymers. Recently, a diblock
copolymer approximation of a piece of DNA has been used in order to
understand how the dishomogeneities in the flexibility affect
the localization of knots on a piece of circular DNA
\cite{orlandinibaiesizontaworkonstiffness,daietal}.
The study of diblock copolymers can be helpful also in technological
applications. For instance, it is already known that the presence of
knots affects the behavior of
polymer materials.
Indeed, the
elasticity response of elastomers cannot be understood without
considering the fact that the polymer chains inside these materials
form knots and links.
The effects of the presence of knots in the conformational properties of ring $AB$ diblock-copolymers have already been noted for instance in Ref.~\cite{vlahosetal}.
The statistical mechanics of open or circular diblock
copolymers has been thouroghly investigated in the past, see e.~g.
\cite{marko,vilgis,holystvilgis,huber,metzler}. Polyelectrolytes similar to those treated here in Setup~I (see below) have been considered in \cite{velyaminov}. Setup~II has some similarities with the Hydrophilic-Polar (HP) protein model \cite{dill}, however in our case the $B$ monomers are subjected to attractive forces. The HP model has been studied using the Wang-Landau algorithm in \cite{wuest,wuest2}.
More recently, there has been some interest on circular
diblock copolymers with non-trivial topologies
\cite{orlandinibaiesizontaworkonstiffness,daietal,vlahosetal,najafi,
kuriatasikorski,benahmed,tagliabueetal,kumar}. For example, in \cite{tagliabueetal} it has been
investigated how the stiffness heterogeneity or the presence of charges
influence the localization of the knot. The role of stiffness and heterogeneity in knot production has been explored in Ref.~\cite{cardellietal}. Other aspects of the topology of diblock-copolymers have been treated in \cite{kumar}.
With the help of the Wang-Landau algorithm \cite{wl}, the statistical
mechanics of knotted diblock copolymers has been studied in
Refs.~\cite{wang,swetnam,FFBlockcopolymerknots-preprint}.
The goal of the present work is to extend the results of \cite{wang,FFBlockcopolymerknots-preprint}
to longer polymers, showing that remarkable properties emerge in this
case.
Knotted copolymers
are defined here on a simple cubic lattice. Their monomers are
subjected to different kinds of very short-range interactions
reproducing different physical setups. In one setup, called hereafter
Setup I, a charged polymer is fluctuating in an ion solution that
screens the long-range Coulomb interactions.
Monomers of type $A$ have
a positive charge, while monomers of type $B$ are negatively charged.
The upshot is that
monomers of the same kind repel themselves, while the interactions
between the $A$ and $B$ monomers are attractive.
Setup I is also relevant for
the case of polymers in water. In water at room temperature, in
fact, the Bjerrum length $l_B$ amounts to just $7\AA$. Let us recall
that the constant $l_B$
measures
the length scale at which the strength of the Coulomb
interactions in a dielectric medium becomes equal to the thermal
energy $k_BT$, where $k_B$ is the Boltzmann constant and $T$ is the
temperature \cite{Dobrynin}.
In the second studied setup, that will be named Setup II, the solvent
is good for
the monomers of type $A$, which thus repel themselves, while
monomers of type $B$ are below the theta point and attract themselves.
Between monomers of type $A$ and $B$ we suppose that only excluded
volume forces are acting.
Fig.~\ref{scheme-setups} summarises the main features of both setups.
Other setups are possible, see for
instance \cite{vilgis}.
\begin{figure}
\begin{center}
\includegraphics[height=5cm]{scheme-for-setup1.pdf}\hspace*{3cm}
\includegraphics[height=5cm]{scheme-for-setup2.pdf}
\caption{This figure summarises the main features of the Setups I and II considered in this paper. Particles in Setup I have a charge $Q$. Apart from a proportionality factor, we have that $Q\propto\sqrt{\varepsilon}$. The exact value of the proportionality constant is not relevant for performing the calculations. Arrows explain if the interactions are repulsive or attractive.
In Setup II the $A$ and $B$ monomers are subjected only to excluded-volume interactions, so that arrows are not necessary in this case.
}\label{scheme-setups}
\end{center}
\end{figure}
Multiblock copolymers with different monomer distributions and
interactions are considered.
We construct knots containing alternating units with $n_A$ monomers of type $A$
and $n_B$ monomers of type $B$ until the total
number of monomers $N$ is obtained. If $N$ is not a multiple of
$n_A+n_B$, a slight excess of monomers of type $A$ is allowed.
In the following, it will be convenient to introduce the total
number $N_A$ of $A-$monomers and the total number $N_B$ of
$B-$monomers. Of course
$N_A+N_B=N$.
Multiblock copolymers of this kind will be denoted with the symbols
$M_I(N,n_A,n_B)$ and $M_{II}(N,n_A,n_B)$.
The subscripts $I$ and $II$ refer the two setups discussed before.
A particularly interesting subcase is that
of
the $AB-$diblock
copolymers composed by two segments, one with $A$ monomers and the other with $B$ monomers. $AB-$diblock copolymers will be distinguished within the more general class of multiblock copolymers introducing the new symbols
$D_I(N_A,N_B)$ and $D_{II}(N_A,N_B)$. Of course, $D_{I/II}(N_A,N_B)=M_{I/II}(N,N_A,N_B)$, where $I/II$ means Setup I or II.
Following the above conventions, uncharged homopolymer knots with
$N$ monomers can be listed under Setup II. A homopolymer knot in a good
solvent corresponds to the $AB-$diblock copolymer $D_{II}(N,0)$ with zero monomers of type $B$,
while a homopolymer knot in a bad solvent can be described with the
symbol $D_{II}(0,N)$.
To further characterize the analyzed knots, also their
monomer composition
$f=N_A/(N_A+N_B)$ will be used.
The rationale for investigating knots made by copolymers is to obtain
macromolecules with different properties by changing the knot
topology, the $A/B$ monomer ratio and the monomer distribution along
the chain.
We show here that this goal can indeed be achieved and that copolymer
knots exhibit
a variety of behaviors that are absent in knots formed by homopolymers.
In a nutshell, it turns out that the relevant parameters of a polymer ring, like gyration radius, heights and temperatures of the peaks of the heat capacity, specific energy and number of contacts (non-contiguous monomers that are at the distance of one lattice unit, see below for a more precise definition), are highly affected by the monomer distribution.
Topology has strong effects on the behaviour of short polymers. In the case of longer polymers, these effects fade out, but still the properties of the knot may be tuned by choosing knots with the same length and monomer distribution, but different topology.
Despite the vanishing influence of topology with increasing polymer lengths, we have observed that, in particular cases, the thermal behaviour may drastically change depending on the type of the knot even in longer polymers.
Concluding this Introduction, we would like to stress that the present
analysis requires the sampling of knot conformations that are
very compact, so that the density of monomers is very high. These
compact states often include conformations that are extremely rare
and thus very difficult to be sampled using Monte Carlo
algorithms. Such conformations may act as bottlenecks in a Monte Carlo
simulation, relevantly increasing the computation time. The problem
of handling rare events is not only related to the case of polymer
systems. In the Wang-Landau Monte Carlo algorithm, which we will adopt
for our simulations, this issue has been already treated in several
previous publications, see
e. g. \cite{rarestatesprevioispublications,dellago}.
Following \cite{publicationoflandauwithasimilaraccelerationtechnique},
in this work we have used parallelization techniques to speed up the Wang-Landau algorithm
allowing to study the lowest energy
conformations. The latter are important because they are dominating at extremely low temperatures. Some
of these techniques
have been discussed in more details in
\cite{YZFFPhysicaA2017}.
In large systems the
number of states to be sampled is enormous. In a knot with $N=500$, for
instance, a conformation with the lowest observed energy value can
appear once in a set of $10^{11}$ samples. In the case of a knot with
$N=1000$, the Wang-Landau sampling process requires a few months.
With the inclusion in our calculations of extremely rare configurations, it has been
possible to show that the phase diagram of knots
formed by block copolymers is more complex than that of homolymers. New peaks appear in the heat
capacity corresponding to different transition processes.
Particularly interesting is the situation in which most of the monomers are subjected to repulsive interactions apart form a small number
of monomers that are able to form contacts with each other. At the lowest temperatures, such knots are found in compact conformations which get soon destroyed upon heating leading to a fast expansion of the knot. With growing temperatures, this expansion continues at a lower pace in Setup I. Finally, at high temperatures these knots behave as their homopolymer counterparts in a good solvent.
In some longer knots,
metastable compact states have been observed at low temperatures, signalising that knots can be subjected to relevant rearrangements of their structure
when heated.
The material presented in this paper is organised as
follows.
In Section~\ref{method} the used methodology is briefly explained.
The obtained results are discussed in Section~\ref{thermalprop}.
The thermal properties of knots in Setup I and Setup II are presented separately in Subsections \ref{thermalsetupI} and \ref{thermalsetupII} respectively.
Finally, the conclusions and open problems are the subject of Section~\ref{conclusions}.
\section{Methodology}\label{method}
Polymer rings are modeled as self avoiding loops on a simple cubic lattice. Monomers are located on the lattice sites and each lattice
side can be occupied by at most one monomer. Two consecutive monomers on the loop are linked by one lattice bond,
so that the total length of the knot in lattice
units is equal to $N$.
The energy of a given knot conformation $X$ is expressed in Setup I
and Setup II by the
following Hamiltonians respectively:
\begin{equation}
H_I(X)=\varepsilon(m_{AA}+m_{BB}- m_{AB})
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{Setup I}
\label{hamI}
\end{equation}
and
\begin{equation}
H_{II}(X)=\varepsilon(+m_{AA}-m_{BB})
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{Setup II}\label{hamII}
\end{equation}
In Eqs.~(\ref{hamI}) and (\ref{hamII}) the quantities
$m_{MM'}$'s count the numbers of contacts between monomers of the kind
$M$ and $M'$, where
$M,M'=A,B$.
Let
$\boldsymbol R_1,\ldots,\boldsymbol R_N$ denote the locations of the $N$ monomers.
Two monomers $i$ and $j$ are said to be in contact if
$i\ne j\pm1$ and $|\boldsymbol R_i-\boldsymbol R_j|=1$.
$\varepsilon>0$ is an energy scale measuring the cost of one contact,
which can be positive or negative depending on the setup and on the
monomer types. We note that the Hamiltonian $H_{II}(X)$ of setup II is a variation of the HP protein model \cite{dill} with the $B$ monomers being identified with the polar (P) aminoacid residues. The difference is that in setup II we have that the $A$ monomers are repelling themselves due to the short-range interaction $+\varepsilon m_{AA}$, while in the HP model they are only subjected to excluded volume interactions.
For convenience, we will introduce the rescaled temperature
$\boldsymbol T=\frac {k_BT}\varepsilon$.
To go back from $\boldsymbol T$ to the usual temperature $T$ measured in Kelvins some assumptions on $\varepsilon$
are needed.
For instance, we suppose that the
strength $\varepsilon$ of the interactions is a multiple of the energy
associated with thermal fluctuations at
room temperature $T_0$, i.~e. $\varepsilon=qk_BT_0$, where $T_0\sim
298K^{\circ}$ and $q$ is a positive real constant. At this point
it is easy to see that the temperature $T$ is expressed in terms of $\boldsymbol T$ as follows:
$q\boldsymbol T T_0=T$. For example, if $q\sim1.5$, the point $\boldsymbol T=1$
corresponds to the temperature $T=1.5T_0\sim 447K^{\circ}$.
After the passage $T\longrightarrow \boldsymbol T$, it is possible to eliminate the $\varepsilon$ factor in the Hamiltonians of Eqs.~(\ref{hamI}) and
(\ref{hamII}). The upshot is that we obtain the following rescaled Hamiltonians: $\boldsymbol
H_{I,II}(X)=\frac{H_{I,II}(X)}\varepsilon$.
Here
$H_{I,II}$ can be either of the two Hamiltonians defined in
Eqs.~(\ref{hamI}) and (\ref{hamII}).
The simulations are performed using the Wang-Landau Monte Carlo
algorithm \cite{wl}.
The initial knot conformations are obtained by elongating
the existing conformations of minimal length knots \cite{rechnitzer,sharein} until the desired final length is attained.
Knots up to six crossing according to the Rolfsen table are studied, though there is no restriction against including more complicated knots.
The details on the sampling and the treatment of the
topological constraints can be found in
Refs.~\cite{yzff} and \cite{yzff2013}.
The random transformations that are necessary for sampling the different knot conformations are the pivot moves of Ref.~\cite{madrasetal}.
In order to preserve the topological state of the system, the pivot algorithm and excluded area (PAEA) method of Ref.~\cite{yzff} is applied.
The partition function of the polymer knot is
given by:
\begin{equation}
Z(\boldsymbol T)=\sum_{ E= E_{min}}^{ E_{max}}e^{- E/\boldsymbol
T}g(E)
\end{equation}
where $g(E)$ denotes the density of
states:
\begin{equation}
g(E)=\sum_X\delta(\boldsymbol H_{I,II}(X)-{E})
\end{equation}
$g(E)$ is
the quantity to be evaluated via Monte Carlo methods. $ E_{min}$
and $ E_{max}$ represent respectively the minimum and maximum
values of the energy. The whole energy range ${\cal I}=[
E_{min}, E_{max}]$ over which the
sampling is
performed depends on the used setup, the length of the knot, its topology and the selected monomer distribution.
To determine the values of $E_{min}$ and $E_{max}$, a preliminary run without specifying any energy limit is performed. In doing that we exploit the fact that
the Wang-Landau algorithm is very efficient in exploring the whole energy range of the system. The preliminary run is stopped when no new values of the energy are found. After that, the averages of the observables are computed
by a second run with the values of $E_{min}$ and $E_{max}$ calculated from the preliminary run. Also in this second run the energy range is kept open, but
for the convergence of the Wang-Landau algorithm only the energy values in the interval $[E_{min},E_{max}$ are considered.
For the convergence of the Wang-Landau algorithm the sampling of an order of $10^{12}$ conformations is necessary. If new values of $E_{min}$ and $E_{max}$ appear during the sampling, the run is repeated with the new, extended energy range. In the case of long polymers with $N\ge 300$, cuts in the energy range are necessary in order to obtain the corvergence of the Wang-Landau algorithm in a reasonable time. In this case, several runs are repeated by slightly changing the energy range to check that the results are independent of the energy range despite these small variations. It turn out that the Wang-Landau algorithm is very robust in this sense. For instance, small variations of the energy range do not have relevant influences on the height and the position of the peaks of the specific heat capacity. The averages of the observables are particularly insensitive under changes of $E_{max}$, while variations of a few percent occur at very low temperatures by changing $E_{min}$ in the case of block copolymers in setup I, which is the most critical with respect to the computational time.
The
expectation values of any observable $\cal O$ may be computed
using the formula:
\begin{equation}
\langle{\cal O}\rangle(\boldsymbol T)=\frac 1{Z(\boldsymbol T)}\sum_{ E=E_{min}}^{E_{max}}e^{-E/\boldsymbol T}g(E){\cal O}_{ E}
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-typeII-N45-B45-3.1-N90.pdf}
\includegraphics[width=0.48\textwidth]{fig-typeII-NA71-B19-5.1-N90.pdf}
\caption{
While the specific energy is relevant for understanding the behavior of homopolymers, in the case of copolymers the numbers of contacts formed by the $A$ and $B$ monomers and the total number of contacts $n_{tot}=m_{AA}+m_{BB}+m_{AB}$ are very useful quantities too. The pictures show how the numbers $n_{tot},m_{AA},m_{AB}$ and $m_{BB}$ change with the temperature in the case of a knot $3_1$ with monomer distributions $D_{II}(45,45)$ (left panel) and $D_{II}(71,19)$ (right panel). }\label{fig0a-ncont}
\end{center}
\end{figure}
Here ${\cal O}_E$ denotes the average of $\cal O$ over all
sampled states with rescaled energy $E$. The observables that
will be considered in this work are the mean specific energy
\begin{equation}
\frac {\langle E(\boldsymbol
T)\rangle}N=\sum_{E= E_{min}}^{E_{max}}Ee^{-E/\boldsymbol
T}g(E)
\end{equation}
the specific heat capacity $C/N=\frac
1N\frac{\partial \langle E(\boldsymbol T)\rangle}{\partial \boldsymbol T}$ and the
mean square
average of the gyration radius $R_G^2$.
Additional information on the shape of the knot at different temperatures and energies has been gathered studying the number of contacts formed by the monomers and by closed inspection of the generated conformations.
\section{Thermal properties of knotted diblock copolymer
rings}\label{thermalprop}
Both Setups I and II are characterised by the coexistence of the attractive and repulsive interactions. The monomers of type $A$ or $B$ may attract or repel themselves or the monomers of the other type. In Setup II only the excluded-volume forces are acting between the $A$ and $B$ monomers.
In this situation, depending on the temperature, a particular interaction can become more
relevant than the others in shaping the behavior of knotted block copolymers and lead to different regimes.
The averages of the numbers $m_{AA},m_{BB},m_{AB}$ of the contacts formed by the monomers of a specific type with themselves or with the other type together with the averaged total number of contacts $n_{tot}=m_{AA}+m_{BB}+m_{AB}$ provide some hint about the knot's conformations in these regimes.
An example of such plots is given in Fig.~\ref{fig0a-ncont}, in which a knot $3_1$ in Setup~II with different monomer distributions is considered.
Another quantity that is important is the specific energy. It turns out that, as it is expected, at any given temperature the specific energy of a copolymer knot
never exceeds the specific energy of a homopolymer knot of equal length and
topology in a good solvent, where all monomers repel themselves. On
the other side, it will always be bigger than that of a homopolymer in
bad solvents.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{Fig1a-2.eps}
\caption{The gyration radius of the knot $3_1$ with $N=90$ in various
monomer distributions is plotted as a function of $\boldsymbol
T$. }\label{fig1a}
\end{center}
\end{figure}
A similar trend is observed in the case of the gyration radius as shown in Fig.~\ref{fig1a}:
At any given temperature, the gyration radii of $3_1$ knots with
different monomer compositions
range between a minimum provided by the
the gyration radius of the $3_1$
homolymer knot $D_{II}(0,90)$ and an upper limit given by the $3_1$
homopolymer knot $D_{II}(90,0)$.
Another feature which is visible in Fig.~\ref{fig1a} is that
the swelling process under increasing temperatures may become much more rapid
when the knot contains monomers of different kinds. For example, the
gyration radius of the knot $3_1$ with monomer distribution
$D_I(75,15)$ changes faster than that of the homopolymer
$D_{II}(0,90)$ and the diblock copolymer $D_{II}(15,75)$.
We also note that the diblock
copolymer $D_I(75,15)$ exhibits a
swelling phase at lower temperatures followed by a mild shrinking phase at higher
temperatures. This behavior becomes stronger with growing topological
complexity.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-prob-N90.pdf}
\caption{The probability $P(E)$ of obtaining a state of energy $E$
for a knot $3_1$ with $N=90$ in various
monomer distributions is plotted as a function of the
energy $E$. }\label{fig-prob}
\end{center}
\end{figure}
In the following, we will discuss the results obtained for knots in
Setup I and Setup II separately.
To conclude the discussion on the general features of knots
formed by copolymers, we stress
that the Monte Carlo sampling is much more difficult for knots
in Setup I than
in Setup II.
The reason can be understood by looking at
Fig.~\ref{fig-prob} where the probability $P(E)$ of states of energy
$E$ has been computed for a few knots in both setups.
That figure shows that knots admit in Setup I
conformations in the
lowest part of the energy spectrum that are much more rare than any knot conformation in Setup II.
This makes the sampling of the lowest energy states in Setup I very difficult, especially for long knots.
Of course, the energy spectrum, and thus also the existence of rare ultralow energy conformations, strongly depends on the monomer composition of a knot. This is visible by looking in Fig.~\ref{fig-prob} at the differences between the same knot $3_1$ with $N=90$ in the monomer distributions $D_I(80,10)$ and $D_I(70,20)$.
However, even in the case in which the monomer distribution is fixed, the energy range grows considerably when passing from Setup I to Setup II.
For example, the most compact state of
a $3_1$ knot with $N=90$ and monomer distribution $D_I(70,20)$
has an energy $E<-60$ and
a
probability that is below $10^{-30}$. In the same knot $3_1$ with
$N=90$ and the similar monomer distribution $D_{II}(71,19)$, the probability
of the lowest energy state is higher of more than $15$ orders.
In a knot $4_1$ with $N=1000$ with monomer distribution $D_I(800,200)$ the
lowest energy state has a probability lower than $10^{-256}$.
\subsection{Results for Setup I}\label{thermalsetupI}
The variety of behaviors that it is possible to obtain in the case of
short copolymer
knots is shown
in Figs.~\ref{fig-local-N90-b} and \ref{fig-local-N90-c}~\footnote{Let
us note that in the present case the knot has length $N=90$. Due to
the fact that $90$ is not divisible by four, the monomer sequence is
obtained by joining together $\nu=22$ basic units of the kind $AABB$
and putting at the end of the sequence (monomers 89 and 90) two
monomers of the $A$ type. As a consequence, the
sequence looks as follows: $AABB-AABB-\cdots-AABB-AA$.}.
Both
figures refer to the same
trefoil knot $3_1$ of length
$N=90$.
In Fig.~\ref{fig-local-N90-b} the range of temperatures has been
restricted to the interval $0.00\le\boldsymbol T\le 2.00$ in order to display
in details
the features of the peaks of the specific heat capacity $C/N$.
The different plots of the specific heat capacity and the mean
square gyration radius $R^2_G$ have been obtained only by varying the
distribution of the $A$ and $B$ monomers.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-local-spec-hc-N90-a.pdf}
\includegraphics[width=0.48\textwidth]{fig-local-spec-hc-N90-b.pdf}
\caption{The specific heat capacity $C/N$ of a knot $3_1$ with $N=90$
in various
monomer distributins is plotted as a function of $\boldsymbol T$.
The peak of $C/N$ is highest in the case of the monomer distribution
$M_I(90,2,2)$ (panel b)
and lowest in the case of the homopolymer $D_{II}(90,0)$
(plot with crosses at the bottom of the picture in panel a).
}\label{fig-local-N90-b}
\end{center}
\end{figure}
It is possible to realize from Fig.~\ref{fig-local-N90-b} that the
heights of the peaks of the specific heat capacity and the temperatures
at which this peak occurs are very sensitive
to the momomer distribution.
The peaks' heights range in fact from about $0.11$ for the
homopolymer $D_{II}(90,0)$ (see plot with crosses at the bottom of Fig.~\ref{fig-local-N90-b}, panel a, and
comments in the caption) up to
about $4.00$ in the case of the multiblock copolymer $M_I(90,2,2)$ in Fig.~\ref{fig-local-N90-b}, panel b.
In general, we observe that the swelling process when the temperature increases is much more abrupt in knots with monomer distributions $M_I(N,2,2)$
than in those with any other monomer distribution. Consequently, the specific heat capacities in knots with monomer distribution $M_I(N,2,2)$ are characterised by high and narrow peaks as it can be seen by comparing the plot of $C/N$ in Fig.~\ref{fig-local-N90-b}, panel b for the distribution
$M_I(90,2,2)$ with the plots of the other distributions
in panel a.
By gradually increasing the size of the basic unit in the multiblock copolymer, for instance choosing $M_I(90,4,4)$,
$M_I(90,8,8)$ etc., the peak of the heat capacity becomes gradually lower and
wider.
We also note that
the temperature at which the peak of the specific heat appears can be
fine-tuned by choosing the monomer distribution.
The temperatures and the height of the peaks related to different
distributions are displayed in Table~\ref{TableI}.
\begin{table}
\centering
\caption{In the second column of this table are reported the values
of the height of the
peaks of the specific heat capacity for a knot $3_1$ with different
monomer distributions. In the third column it is shown the
temperature $\boldsymbol T_{MAX}$,
at which the heat capacity is at its maximum. The plots of the
specific heat capacity are displayed in Fig.~\ref{fig-local-N90-b}.}
\label{TableI}
\begin{tabular}{ccc}
\toprule\\
Monomer\hspace*{1cm}& Peak\hspace*{1cm}& Temperature\\
distribution\hspace*{1cm}& height \hspace*{1cm}& $\boldsymbol T_{MAX}$ \\\toprule
$D_{II}(0,90)$\hspace*{1cm}& 0.33\hspace*{1cm} & 1.22 \\
$M_I(2,2)$\hspace*{1cm}&3.99\hspace*{1cm} & 1.03 \\
$D_I(45,45)$\hspace*{1cm}& 0.87\hspace*{1cm} & 0.94 \\
$D_I(60,30)$\hspace*{1cm}&0.72 \hspace*{1cm} &0.94 \\
$D_{II}(90,0)$\hspace*{1cm} &0.11\hspace*{1cm} & 0.69 \\
$D_I(70,20)$\hspace*{1cm} &0.85\hspace*{1cm} & 0.56\\
$D_I(80,10)$\hspace*{1cm}&0.49\hspace*{1cm}&0.51\\
$D_I(75,15)$\hspace*{1cm}&0.70 \hspace*{1cm} & 0.48
\end{tabular}
\end{table}
The data in the table are ordered according to decreasing
temperatures $\boldsymbol T_{MAX}$, whose values range in the wide interval $0.48\le
\boldsymbol T_{max}\le 1.22$.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-local-spec-Rg-N90-2a.pdf}
\caption{The mean square gyration radii $R_G^2$ of a knot $3_1$ with
$N=90$ in various
monomer configurations is plotted as a function of $\boldsymbol T$. Going
from the top to the bottom it is possible to distinguish the plots of
the gyration radii for the
following monomer distributions: $D_{II}(90,0)$, $D_I(80,10)$,
$D_I(75,15)$ and $D_I(70,20)$. Below, there is a partial overlapping
of the plots of the mean square
gyration radii for the distributions $D_I(60,30)$ (line with black squares),
$M_I(90,2,2)$ (line with white circles) and $D_I(45,45)$ (line with
black circles). The plot in the bottom part of the figure is that of the
homopolymer in a bad solvent
$D_{II}(0,90)$ (white triangles). }\label{fig-local-N90-c}
\end{center}
\end{figure}
The distribution of the $A$ and $B$ monomers greatly influences also the allowed
range of possible sizes as shown in Fig.~\ref{fig-local-N90-c}. For example,
in
the $3_1$ copolymer knot in the setup
$D_I(80,10)$ the values of the mean square gyration radius are
restricted to the narrow interval $13\le R_G^2\le 16$. By passing to
the $D_I(75,15)$ distribution, a change that requires just the
substitution of five monomers of type $A$ with monomers of type $B$,
the new range in which $R_G^2$ can take its values is $8.86\le
R_G^2\le14.89$. Thus even little variations in the monomer distribution are
able to introduce significant changes in the expectation values of the gyration radius.
Let us note that all gyration radii
in Fig.~\ref{fig-local-N90-c} converge to a common limit at very high
temperatures as it is expected, because at high temperatures the
interactions between the monomers are no longer relevant due to the
strong thermal fluctuations.
From Fig.~\ref{fig-local-N90-c} it turns also out that, as expected, at any temperature
the self-attracting homopolymer $3_1$ knot
$D_{II}(0,90)$ is always much smaller than all other trefoil
knots in which repulsive interactions are present.
One goal of this work is to investigate
how the topology of a knot influences its thermal behavior.
In the case of homopolymers it is known that the topological effects
are particularly strong in short knots and gradually fade out with
increasing length, see e.~g. \cite{yzff2013}.
This is true also in the case of copolymers.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-top-rg-N90.pdf}
\caption{
Presented are the plots of $R_G^2$ in the case of the knots $3_1,4_1$
and $5_1$ with $N=90$. For each knot two values of $f$ are
considered: $f=0.50$, corresponding to the monomer
distribution $D_I(45,45)$ and $f\sim 0.83$, corresponding to the
monomer distribution $D_I(75,15)$.}\label{fig-top-N90}
\end{center}
\end{figure}
The plots in Fig.~\ref{fig-top-N90} show the gyration radii of
a few $AB-$diblock copolymers of knot types $3_1,4_1$ and $5_1$.
For each knot type two different monomer distributions have been
taken into account, namely $D_I(45,45)$ and $D_I(75,15)$.
Fixing the monomer
distribution and the knot length, which is equal to $N=90$, we expect
the differences in the plots to be due to pure topological effects. The
latter are quite evident in the figure. For instance, by looking at the
gyration radii of the knots with monomer distribution
$D_I(45,45)$, it is clear that the sizes of these knots are
changing with the topology. The same conclusion is valid if we look at
the plots with monomer
distribution $D_I(75,15)$.
It should be noticed that topological effects are less marked than
those connected with the modification of the monomer distribution.
This fact is well illustrated also by the plots of the specific heat
capacity. Fig.~\ref{fig-top-shc-N90} shows that by changing
the topology of a knot while keeping its monomer distribution fixed,
it is possible to shift significantly the
peaks of the specific heat capacity but to a smaller extent than in
the case in which the topology is fixed
and the monomer distribution is modified.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-top-shc-N90.pdf}
\caption{Dependence of the specific heat capacity $C/N$ on the
topology and on the monomer composition $f$ for $AB-$diblock
copolymers.
Presented are the plots of $C/N$ in the case of the knots $3_1,4_1$
and $5_1$ with $N=90$. For each knots two values of $f$ are
considered: $f=0.50$ (case $D_I(45,45)$) and $f\sim 0.83$ (case
$D_I(75,15)$).}\label{fig-top-shc-N90}
\end{center}
\end{figure}
The above comments concerning the topological effects are valid also
in the case of longer polymers.
The plots in
Figs.~\ref{fig-top-rg-N200} and~\ref{fig-top-shc-N200} of the
gyration radius and heat capacity respectively
illustrate the influence of both topology
and monomer distribution on the thermal behaviour
of various knots of length $N=200$.
Fig.~\ref{fig-top-rg-N200} shows that topology is affecting
the knot size. Moreover, the shape of
the peaks of the specific heat capacity changes with
the topology of the knot, as it is possible to see in Fig.~\ref{fig-top-shc-N200}. These changes cannot be attributed to statistical errors, as it has been verified by repeating the simulations starting from different seeds.
Of course, with increasing polymer lengths the topological effects
start to fade out.
In particular, the range of temperature in which the peaks of the specific heat capacity
are appearing is approximately the same independently of topology for all
knots considered
in Fig.~\ref{fig-top-shc-N200}.
This is not the case of the shorter knots with $N=90$, see
Fig.~\ref{fig-top-shc-N90}.
The novelty when $N=200$ with respect to shorter polymers
is the appearance of double peaks or of a peak with a shoulder in the specific
heat capacities of the knots with monomer
distribution $D_I(167,33)$, see Fig.~\ref{fig-top-shc-N200}. A double peak
is characterising also the specific heat capacity of the knots
$0_1$ and $6_1$ with $N=200$ and monomer distribution $D_I(167,33)$ (not
shown in the figure).
This phenomenon
is accompanied by a pattern that is visible in the
behaviour of the mean square gyration
radius $R_G^2$ at low temperatures.
An example of this behaviour
is shown in the inset of
Fig.~\ref{fig-top-rg-N200}, in
which the plot of $R_G^2$ of
a knot $4_1$ with monomer distribution $D_I(167,33)$ is displayed
in greater detail. As it is possible to see, there is a rapid
increase of $R_G^2$
in the range of temperatures $0.4\le \boldsymbol
T\le 0.6$. This range coincides approximately with that in which the
peak of the specific heat
capacity of the knot $4_1$ centered at about
$\boldsymbol T=0.50$ is appearing, see the line with empty squares in
Fig.~\ref{fig-top-shc-N200}.
The appearance of double peaks and peaks with
shoulders in the plots of the heat capacity of chains has been shown to be
related to the occurrence of
two different phase transitions in a small interval of
temperatures, see
\cite{rampf,janke1,janke2,proceedings} and \cite{velyaminov} in the case of polyelectrolytes.
To understand what happens in the present context,
we recall the behaviour of homopolymers upon heating.
In homopolymers in a good solvent all monomers are subjected to purely repulsive interactions. At the lowest temperatures, the
number of contacts between the monomers is minimal and the knot attains its maximally swollen size.
When the temperature is rising, the increasingly strong thermal fluctuations become energetic enough to break the potential barrier that prevents the formation of contacts between the monomers. As a consequence, the size of the knot moderately shrinks with growing temperatures. This shrinking is moderate and the related process causes just a small peak in the heat capacity, see the examples of
Fig.~\ref{fig-top-shc-N200}, right panel.
In homopolymers in a bad solvent, on the contrary, the minimal size conformations are found when the temperature is very low.
The compact states that arise in this case consist of a large number of contacts. With growing temperatures, the number of contacts decreases and the knot continues to swell up to high temperatures until the thermal fluctuations dominate over the interactions. At this point, the gyration radius and the number of contacts of the ring are completely determined by entropy and the knot's topology. This swelling process in a bad solvent generates in the plots of the specific heat capacity a broad peak
extending over a large interval of temperatures.
In block copolymers, the situation is different because there are simultaneously both attractive and repulsive interactions.
Knotted diblock copolymers with $N_A\sim N_B$ have a behaviour similar to that of knotted homopolymers in a bad solvent.
The reason is that, when $N_A\sim N_B$, the attractive forces are very strong because there is a large number of $A$ and $B$ monomers that may build contacts with each other. As a consequence, for energy reasons, at very low temperatures the number of contacts between the $A$ and $B$ monomers is at its maximum and the volume occupied in space by the knot is very small, though not so small as in the homopolymer case. With growing temperatures, similarly to what happens for knotted homopolymers in a bad solvent, the number of contacts decreases and the swelling process continues also when the temperature is high. In knots of length $N=90$ and $N=200$ the expansion is completed at a temperature of $\boldsymbol T\sim 3$ when $N_A=N_B$.
The result is a broad peak in the specific heat capacity with the maximum of the peak at a temperature of $\boldsymbol T\sim 1$ (see Figs.~\ref{fig-top-shc-N90} and \ref{fig-top-shc-N200}) or higher like in the case of the knot $5_1$ with length $N=500$ and monomer distribution $D_I(250,250)$ of Fig.~\ref{fig-N500}.
A thorough investigation with a knot $4_1$ of length $N=200$ has shown that the behaviour of knots with $N_A\sim N_B$ described before still persists at least until $N_A=150$ and $N_B=50$. This means that the heat capacity is characterised only by a single peak. Of course, the heights of this peak will decrease with decreasing values of $N_B$ because less and less contacts between the $A$ and $B$ monomers are possible.
Below a certain threshold of the number $N_B$ of available $B$ monomers, knots depart significantly from the behaviour described above.
To fix the ideas, we will say that such knots have
monomer configurations of the type $N_A>>N_B$.
In this case
the number of $B$ monomers that are able to form contacts with the $A$ monomers is extremely limited. This explains why the heights of the peaks of the specific heat capacity are decidedly lower than those of knots in which the monomer distribution is $N_A\sim N_B$, see Fig.~\ref{fig-top-shc-N200}.
Despite the fact that $N_B<<N_A$, at very low temperatures the number of contacts $m_{AB}$ between the monomers of types $A$ and $B$ is quite high in order to minimise the energy according to the Hamiltonian~(\ref{hamI}).
Indeed, a knot $4_1$ with $N=200$ and monomer distribution $D_I(167,33)$ has been studied in details in order to understand the origin of the double peak when $N_A>>N_B$. It turns out that at the minimum energy value $E_{min}=-111$, the average total number of contacts formed by this knot is $\langle n_{tot}\rangle=122$. The overwhelming majority of these contacts is due to the $A$ and $B$ monomers, because at $\boldsymbol T\sim 0$ we have that $\langle m_{AB}\sim 118$, $\langle m_{AA}\rangle\sim 4$ and $\langle m_{BB}\rangle =0$.
Of course, with rising temperatures $m_{AB}$ will decrease similarly as in the case $N_A\sim N_B$ illustrated before. The striking difference between the cases $N_A\sim N_B$ and $N_A>>N_B$ is however that
knots with $N_A>>N_B$ undergo an extra rearrangement of their structures at very low temperatures.
More precisely, knots with $N_A>>N_B$ are approaching a very compact state when the temperature becomes very low like their counterparts with $N_A\sim N_B$.
In the case of the prototype knot $4_1$ mentioned above, this state contains $\langle m_{AB}\rangle\sim 103$ contacts between the $A$ and $B$ monomers. The breaking of these contacts causes a swelling process that produces in the plot of the specific heat capacity a peak around the temperature $\boldsymbol T\sim 0.7$, see Fig.~\ref{fig-top-shc-N200}, left panel, plot with white circles. This is the origin of the second peak in the heat capacities of the knots with monomer distribution $D_I(167,33)$ in Fig.~\ref{fig-top-shc-N200}.
When the temperature further decreases, the quantity $R_G^2$ of the $4_1$ knot changes very rapidly from
$30$ to $24$, see the inset of Fig.~\ref{fig-top-rg-N200}.
This points to the fact that a rearrangement of the structure of the knot has taken place.
This rearrangment is connected with the formation of only a small number of new contacts.
Unfortunately, the close inspection of very rare conformations at the lowest energies
is very difficult, but it has been possible to capture conformations at the energies $E=-105$ and $E=-100$ of the knot $4_1$ with $N=200$ and monomer distribution $D_I(167,33)$, see Fig.~\ref{comparison-115-103}.
Basing on the available data, our interpretation of the phenomena associated with the presence of the two peaks in the specific heat capacity of knots in which $N_A>>N_B$ is as follows.
The second peak, appearing in all observed cases at temperatures $\boldsymbol T>0.5$, is due to the breaking of the contacts that are responsible for a bulk compact state that arises due to the attractive forces between the $A$ and $B$ monomers.
The first peak, observed at temperatures $\boldsymbol T< 0.5$, is due to a rearrangement of the knot, or at least a part of it, involving the formation of just a few additional contacts. We suspect that the rearrangement is mostly concerning the segment with the $A$ monomers. Above the transition, there is a bulk compact state which is held together by the portion of the $A$ monomers that are in contact with the $B$ monomers. The other $A$ monomers cannot form contacts because of the few $B$ monomers available. As a consequence, these $A$ monomers form a long tail departing from the bulk compact state and fluctuating almost freely. Below thetransition, a rearrangement leading to the formation of additional contacts with more $A$ monomers becomes possible and the tail disappears or is reduced.
A close inspection of the conformations of the knot $4_1$ that have been captured confirms the above claim, see Fig.~\ref{comparison-115-103} and comments in the related captions.
Other rearrangements could become possible in longer polymers.
They give rise to metastable states that may appear when the temperature is low, so that the formation
of contacts between the $A$ and $B$ monomers is energetically convenient.
Such metastable states and also the double peak are not observed in short knots,
like for instance those with $N=90$.
The main reason is that
in a short knot, the interactions between the monomers are more
frequent than in a longer one. As a consequence, the
rearrangements such that more monomers of the $A$ type will be in contact with monomers of the $B$ type will lead unavoidably also to contacts between the $A$ and $B$ monomers with themselves. This will increase the energy of the obtained conformation due to the repulsive interactions to which monomers of the same type are subjected.
One additional transition is related to a
process analogous to the shrinking taking place at high temperatures in the case of
homopolymers in a good solvent.
Indeed, since the $A$ monomers are numerous in the case $N_A>>N_B$ and they are
subjected to repulsive interactions,
we expect that, at high temperatures, the knot will slightly shrink as
homopolymers do. This shrinking process causes in homopolymers just a small peak in the heat capacity, see
Fig.~\ref{fig-top-shc-N200}, right panel, so that in knotted diblock copolymers the effect on the plots of the heat capacity will be limited. However, the shrinking is visible in the plots of the gyration radius.
The general picture presented above fits very well with
the results obtained for knots of length $N=200$.
The plots of the specific heat capacity of
Fig.~\ref{fig-top-shc-N200}, left panel,
clearly show that the knots with monomer composition $f\sim 0.83$ ($N_A>>N_B)$ have
much lower peaks
than those with $f=0.50$ ($N_A\sim N_B$).
Moreover, the peaks of the specific heat capacity of the knots
with $f\sim0.83$ occur at significantly lower temperatures ($T\leq 0.50$ the first peak and $\boldsymbol
T\leq 0.70$ the second peak) than the
peak of the specific heat capacity of the knots with $f\sim 0.50$
($\boldsymbol T\sim 1.00$).
The sharp increase of the gyration radius following the
Interestingly, out of all investigated knots $0_1,3_1,4_1,5_1$ and
$6_1$ with monomer composition $f\sim 0.83$ (the plots of $0_1$ and
$6_1$ are not reported), the second peak
has been replaced by a
shoulder only in the heat capacity of the knot
$5_1$.
Let's now discuss the results of the knots with monomer distribution
$D_I(100,100)$.
As previously discussed, in this case
double peaks and shoulders are absent from
the plots of diblock copolymers when $N_A\sim N_B$. The plots with the
heat capacities of knots with $D_I(100,100)$ pointed out in
Fig.~\ref{fig-top-shc-N200} confirm this expectation.
Moreover, the peaks of the specific heat capacity are much higher than
those of knots with $D_I(167,33)$.
The data
of longer knots with $N=300$ and $N=500$ agree with the previous conclusions, see Figs.
\ref{fig-longer-shc}--\ref{fig-N500}.
In longer polymers, a third small peak appears in the
case of knot $4_1$ with $N=300$ and monomer distribution $D_I(250,50)$, see Fig.~\ref{fig-new-events}, left panel. As
explained before, this extra peak could be related to the presence of a metastable state.
In the inset of Fig.~\ref{fig-longer-shc} it is shown that
the rapid growth of the gyration radius at about $\boldsymbol T\sim 0.35$ corresponds to the first peak in the specific heat capacity.
The middle peak at about $\boldsymbol T\sim 0.6$ corresponds to a temperature in which the swelling rate is slowing down considerably, see the inset.
We interpret this with the fact that the knot is captured into a metastable state, which stabilises the size of the knot over a small interval of temperatures.
One should keep in mind that the detection of metastable states in long knots
is particularly difficult.
Indeed, there are hints that the energy landscape for long polymers
could be funnel-like like in proteins~\cite{FFYZRMP}.
Thus, long knots are complex systems and the search for metastable states requires an extended sampling before they are found.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-top-rg-N200.pdf}
\caption{Dependence of the mean square gyration radius $R_G^2$ on the
topology and on the monomer composition $f$ in the case of
$AB-$diblock copolymers.
Presented are the plots of $R_G^2$ in the case of the knots $3_1,4_1$
and $5_1$ with $N=200$. For each knots two values of $f$ are
considered: $f=0.50$, corresponding to the monomer
distribution $D_I(100,100)$ and $f\sim 0.83$, corresponding to the
monomer distribution $D_I(167,33)$. In the inset it is shown the detail
of the behavior of $R_G^2$ for the knot $4_1$ with monomer
distribution $D_I(167,33)$.}\label{fig-top-rg-N200}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-spec-hc-N200.pdf}
\includegraphics[width=0.48\textwidth]{fig-local-spec-hc-N200.pdf}
\caption{The left panel shows the dependence of the specific heat
capacity $C/N$ on the
topology and on the monomer composition $f$ for $AB-$diblock
copolymers. The plots of $C/N$ are presented in the case of the knots $3_1,4_1$
and $5_1$ with $N=200$. For each knot two values of $f$ are
considered: $f=0.50$ (case $D_I(100,100)$) and $f\sim 0.83$ (case
$D_I(167,33)$). The right panel shows the specific heat capacity of
the homopolymer version of the knots $4_1$ and $5_1$ with $N=200$
fluctuating in a good solvent.
}\label{fig-top-shc-N200}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-TRY1-conf-100-enhaced.png}
\includegraphics[width=0.48\textwidth]{fig-TRY1-conf-105-enhaced.png}
\end{center}
\caption{ This figure shows two sample conformations of a knot $4_1$ with length $N=200$ and monomer distribution $D_I(167,33)$. The left conformation has energy $E=-100$. It is characterised by a partially ordered portion located in the upper part of the knot in which some of the $A-$monomers are in contact with the $B-$monomers. Other $A-$monomers form a "tail" on the bottom of the knot.
At the just slightly higher energy of $E=-105$, the sample conformation of the knot appears very different. The chain containing the $B-$monomers is more stretched than in the case $E=-100$. This allows the formation of two partially ordered portions of the knot concentrated at both ends of the chain with the $B-$monomers. In both pictures the knots have been rescaled in the same way to fit into the page and no different rescaling has been applied to different axes.}\label{comparison-115-103}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-local-spec-Rg-N300.pdf}
\caption{The data of the gyration radius $R_G^2$ of knots $0_1$, $4_1$, $5_1$ and $6_2$ of length
$N=300$ and monomer distribution
$D_I(250,50)$ are presented. In the case of knot $5_1$ it is shown the plot of $R_G^2$ also for the monomer distribution $D_I(200,100)$. In the inset the behaviour of the gyration radius of knots $0_1$ and $4_1$ is displayed in more details at low temperatures. Note the characteristic saddle point in the plot of the gyration radius of knot $4_1$ at $\boldsymbol T\sim 0.45$.\label{fig-longer-shc}}\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-local-spec-hc-N300.pdf
\includegraphics[width=0.48\textwidth]{fig-local-spec-hc-N300-2-2.pdf}
\caption{Left panel: Plots of the specific heat capacity of diblock copolymers
with $N=300$ for knots: $0_1$, $4_1$, $5_1$ and $6_2$ with monomer
configuration $D_I(250,50)$. The plot of knot $5_1$ with
configuration $D_I(200,100)$ (black squares) has been added to show the differences when the number of monomers of type $A$ and $B$ become comparable. As we see, the specific heat capacity of this knot exhibits just a single peak that is higher than those of the knots with
$D_I(250,50)$. Moreover, the peak appears at a much higher
temperature with respect to all other knots in which $N_A=250$ and $N_B=50$.
The strongest observed compact states have been observed in the case of the monomer distribution
$M_I(N,2,2)$ (right panel).}\label{fig-new-events}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-Rg-N500.pdf}
\includegraphics[width=0.48\textwidth]{fig-spec-hc-N500.pdf}
\caption{Plots of the gyration radius (left panel) and specific heat capacity (right panel)
of a few knots with length $N=500$ in three different monomer
distributions.}\label{fig-N500}
\end{center}
\end{figure}
Finally, also the data of the gyration radius in the case of polymers
with $N=500$
confirm the general picture presented before, see Fig.~\ref{fig-N500}. Again, knots with $f\sim 0.50$ ($N_A\sim N_B$) behave differently from those with
$f\sim1$ ($N_A>>N_B$) or, equivalently, $f\sim 0$ ($N_B>>N_A$).
As a curiosity, from the performed numerical experiments it turns out
that the
minimal energy state created by a knot at the lowest
temperature is not always the most compact one. Indeed,
the size of the knot in two cases (knots $5_1$
with monomer distribution
$D_I(420,80)$ and $3_1$ with monomer distribution $D_I(400,100)$)
decreases at about $\boldsymbol T\sim 0.6$.
This is probably due to the excess of monomers of a
given type. Indeed, when temperatures are low
there are two competing conditions that should be fulfilled in order
to minimise the energy.
First, the largest possible number of
$A$ monomers should be in contact with the few avaliable $N_B$
monomers. At the same time, however,
the $A$ monomers cannot get near to each other, as this
is energetically expensive due to the repulsive interactions between
monomers of equal type. This last requirement is responsible for the
fact that the minimal energy state could be not the most compact one.
If the temperature is rising, in fact, more energy will be available to the system,
so that conformations of the knot in which more $A$ monomers are in
contact with themselves become possible. As a consequence,
certain knots
may attain at higher temperatures a total gyration radius which is
smaller than that of the lowest energy state.
To conclude this subsection,
we would like to stress that,
while homopolymers are simple
systems whose size
steadily increases (in bad solvents) or decreases (in good solvents)
with growing temperatures, diblock copolymers with $f\sim 1$ (or $f\sim 0$),
exhibit a more complex behavior. Their
mean square gyration radius is smallest at low temperatures and
increases up to its maximum value at intermediate temperatures. After
that, it starts to decrease and finally stabilizes
to some value between the maximum and the minimum at high
temperatures.
The presence of three different
regimes, compact, ultra swollen and swollen is strongly dependent on
the monomer composition.
\subsection{Results for Setup II}\label{thermalsetupII}
In Setup II the monomers of type $A$ repel themselves, while monomers
of type $A$ and $B$ are subjected to excluded volume forces. Only the monomers of type $B$
attract themselves. For that reason,
it could be expected that the behavior of
knots in Setup II will be similar to that of the purely repulsive case (i. e. they attain the largest size at very low temperatures to minimize the energy and then shrink upon heating)
unless the number $N_B$ of monomers $B$ will be sufficiently high
to trigger some behavior typical of attractive interactions (i. e. they swell when heated).
This
expectation is only partially true. For instance, knots formed by diblock-copolymers
with $N_B<<N_A$ exhibit a phase of fast, but moderate expansion when heated, so in this sense they share some properties of knots in a bad solvent despite the fact that the monomers of type $A$ subjected to repulsive forces constitute an overwhelming majority.
On the contrary, when $N_A= N_B$ the size of the knot does not exhibit any significant change.
If instead $N_B>>N_A$, attractive interactions are overwhelming and
the behavior of the knot becomes similar to that of a polymer in a bad solvent.
In all cases,
including the monomer distribution $M_{II}(N,2,2)$,
the swelling is much less marked than in Setup I. It takes place in a limited range of temperatures and it is soon followed by the shrinking which is typical
of homopolymers in a good solvent.
The situation is well summarized by
Fig.~\ref{5.1-setupII}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-typeII-Rg-N90-5.1.pdf}
\includegraphics[width=0.48\textwidth]{fig-typeII-CV-N90-55.pdf}
\caption{The mean square gyration radius $R_G^2$ (left panel) and
the heat capacity (right panel) for a knot $5_1$ in Setup II
with $N=90$ and different types of monomer distributions.}\label{5.1-setupII}
\end{center}
\end{figure}
in which the mean square gyration radii and the specific
heat capacities of a knot $5_1$ of length $N=90$ are displayed for different monomer distributions.
A possible explanation of the different behaviours between the cases $N_A>>N_B$
and $N_A\sim N_B$
is that in the latter case the $B$ monomers form a larger number of contacts. The compact state that arises in this way is stable under temperature changes and it starts to melt only at high temperatures, i. e. when $\boldsymbol T\sim 1.00$ similarly as the powerful compact states built by homopolymers in a bad solvent. At such high temperatures the melting process and the shrinking process are no longer well distinguishable in the plots of the specific heat capacity, while this is possible when the monomer distribution is such that $N_A>>N_B$ and the melting of the compat state takes places at much lower temperatures.
It is worth noticing that in the knot $5_1$
the swelling and shrinking phases are well recognizable only when the number of $B$ monomers is small. This is the case of the monomer distributions $D_{II}(71,19)$ or $D_{II}(60,30)$, see left panel of Fig.~\ref{5.1-setupII}. This fact is also visible in the plots of the specific heat capacity of this knot (lines with black squares and black triangles in the right panel of Fig.~\ref{5.1-setupII} which are characterised by a double peak. The first peak can be associated to the swelling process with the melting of the compact state formed by the $B$ monomers and the second, at higher temperatures, to the shrinking process. Indeed, the first peaks in the case of the monomer distributions $D_{II}(71,19)$ and $D_{II}(60,30)$ are centered more or less at the temperatures in which the swelling phase is taking place. The second peaks are instead much broader and appear in the range of temperatures in which shrinking takes place. Finally, the heights of the peaks are of the expected order, with higher peaks in the case of swelling and lower in the case of shrinking.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{cartoon.pdf}
\caption{This figure summarises the situation of knotted diblock copolymers in Setup II: Two chains, one with $A-$type monomers and the other with $B-$type monomers, are attached together at their ends to form a knot. Both chains share the same end-to-end distance $\boldsymbol R_0$. It turns out that
the equilibrium value of the length $|\boldsymbol R_0|$ of $\boldsymbol R_0$ is determined not only by the monomer distribution, but also by topology. The
$A$ monomers contribute to the increase of $|\boldsymbol R_o|$ because they are subjected to repulsive interactions. On the contrary, the $B$ monomers, subjected to attractive interactions, and increasing topological complexities of the knot, both contribute to the decrease of $|\boldsymbol R_0|$. }\label{cartoon}
\end{center}
\end{figure}
To check the effects of topology, in Fig.~\ref{fig11} we have displayed the gyration radii of different knots with two monomer compositions, namely
$f=0.50$ ($D_{II}(45,45)$ and $f=0.79$ ($D_{II}(71,19)$).
As it is possible to see, also the knots $3_1$ and $4_1$ exhibit the same behaviors already observed in Fig.~\ref{5.1-setupII} in the case of knot $5_1$. However, as a general trend, it turns out that
the swelling process taking place in knots with monomer composition $f=0.79$ becomes increasingly more
abrupt and start at lower temperatures with growing topological complexity.
This influence of topology is also visible in the plots of $C/N$
of Fig.~\ref{fig12}.
As a matter of fact, when $f=0.79$,
the heights of the peaks of the specific heat capacity is gradually
rising passing from the knot $3_1$ to the knot $6_1$. Moreover, the
temperature around which the peak of $C/N$ is centered is decreasing with increasing knot complexity.
We notice in Fig.~\ref{fig12} that knot $4_1$ with monomer distribution $M_{II}(90,2,2)$ undergoes a swelling process that is much milder than that of the same knot in Setup I.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{fig-typeII-Rg-N90.pdf}
\caption{The mean square gyration radius $R_G^2$ of knots of
different topology and monomer distributions in Setup II. The length of all
knots is $N=90$. The plots with black points (black squares, circles and triangles) correspond to knots with monomer conformations such that $N_A>>N_B$, while those with white points (white squares, circles and triangles) correspond to knots in which $N_A\sim N_B$. In the former case, it is possible to distinguish an expansion of the knot followed by a shrinking phase.
When $N_A\sim N_B$, only shrinking is observed or, at most, small size fluctuations (knot $3_1$, white squares). There are strong effects of topology
that may be easily detected by looking separately at the plots with black and white points. }\label{fig11}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.50\textwidth]{fig-typeII-CV-N90-2.pdf}
\caption{The specific heat capacity $C/N$ of knots of
different types and monomer distributions. The length of all
knot is $N=90$. This picture shows that there are important effects of topology in the behaviour of the knots.}\label{fig12}
\end{center}
\end{figure}
Going to longer knots, we see in general a fading out of the effects of topology. This is for instance visible in the fact that the knots with monomer distributions $D_{II}(100,100)$ in Fig.~\ref{fig-setup-II-N200} have more or less the same behaviour. A remarkable exception is the knot $3_1$ with monomer distribution $D_{II}(167,33)$ whose values of the gyration radius are decidedly greater than those of knots $4_1$ and $5_1$ with the same monomer distribution. This effect is certainly due to topology
and it has been observed also in the case of the knot $3_1$ with length $N=90$ and monomer distribution $D_{II}(60,30)$.
A temptative explanation of this phenomenon can be the following. Looking at the picture in Fig.~\ref{cartoon}, we see that in Setup II knots consist into two open chains, one with monomers of type $A$ and one with monomers of type $B$, joined together at their ends. For this reason, the end-to-end distance $R_0$ is common for both chains.
$R_0$ determines to some extent also the gyration radii of these chains and eventually the gyration radius of the whole knot.
Clearly, the segment with the $A$ monomers, which are subjected to repulsive interactions, will try to increase the value of $R_0$. On the contrary, the segment with the $B$ monomers which are attracting themselves, will tend to have smaller values of its end-to-end distance. If the monomer distribution is $D_{II}(167,33)$, then the repulsive interactions will certainly be dominating because of the large number of $A$ monomers. This would imply that the value of $R_0$ will be mainly determined by the part with the $A$ monomers. However, if the topology of the knot is complex, then the numbers of turns made by the path of the segment with the $A$ monomers will be high. This will make the knot more compact and thus also $R_0$ will be relatively smaller than in simpler topologies. In this situation it is very likely that the effects of the fluctuations to which the $A$ monomers are subjected will be hampered by the topological constraints and will not be able to destroy the contacts made by the $B$ monomers.
If this happens, there is a chance that the the $B$ monomers will prevail and succeed to keep the value of $R_0$ small as required by the energy and entropy considerations for segment $B$.
As far as it is possible to see from our simulations, this topological mechanism to keep together the knot in a compact state is
working when the topology is more complex than that of a knot $3_1$.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{Rg2-N=200.pdf}
\includegraphics[width=0.48\textwidth]{CV-N=200.pdf}
\caption{The mean square gyration radius $R_G^2$ of knots of
different types and monomer distributions. The length of all
knots is $N=200$.}\label{fig-setup-II-N200}
\end{center}
\end{figure}
\section{Conclusions}\label{conclusions}
The Wang-Landau algorithm has been applied here to study the thermal properties of knotted copolymers in a solution.
Two different types of monomers have been considered, called type $A$ and type $B$. The monomers are subjected to very short-range interactions. Two different setups have been investigated. Setup~I corresponds to the case of charged monomers in an ion solution. The $A-$type monomers carry a positive charge and $B-$type monomers a negative one. In Setup~II the monomers are not charged, but the solvent is good for type $A$ monomers and bad for the $B$ monomers.
Block copolymers knots exhibit a more complex behaviour than knotted homopolymers. The latter are in fact simple
two-state systems. When they are in a bad solvent, they are found at
very low temperatures in extremely compact and ordered conformations,
called crystallites following \cite{binder}. With the increasing of
the temperature, these knots start to swell until the state of
expanded coils is reached. The swelling process is much less violent
than in the case of linear polymer chains studied in \cite{binder},
whose specific heat capacity is characterized by a very sharp peak. In
knotted polymer rings, the peak caused by swelling is much broader.
When the solvent is good,
instead,
knots made by homopolymers are in an expanded state
at low temperatures and slightly shrink with increasing temperatures, see \cite{yzff,yzff2013}.
The introduction of monomers of two kinds drastically changes this situation.
For example, for suitable values of the monomer composition $f$,
knots formed by $AB-$diblock copolymers in an ion solution (Setup I)
perform as homopolymers in a bad solvent at low temperatures, but
exhibit features typical of homopolymers in a good solvent at high
temperatures.
In practice, they become systems with several states in which at least three different states can be distinguished.
At low temperatures these knots are found
in the compact and ordered state of crystallites. With growing temperatures, they are swelling
like homopolymers in a bad solvent, but after a maximum gyration
radius is reached, they start to shrink like knotted homopolymer rings
in a good solvent.
Examples of this behaviour are in Setup I the knot $3_1$ with monomer distribution $D_I(75,15)$ of Fig.~\ref{fig1a} and in Setup II the knot $5_1$ with monomer distribution $D_{II}(60,30)$ of Fig.~\ref{5.1-setupII}.
By choosing the topology and the monomer composition of the knot, it is
possible to tune both its size at different
temperatures and the temperature at which the maximum value of the
gyration radius is attained. Also the range in which the gyration
radius is allowed to vary can be determined to some extent.
Knots with such features are clearly an advantage with respect to homopolymers
in potential medical applications and in the production of intelligent
polymer materials containing knots.
In Setup II, knots formed by $AB-$diblock
copolymers have a behavior that is strongly dependent on the
monomer composition $f$. Following the intuition,
if the number of $A$ monomers largely exceeds that of $B$ monomers, i.~e. $N_A>>N_B$, then it could be expected that
knots behave like knotted homopolymer rings in a good solvent, since
the $A$ monomers are subjected to repulsive interactions.
Conversely, if $N_B>>N_A$ and the $B$ monomers are below the theta point, we would rather expect a behavior typical of a homopolymer knot in a bad
solvent. The performed simulations show however that the situation is more complicated than that.
For instance, in the case $N_A>>N_B$ we see from Fig.~\ref{fig11} that,
in the case of the knots $3_1,4_1$ and $5_1$ with monomer distribution $D_{II}(71,19)$,
the less numerous $B$ monomers play the dominant role at very low temperature. Indeed, these knots
exhibit features typical of homopolymers in a bad solvent, i.~e. they swell
with growing temperatures.
Multiblock copolymers in the classes $M_I(N,n_A,n_B)$ and $M_{II}(N,n_A,n_B)$, where $n_A$ and $n_B$ are small in comparison to $N$, have remarkable
properties too. These properties are not easily predictable by simply looking at
the polymer composition.
For example, we observe a
transition from the compact state to the swollen state which is much more abrupt
than that of knots realized using monomers of the same type or diblock
copolymers.
The main conclusions of this work can be summarised as follows:
\begin{enumerate}
\item The strongest compact states due to the
contacts formed by the monomers subjected to attractive interactions have been observed in knots of various lengths with
monomer distribution $M_{I}(N,2,2)$, see Figs.~\ref{fig-local-N90-b}, right panel and \ref{fig-new-events}, right panel. The specific heat capacity
of these knots is characterised by a high peak concentrated in a very narrow range of temperatures corresponding to the
melting of these bound states. This interpretation is corroborated by the fact that, exactly in the same range of temperatures, the knot undergoes a
sudden and rapid swelling process.
When monomer distributions of the kind $M_{I}(N,n_A,n_B)$ are considered with increasing values of $n_A$ and $n_B$
($n_A,n_B=4,8,...$), the peak widens and its height becomes lower, implying that
strong compact states are still present at least up to the tested value of $n_A=n_B=8$, but they become weaker and weaker.
\item Topological effects strongly influence the behaviour of knots in the case of short polymers ($N\sim 90$). Examples of these effects in Setup I can be observed in
the plots of the gyration radius
and specific heat capacity reported
in Fig.~\ref{fig-top-N90} and in Fig.~\ref{fig-top-shc-N90} respectively. Fig.~\ref{fig12} displays the changes due to topology of the heights and temperatures of the peaks of the specific heat capacity in the case of Setup II.
With increasing polymer lengths these effects fade out and
the choice of the monomer distribution becomes the main factor influencing
the properties of the knots. Yet, topology is still relevant for longer polymers because it provides a way to fine tune their behaviour.
For instance, the effects of topology when $N=200$ on the way in which the gyration radius
changes with different temperatures are reported in Fig.~\ref{fig-top-rg-N200}. A certain dependence on topology of
the heights of the peaks of the specific heat capacity
can also be spotted in the plots of Fig.~\ref{fig-top-shc-N200} where
knots of length $N=200$ are considered.
However, in general the influence of topology is more visible in the plots of the gyration radius. Let us notice that topology plays some role even in the case of the longest polymers that have been investigated here.
For instance, the knot $3_1$ with $N=500$ and monomer distribution $D_I(400,100)$ is suddenly shrinking at a temperature of $\boldsymbol T\sim 0.5$ (the value of $R_G^2$ goes from $75$ to $52$), a feature that is not so marked in the case of the other knots whose plots have been displayed in Fig.~\ref{fig-N500}, left panel, including the knot $5_1$ with the same monomer distribution.
\item Several exceptions to the rule that the influence of topology should fade out with increasing polymer lengths have been observed.
For instance, the gyration radius of the knot $3_1$ with length $N=200$ and monomer distribution $D_{II}(167,33)$ is much larger than
the gyration radii of knots $4_1$ and $5_1$ with the same lengths and monomer distribution, see Fig.~\ref{fig-setup-II-N200}. This striking difference can be explained by the strong entropic effects induced in Setup II
by the fact that the paths of the knots are subjected to topological constraints, see Fig~\ref{cartoon} and related comments. Also when $N=90$, it is possible to see from Fig.~\ref{fig11} that, when the monomer distribution is $D_{II}(71,19)$, the swelling of the knot $3_1$ with rising temperatures is much more limited than that of knots $4_1$ and $5_1$.
\item A characteristics that emerges in knotted block copolymers
and is not present in the case of homopolymers,
is the existence of rearrangements of the knot structures at low temperature. In some cases, this leads to intermediate states. These rearrangements can be detected by the appearance of extra peaks or shoulders in the specific heat capacity of longer knots in Setup~I.
Metastable intermediate states have been observed for instance in knot $3_1$ with $N=500$ and monomer distribution $D_I(400,100)$, see Fig.~\ref{fig-N500}, right panel, and knot $4_1$ with $N=300$ and monomer distribution $D_I(250,50)$, see Fig.~\ref{fig-new-events}.
\end{enumerate}
The
simulations presented in this paper require the sampling of an
extensive amount of knot conformations. Despite major improvements
in the sampling procedure, that of rare events is still a problem in the case of very long polymers.
Some of the
conformations
appear after several hundred billions of trials and their inclusion
extends enormously the calculation times.
Moreover, in this work very short-range interactions have been considered. This is enough to study the cases of flexible knots in a good or bad solutions, but it would be interesting to add more complicated interactions.
In this way it would be possible to consider for instance also the polymer rigidity and the transition from bad to good solvents at the theta point.
Work is in progress to implement in our code the backbone rigidity and the Lennard-Jones interactions.
\begin{acknowledgments}
The simulations reported in this work were performed in part using the HPC
cluster HAL9000 of
the University of Szczecin.
The research presented here has been supported by the Polish National Science Centre under
grant no. 2020/37/B/ST3/01471.
This work results within the collaboration of the COST
Action CA17139 (EUTOPIA). The use of some of the facilities of the Laboratory of
Polymer Physics of the University of Szczecin, financed by
a grant of the European Regional Development Fund in the frame of the
project eLBRUS (contract no. WND-RPZP.01.02.02-32-002/10), is
gratefully acknowledged. \end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,831
|
Q: If inline JavaScript code will be executed before document load event? I have a page with <form> tag. Inside a form there is a lot of html plus some inline JavaScript at the very end of the <form> tag.
I listen to document load event. Can I be 100% sure, that when document load will be fired, all this inline JavaScript code has been already executed?
Example of markup:
<body>
<form>
--html controls---
<script type="text/javascript">
--some code to run here--
</script>
</form>
</body>
My thoughts that answer is yes, inline JavaScript will be executed before document load, but I want to find evidence.
edit
live demo
Document load fires only when all html controls loaded and JavaScript (inline or with src attribute)loaded and interpreted. Am I correct with this statement?
A: Unless you put the code you want to execute in the domready callback function, your inline javascript code will be executed immediately when you load the page (before the the domready).
A: Inline script will execute immediately as soon as the script tag finishes parsing, so you won't be able to access the rest of the document yet. On the other hand, it allows you to write additional HTML at that point in the document.
Note that Firefox 3.5 had a bug whereby you could set the defer attribute on the inline script and it would not execute immediately. This non-standard behaviour was fixed in Firefox 3.6.
A: I think you have no guarantees
If it is a slow javascript (emscripten) i think it is possible that the code is still beeing executed while the onload fires.
but i could not find clear documentation:
https://developer.mozilla.org/en/DOM/window.onload
http://msdn.microsoft.com/en-us/library/ie/cc197055%28v=vs.85%29.aspx
http://www.w3.org/TR/html4/html40.txt :
onload = script [CT]
The onload event occurs when the user agent finishes loading a
window or all frames within a FRAMESET. This attribute may be
used with BODY and FRAMESET elements.
so I can't find guarantees, you can either test (with something heavy, e.g. a demo from here https://github.com/kripken/emscripten/wiki), hpe for the best, or build in a savequard of your on that checks weather your inline script completed
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,146
|
Bobby Bowden dies: Legendary coach built Florida State into college football powerhouse
Jim Henry
Bobby Bowden, who built Florida State football into a national powerhouse and directed the program with a folksy, southern charm, died early Sunday morning.
On July 23, Terry Bowden – son of the Hall of Fame coach – revealed his father was suffering from pancreatic cancer. This came a day after his family released a statement through the Democrat indicating he had been diagnosed with a terminal medical condition.
Bowden was surrounded by his family — wife Ann and their six children — when he passed away peacefully at 5:08 a.m. at his Killearn Estates home, daughter Ginger Bowden told the Democrat Sunday morning.
"He passed peacefully," Ginger said. "His family was with him during the night."
Funeral arrangements:Bobby Bowden to lie in honor at Florida Capitol; Public service set for Tucker Civic Center
NEWS OBITUARY:His name shall endure: Bobby Bowden took FSU from 'nowhereland to splendor' | Gerald Ensley
From the sports editor:Early to bed, early to rise. Morning calls with Coach Bowden were special | Jim Henry
"I couldn't have asked for a better personal mentor than my father," Terry Bowden, the first-year coach at the University of Louisiana at Monroe, said in a statement Sunday.
"He was a wonderful husband and father, who relied on his strong Christian faith to provide the foundation for his life. I also was fortunate to be raised by a football coach who had a reputation for coaching the right way his entire career. He was admired by everyone who played for him or coached against him. As a family, we will embrace all of those wonderful memories and celebrate a life well lived."
Bowden was being treated at his Killearn Estates home by caretakers and family. Upbeat and optimistic, he also felt well enough at times during his final weeks to welcome visitors and take telephone calls.
"I feel fine but I can't do much," Bowden told the Democrat in early July.
Bowden — a devout Baptist — made his last public appearance on stage in early June as the guest speaker at the Send Luncheon, hosted by the North American Mission Board (NAMB), in Nashville.
Bowden's health had deteriorated after he tested positive for COVID-19 in October 2020. He was hospitalized in late June for five days for fatigue and additional medical tests.
Recent talks with Bobby Bowden
Seminoles & Gators together:Former UF stars Yancey Sutton, Scot Brantley visit Bobby Bowden
'I had never heard of them':Bobby Bowden was Alabama fan when Florida State was a girls school
Conversation with Bobby:Ever wonder how Bobby Bowden became a football coach? New podcast reveals this and more
Bobby Bowden won two national championships and built FSU into a national power
Bowden arrived in Tallahassee in 1976, never to leave and becoming one of college football's most successful coaches and patriarch of a well-known football family.
Bowden posted a 316-97-4 record with two national titles (1993 and 1999) in 34 years at FSU. He had one losing season – 5-6 during his first year at the school in 1976 – and was forced into retirement following a 7-6 record in 2009.
Bowden ended his career with a 33-21 victory in the Gator Bowl over West Virginia on Jan. 1, 2010.
More:FAMU football coaches share fond ties to famed FSU legend Bobby Bowden
Bowden boasted an overall coaching record of 377-129-4 to rank second all-time in major college football history behind Joe Paterno (409 wins). He ranks fourth all-time across all divisions in college football.
Between 1987 and 2000, Bowden guided the Seminoles to 14 consecutive 10-win seasons and top-five finishes in the Associated Press poll. That streak earned the program Dynasty status by the NCAA.
Two of his FSU players (quarterbacks Charlie Ward and Chris Weinke) won the Heisman Trophy and three (cornerback Deion Sanders, linebacker Derrick Brooks and offensive lineman Walter Jones) went onto NFL greatness and are in the Pro Football Hall of Fame. Twenty-six Seminoles were named consensus All-Americans under Bowden and his players also earned major awards.
When he received the inaugural Governor's Medal of Freedom from Gov. Ron DeSantis in April, Bowden was quick to share the success.
"I get the credit, the head coach gets the credit, but it's the coaches who do all the coaching," Bowden said. "I had great coaches and I had some great players... They get you there."
Bobby Bowden was married 72 years to his wife, Ann
Bowden, a native of Birmingham, Alabama, was married 72 years to childhood sweetheart Ann Estock. Bowden was 19, Ann 16 when they married at the home of the Justice of the Peace in Rising Fawn, Georgia.
They lived in the same Killearn Estates home they moved into when Bowden was hired by FSU from West Virginia 45 years ago. Bowden was such a fixture in the community that his phone number was even listed in the phone book. He also allowed fans to drop off memorabilia in his carport that he'd sign and leave for pickup.
The couple raised six children – four boys and two girls – and their family reunion at their beach home on Panama City Beach in the spring of 2020 featured more than 40 family members.
Sons Tommy, Terry and Jeff each coached at the collegiate level. Tommy was previously the head coach at Tulane and Clemson; Terry was the head coach at Salem, Samford, Auburn, North Alabama, Akron and last December was hired by Louisiana-Monroe. He left the team Friday to be with his ailing father in Tallahassee. Jeff was an assistant at FSU under his father for 13 seasons.
Steve Bowden has worked most of his career in academia. Daughter Ginger Bowden Madden is the state attorney for the First Judicial Circuit of Florida. Robyn – the oldest child – was a school teacher for many years but retired early to help her parents.
Bobby Bowden credited his success to strong Christian faith
Bowden credited his football success to his strong faith, often sharing his Christian testimony from the church pulpit over the years.
Bowden was a creature of habit – early to bed, early to rise.
He often arose at 4 a.m., reading the Bible, skimming through a book and the Tallahassee Democrat newspaper with his coffee. Bowden also loved to golf – his home is off the seventh hole at Killearn Country Club – and watch World War II documentaries. He also had a noted sweet tooth and was diagnosed later in life with Type 2 diabetes.
Bobby Bowden's late health issues included cancer, COVID
The past few years Bowden was slowed by lingering, painful back and hip issues that kept him off the golf course and from walking his neighborhood. At one time he was one of the country's most sought-after motivational speakers, sharing football stories and his faith.
Bowden's health issues were magnified in mid-September 2020, when he was hospitalized at Tallahassee Memorial HealthCare for nearly 10 days with a leg infection following the removal of skin cancer spots.
Bowden was informed he tested positive for COVID-19 – the infectious disease caused by the coronavirus – on Oct. 3, two days after his release from the hospital's rehab facility.
Bowden was readmitted to the hospital for fatigue on Oct. 6. He underwent treatment for COVID-19 for 10 days before being released on Thursday, Oct. 15.
Regaining his strength proved to be a difficult challenge for Bowden during his final months, which included the announcement that he was fighting pancreatic cancer.
As family gathered by his side, an outpouring of well-wishes flooded social media and the airwaves, including from FSU football coach Mike Norvell. Norvell, 39, is entering his second season and trying build his own legacy with the Seminoles' once-mighty program.
"We're grateful for the example of Coach Bobby Bowden," Norvell said Sunday morning following practice. "And we're going to honor him in everything that we do, each and every day. Because he helped build this place into something that is incredibly special -- with all of his heart and all of his life. And we're grateful for him."
Reach Jim Henry at jjhenry@tallahassee.com.
No one covers the 'Noles like the Tallahassee Democrat. Subscribe using the link at the top of the page.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,253
|
package game.ui;
/** #define DEFAULT_LABEL_MARGIN Vector2(15, 15) */
import com.badlogic.gdx.math.Vector2;
import game.CGame;
import game.tilesets.CFontTileset;
/**
* @brief A UI element that renders text
*/
public class CLabel extends CUIElement {
/**
* @brief Create a new label
* @param text
* The text to show
* @param margin
* The margin around the text for spacing
* @param font
* The font to use
*
* @brief The margin around the text
*/
protected Vector2 DMargin = new Vector2();
/**
* @brief The text to show
*/
protected String DText;
/**
* @brief Font used to draw
*/
protected CFontTileset DFont;
public CLabel(CGame game, String text, Vector2 margin) {
this(game, text, margin, null);
}
public CLabel(CGame game, String text) {
this(game, text, new Vector2(15, 15), null);
}
public CLabel(CGame game, String text, Vector2 margin, CFontTileset font) {
DFont = font != null ? font : game.Resources().DTilesets.DBlackFont;
Text(text);
Margin(new Vector2(margin));
}
/**
* @brief Sets the text and updates the size
* @param text
* The new text
*/
public final void Text(String text) {
DText = text;
UpdateSize();
}
/**
* @brief Sets the margin and updates the size
* @param margin
* The new margin
*/
public final void Margin(Vector2 margin) {
DMargin = new Vector2(margin);
UpdateSize();
}
/**
* @brief Draws the text into the game
* @param game
* The game to draw in
* @param translation
* The computed translation
*/
public void Draw(CGame game, Vector2 translation) {
DFont.DrawText(game, new Vector2(Margin().x + translation.x + Translation().x, Margin().y + translation.y
+ Translation().y), Text());
CUIElementDraw(game, new Vector2(translation));
}
/**
* @brief The margin
* @return The margin
*/
public final Vector2 Margin() {
return DMargin;
}
/**
* @brief The text
* @return The text
*/
public final String Text() {
return DText;
}
/**
* @brief Used to recompute the size of the text and update the size of the label
*/
protected void UpdateSize() {
int Width;
int Height;
Vector2 dimensions = DFont.MeasureText(Text());
Size(new Vector2(dimensions.x, dimensions.y).add(Margin().cpy().scl(2f)));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,670
|
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