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{"url":"https:\/\/www.physicsforums.com\/threads\/a-question-about-relativity-model.81010\/","text":"# A question about relativity model\n\n1. Jul 3, 2005\n\n### Ar edhel\n\nI read a recent article in the globe and mail about Einstein theory of motion. it suggested a comparison between a observer's onboard and offboard a moving object, in this case a train. They hinted a very similar model to a ball bouncing on a moving train in my other studies. onboard the train they represent light as traveling a straight perpendicular vertical up and down motion. relative to an observer onboard, the light was required to travel no additional horizontal motion. Then they described how an observer offboard the train would observe the same motion, but also spread across the horizontal distance covered by the motion of the train.\n\nI hope i described the example that was printed well enough. I have a question, mostly to ask if this theory still holds true, and is this still an accurate model... (long question ahead)\n\nI believe this description is very similar to a ball bouncing model we learned about in physics. while a ball bouncing on a train, is not dealing with the nature of light, i saw a few perplexing similarities. As a ball was described, this example deduces using the similar principles. how is it a ball, who's actual speed is not fixed by a constant, be accurately compared against a constant such as light. in order for an observer onboard to bounce a ball up and down (it could be described) as having no additional horizontal motion. and the observer onboard would reflect that it traveled a shorter distance...\n\nBut how would this be in any way plausible if the ball was never boosted by the horizontal motion of the train?\n\nOh and later on in the article it clearly states that motion can neither slow or boost the speed of light. as per Einsteins very words.\n\nThus i consider it highly improbable that light would take less time to travel, for an observer onboard. because the mere motion alone would not increase the horizontal motion without reducing the amount of motion capable of also exerting vertical... regardless no matter what fraction of time is experienced onboard the train. it is utterly strange for me to conceive that the train would not similarly move forward in the time between up and down intervals, regardless of what smaller accountable time is experienced onboard.\n\nLast edited: Jul 3, 2005\n2. Jul 3, 2005\n\n### JesseM\n\nThe example you're talking about with a light beam moving up and down is known as a \"light clock\"--see the explanation here--and it's used to show that in order for both observers to measure the light moving at the same speed c, it must be true that the observer on the tracks sees clocks onboard the train slowed down by a factor of $$\\sqrt{1 - v^2\/c^2}$$, where v is the velocity of the train as measured by the observer on the tracks.\n\nFor a bouncing ball, of course it's no longer true that both observers will measure it travelling at the same speed. If the observer on the train sees the vertical velocity as y and the horizontal velocity as u (u=0 if the ball is bouncing straight up and down), then if the observer on the tracks sees the train going by at velocity v, according to Newtonian physics you'd expect this observer to see the vertical velocity as y and the horizontal velocity as u+v, so using the pythagorean theorem he'd see the total velocity as $$\\sqrt{y^2 + (u+v)^2}$$. But in relativity velocities don't add in this simple way--the observer on the tracks will see the vertical velocity as $$y \\sqrt{1 - v^2\/c^2 }$$ (because if the observer on the train sees the object moving up a distance y in a time 1, the observer on the tracks sees it move up a distance y in a time $$1\/\\sqrt{1 - v^2\/c^2}$$, due to the fact that he sees the train's clocks slowed down by $$\\sqrt{1 - v^2\/c^2}$$), and he won't see the horizontal velocity as u+v, instead you must use the formula for addition of relativistic velocities (that site is down at the moment so you can also look here for more info), $$(u+v)\/(1 + uv\/c^2)$$. So, using the pythagorean theorem the observer on the tracks will see the total velocity as $$\\sqrt{[y^2 (1 - v^2\/c^2)] + [(u+v)^2 \/ (1 + uv\/c^2)^2 ] }$$. You can see that if you apply this formula to y=c and u=0 (a light beam bouncing straight up and down, as seen by an observer on the train), then the observer on the tracks will see the total velocity as $$\\sqrt{c^2 (1 - v^2\/c^2) + v^2 }$$, or $$\\sqrt{ c^2 - v^2 + v^2}$$, which of course reduces to c. On the other hand, if you apply it to a bouncing ball with y equal to a velocity smaller than c, then the two observers will get different answers about its total velocity.\n\nLast edited: Jul 3, 2005\n3. Jul 3, 2005\n\n### Ar edhel\n\nOk it seems straight forward thanks. i may have misunderstood a simple concept about light clocks, i do however have a question, again possibly a long one. (note if you care to read brush up on jesseM's first example) it is undeniable the model you should choose, mine resulting from idle speculation.\n\ni do not see how the motion of the light would reflect in any regard horizontal motion, once emitted it would travel directly upwards. the horizontal motion of the skateboard would not boost the light into following a horizontal path. the reason why it does is because it is chained to a pre guided path within a cylinder. as the cylinder contains horizontal motion, the light within travels an artificially forced increase in horizontal motion, as it may not deviate from its forced route. this is no longer a constant speed, it now contains an artificial method for inducing horizontal motion as well as being required to travel full vertical speed. measuring the distances light traveled is fine... but can you use a constant for light in both equations?\n\nBefore you answer. in order to conclude this test the light must also be given the horizontal motion required to remain at the same rate of speed and distance as the observer onboard. by whatever means does it seem likely that light would have traveled horizontal motion by itself? noting that motion may not boost the speed of light.\n\nits hard for me to comprehend, i tend to examine light the instant it is emitted upwards, as a frame... i understand that pausing that moment. light emitted at that point in time would not suddenly change from a purely vertical, towards a vertical and horizontal linear angle. without atleast a force or condition to shape its path horizontal. i wonder if unconditioned to follow a certain path, the light would be required to be sent at an angle to compensate for horizontal motion.\n\nit is not my intent to prove Einstein wrong, in fact i adore his opinions and his life. i have always contained a fond interest of his theories, when i came of the age to understand and fully contemplate them. i however hold back an opinion i believe strongly. it may seem like i blatantly attack peoples opinions, but i do often think best when I'am challenged by other people. I mean no offence to you or anyone that most likely understand's this as fact. i merely challenge other peoples beliefs, cause I'am often not educated enough to understand their explanation... well thanks\n\n4. Jul 3, 2005\n\n### JesseM\n\nIf you have a light source which emits light only in a certain direction, like a flashlight or a laser, then it will always emit the light in the same direction in its own rest frame, and if it always emits light vertically in its own rest frame, then from the point of view of an observer who sees the source moving horizontally, the light beam will move diagonally. You shouldn't need relativity to show this, presumably it can be derived as a consequence of Maxwell's laws of electromagnetism which predate relativity.\n\n5. Jul 5, 2005\n\n### Ar edhel\n\nok im a dullard...\n\n$$\\sqrt{1 - v^2\/c^2}$$\n$$1\/\\sqrt{1 - v^2\/c^2}$$\n\nthese formulas represent the increase of distance. to solve or work it out, do they not exactly also measure the amount of vertical motion lost by the exact nature of measuring the total increase as a result of horizontal motion?\n\ndeeming that it is easier to discribe that time slows onboard rather then our time increases\n\nLast edited: Jul 5, 2005\n6. Jul 5, 2005\n\n### JesseM\n\nThe Lorentz contraction formula $$L = L' \\sqrt{1 - v^2\/c^2}$$ tells you the decrease in the length of a moving object in your frame (L' is the length in the object's rest frame, L is the length measured in your frame), but only along the axis the object is moving. So if something is moving horizontally, its horizontal length will appear to shrink but the vertical length will be unchanged.\n\n7. Jul 5, 2005\n\n### Ar edhel\n\ni think that is simple to understand and straight forward. is that because light of the observer ofboard measures an angle vrs a straight line vertically?\n\nas for the actual length, the theory you put forward simple measures the distances as though light was not forced to compensate for additional horizontal motion, is this the correct answer?\n\nLast edited: Jul 5, 2005\n8. Jul 5, 2005\n\n### Ar edhel\n\nsorry to refer to starwars, but if you have a lightsource that emits light slow enough, by this definition you could wave it around like a lightsaber. it would only grow longer. hehe\n\n9. Jul 5, 2005\n\n### JesseM\n\nnot sure what you mean by that...just think of a hose spraying water out, it always sprays water straight out from the hose in the rest frame of the hose, but that means someone moving relative to the hose will see the the stream of water moving diagonally. For example, if the hose is pointing down from the ceiling of a train car, an observer on board the train will see the stream of water go straight down to the spot on the floor directly underneath the hose, but an observer on the track sees the train moving, so any water droplet that's part of the stream will have a horizontal velocity as well as a vertical velocity and thus move diagonally in his frame.\n\nLast edited: Jul 5, 2005\n10. Jul 5, 2005\n\n### JesseM\n\nWhat do you mean by \"forced to compensate for the additional horizontal motion\"? The length of a moving object can be measured using rulers and clocks--if I have two clocks which are 1 meter apart in my frame and which are also synchronized and at rest in my frame, then if a rod is flying by these clocks and the back end of the rod passes the back clock at the same time the front end passes the front clock (with 'same time' meaning both clocks show the same reading at the moment of these two events), then I say this rod is 1 meter long in my frame. But if it's moving at 0.866c, for example, then in its own rest frame it would be measured to be 2 meters long.\n\n11. Jul 5, 2005\n\n### Ar edhel\n\nthe water carrys the inertia of the motion. if light does the same, if you wave a flashlight horizontal is it likely that if it was slowed to a visible state. it would move as as something similar to a lightsaber. in other words, if you imagined you held a slowed lightsource in your hand and slowly moved it left to right. at several different points amongst that motion, would the light all eventually arive at the same location? everything i read supports that light is effected by the inertia of the motion, otherwise the light emitted, early in the experiment, would have gone directly vertical and the experiment would of kept going horizontal without it.\n\nI cant qoute this *i cant even remember whos side of the coin this side lands on* If something has inertia does it not also have speed?\\\n\nIt would seem that older light would forsake its original path throughout the motion, in order to conduct the test in a different location then when the light was emitted, it would seem that light thoughout the experiment would be required to contain inertia in order go vertical, without being left behind the train.\n\nLast edited: Jul 5, 2005\n12. Jul 5, 2005\n\n### JesseM\n\nSame location where, and in whose frame? Like the same spot on the ceiling of a moving train car where the light source was bolted to the floor? Also, what properties of the \"lightsaber\" are you thinking of that an ordinary flashlight wouldn't also have?\n\n13. Jul 5, 2005\n\n### Ar edhel\n\nhehe sorry, i am kinda poorly worded in nature. to remain at the same onboard frame, is it likely that as the light moves up and down. horizontal motion is attained, now in order for the light to go up and down. the offboard frame would view that light had to cover enough horizontal motion in order to remain at the same horizontal position as the onboard observer.\n\ni believe it is inertia that causes this because the amount of horizontal distance light travels would almost directly corilate with the speed of the train. so in order to travel the same distance as the scientist onbaord, is it plausible that light would be required to expend a portion of its available vertical speed to remain at the same rate of horizontal movement of the train? given that light may be a constant?\n\nWith the increased distance light travels, from the frame of an offboard observer. a stationary clock with a total vertical distance equal to the hypotenuse angle would seem to get the same results...\n\nLast edited: Jul 5, 2005\n14. Jul 5, 2005\n\n### JesseM\n\nYeah, exactly. Did you look at the link to the explanation of the \"light clock\" thought-experiment in my first post? It's all about a beam of light that's bouncing up and down in one frame, so in another frame it must be moving diagonally.\nIts vertical speed $$s_v$$ will have to be less than c, since the total speed of light must be c in all frames, and by the pythagorean theorem the total speed will be $$\\sqrt{ {s_v}^2 + {s_h}^2 }$$, where $$s_h$$ is the horizontal speed. Of course, this equation applies for any object that's moving both vertically and horizontally, not just light; it's just that for other objects, their total speed can be different in different frames.\nRight, assuming by \"same results\" you just mean the light will take the same amount of time to go from the bottom mirror to the top mirror.\n\nLast edited: Jul 5, 2005","date":"2017-03-28 14:24:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7034074068069458, \"perplexity\": 613.1849995100231}, \"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-2017-13\/segments\/1490218189771.94\/warc\/CC-MAIN-20170322212949-00244-ip-10-233-31-227.ec2.internal.warc.gz\"}"}
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\section{Introduction}
The study of congruences between automorphic forms has a long and rich tradition.
A paradigm shift occurred when Deligne's construction of Galois representations attached to classical holomorphic Hecke eigenforms opened the door to the study of congruences of automorphic forms through congruences of Galois representations.
In fact, conjectures of Fontaine--Mazur--Langlands and Serre suggest that these are really two sides of the same coin.
Let $p$ be a prime.
Recall that Serre's conjecture asserts that every continuous, odd, and irreducible Galois representation $\rhobar: G_{\Q} \ra \GL_2(\overline{\F}_p)$ of the absolute Galois group of $\Q$ is \emph{modular}, i.e.~arises from the reduction of a Galois representation attached to a classical modular form.
It furthermore asserts a refinement which specifies the minimal weight and level at which one may find such a modular form in terms of local properties of $\rhobar$.
Turning the perspective around, if one begins with a modular $\rhobar$, then the refinement predicts congruences between modular forms of many different levels and weights.
While the level part generalizes quite naturally, the weight part is subtler, because it turns out to be inextricably linked to integral $p$-adic Hodge theory. The goal of this survey is to describe the circle of ideas surrounding recent developments on the weight part of Serre's conjecture.
\subsection{Overview}
In \S \ref{sec:coh}, we articulate the main questions of interest in the context of cohomological automorphic forms, especially motivating the representation theoretic perspective on congruences.
In \S \ref{sec:reptheory} and \ref{sec:galrep}, we briefly discuss background on modular representation theory and Galois representations. This will be necessary to state the higher dimensional generalizations of the weight part of Serre's conjecture.
In \S \ref{sec:conj}, we state generalizations of Serre's conjecture due to Ash, Gee, Herzig, and Savitt among many others which provide conjectural answers to these questions in many cases. In \S \ref{sec:result}, we narrow our focus to two cases, definite unitary groups in joint work with B.~Levin and S.~Morra and $\GL_n$ over CM fields, where we established some conjectures of \cite{GHS} when $\rhobar$ is tame and sufficiently generic at $p$.
In the proofs, the Kisin--Taylor--Wiles method plays a vital role in reducing a global problem to a local one.
As the internal details of the Kisin--Taylor--Wiles method are orthogonal to our goals, we have chosen to axiomatize its essential output and explain how it is used.
Finally, \S \ref {sec:LM} summarizes the key results on local models for local deformation rings which are the essential ingredients to prove the results on the weight part of Serre's conjecture.
\subsection{Acknowledgments}
This survey is based on two talks given by the authors at the International Colloquium on Arithmetic Geometry at TIFR in January 2020.
We thank the organizers for the invitation to this wonderful mathematical and cultural event.
D.L.~was supported by the National Science Foundation under agreement No.~DMS-1703182. B.LH.~acknowledges support from the National Science Foundation under grant Nos.~DMS-1128155, DMS-1802037 and the Alfred P.~Sloan Foundation.
\section{Cohomology of arithmetic manifolds} \label{sec:coh}
Let $\bG$ be a connected reductive group over $\Q$.
Let $A_\infty^\circ$ be the connected component of the group of $\R$-points of a maximal $\Q$-split torus in the center of $\bG$, and let $K_\infty^\circ$ be a maximal connected compact subgroup of $\bG(\R)$.
For a compact open subgroup $K_f \subset \bG(\A)$, consider the adelic double quotient
\[
Y(K_f) \defeq \bG(\Q)\backslash \bG(\A)/A_\infty^\circ K_\infty^\circ K_f.
\]
In various places, we require technical properties of $K_f$ that can always be attained by passing to a finite index subgroup.
Furthermore, these required properties are preserved under passing to a finite index subgroup (away from a finite set of places).
We assume throughout that $K_f$ is sufficiently small (cf.~\cite[\S 9.1]{MLM}).
In particular, $K_f$ is neat so that $Y(K_f)$ is naturally a real manifold.
Moreover, $Y(K_f)$ can be rewritten as a finite disjoint union of quotients of the symmetric space $\bG(\R)/A_\infty^\circ K_\infty^\circ$ by subgroups of $\bG(\Q)$ which are discrete and of finite covolume in $\bG(\R)$.
Finally, $Y(K_f)$ is homotopy equivalent to its Borel--Serre compactification (cf.~\cite{Borel-Serre}), a compact real manifold with corners and, in particular, a finite CW complex.
\subsection{Rational cohomology} \label{sec:ratcoeff}
Let $V$ be an algebraic representation of $\bG$ over $\Q$.
Then let $\cV_{\Q}$ be the $\Q$-local system
\[
\bG(\Q) \Big\backslash (\bG(\A) /A_\infty^\circ K_\infty^\circ K_f) \times V(\Q),
\]
where $\bG(\Q)$ acts diagonally.
Then the (finite-dimensional) sheaf cohomology groups $H^*(Y(K_f),\cV_{\Q})$ have a convolution action by the double coset (Hecke) algebra $\Q[K_f\backslash \bG(\bA_f) /K_f]$.
(We will consider sequences of cohomology groups as objects in appropriate bounded derived categories. In most instances, little is lost if $H^*(Y(K_f),\cV_{\Q})$ is replaced by $\oplus_{i\in \Z} H^i(Y(K_f),\cV_{\Q})$.)
The significance of these cohomology groups stems from the fact that the Hecke module $H^*(Y(K_f),\cV_{\Q}) \otimes_{\Q} \C$ can be computed by automorphic forms by work of Matsushima and Franke \cite{matsushima,franke}. This is essentially Hodge theory for the locally symmetric manifolds $Y(K_f)$.
Suppose that $A_\infty^\circ$ acts on $V(\C)$ through a character.
If we let $\cA(K_f)$ denote the space of automorphic forms on $Y(K_f)$, then
\begin{equation}\label{eqn:matsushima}
H^*(Y(K_f),\cV_{\Q}) \otimes_{\Q} \C \cong H^*(\mathfrak{g},\mathfrak{p},(\cA(K_f) \otimes_{\C}V(\C))^{A_\infty^\circ}),
\end{equation}
where $\mathfrak{g}$ is the Lie algebra of the group of real points of the intersection of all kernels of rational characters of $\bG$.
Let $\cV_{\C}$ be the local system
\[
(\bG(\Q)\backslash \bG(\A)/K_f) \times V(\C) \Big/ A_\infty^\circ K_\infty^\circ,
\]
where the right action of $A_\infty^\circ K_\infty^\circ$ is diagonal with the inverse of the left action on $V(\C)$.
Then the bijection
\begin{align}
\bG(\A) \times V(\C) &\ra \bG(\A) \times V(\C)\\
(g,v) &\mapsto (g,g_\infty^{-1}v)
\end{align}
induces an isomorphism $\cV_{\Q} \otimes_{\Q} \C \risom \cV_{\C}$.
This is the first step in establishing \eqref{eqn:matsushima}.
We will assume that we can write $K_f$ as $K_\Sigma K^\Sigma$ where $\Sigma$ is a finite set of finite places and $K^\Sigma = \prod_{\ell\notin \Sigma} K_\ell$ where for all $\ell\notin \Sigma$, $K_\ell$ is a hyperspecial subgroup of $\bG(\Q_\ell)$ (in particular, we assume that $\bG$ is unramified at places $\ell\notin \Sigma)$.
Then $T^\Sigma_{\Q} \defeq \Q[K^\Sigma \backslash \bG(\A^\Sigma)/K^\Sigma]$ is commutative (see \S \ref{sec:satake}). Since $H^*(Y(K_f),\cV_{\Q})$ are finite dimensional $\Q$-vector spaces, the eigenvalues of the $T^\Sigma_{\Q}$-action on the part of $\cA(K_f)$ which contributes to $H^*(Y(K_f),\cV_{\C})$ (the \emph{cohomological automorphic forms}) are algebraic numbers.
\subsection{Classical modular forms} \label{sec:classical}
If $\bG = \GL_2$, then $Y(K_f)$ is a modular curve and has the additional structure of a variety defined over $\Q$.
Any irreducible algebraic representation of $\GL_2$ is of the form $V(a,b) \defeq \Sym^{a-b} (\Q^2) \otimes_\Q \det^b$.
Let $T_\ell \in \Q[\GL_2(\Z_\ell) \backslash \GL_2(\Q_\ell)/\GL_2(\Z_\ell)]$ be the double coset operator \[\GL_2(\Z_\ell) \begin{pmatrix}
\ell & 0 \\
0 & 1
\end{pmatrix}
\GL_2(\Z_\ell).\]
A well-known incarnation of \eqref{eqn:matsushima} is that there is a normalized Hecke eigenform
\[
f(z) = \sum_{n=0}^\infty a_n q^n, \qquad \textrm{with } q = e^{2\pi i z}
\]
(i.e.~$a_1=1$) of weight $k \geq 2$ and level $K_f$ (or $K_f \cap \SL_2(\Z)$) if and only if there is a $T^\Sigma_{\Q}$-eigenvector in $H^1(Y(K_f),\cV(b+k-2,b))$ such that $T_\ell$ acts by $\ell^b a_\ell$ for all $\ell \notin \Sigma$.
It is well-known that the space of modular forms has a basis with integral $q$-expansions whose $\Z$-span is Hecke stable.
In particular, $(a_\ell)_\ell$ are not just algebraic numbers, but are in fact algebraic integers. This gives one way to make the notion of congruences between eigenforms precise: one asks for a congruence between the (integral) Fourier coefficients.
It turns out that there are a lot of congruences between $q$-expansions of integral Hecke eigenforms.
A basic example comes from the Eisenstein series
\[
G_k(z) \defeq -\frac{B_k}{2k} + \sum_{n\geq 1} \sigma_{k-1}(n)q^n, \qquad k \geq 4 \textrm{ even},
\]
where $B_k$ is the $k$-th Bernoulli coefficient.
Fixing a (rational) prime $p$, the mod $p$ $q$-series $G_k \pmod p$ depends only on $k \pmod {p-1}$.
Another well-known example is the congruence
\begin{equation}\label{eqn:cuspcong}
\Delta(z) \defeq q\prod_{m=1}^\infty (1-q^n)^{24} \equiv q\prod_{m=1}^\infty (1-q^n)^2 (1-q^{11n})^2 \pmod {11}
\end{equation}
between the unique normalized cuspforms of level $\Gamma(1)$ and weight $12$ and of level $\Gamma_0(11)$ and weight $2$, respectively.
The above notion of congruences between eigenforms is essentially equivalent to congruences between the system of Hecke eigenvalues on rational cohomology, and thus can also be detected by contemplating the action of (suitably integral) Hecke operators on cohomology with integral coefficients. It turns out that this shift of perspective from $q$-expansion to integral cohomology (initiated by Ash--Stevens) will give a systematic mechanism to explain congruences between automorphic forms via representation theory.
\subsection{Integral structure}\label{sec:integral}
Fix a prime $p$ and suppose that $K_f$ factors as the product $K_f^p K_p$.
We fix an algebraic closure $\ovl{\Q}_p$ of $\Q_p$ and let $E$ be a subfield of $\ovl{\Q}_p$ of finite degree over $\Q_p$.
By replacing $E$ if necessary, we will assume that $E$ is sufficiently large.
Let $\cO$ be the ring of integers of $E$ with uniformizer $\varpi$ and $\F$ be the residue field.
We define $\cV_{E}$ to be the nonarchimedean analogue
\[
(\bG(\Q)\backslash \bG(\A)/A_\infty^\circ K_\infty^\circ K_f^p) \times V(E)\Big/ K_p
\]
of $\cV_{\C}$ in \S \ref{sec:ratcoeff}, where $K_p$ acts diagonally (using the natural right action on $\bG(\Q)\backslash \bG(\A)/A_\infty^\circ K_\infty^\circ K_f^p$ and the inverse of the natural left action on $V(E)$).
Then as before, the map
\begin{align}
\bG(\A) \times V(E) &\ra \bG(\A) \times V(E)\\
(g,v) &\mapsto (g,g_p^{-1}v)
\end{align}
induces an isomorphism $\cV_{\Q} \otimes_{\Q} E \risom \cV_{E}$.
As $K_p$ is a compact group, there exists a $K_p$-stable $\cO$-lattice $W$ in $V(E)$.
If we let
\begin{equation}\label{eqn:Zplocsys}
\cW\defeq (\bG(\Q)\backslash \bG(\A)/A_\infty^\circ K_\infty^\circ K_f^p \times W)/K_p,
\end{equation}
then the map
\begin{equation} \label{eqn:integral}
H^*(Y(K_f),\cW) \ra H^*(Y(K_f),\cV_{E}) \cong H^*(Y(K_f),\cV_{\Q}) \otimes_{\Q} E
\end{equation}
gives a natural integral structure on $H^*(Y(K_f),\cV_{\Q}) \otimes_{\Q} E$.
In fact, the definition \eqref{eqn:Zplocsys} makes sense for any $\cO[\![K_p]\!]$-module and defines a functor from $\cO[\![K_p]\!]$-modules to local systems on $Y(K_f)$.
We caution that \eqref{eqn:integral} may not be injective in any given degree.
Indeed, $H^*(Y(K_f),\cW)$ may contain torsion, and in fact this torsion is expected to be abundant and to play an important role in connecting cohomological automorphic forms and Galois representations.
\subsection{Congruences between Hecke eigensystems}\label{sec:congruences}
Let $T^\Sigma_{\cO}$ be the Hecke algebra $\cO[K^\Sigma\backslash \bG(\A^\Sigma)/K^\Sigma]$, which acts naturally on $H^*(Y(K_f),\cW)$.
As $H^*(Y(K_f),\cW)$ is a finite $\cO$-module, there are only finitely many maximal ideals $\fm \subset T^\Sigma_{\cO}$ for which the localization $H^*(Y(K_f),\cW)_{\fm}$ is nonzero.
These localized modules record congruences between systems of Hecke eigenvalues:
a Hecke eigenclass $H^*(Y(K_f),\cW)[\frac{1}{p}]$ survives in the localization $H^*(Y(K_f),\cW)_{\fm}[\frac{1}{p}]$ if and only if its (automatically integral) system of Hecke eigenvalues lifts the mod $p$ system given by $\fm$.
However, for a fixed $\fm$, there may be various local systems $\cW$ for which $H^*(Y(K_f),\cW)_{\fm}$ is nonzero.
Indeed, we saw in \S \ref{sec:classical} that if $\fm_{G_k}$ corresponds to (the system of Hecke eigenvalues of) the Eisenstein series $G_k \pmod p$ for $4\leq k \leq p+1$ and $\cW(a,b)$ corresponds to the lattice $W(a,b) \defeq \Sym^{a-b} \Z_p^2 \otimes \det^b$ for $a \geq b$, then $H^*(Y(\GL_2(\widehat{\Z})),\cW(k'-2,0))_{\fm_{G_k}}$ is nonzero for all $k' \equiv k \pmod {p-1}$.
(Since $\GL_2(\widehat{\Z})$ is not neat, these cohomology groups should be interpreted as the cohomology groups of an orbifold.)
Furthermore, if $\fm_\Delta$ corresponds to the Ramanujan Delta function mod $11$, then both $H^*(Y(K_f),\Z_p)_{\fm_\Delta}$ and $H^*(Y(\GL_2(\widehat{\Z})),\cW(10,0))_{\fm_\Delta}$ are nonzero where $K_f$ corresponds to the congruence subgroup $\Gamma_0(11)$.
The upshot of our discussion above is that congruences between eigenforms can be thought as the non-vanishing of localized cohomology for many different coefficient sheaves.
Thus a complete classification of such congruences is equivalent to the following question:
\begin{ques}\label{ques:Serre1}
Given a mod $p$ Hecke eigensystem $\fm$, for which $\cO$-local systems $\cW$ on $Y(K_f)$ is $H^*(Y(K_f),\cW)_{\fm}$ nonzero?
\end{ques}
Serre studied this question extensively in the case of $\GL_2$ \cite{serre-duke}.
This perspective of cohomology actually gives a natural explanation for the congruences for $\GL_2$ in \S \ref{sec:classical}. We explain how it naturally leads to considerations in modular representation theory.
From the short exact sequence $0 \ra W \overset{\cdot p}{\ra} W \ra W\otimes_{\Z_p} \F_p \ra 0$, we see that $H^*(Y(K_f),\cW)_{\fm}$ is nonzero if and only if $H^*(Y(K_f),\cW\otimes_{\Z_p}\F_p)_{\fm}$ is.
While $W(a,b) \otimes_{\Z_p} \Q_p$ is an irreducible $\GL_2(\Z_p)$-module, $W(a,b) \otimes_{\Z_p} \F_p$ is irreducible if and only if $a-b \leq p-1$, in which case, we let $F(a,b)\defeq W(a,b) \otimes_{\Z_p} \F_p$.
All (absolutely) irreducible $\GL_2(\Z_p)$-modules over $\F_p$ arise in this way, and $F(a,b) \cong F(c,d)$ if and only if $a-c = b-d \in (p-1)\Z$.
Let $\cF(a,b)$ be the corresponding local system.
Let us first revisit the congruences between Eisenstein series.
For any $a' > b$, the submodule of $W(a',b) \otimes_{\Z_p} \F_p$ generated by a (nonzero) highest weight vector is isomorphic to $F(a,b)$ where $a$ is the unique integer such that $0<a - b<p$ and $a \equiv a' \pmod{p-1}$.
This gives a map
\[
H^*(Y(\GL_2(\widehat{\Z})),\cW(k-2,0) \otimes_{\Z_p} \F_p)_{\fm_{G_k}} \ra H^*(Y(\GL_2(\widehat{\Z})),\cW(k'-2,0)\otimes_{\Z_p} \F_p)_{\fm_{G_k}}
\]
where $2 < k < p+2$ and $k' > 2$ with $k' \equiv k \pmod{p-1}$.
It can be shown (for example by applying Hida's ordinary projector) that these maps are injective in each degree.
This illustrates how modular representation theory can be used to produce infinite families of congruences between Hecke eigensystems.
The congruence \eqref{eqn:cuspcong} between cuspforms is simpler.
Recall that here $p=11$ and that $\bG = \GL_2$.
Then the fact that $\fm_\Delta$ is \emph{non-Eisenstein} implies that for all local systems $\cW$, $H^*(Y(K_f),\cW)_{\fm_\Delta}$ is zero unless $* = 1$.
First, Shapiro's lemma now implies that $H^*(Y(K_f),\cW)_{\fm_\Delta} \cong H^*(Y(\GL_2(\widehat{\Z})),\cW')_{\fm_\Delta}$, where $\cW'$ corresponds to the principal series representation $\Ind_{B(\F_p)}^{\GL_2(\F_p)} W$.
Second, the functor $W \mapsto H^1(Y(\GL_2(\widehat{\Z})),\cW)_{\fm_\Delta}$ is an exact functor from the category of finite $\Z_{11}[\![\GL_2(\Z_{11})]\!]$-modules to the category of finite $\Z_{11}$-modules.
Now $\Ind_{B(\F_{11})}^{\GL_2(\F_{11})} \bf{1}$ is naturally identified with the space of $\F_{11}$-valued functions on $\bP^1(\F_{11})$ and decomposes as $\Sym^{10} \F_{11}^2 \oplus \bf{1}$.
Then the injection
\[
H^*(Y(\GL_2(\widehat{\Z})),\cF(10,0))_{\fm_\Delta} \into H^*(Y(K_f),\F_p)_{\fm_\Delta}
\]
provides the desired congruence.
This example illustrates the important phenomenon of how the weight and level of modular forms can interact mod $p$.
With these representation theoretic arguments in mind, Ash, Stevens, and others have suggested that one should narrow the focus of Question \ref{ques:Serre1} to when $K_p$ is a maximal compact open subgroup and $\cW$ is an irreducible $\F$-local system.
\begin{ques}[The weight part of Serre's conjecture] \label{ques:Serre2}
Suppose that $K_p$ is a maximal compact open subgroup. Given a mod $p$ Hecke eigensystem $\fm$, for which irreducible $\F$-local systems $\cW$ on $Y(K_f)$ is $H^*(Y(K_f),\cW)_{\fm}$ nonzero?
\end{ques}
\noindent
While this is a substantial reduction since there are only finitely many such $\F$-local systems up to isomorphism, little is expected to be lost as we now explain.
The following proposition is immediate.
\begin{prop}\label{prop:JHnonzero}
If $H^*(Y(K_f),\cW)_{\fm}$ is nonzero, then $H^*(Y(K_f),\cF)_{\fm}$ is nonzero for some irreducible subquotient $\cF$ of $\cW \otimes_{\cO} \F$.
\end{prop}
\noindent The converse to Proposition \ref{prop:JHnonzero} holds in non-Eisenstein cases for $\bG$ a Weil restriction of $\GL_n$ if expected vanishing conjectures hold (see \S \ref{sec:vanish}).
These vanishing conjectures generalize the vanishing outside of degree $1$ for $\bG = \GL_2$.
Question \ref{ques:Serre2} turns out to be quite subtle.
Nonisomorphic irreducible $\F$-local systems may contain the same Hecke eigensystems, i.e.~not all congruences arise from modular representation theory.
For example, with $p=23$, both $H^*(Y(\GL_2(\widehat{\Z})),\cF(10,0))_{\fm_\Delta}$ and $H^*(Y(\GL_2(\widehat{\Z})),\cF(21,11))_{\fm_\Delta}$ are nonzero.
If we write
\[
\Delta(z) = \sum_{n=1}^\infty \tau(n) q^n,
\]
then $T_{\ell}$ acts on $H^*(Y(\GL_2(\widehat{\Z})),\cF(21,11))_{\fm_\Delta}$ by $\ell^{11}\tau(\ell)$ for all primes $\ell \neq 23$ by \eqref{eqn:matsushima} (see \S \ref{sec:classical}). This implies that
\begin{equation} \label{eqn:taucong}
\tau(n) \equiv \big(\tfrac{n}{23}\big)\tau(n) \pmod{23}
\end{equation}
for all $n$ coprime to $23$.
In other words, $23\mid \tau(n)$ if $\big(\tfrac{n}{23}\big) = -1$.
\section{An interlude on representation theory}\label{sec:reptheory}
\subsection{Serre weights} \label{sec:serrewt}
In order to explore Question \ref{ques:Serre2}, the natural first step is to ask for a classification of simple $\F[\![K_p]\!]$-modules.
If $K_p(1) \subset K_p$ is a normal finite index pro-$p$ subgroup and $W$ is a finite $\F[\![K_p]\!]$-module, then $W^{K_p(1)}$ is an $\F[\![K_p]\!]$-submodule of $W$ which is \emph{nonzero} (since the action of a $p$-group on a finite-dimensional $\F$-vector space must have a nonzero fixed vector).
Then the action of $K_p$ on a simple $\F[\![K_p]\!]$-module factors through the finite quotient $K_p/K_p(1)$, which can often be arranged to be a finite group of Lie type.
In this section, we discuss the (modular) representation theory of these groups.
Let $G$ be a connected reductive group over $\F_p$ which splits over $\F$.
(We will eventually take $G$ to be the mod $p$ reduction of an integral model of $\bG$.)
An isomorphism class of a simple $\F[G(\F_p)]$-module is known as a \emph{Serre weight for} $G(\F_p)$.
Our goal now is to describe the (finite) set of Serre weights for $G(\F_p)$.
Let $B$ be an $\Fp$-rational Borel subgroup in $G$ with Levi subgroup $T$.
We denote by $W$ the Weyl group $N(T)/T$, which has a Bruhat partial order with a unique longest element $w_0$.
We write $X(T)$ for the character group of $T$, which has an action of $W$ and an induced action from the relative Frobenius $F$ acting on $T$.
This group has a subset
\[
X_1(T) \defeq \{ \lambda \in X(T) : 0 \leq \langle \lambda,\alpha^\vee \rangle \leq p-1,\, \textrm{for all simple }\alpha \}
\]
which plays an important role in the modular representation theory of $G$.
We let
\[
X^0(T) \defeq \{ \lambda \in X(T) : \langle \lambda,\alpha^\vee \rangle =0\, \textrm{ for all roots }\alpha \}.
\]
For any character $\lambda \in X(T)$, we can consider the algebraic induction $W(\lambda) \defeq \Ind_B^G w_0\lambda$ (also known as the dual Weyl module), which is nonzero if and only if $\lambda$ is dominant with respect to $B$.
We let $L(\lambda)$ denote the socle of $W(\lambda)$, which is the simple submodule generated by a nonzero highest weight vector.
Then we have the following result about Serre weights for $G(\F_p)$.
\begin{thm}
The map
\begin{align}
\frac{X_1(T)}{(F-1)X^0(T)} &\ra \{\textrm{Serre weights for } G(\F_p)\}\\
\lambda &\mapsto L(\lambda)(\F)|_{G(\F_p)}
\end{align}
is a bijection.
\end{thm}
\noindent We denote $L(\lambda)(\F)|_{G(\F_p)}$ by $F(\lambda)$.
(To avoid conflicts with the $F$-action on $X(T)$, we will write this action without parentheses.)
\subsection{Deligne--Lusztig representations}
While Question \ref{ques:Serre2} only involves $\F$-local systems, we will see that it is inextricably linked to $\cO$-torsion free local systems.
It is then natural to ask for a classification of irreducible $G(\F_p)$-representations in characteristic $0$.
We now recall such a classification, provided by work of Deligne and Lusztig \cite{DeligneLusztig}.
For an element $w \in W$, there exists $g_w \in G(\ovl{\F}_p)$ such that $g_w^{-1} F(g_w) \in N(T)(\ovl{\F}_p)$ represents $w$.
Then we let $T_w$ be the $F$-stable torus $g_w T g_w^{-1}$.
Let $\tld{W}$ denote the extended affine Weyl group which is the semidirect product $X(T) \rtimes W$.
For an element $\tld{w} = (\mu, w) \in \tld{W}$, we define $\theta_{\tld{w}}: T_w(\F_p) \ra E^\times$ to be the restriction of the character
\begin{align*}
T_w(\ovl{\F}_p) &\ra \ovl{\Q}_p^\times\\
g &\mapsto [\mu](g_w^{-1}g g_w)
\end{align*}
to $T_w(\F_p)$ (here, $[\mu]$ denotes the Teichm\"uller lift of $\mu$).
To a character $\theta_{\tld{w}}$ of a maximal rational torus $T_w(\F_p)$ of $G(\F_p)$, Deligne and Lusztig associate a virtual (Deligne--Lusztig) representation over $E$ which they denote $\epsilon_G \epsilon_{T_w} R_{T_w}^{\theta_{\tld{w}}}$.
We will instead denote this virtual representation by $R(\tld{w})$ and say that $\tld{w}$ is a presentation for $R(\tld{w})$.
The map $\tld{w} \mapsto R(\tld{w})$ is not injective---two elements map to the same virtual representation if and only if they lie in the same orbit of the action of $\tld{W}$ on itself given by
\[
(\nu,s)\cdot(\mu,w) = (s\mu+F\nu-swF(s)^{-1}(\nu),swF(s)^{-1}).
\]
The simplest case of the above construction occurs when $w$ is the identity.
Then $T_w = T$, $\theta_{\tld{w}}$ is a character of $T(\F_p)$ and by inflation a character of $B(\F_p)$, and $R(\tld{w})$ is the principal series representation $\Ind_{B(\F_p)}^{G(\F_p)} \theta_{\tld{w}}$.
Nonuniqueness of presentations can be seen from the existence of intertwiners between principal series representations.
The group $\tld{W}$ acts on $X(T)$ in the usual way---$W$ acts on $X(T)$ by group automorphisms and $X(T)$ acts on itself by translation. Let $m$ be a nonnegative integer and let $0 \in X(T)$ denote the trivial character.
We say that $\tld{w} \in \tld{W}$ is (lowest alcove) $m$-generic if $\langle \tld{w}(0),\alpha^\vee \rangle > m$ for all simple roots $\alpha$ and $\langle \tld{w}(0),\alpha^\vee \rangle < p-m$ for all roots $\alpha^\vee$.
We say that a Deligne--Lusztig representation $R$ is $m$-generic if $R = R(\tld{w})$ for some $m$-generic $\tld{w}$.
An $m$-generic $\tld{w}$ or $R$ exists only if $m h < p$, where $h$ denotes the Coxeter number of $G$.
\cite[Proposition 10.10]{DeligneLusztig} implies that if $R$ is $0$-generic, then $R$ is in fact a genuine representation.
Let $\ovl{R}$ denote the semisimplification of the reduction of any $G(\F_p)$-stable $\cO$-lattice in a genuine $G(\F_p)$-representation $R$ over $E$ ($\ovl{R}$ does not depend on the choice of lattice).
In relation to Question \ref{ques:Serre2}, it is important to have an understanding of $\ovl{R}$ for Deligne--Lusztig representations $R$.
This is provided by Jantzen's formula for the reductions of Deligne--Lusztig representations in terms of virtual linear combinations of dual Weyl modules \cite{jantzenDL}.
If $R$ is sufficiently generic, then the Jordan--H\"older factors of $\ovl{R}$ admit the following description in terms of alcove geometry, which is in a sense independent of $p$.
For convenience, we assume that $G$ admits a \emph{twisting element} $\eta \in X(T)$, defined up to $X^0(T)$, which by definition has the property that $\langle \eta,\alpha^\vee\rangle = 1$ for all simple roots $\alpha$.
The existence of an $\eta$ can always be arranged by passing to a central extension of $G$ by $\bG_m$ (see \cite[Proposition 5.3.1(a)]{BG}).
We write $\cdot$ for the $p$-dot action so that $(\nu,w)\cdot \lambda = w(\lambda+\eta) - \eta + p\nu$.
See \cite[\S 2]{MLM} for any unexplained notation below.
\begin{prop} \label{prop:JH} \cite[Proposition 2.3.6]{MLM}
Let $h$ be the Coxeter number of $G$.
If $\tld{w} \in \tld{W}$ is $2h$-generic, then the Jordan--H\"older factors of $\ovl{R}(\tld{w})$ are precisely the Serre weights of the form
\[
F(\pi^{-1}(\tld{w}_1)\cdot(\tld{w}\tld{w}_2^{-1}(0)-\eta))
\]
with $\tld{w}_1\in \tld{W}$ restricted and dominant, $\tld{w}_2 \in \tld{W}$ dominant, and $\tld{w}_1 \uparrow (\eta,w_0) \tld{w}_2$.
\end{prop}
\begin{rmk}
Of course, the description in Proposition \ref{prop:JH} does not depend on the choice of twisting element $\eta$ and could in fact be rephrased without any reference to $\eta$.
\end{rmk}
\section{Relations to Galois representations}\label{sec:galrep}
In order to address Question \ref{ques:Serre2} (and to explain the congruence \eqref{eqn:taucong}), we introduce some conjectures and results concerning the relationship between cohomological automorphic forms and Galois representations.
We follow the approach in \cite{gross}, which seems to be more standard when $\bG$ is a general linear group or a unitary group.
For a more canonical approach to conjectures concerning Galois representations attached to cohomological automorphic forms, see \cite{BG}.
\subsection{Twisting element}
For a field $F$, let $G_F$ denote the absolute Galois group $\Gal(F^{\mathrm{sep}}/F)$ where we fix some separable closure $F^{\mathrm{sep}}$.
Fix a maximal torus $T$ and Borel subgroup $B$ in $\bG_{/\ovl{\Q}}$.
We assume now that $\bG$ has a \emph{twisting element} $\eta$ which is by definition an element of $X(T)^{G_{\Q}}$ such that $\langle \eta,\alpha^\vee\rangle=1$ for all simple roots $\alpha$.
If $\bG$ is $\GL_n$, we can take $\eta$ to be $(n-1,n-2,\ldots,1,0)$.
As before, a twisting element always exists if we replace $\bG$ by a central extension of $\bG$ by $\bG_m$.
The effect of this on the constructions and questions in \S \ref{sec:coh} is minimal.
A twisting element is only unique up to $X^0(T)^{G_{\Q}}$.
\subsection{Satake parameters}\label{sec:satake}
Fix a prime $\ell$ and suppose that there is a reductive model $G$ over $\Z_\ell$ for $\bG$ such that $G(\Z_\ell) = K_\ell$.
Let $T\subset B \subset G$ be a maximal torus and Borel subgroup, respectively.
We have the Satake isomorphism (see e.g.~\cite[\S 4.2]{cartier})
\[
\cS: \Z[1/\ell][K_\ell \backslash \bG(\Q_\ell) / K_\ell] \risom \Z[1/\ell][T(\Z_\ell) \backslash T(\Q_\ell) / T(\Z_\ell)]^{W_s}
\]
normalized using the choice of twisting element $\eta$ as in \cite[Proposition 3.6]{gross-satake}, where $W_s$ is the Weyl group of the maximal split torus in $T$.
If $T$ is split, then $T(\Z_\ell) \backslash T(\Q_\ell) / T(\Z_\ell) \cong Y(T)$ (here $Y(T)$ denotes the cocharacter group of $T$) and $E$-valued characters of
\[
\Z[1/\ell][K_\ell \backslash \bG(\Q_\ell) / K_\ell]\cong \Z[1/\ell][Y(T)]^W \cong \cO(\widehat{T}/\!\!/W)
\]
are in bijection with semisimple conjugacy classes in the dual group $\widehat{G}(E)$ for any coefficient field $E$ of characteristic not equal to $\ell$.
In general, $E$-valued characters $\chi$ of $\Z[1/\ell][K_\ell \backslash \bG(\Q_\ell) / K_\ell]$ are in bijection with semisimple conjugacy classes $C_\chi$ of $^L G(E)$, where $^L G$ denotes the Langlands dual of $G$ (see \cite[\S 16]{gross}).
\subsection{Conjectures on Galois representations associated to cohomological automorphic forms}
We fix a prime $p$ and a sufficiently large subfield $E \subset \ovl{\Q}_p$ of finite degree over $\Q_p$.
\begin{conj}\label{conj:galrep}
Let $V(\lambda)_E$ denote the irreducible representation of $\bG_{/E}$ of highest weight $\lambda$.
Suppose that $\fp \subset T_{\Q}^\Sigma \otimes_{\Q} E$ is a maximal ideal such that the $\fp$-torsion $H^*(Y(K_f),\cV(\lambda)_E) [\fp]$ is nonzero.
Then there exists a continuous homomorphism
\[
\rho: G_{\Q} \ra\, ^L\bG(E)
\]
such that
\begin{enumerate}
\item the composition of $\rho$ with the projection $^L\bG(E) \ra G_{\Q}$ is the identity on $G_{\Q}$;
\item for $\ell \notin \Sigma$ and $\ell \neq p$, $\rho$ is unramified at $\ell$ and $\rho(\Frob_\ell)$ is in $C_\chi$ where $\chi$ is the character
\[
\Z[1/\ell][K_\ell \backslash \bG(\Q_\ell) / K_\ell] \subset T_{\Q}^\Sigma \otimes_{\Q} E \ra (T_{\Q}^\Sigma \otimes_{\Q} E)/\fp \cong E
\] and $\Frob_\ell$ is an arithmetic Frobenius element at $\ell$;
\item $\rho|_{G_{\Q_p}}$ is de Rham with Hodge--Tate cocharacter $\lambda+\eta$ (see \cite[\S 2.4]{BG}; in our normalization the cyclotomic character corresponds to the cocharacter $\id_{\bG_m}$) and is moreover crystalline if $p \notin \Sigma$; and
\item $\rho$ is odd in the sense of \cite[Conjecture 17.2(a)]{gross}.
\end{enumerate}
\end{conj}
There is an analogous conjecture with torsion coefficients. As before, let $\F$ denote the residue field of $E$.
\begin{conj}\label{conj:modp_galrep}
If $W$ is a finite $\F[\![K_p]\!]$-module, let $\cW$ denote the $\F$-local system on $Y(K_f)$.
Suppose that $\fm \subset T_{\cO}^\Sigma$ is a maximal ideal such that $H^*(Y(K_f),\cW)_{\fm}$ is nonzero.
Then there exists a continuous homomorphism
\[
\rhobar: G_{\Q} \ra \, ^L\bG(\F)
\]
such that
\begin{enumerate}
\item the composition of $\rhobar$ with the projection $^L\bG(\F) \ra G_{\Q}$ is the identity on $G_{\Q}$;
\item for $\ell \notin \Sigma$ and $\ell \neq p$, $\rhobar$ is unramified at $\ell$ and $\rhobar(\Frob_\ell)$ is in $C_\chi$ where $\chi$ is the character \[
\Z[1/\ell][K_\ell \backslash \bG(\Q_\ell) / K_\ell] \ra T_{\Q}^\Sigma \otimes_{\Q}\F \ra T_{\Q}^\Sigma \otimes_{\Q} \F/\fm \cong \F
\] and $\Frob_\ell$ is an arithmetic Frobenius element at $\ell$; and
\item $\rhobar$ is odd in the sense of \cite[Conjecture 17.2(a)]{gross}.
\end{enumerate}
\end{conj}
\begin{rmk}\label{rmk:galrep}
\begin{enumerate}
\item One also expects that $\rho$ satisfies a compatibility with the conjectural local Langlands correspondence at places in $\Sigma$.
\item Comparing the two conjectures, observe that there is no property at $p$ for $\rhobar$.
Such a property would be closely related to Questions \ref{ques:Serre1} and \ref{ques:Serre2}.
\item \label{item:cheb} It is clear that $\rho$ and $\rhobar$ determine $\fp$ and $\mathfrak{m}$, respectively.
On the other hand, the properties of $\rho$ described in Conjectures \ref{conj:galrep} and \ref{conj:modp_galrep} do not characterize $\rho$ in general.
When $\bG$ is a Weil restriction of $\GL_n$, the first two properties characterize the isomorphism class of the semisimplification of $\rho$ by the Chebotarev density theorem and the Brauer--Nesbitt theorem.
But even for tori, these properties do not characterize $\rho$ (see \cite[Remark 3.2.4]{BG}).
\item The existence of torsion cohomology classes means that Conjecture \ref{conj:galrep} does not immediately imply Conjecture \ref{conj:modp_galrep}.
A version of Conjecture \ref{conj:modp_galrep} for all torsion coefficients implies Conjecture \ref{conj:galrep} (except for the third property) by taking a limit.
\end{enumerate}
\end{rmk}
There are two cases, relevant to what follows, when both conjectures are known:
\begin{enumerate}
\item the Weil restriction of a definite unitary group relative to a CM extension of a totally real field not equal to $\Q$ \cite{kottwitz,HT,labesse,shin,CH}, and \\
\item the Weil restriction of $\GL_n$ over a CM field \cite{scholze}.
\end{enumerate}
In these cases, the attached Galois representations are determined up to semisimplification by Remark \ref{rmk:galrep}\eqref{item:cheb}.
We say that $\rhobar$ (or $\fm$) is non-Eisenstein if $\rhobar$ does not factor through a proper parabolic subgroup after any finite extension of $\F$.
\subsection{Modular Serre weights}
Fix $p$ and $E$ as before, and let $\F$ be the residue field of $E$.
Suppose from now on that $\bG_{/\Q_p}$ has an integral model $G_{/\Z_p}$.
Having classified irreducible representations of $G(\Z_p)$ and introduced Galois representations, we now revisit Question \ref{ques:Serre2} through that lens.
We let $K_p$ be $G(\Z_p)$.
An irreducible $G(\Z_p)$-representation over $\F$ factors through the reduction map $G(\Z_p) \surj G(\F_p)$ (whose kernel is pro-$p$), i.e.~is the inflation of a Serre weight for $G(\F_p)$.
For a Serre weight $\sigma$, let $\cF_\sigma$ denote the corresponding $\F$-local system on $Y(K_f)$.
Fix an $\F$-valued Hecke eigensystem $\fm$.
To understand Question \ref{ques:Serre2} is to understand the following (finite) set.
\begin{defn}
Let $W(\fm)$ be the set of isomorphism classes of Serre weights $\sigma$ for $G(\F_p)$ for which $H^*(Y(K_f),\cF_\sigma)_\fm$ is nonzero.
If $\sigma \in W(\fm)$, then we say that $\sigma$ is a \emph{modular Serre weight} (for $\fm$).
\end{defn}
\noindent If there is a Galois representation $\rhobar: G_{\Q} \ra \, ^L\bG(\F)$ satisfying the properties in Conjecture \ref{conj:modp_galrep} for $\fm$, then we also write $W(\rhobar)$ for $W(\fm)$ and say that $\sigma$ is a modular Serre weight for $\rhobar$ when $\sigma \in W(\rhobar)$.
\subsection{The case of $\GL_2$} \label{sec:GL2}
Fix $p$ and $E$ as before.
The first result on Conjecture \ref{conj:galrep} for nonabelian $\bG$ was work of Deligne \cite{deligne} in the case of $\bG=\GL_2$ (building on work of Eichler and Shimura).
In this case there is no torsion in cohomology, and so Conjecture \ref{conj:galrep} implies Conjecture \ref{conj:modp_galrep}.
Deligne constructed $\rho$ satisfying all the properties of Conjecture \ref{conj:galrep} except the third, which follows from subsequent work of Faltings on $p$-adic comparison theorems.
(We set $\eta = (1,0)$ here.)
Let $K_p = \GL_2(\Z_p)$.
Fix a mod $p$ Hecke eigensystem $\fm$.
Assume that $\fm$ is non-Eisenstein, i.e.~that the attached Galois representation $\rhobar: G_{\Q} \ra \GL_2(\F)$ is absolutely irreducible.
(The data of $\fm$ is equivalent to that of the isomorphism class of $\rhobar$ by Remark \ref{rmk:galrep}\eqref{item:cheb}.)
Since cohomology groups outside degree $1$ do not admit non-Eisenstein mod $p$ Hecke eigensystems, the functor $W \ra H^1(Y(K_f),\cW)_{\fm}$ is exact, as explained in \S \ref{sec:congruences}.
In particular, Question \ref{ques:Serre1} reduces to Question \ref{ques:Serre2}, namely an investigation of $W(\rhobar)$.
If $H^1(Y(K_f),\cF(a,b))_\fm$ is nonzero, then so is $H^1(Y(K_f),\cW(a,b))_\fm$.
A necessary condition for $F(a,b)$ to be in $W(\rhobar)$ is that $\rhobar$ is the reduction of a representation $\rho: G_{\Q} \ra \GL_2(E)$ which is unramified outside of $\Sigma$ and $p$ and is crystalline at $p$ of Hodge--Tate weights $a+1$ and $b$.
(Since $\rhobar$ is irreducible, a $G_{\Q}$-invariant $\cO$-lattice in $\rho$ is unique up to scaling.)
In particular, the restriction $\rhobar|_{G_{\Q_p}}$ is the reduction of (a lattice in) a crystalline representation $\rho_p: G_{\Q_p} \ra \GL_2(E)$ of Hodge--Tate weights $a+1$ and $b$.
The following result, known as the weight part of Serre's conjecture, is a local-global principle (i.e.~$W(\rhobar)$ only depends on $\rhobar|_{G_{\Q_p}}$) that asserts that the necessary condition is in fact sufficient.
\begin{thm}[\cite{gross,edixhoven,CV}]\label{thm:SWC_GL2}
Suppose that $\fm$ is non-Eisenstein.
Then $F(a,b)\in W(\rhobar)$ if and only if $\rhobar|_{G_{\Q_p}}$ is the reduction of a crystalline representation of Hodge--Tate weights $a+1$ and $b$.
\end{thm}
\begin{rmk}\label{rmk:SWC_GL2}
\begin{enumerate}
\item The above formulation is slightly different, albeit equivalent, from Serre's original formulation in \cite{serre-duke}. In \emph{loc.cit.}, the recipe for $W(\rhobar)$ is completely explicit in terms of the ``inertial weights" of $\rhobar|_{G_{\Q_p}}$ when $\rhobar|_{G_{\Q_p}}$ is semisimple, with additional modfications in terms of ramification properties of an extension class in general.
In particular, in the semisimple case, there is a simple combinatorial formula for $a$ and $b$ solely in terms of the inertial weights.
Early generalizations of Serre's conjecture beyond $\GL_{2/\Q}$, e.g.~\cite[Conjecture 3.1]{ADP}, involved similar formulas (see \cite[\S 7]{GHS} for more recent formulas for a larger list of Serre weights).
On the other hand, while \cite{BDJ} also contains formulas \`a la Serre, it emphasizes the above ``crystalline lifts" perspective.
\item Theorem \ref{thm:SWC_GL2} was generalized to (the Weil restriction of) the unit groups in quaternion algebras over totally real fields split at no more than one archimedean place and definite unitary groups over a totally real field when $p>2$ (and under mild additional hypotheses) in a series of works by Gee, Newton, Kisin, Liu, and Savitt \cite{newton13,gee-kisin,GLS}.
Some of these build on earlier work of Gee that introduced the Taylor--Wiles method to produce proofs rather different from the original proofs of Theorem \ref{thm:SWC_GL2}.
\end{enumerate}
\end{rmk}
We now revisit the mod $23$ congruence for the Ramanujan Delta function.
Let $\rhobar: G_{\Q} \ra \GL_2(\F_{23})$ be the associated Galois representation.
In this case, $\rhobar|_{I_{\Q_{23}}} \cong \omega^{11} \oplus \bf{1}$, where $\omega$ denotes the reduction of the $23$-adic cyclotomic character $\chi$ and $I_{\Q_{23}}\subset G_{\Q_{23}}$ denotes the inertial subgroup.
Then $\rhobar|_{G_{\Q_{23}}}$ is the reduction of both $\chi^{11} \oplus \bf{1}$ and $\chi^{11} \oplus \chi^{22}$ (up to unramified twists).
Theorem \ref{thm:SWC_GL2} implies that $\{F(10,0),F(21,11)\} \subset W(\rhobar)$.
In fact, this is an equality.
The behavior of the Ramanujan Delta function modulo $23$ illustrates a rare phenomenon.
For example, if we instead take $p = 19$ and $\rhobar: G_{\Q} \ra \GL_2(\F_{19})$ the corresponding representation, then $\rhobar|_{I_{\Q_{19}}}$ is a \emph{nontrivial} extension of $\bf{1}$ by $\omega^{11}$.
Moreover, $\rhobar|_{G_{\Q_{19}}}$ is the reduction of a crystalline representation which after restriction to inertia is a nontrivial extension of $\bf{1}$ by $\chi^{11}$.
However, any nontrivial extension of $\chi^{18}$ by any unramified twist of $\chi^{11}$ is \emph{not} crystalline.
In fact, we have $\{F(10,0)\} = W(\rhobar)$ in this case.
One expects more generally that $W(\rhobar)$ is larger when $\rhobar|_{G_{\Q_p}}$ is semisimple.
Indeed, if $\rhobar|_{G_{\Q_p}}$ is the reduction of an $\cO$-lattice in $r: G_{\Q_p} \ra \GL_2(E)$, then there is another $\cO$-lattice (possibly after enlarging $E$) whose reduction is the semisimplification of $\rhobar|_{G_{\Q_p}}$.
One approach to Theorem \ref{thm:SWC_GL2} is as follows.
If $\rhobar|_{G_{\Q_p}}$ admits a local crystalline lift of Hodge--Tate weights $a+1$ and $b$, show that $\rhobar$ admits a global lift $\rho$ which is crystalline (at $p$) of Hodge--Tate weights $a+1$ and $b$.
Then show that $\rho$ comes from a modular form as predicted by the Fontaine--Mazur conjecture \cite{fontaine-mazur}.
These steps can be executed using a combination of tools in the Taylor--Wiles method independently discovered by Khare--Wintenberger and Gee \cite{KW_Serre_2,gee-annalen}.
\subsection{Vanishing conjectures for cohomology}\label{sec:vanish}
In the previous section, we were in the advantageous situation where $H^*(Y(K_f),\cW)_{\fm}$ vanished outside of degree one for all $\cO$-local systems $\cW$ when $\fm$ is non-Eisenstein.
While this is not true in general, this property does nevertherless admit a conjectural generalization.
Let $d_Y$ be the dimension of $Y(K_f)$.
Define the integers $\ell_0\defeq \rk \bG(\R) - \rk A_\infty^\circ K_\infty^\circ$ and $q_0 = \frac{1}{2}(d_Y-\ell_0)$.
(The group $\bG$ admits discrete series if and only of $\ell_0 = 0$.)
\begin{conj}[\cite{CG}] \label{conj:vanish}
Suppose that $\fm$ is non-Eisenstein.
If $H^i(Y(K_f),\cW)_{\fm} \neq 0$, then $i \in [q_0,q_0+\ell_0]$.
\end{conj}
\noindent \cite{GN} shows that if Conjecture \ref{conj:vanish} holds, then so does the converse to Proposition \ref{prop:JHnonzero} when $\bG$ is a Weil restriction of $\GL_n$, so that, as in our discussion above, it suffices to analyze Question \ref{ques:Serre1} for irreducible mod $p$ local systems (note that this is far from obvious in general, as it is \emph{a priori} possible for the cohomology complex of irreducible local system to cancel each other out when they spread to several cohomological degrees). More seriously, Conjecture \ref{conj:vanish} plays a prominent role in the Taylor-Wiles patching method, which is the main tool to attack Question \ref{ques:Serre1} in general, cf.~ Remark \ref{rmk:patch} below.
Unfortunately, there are few cases where Conjecture \ref{conj:vanish} is known.
One trivial case is that of groups which are anisotropic modulo their center, e.g.~definite unitary groups, when $Y(K_f)$ is a finite set of points.
\cite{CS} have shown that for certain unitary groups ($\ell_0 = 0$) Conjecture \ref{conj:vanish} holds under some additional hypotheses.
The case of the Weil restriction of $\GL_n$ (for $n>1$) over a number field $F$ (even CM fields) is open beyond the case $n=2$ and $F$ either totally real or imaginary quadratic.
\section{Conjectures and results on the weight part of Serre's conjecture}\label{sec:conj}
\subsection{Taylor--Wiles patching}\label{sec:TW}
Suppose for the moment that we are in a context where $\ell_0= 0$ and Conjectures \ref{conj:galrep} and \ref{conj:vanish} hold (e.g., definite unitary groups).
We fix $p$ and $E$ as before.
We assume that a reductive integral model $G_{/\Z_p}$ of $\bG_{/\Q_p}$ exists and continue to let $K_p=G(\Z_p)$.
If $F(\lambda) \in W(\rhobar)$, then $H^*(Y(K_f),\cW(\lambda))_\fm$ is nonzero (where $W(\lambda)$ is an $\cO$-lattice in the irreducible algebraic representation $V(\lambda)$).
In particular, $\rhobar$ (attached to $\fm$) is the reduction of a crystalline representation of Hodge--Tate cocharacter $\lambda+\eta$.
In light of Theorem \ref{thm:SWC_GL2}, it is tempting to guess that the converse holds.
However, counterexamples to this have been found for definite unitary groups in three variables \cite[Proposition 7.18]{LLLM}.
The reason is that in contrast to the case of $\GL_2$, $W(\lambda)$ may include many Jordan--H\"older factors other than $F(\lambda)$.
In fact, $F(\lambda)$ as a $\cO[\![K_p]\!]$-module often does not lift to an $\cO$-torsion-free module, which makes it more difficult to use the (expected) $p$-adic Hodge theoretic properties of Galois representations attached to automorphic forms.
However, $F(\lambda)$ does lift \emph{virtually}.
Suppose that $[F(\lambda)] = \sum_W c_W [W]$ in the Grothendieck group of $\F[\![K_p]\!]$-modules, where each $W$ in the sum lifts to characteristic $0$ (for example, one can take $W$ running over the reductions of various $\cO[G(\F_p)]$-modules using \cite[Theorem 33]{serre-book}).
Then exactness gives us
\[
[H^*(Y(K_f),\cF(\lambda))_{\fm}] = \sum_W c_W [H^*(Y(K_f),\cW)_{\fm}].
\]
Since the $\F$-vector spaces on the right hand side lift, their dimensions can in principle be computed in characteristic zero.
However, they are of a global nature, and thus difficult to access.
In contrast, we still expect that $W(\rhobar)$ depends only on $\rhobar|_{G_{\Q_p}}$.
The Taylor--Wiles method ``patches" together cohomology functors (or rather, the total cohomology complex computing) $H^*(Y(K_f),\cF_-)_{\fm}$ (for varying $K_f$) to obtain a functor $M_\infty(-)$ that plausibly depends (roughly speaking) only on $\rhobar|_{G_{\Q_p}}$.
Moreover, a control theorem guarantees that $M_\infty(\sigma)$ is nonzero if and only if $H^*(Y(K_f),\cF_\sigma)_{\fm}$ is.
For a Galois representation $\rbar: G_{\Q_p}\ra \, ^L \bG(\F)$, let $R_{\rbar}$ denote the (framed) deformation ring parametrizing lifts $r: G_{\Q_p}\ra \, ^L \bG(R)$ of $\rbar$ for complete local Noetherian $\cO$-algebras $R$ with residue field $\F$.
Building on work of Kisin \cite{KisinPSS} in the case when $\bG_{/\Q_p}$ is a Weil restriction of $\GL_n$, Balaji \cite{balaji} in particular defined a family of (reduced) semistable deformation rings $R^{\lambda+\eta,\tau}_{\rbar}$ whose $\ovl{\Q}_p$-points correspond to potentially semistable Galois representations of Hodge--Tate cocharacter $\lambda+\eta$ and Galois type $\tau$.
In certain contexts where (enough of) an inertial local Langlands correspondence is known, one can define a finite dimensional locally algebraic $E[K_p]$-module $\sigma(\lambda,\tau) \defeq V(\lambda) \otimes_E \sigma(\tau)$.
For example, when $\tau$ is tame, $\sigma(\tau)$ can be taken to be a certain combinatorially defined Deligne--Lusztig representation.
For a ring $A$, let $A\textrm{-mod}^{\textrm{fg}}$ denote the full subcategory of $A$-mod of finitely generated $A$-modules.
\begin{ax}\label{ax:patch}
There is an exact functor $M_\infty(-): \cO[\![K_p]\!]\textrm{-mod}^{\textrm{fg}} \ra R_{\rhobar|_{G_{\Q_p}}}\textrm{-mod}^{\textrm{fg}}$ with the following properties.
\begin{enumerate}
\item \label{item:CM} For a Serre weight $\sigma$, $M_\infty(\sigma)$ is a maximal Cohen--Macaulay module on $R^{\lambda+\eta,\tau}_{\rbar} \otimes_{\cO} \F$ for some $\lambda$ and $\tau$.
\item \label{item:control} $M_\infty(\sigma)$ is nonzero if and only if $H^*(Y(K_f),\cF_\sigma)_{\fm}$ is nonzero.
\item \label{item:LGC} If $\sigma(\lambda,\tau)^\circ$ is an $\cO$-lattice in $\sigma(\lambda,\tau)$, then $M_\infty(\sigma(\lambda,\tau)^\circ)$ is a maximal Cohen--Macaulay $R^{\lambda+\eta,\tau}_{\rbar}$-module of generic rank at most $1$.
\item \label{item:fullsupport} If $\sigma(\lambda,\tau)^\circ$ is an $\cO$-lattice in $\sigma(\lambda,\tau)$, then $M_\infty(\sigma(\lambda,\tau)^\circ)[1/p]$ is a generically free $R^{\lambda+\eta,\tau}_{\rbar}[1/p]$-module of rank $1$.
\end{enumerate}
\end{ax}
\begin{rmk}\label{rmk:patch}
\begin{enumerate}
\item It may be impossible to arrange for the stringent rank conditions in Axiom \ref{ax:patch}, but the ranks can still be controlled and many arguments below can be successfully modified.
\item If Conjecture \ref{conj:vanish} holds, then it is often possible to use the Taylor--Wiles method to construct a functor $M_\infty(-)$ satisfying the first three items of Axiom \ref{ax:patch} with $\rbar \defeq \rhobar|_{G_{\Q_p}}$ \cite{CEGGPS,GN}, at least after adding formal variables to $R^{\lambda+\eta,\tau}_{\rbar}$ which we ignore.
In general, the total (localized) cohomology complex may have non-vanishing cohomology groups in several degrees. The role of Conjecture \ref{conj:vanish} is to guarantee that after Taylor-Wiles patching, these complexes becomes concentrated into a single cohomological degree, i.e.~they turn into usual modules. This concentration effect relies on the ``numerical coincidence'' that powers the Taylor-Wiles method.
\item \label{item:domaincase} It seems to be difficult to guarantee the last item (for all choices of $(\lambda,\tau)$) except when $\bG = \GL_2$ \cite{kisin-fontaine-mazur}.
Indeed, it is essentially equivalent to a modularity lifting result with very general $p$-adic Hodge theoretic hypotheses.
However, there are some instances of specific $(\lambda,\tau)$ where the final item follows from the third: when $R^{\lambda+\eta,\tau}_{\rbar}$ is zero and when $R^{\lambda+\eta,\tau}_{\rbar}$ is a domain and $M_\infty(\sigma(\lambda,\tau)^\circ)$ is nonzero for some $\cO$-lattice $\sigma(\lambda,\tau)^\circ\subset \sigma(\lambda,\tau)$.
\end{enumerate}
\end{rmk}
Admitting Axiom \ref{ax:patch} for the moment, there is the following strategy for determining $W(\rhobar)$.
Let $d = \dim_{E} G_{/E}+\dim_{E} G_{/E}/B_{/E}$ where $B_{/E} \subset G_{/E}$ is a Borel subgroup.
By a result of Kisin (see Theorem \ref{thm:local_properties}), $d$ is the dimension of $R^{\lambda+\eta,\tau}_{\rbar}$ over $\cO$ for any $\lambda$ and $\tau$.
For each Serre weight $\sigma$, we write a presentation
\begin{equation}\label{eqn:present}
[\sigma] = \sum_{(\lambda,\tau)} c_{\lambda,\tau} [{\ovl{\sigma(\lambda,\tau)}}]
\end{equation}
in the Grothendieck group of $\F[\![K_p]\!]$-modules.
Then Axiom \ref{ax:patch} guarantees that
\begin{equation}\label{eqn:BM}
Z(M_\infty(\sigma)) = \sum_{(\lambda,\tau)} c_{\lambda,\tau} Z(R^{\lambda+\eta,\tau}_{\rbar} \otimes_{\cO} \F),
\end{equation}
where $Z(-)$ corresponds to taking the $d$-dimensional support cycle of the $R_{\rbar}$-module (note that all modules involved have support of dimension $\leq d$).
Since the right hand side of \eqref{eqn:BM} only depends on $\rbar \defeq \rhobar|_{G_{\Q_p}}$, so does the left hand side.
In particular, this implies $W(\rhobar)$, being the set of $\sigma$ with $Z(M_\infty(\sigma))\neq 0$ by the Cohen--Macaulay property, depends only on $\rbar$ as expected.
Another feature of the situation is that there are many possible choices of expressions (\ref{eqn:present}), even if we restrict to the case when $c_{\lambda,\tau} =0$ for all $\lambda \neq 0$.
Since the left hand side of (\ref{eqn:BM}) involves only $\sigma$, we get the surprising conclusion that the right hand side must be independent of the choice of expressions (\ref{eqn:present}). In other words, there are many non-trivial relations between cycles of special fibers of potentially crystalline deformation rings.
We summarize the above arguments as the following conjecture.
\begin{conj}\label{conj:GHS}
\begin{enumerate}
\item \label{item:BM} \cite{BM,EG} The left hand side of \eqref{eqn:BM} is independent of the presentation in \eqref{eqn:present}.
\item \label{item:BMSW} \cite{GHS} For any presentation as in \eqref{eqn:present}, $\sigma \in W(\rhobar)$ if and only if the right hand side of \eqref{eqn:BM} is nonzero.
\end{enumerate}
\end{conj}
\begin{rmk}
Conjecture \ref{conj:GHS}\eqref{item:BM} is purely local in the sense that it only involves $\bG_{/\Q_p}$ and $\rbar$ while \eqref{item:BMSW} is global since it involves $\rhobar$.
However, both follow from Axiom \ref{ax:patch}.
\end{rmk}
\subsection{Herzig's recipe}
The complexity of Conjecture \ref{conj:GHS} suggests that Question \ref{ques:Serre2} does not admit a simple answer.
Indeed, the case of $\GL_2$ is perhaps misleading because of the simplicity of the geometry and representation theory involved.
However, the class of semisimple representations $G_\Q \ra \, ^L \bG(\F)$ admits an essentially combinatorial classification, and so one could ask for a combinatorial description of $W(\rhobar)$ when $\rhobar|_{G_{\Q_p}}$ is semisimple which generalizes Serre's recipe (see Remark \ref{rmk:SWC_GL2}).
Herzig's recipe gives such a (conjectural) description.
We assume in this section that $\bG_{/\Q_p}$ is unramified, with reductive integral model $G_{/\Z_p}$.
We let $K_p$ be $G(\Z_p)$.
As before, we fix a twisting element $\eta$ of $\bG$.
A \emph{regular} Serre weight is a Serre weight $F(\lambda)$ with $0 \leq \langle \lambda,\alpha^\vee \rangle< p-1$ for all simple roots $\alpha$.
We define an involution $\cR$ on the set of regular Serre weights $F(\lambda)$ by the formula
\[
\cR(F(\lambda)) = F((-w_0(\eta),w_0)\cdot \lambda).
\]
If $\rbar: G_{\Q_p} \ra \, ^L\bG(\F)$ is semisimple, then the restriction $\rbar|_{I_{\Q_p}}$ to the inertial subgroup factors through a torus.
Its Teichm\"uller lift $[\rbar|_{I_{\Q_p}}]: G_{\Q_p} \ra \, ^L\bG(E)$ is then a tame inertial type.
Recall that to a tame inertial type $\tau$ (defined over $E$), one can attach through a tame inertial local Langlands a Deligne--Lusztig representation $\sigma(\tau)$ of $G(\F_p)$ (defined over $E$).
Then $K_p$ acts on $\sigma(\tau)$ by inflation, and we let $\ovl{\sigma}(\tau)$ denote the semisimplification of the reduction of any $K_p$-stable $\cO$-lattice in $\sigma(\tau)$.
We have the following conjecture.
\begin{conj}[\cite{herzig-duke,GHS}]\label{conj:herzig}
The subset of regular Serre weights in $W(\rhobar)$ is $\cR(\JH(\ovl{\sigma}([\rhobar|_{I_{\Q_p}}])))$.
\end{conj}
\noindent Conjectures \ref{conj:GHS} and \ref{conj:herzig} are of a rather different nature.
For one thing, Conjecture \ref{conj:herzig} only applies to $\rhobar$ which are semisimple locally at $p$, which is when one expects $W(\rhobar)$ is largest.
However, Conjecture \ref{conj:herzig} is rather more explicit than Conjecture \ref{conj:GHS} when combined with Proposition \ref{prop:JH}.
\subsection{Results on the weight part of Serre's conjecture} \label{sec:result}
Conjecture \ref{conj:herzig} and a weakened version of Conjecture \ref{conj:GHS} is known when $\bG$ is $\GL_2$, the Weil restriction of the unit group in a quaternion algebra which is indefinite at no more than one archimedean place, or the Weil restriction of a definite unitary group in two variables under mild hypotheses (see Remark \ref{rmk:SWC_GL2}).
Similar results \cite{LLLM,LLLM2,wild} are known for the Weil restrictions of definite unitary groups in three variables that are unramified at $p$ under an additional \emph{genericity} hypothesis.
Suppose now that $\bG_{/E}$ is a product of $\GL_n$ over a finite set $\cJ$.
For $\tld{w} \in \tld{W}$, $\tld{w}(0)$ is a tuple of elements in $\Z^n$ indexed by $\cJ$.
We write $\tld{w}(0)_j \in \Z^n$ for the element corresponding to $j\in \cJ$.
We say that a semisimple $\rbar: G_{\Q_p} \ra \, ^L\bG(\F)$ is sufficiently generic at $p$ if ${\sigma}([\rbar|_{I_{\Q_p}}]) = R(\tld{w})$ where $0 \leq \langle \tld{w}(0),\alpha^\vee\rangle \leq p$ for all simple roots $\alpha$ and $p \nmid P(\tld{w}(0)_j)$ for an implicit polynomial $P \in \Z[X_1,\ldots,X_n]$ which depends only on $n$ (and not on $p$ or $j$).
If $\rbar$ is not semisimple, we say that it is sufficiently generic if its semisimplification is.
In a precise sense, most $\rbar$'s are sufficiently generic as $p \ra \infty$.
\begin{thm}\label{thm:main}
\begin{enumerate}
\item \label{item:genBM} \cite[Corollary 8.5.2]{MLM} Suppose that $\bG_{/\Q_p}$ is an unramified Weil restriction of $\GL_n$.
Then Conjecture \ref{conj:GHS}\eqref{item:BM}, restricted to presentations coming from Deligne--Lusztig representations, holds for sufficiently generic $\rbar$.
\item \label{item:unitSW} \cite[Theorem 9.1.6]{MLM} Suppose that $\bG$ is the Weil restriction of a definite unitary (over a nontrivial totally real extension of $\Q$) and that $\rhobar|_{G_{\Q_p}}$ is sufficiently generic and semisimple.
Then under mild Taylor--Wiles hypotheses, Conjectures \ref{conj:GHS}\eqref{item:BMSW} (for the restricted presentations in the previous item) and \ref{conj:herzig} hold.
\item \label{item:GLnSW} \cite{LL} Suppose that $\bG$ is the Weil restriction of $\GL_n$ over a CM field and that $\rhobar|_{G_{\Q_p}}$ is sufficiently generic and semisimple.
Suppose moreover that $H^*(Y(K_f),\cW)_\fm \otimes_{\cO} E$ is nonzero for some $\cW$ corresponding to a Deligne--Lusztig representation.
Then under mild Taylor--Wiles hypotheses, Conjectures \ref{conj:GHS}\eqref{item:BMSW} (for the restricted presentations in the previous item) and \ref{conj:herzig} hold.
\end{enumerate}
\end{thm}
A critical tool to prove Theorem \ref{thm:main} is the following.
\begin{thm}\label{thm:domain}
Suppose that $\rbar: G_{\Qp} \ra \, ^L \bG(\F)$ is semisimple.
\begin{enumerate}
\item \cite[Theorem 7.3.2(2)]{MLM} If $\tau$ is a sufficiently generic tame inertial type, then $R_{\rbar}^{\eta,\tau}$ is an integral domain (if it is nonzero).
\item \label{item:pdpatch} \cite[Proposition 6.2.7]{MLM} There exists a functor $M_\infty$ (up to adding formal variables) satisfying Axiom \ref{ax:patch}\eqref{item:CM}-\eqref{item:LGC}, with $\bG$ a Weil restriction of a definite unitary group, such that $M_\infty(\sigma(\tau)^\circ)$ is nonzero if $R_{\rbar}^{\eta,\tau}$ is nonzero for a sufficiently generic tame inertial type $\tau$.
\end{enumerate}
\end{thm}
\begin{rmk}
\begin{enumerate}
\item Theorem \ref{thm:domain}\eqref{item:pdpatch} combines the modularity of obvious weights \cite{LLL} and the coherence conjecture for local models of Shimura varieties \cite{Zhu_coherence}.
\item In fact, Theorem \ref{thm:domain} also holds for any $\lambda+\eta$ with $\lambda$ dominant, though the implicit polynomial defining genericity depends on $\lambda$. See Theorem \ref{thm:main_def_ring}.
\end{enumerate}
\end{rmk}
As alluded to in Remark \ref{rmk:patch}\eqref{item:domaincase}, Theorem \ref{thm:domain} implies the existence of a functor $M_\infty$ satisfying Axiom \ref{ax:patch}\eqref{item:CM}-\eqref{item:fullsupport} restricted to cases when $\lambda = 0$ and $\tau$ is a generic tame inertial type.
The argument from \S \ref{sec:TW} shows that Theorem \ref{thm:domain} implies Theorem \ref{thm:main}\eqref{item:genBM} for sufficiently generic \emph{semisimple} $\rbar$.
The nonsemisimple case follows from a simple argument using the global geometry of the Emerton--Gee stack relying on the fact that there is a semisimple $\rbar$ on every component (see Remark \ref{rmk:BM}(\ref{rmk:suffmany})).
Moreover, Theorem \ref{thm:main}\eqref{item:unitSW} follows from Axiom \ref{ax:patch}\eqref{item:control}.
\begin{rmk}
The final part of Theorem \ref{thm:main} is a bit more subtle.
When $\ell_0=0$, one expects $H^*(Y(K_f),\cW)_\fm$ to have little torsion. In contrast when $\ell_0>0$, one expects cohomology to be dominated by torsion and characteristic $0$ classes to be rare.
This means that the lifting hypothesis, i.e.~that $H^*(Y(K_f),\cW)_\fm \otimes_{\cO} E$ is nonzero in Theorem \ref{thm:main}\eqref{item:GLnSW}, is quite restrictive.
This condition could be removed if one knew Conjecture \ref{conj:vanish} (see Remark \ref{rmk:patch}).
In lieu of this, the lifting hypothesis can be used to make an argument with Euler characteristics of the functor $M_\infty$, whose image is a priori an object in $D^b(R_{\rhobar|_{G_{\Q_p}}}\textrm{-mod}^{\textrm{fg}})$, adopting Taylor's Ihara avoidance trick to this setting \cite{10author}.
\end{rmk}
\section{Local models for potentially crystalline deformation rings}\label{sec:LM}
The heart of the proof of Theorem \ref{thm:main} reduces, via the Taylor-Wiles method, to understanding the support of the patched modules $M_\infty(\sigma)$ in Axiom \ref{ax:patch}, and ultimately to geometric properties of the potentially crystalline deformation rings $R^{\lambda,\tau}_{\rbar}$. We achieve this by introducing and analyzing certain (finite type) group-theoretic moduli spaces which algebraize these deformation rings.
Fix a prime $p$.
In this section, we will restrict to the case $\bG_{/\Qp}=\mathrm{Res}_{K/\Qp}\GL_n$ for an unramified extension $K=\bQ_{p^f}$ of $\bQ_p$. In particular, we have a reductive integral model $G_{/\Zp}=\mathrm{Res}_{\cO_K/\Zp}\GL_n$ of $\bG$.
Recall from \S \ref{sec:integral} that $E$ is a sufficiently large finite extension of $\Q_p$ with ring of integers $\cO$, uniformizer $\varpi$, and residue field $\F$.
\subsection{Potentially crystalline deformation rings}
Let $\rbar: G_K\to \GL_n(\F)$ be a mod $p$ local Galois representation. Recall that we have $R_{\rbar}$ the (framed) deformation ring that classifies lifts of $\rbar$, and the $\Qpbar$-points of $R_{\rbar}$ correspond to $p$-adic Galois representations of $G_K$ (lifting $\rbar$). Given a Hodge--Tate cocharacter $\lambda$ and inertial type $\tau$,
one has the potentially crystalline deformation ring $R_{\rbar}^{\lambda,\tau}$ which is characterized as the unique reduced $p$-flat quotient of $R_{\rbar}$
whose $\ovl{\Q}_p$-points correspond to lifts $r: G_K \ra \GL_n(\ovl{\Q}_p)$ which are potentially crystalline of type $(\lambda,\tau)$ (i.e.~the Hodge--Tate weights of $r$ are given by $\lambda$ and $\mathrm{WD}(r)$ induces the inertial type $\tau$).
In the setting of $\GL_n$, these rings were first constructed by Kisin, who also established their basic properties \cite{KisinPSS}.
\begin{thm}[Kisin]\label{thm:local_properties}
\begin{enumerate}
\item $R_{\rbar}^{\lambda,\tau}[\frac{1}{p}]$ is regular.
\item $\dim_{\cO} R_{\rbar}^{\lambda,\tau}=\dim_{E} G_{/E}+\dim_{E} G_{/E}/P_{\lambda/E}$,
where $P_{\lambda}$ is the parabolic subgroup corresponding to $\lambda$. In particular $\dim_{\cO}R^{\lambda,\tau}$ is a constant $d$ as $\lambda$ varies over regular dominant cocharacters.
\end{enumerate}
\end{thm}
When $\lambda$ is \emph{regular dominant}, the rings $R_{\rbar}^{\lambda,\tau}$ play a pivotal role in the Taylor-Wiles method: they act on patched spaces of automorphic forms $M_{\infty}(\sigma(\lambda-\eta,\tau))$, which govern questions about modularity and congruences (cf.~Axiom \ref{ax:patch}\eqref{item:LGC}). Even better, they are maximal Cohen--Macaulay modules, and hence must be supported on a union of irreducible components.
For global applications, it is essential to understand global properties of $R_{\rbar}^{\lambda,\tau}$ such as irreducibility. This turns out to be a notoriously difficult problem. There are roughly two reasons for this.
\begin{itemize}
\item Outside some special cases, $R_{\rbar}^{\lambda,\tau}$, being characterized by its $\Qpbar$-points, has no known moduli interpretation. This is related to the fact that integral $p$-adic Hodge theory is much less well understood than rational $p$-adic Hodge theory.
\item The internal structure of $R_{\rbar}^{\lambda,\tau}$ is intrinsically complicated in general. Thus, one can not expect to have completely explicit descriptions for all $\lambda$ and $\tau$.
\end{itemize}
The second point is best illustrated by the Breuil--M\'ezard conjecture, which quantifies the complexity of the special fibers of $R_{\rbar}^{\lambda+\eta,\tau}$ as $\lambda$ and $\tau$ vary in terms of the mod $p$ representation theory of $\GL_n(\cO_K)$ (we shift from $\lambda$ to $\lambda+\eta$ for the rest of this subsection to be consistent with \S \ref{sec:TW}).
We let $Z(R_{\rbar}^{\lambda+\eta, \tau}/\varpi)$ denote the $d$-dimensional cycle of $R_{\rbar}^{\lambda+\eta, \tau}/\varpi$, which counts the irreducible components with appropriate multiplicities.
For a $\GL_n(\cO_K)$-representation $V$ over $E$, recall that $\ovl{V}$ denotes the $\GL_n(\cO_K)$-representation over $\F$ which is the semisimplification of the reduction modulo $\varpi$ of any $\GL_n(\cO_K)$-stable $\cO$-lattice in $V$.
The following is a reformulation of Conjecture \ref{conj:GHS}\eqref{item:BM}.
\begin{conj}[Breuil--M\'ezard, Emerton-Gee] \label{conj:introBM} There exist $d$-dimensional cycles $\cZ_{\sigma}(\rbar)$ in $\Spec R_{\rbar}/\varpi$ for each irreducible $\GL_n(\cO_K)$-representation $\sigma$ over $\F$ \emph{(}i.e.~a Serre weight for $G(\F_p)$\emph{)} such that for all $\tau$ and $\lambda$,
\[
Z(R_{\rbar}^{\lambda+\eta, \tau}/\varpi) = \sum_\sigma m_{\lambda,\tau}(\sigma) \cZ_\sigma(\rbar),
\]
where $m_{\lambda,\tau}(\sigma)$ denotes the multiplicity of $\sigma$ in $\ovl{\sigma(\lambda,\tau)}$.
\end{conj}
In other words, the special fibers $R_{\rbar}^{\lambda+\eta, \tau}/\varpi$ are built out of a \emph{finite list} of basic cycles $\cZ_\sigma(\rbar)$, with multiplicities governed by the purely representation theoretic quantities $m_{\lambda,\tau}(\sigma)$.
Conjecture \ref{conj:introBM} is known when $n=2$ and $\lambda = 0$ by work of Gee and Kisin \cite{gee-kisin}.
When $\tau$ is a generic tame type, $m_{0,\tau}(\sigma) = 1$ for $2^f$ Serre weights $\sigma$ and is zero otherwise.
In general, the quantities $m_{\lambda,\tau}$ are very complicated: if $\lambda=0$ and $\tau$ is tame, $m_{\lambda,\tau}(\sigma)$ computes the multiplicities of a mod $p$ Deligne--Lusztig representation, which for generic $\tau$ is given by periodic Kazhdan--Lusztig polynomials.
In particular, as the rank of $G$ grows, the special fibers $R_{\rbar}^{\lambda+\eta, \tau}/\varpi$ tend to be highly non-reduced.
As explained in \S \ref{sec:TW}, to prove Theorem \ref{thm:main}, one needs to establish Axiom \ref{ax:patch}, particularly the main bottleneck \eqref{item:fullsupport}.
We do this by proving Theorem \ref{thm:domain}.
\begin{thm}\label{thm:main_def_ring}\cite[Theorem 7.3.2(2)]{MLM} Assume that $\rbar$ is \emph{semisimple} and $\tau$ is a tame inertial type which is sufficiently generic relative to $\lambda$ (in the sense of \S \ref{sec:result}). Then $R_{\rbar}^{\lambda+\eta,\tau}$ is a domain (or zero).
\end{thm}
\begin{rmk}
\begin{enumerate}
\item
Explicit computations that suggest that Theorem \ref{thm:main_def_ring} is \emph{false} without the tameness assumption on $\rbar$ when $n > 2$ unless $n=3$ and $\lambda= 0$.
\item If $R^{\lambda+\eta,\tau}_{\rbar} \neq 0$, then sufficient genericity of $\tau$ implies that of $\rbar$ and vice versa (generally with different choices of implicit polynomials). Because of this, the conclusion of Theorem \ref{thm:main_def_ring} also holds if we let $\rbar$ be tame and sufficiently generic but impose no genericity hypothesis on $\tau$.
\end{enumerate}
\end{rmk}
\subsection{The Emerton-Gee stack}\label{sec:EG}
In \cite{EGstack}, Emerton--Gee constructed the moduli stack $\cX_n$ over $\Spf \cO$ of rank $n$ \'{e}tale $(\varphi,\Gamma)$-modules. By its construction, $\cX_n$ interpolates framed deformation rings in the sense that the set $\cX_n(\ovl{\F}_p)$ is in bijection with the set of continuous representations $\rbar:G_K \ra \GL_n(\overline{\F}_p)$, and framed deformation rings $R_{\rbar}$ are versal rings (in the sense of \cite[Definition 2.2.9]{EGschemetheoretic}) for $\cX_n$. Furthermore, for a Hodge--Tate cocharacter $\lambda$ and an inertial type $\tau$, they construct a $p$-flat $p$-adic formal algebraic closed substack $\cX^{\lambda,\tau}$ which is characterized by the property that its points over any finite flat $\cO$-algebra correspond to potentially crystalline representations $r$ of type $(\lambda,\tau)$. Thus $\cX^{\lambda,\tau}$ interpolates the potentially crystalline deformation rings $R_{\rbar}^{\lambda,\tau}$ as $\rbar$ varies.
The basic properties of these stacks are as follows:
\begin{thm}[Emerton--Gee]
\begin{enumerate}
\item \cite[Corollary 5.5.18]{EGstack} $\cX_n$ is a Noetherian formal algebraic stack.
\item \cite[Theorem 4.8.12]{EGstack} $\cX^{\lambda,\tau}$ is a $p$-flat $p$-adic formal algebraic stack of dimension $\dim G_{/E}/P_{\lambda/E}$.
\item \cite[Theorem 6.5.1]{EGstack} The irreducible components of the underlying reduced stack $\cX_{n,\red}$ are in bijection with the Serre weights of $G(\Fp)$.
\end{enumerate}
\end{thm}
We let $\cC_{\sigma}$ be the irreducible component labelled by $\sigma$.
Let $\cZ_{\lambda+\eta,\tau}$ denote the top dimensional cycle of $\cX^{\lambda+\eta,\tau}/\varpi$, which has dimension independent of $\lambda$ since $\lambda+\eta$ is regular dominant. One has the following interpolation of the Breuil--M\'ezard conjecture over $\cX_n$:
\begin{conj}[Conjecture 8.2.2 \cite{EGstack}]\label{conj:intro:BMstack}
For each Serre weight $\sigma$, there exists an effective top-dimensional cycle $\cZ_\sigma$ on $\cX_{n,\mathrm{red}}$ such that for all $\lambda$ and inertial types $\tau$, we have
\[\cZ_{\lambda+\eta,\tau} = \sum_{\sigma} m_{\lambda,\tau}(\sigma) \cZ_\sigma.
\]
\end{conj}
\begin{rmk}\label{rmk:BM}
\begin{enumerate}
\item \label{rmk:suffmany} Conjecture \ref{conj:intro:BMstack} recovers Conjecture \ref{conj:introBM} by taking versal rings at $\rbar$. Conversely, knowledge of Conjecture \ref{conj:introBM} at \emph{sufficiently many} $\rbar$ would imply Conjecture \ref{conj:intro:BMstack}. This gives a mechanism to deduce Conjecture \ref{conj:introBM} for more general $\rbar$ from a few ``basic $\rbar$''. This allows us to reduce Theorem \ref{thm:main}\eqref{item:genBM} to the case of semisimple $\rbar$.
\item
In \cite[\S 8]{MLM}, using Taylor-Wiles patching, we constructed cycles $\cZ_\sigma$ for sufficiently generic $\sigma$, which satisfies a (finite) subset of the equations postulated in Conjecture \ref{conj:intro:BMstack}. As the conjectural cycles in Conjecture \ref{conj:intro:BMstack} is expected to be compatible with Taylor-Wiles patching, the cycles constructed in \emph{loc.~cit.}~should be the ``correct'' ones.
\item (cf.~\cite[Remark 1.4.11]{MLM}) One expects $\cZ_\sigma$ to contain the irreducible component $\cC_{\sigma}$ with multiplicity one. That is, one should have a decomposition
\[\cZ_\sigma= \sum_{\sigma'} b_{\sigma',\sigma} \cC_{\sigma'}\]
with $b_{\sigma',\sigma}\geq 0$ and $b_{\sigma,\sigma}=1$. This is indeed true in the cases studied in \cite[\S 8]{MLM} and \cite{gee-kisin}. For example, in the setting of \cite{gee-kisin}, the cycles $\cZ_\sigma=\cC_{\sigma}$, unless $\sigma$ is a twist of the Steinberg weight (in particular such $\sigma$ would be non-generic), in which case $\cZ_{\sigma}$ is $\cC_{\sigma}+\cC_{\sigma'}$ for a suitable $\sigma'$ (cf.~\cite[Theorem 8.6.2]{EGstack}).
For $n>3$, it is quite difficult to compute $b_{\sigma',\sigma}$, and one does not expect $\cZ_{\sigma}=\cC_{\sigma}$ in general, even for generic $\sigma$. This is analogous to the situation of the locally analytic Breuil-M\'{e}zard conjecture studied in \cite{BHS}.
\end{enumerate}
\end{rmk}
\subsection{Local models and their geometric properties}
Let $L\cG$ be the loop group, which is the ind-group scheme given by $L\cG(R)=\GL_n(R(\!(v+p)\!))$ for any $\cO$-algebra $R$. Consider the positive loop group scheme $L^+\cG$ over $\cO$ sending an
$\cO$-algebra
$R$ to the subgroup of $\GL_n(R[\![v+p]\!])$ consisting of matrices that
are upper triangular mod~$v$. Note that when $p$ is invertible in $R$, $L^+\cG(R)=\GL_n(R[\![v+p]\!])$ is the positive loop group for $\GL_n$, whereas when $p=0$ in $R$, $L^+\cG(R)=\cI(R)$, the standard Iwahori group scheme.
The quotient $L^+\cG\backslash L\cG$ is represented by an ind-proper
$\cO$-ind-scheme $\Gr_{\cG}$. This is a mixed characteristic
version of the degeneration of affine Grassmannians introduced by Gaitsgory: indeed its generic fiber $\Gr_{\cG, E}$ is isomorphic to an affine Grassmannian,
while the special fiber $\Gr_{\cG, \F}$ is isomorphic to the affine flag variety $\mathrm{Fl}$.
The affine Grassmannian has the affine Schubert stratification $\Gr_{\cG,E}=\bigcup_{\lambda}L^+ \cG_E\backslash L^+ \cG_E(v+p)^{\lambda}L^+ \cG_E$, where $\lambda$ runs over dominant coweights of $\GL_n$. Similarly, the affine flag variety $\Fl=\bigcup_{\tld{w}} \cI\backslash \cI \tld{w} \cI$, where $\tld{w}$ runs over the extended affine Weyl group $\tld{W}$.
For dominant $\lambda$, the Pappas--Zhu local model $M(\leql)$ is the Zariski closure of $L^+ \cG_E\backslash L^+ \cG_E(v+p)^{\lambda}L^+ \cG_E$ in $\Gr_{\cG}$, cf.~\cite{PZ}.
Let $\bf{a} \in \cO^n$. We now consider the condition
\[
\tag{$\star$} v \frac{dA}{dv} A^{-1} + A \Diag(\bf{a}) A^{-1} \in
\left(\frac{1}{v+p}\right)\mathrm{Lie} \, L^+\cG \label{eq:monodromy}
\]
for $ A\in L\cG (R)$. This is an approximation to the monodromy condition coming from $p$-adic Hodge theory.
This condition clearly descends to a closed condition on $\Gr_{\cG}$.
\begin{defn}
\label{defn:LMQp}
The local model $M(\lambda, \nabla_{\bf{a}})$ is
the Zariski closure in $M(\leql)$ of the locus cut out by
\eqref{eq:monodromy} in $L^+\cG_E\backslash L^+ \cG_E(v+p)^{\lambda}L^+ \cG_E$.
\end{defn}
Note that right multiplication by the constant diagonal torus $T^\vee$ preserves \eqref{eq:monodromy}.
(Here, $T^\vee$ is a maximal torus in $\GL_n$ which is the dual group $G^\vee$ of the group $G = \GL_n$ which appeared in \S \ref{sec:serrewt}.)
Thus, $M(\lambda, \nabla_{\bf{a}})$ inherits a $T^\vee$-action compatible with the $T^\vee$-action on $M(\leql)$.
By contemplating the interaction of condition \eqref{eq:monodromy} with the affine Schubert stratification, one observes:
\begin{itemize}
\item \cite[Proposition 4.1.1]{MLM} $M(\lambda,\nabla_{\bf{a}})_{/E}$ is isomorphic to $P_{\lambda}\backslash \GL_n$, hence is smooth and irreducible.
\item \cite[Theorem 4.2.4]{MLM} Provided $\bf{a}$ mod $p$ sufficiently regular, the locus cut out by \eqref{eq:monodromy} in each open Schubert cell $\cI\backslash \cI \tld{w} \cI\subset \Fl$ is an affine space, with dimension combinatorially determined by $\tld{w}$.
\end{itemize}
Thus $M(\lambda,\nabla_{\bf{a}})$ is a degeneration of a partial flag variety, and one has control over its \emph{reduced} special fiber.
\begin{exam}
\begin{enumerate}
\item
For example, when $n=2$ and $\lambda=(1,0)$, condition $\eqref{eq:monodromy}$ is empty, and $M(\lambda,\nabla_{\bf{a}})=M(\leql)$ is a degeneration of $\bP^1$ into a union of two $\bP^1$ crossing transversely at a point.
More generally, one has $M(\lambda,\nabla_{\bf{a}})=M(\leql)$ if and only if $\lambda$ is minuscule.
\item Suppose $n=3$ and $\lambda=(2,1,0)$, and $\bf{a}$ mod $p$ is sufficiently regular. Then $\dim M(\leql)=4$, whereas $\dim M(\lambda,\nabla_{\bf{a}})=\dim B\backslash \GL_3 =3$. The special fiber $M(\lambda,\nabla_{\bf{a}})_{\F}$ is reduced and has $9$ irreducible components, six of which are isomorphic to the flag variety $B\backslash \GL_3$, while the remaining three are more complicated rational smooth varieties. Already in this case, the behavior of the intersections among the irreducible components is somewhat elaborate, cf.~\cite{wild}.
\end{enumerate}
\end{exam}
\begin{rmk} \label{rmk:explicit_charts}Around each point $\tld{z}\in \Fl$, one can write down an explicit open neighborhood $\cU(\tld{z})$ of $\Gr_{\cG}$ using the theory of the ``big cell''. This allows us to in principle give explicit coordinate charts for $M(\lambda,\nabla_{\bf{a}})$: the coordinate charts parametrize matrices $A$ with polynomial entries whose degrees are bounded in terms of $\tld{z}$, and one then imposes elementary divisor conditions dictated by $\lambda$ together with the explicit equation \eqref{eq:monodromy}
and takes the $p$-saturation of the result. It is the $p$-saturation operation that makes this description rather difficult to work with.
\end{rmk}
In order to establish the connection between the above models to Galois deformation theory, we have to understand the behavior of $M(\lambda, \nabla_{\bf{a}})$ under completion. The essential difficulty is that an irreducible variety may break up into formal branches in some complicated way after completions: its singularities may not be \emph{unibranch}.
Unfortunately, $M(\lambda, \nabla_{\bf{a}})$ fails to be unibranch in general, and in such situations it is difficult to control the subset of the formal branches that are related to Galois deformation theory. Fortunately, it turns out there is supply of special points where this difficulty does not manifest:
\begin{thm} \label{thm:intro:unibranchMLM}
\emph{(}\cite[Theorem 3.7.1]{LLLM}\emph{)} There exists a nonzero polynomial $P \in \Z[X_1, \ldots, X_n]$ such that if $P(\bf{a}) \neq 0 \mod p$, then for any $T$-fixed point $x \in M(\lambda, \nabla_{\bf{a}})(\ovl{\F}_p)$, the completed local ring $\cO_{ M(\lambda, \nabla_{\bf{a}}), x}^{\wedge}$ is a domain $($i.e., $M(\lambda, \nabla_{\bf{a}})$ is unibranch at its $T$-fixed points).
\end{thm}
This key result, whose proof we now sketch, underlies everything else.
One first observes that the theorem holds (under a mild assumption on the characteristic) for the equal characteristic analogues of $M(\lambda, \nabla_{\bf{a}})$ where $E$ is replaced by $\F(\!(t)\!)$. In this function field setting, there is an additional symmetry: there is an extra $\bG_m$-action given by ``loop rotation'' which scales $t$. This implies that the $T$-fixed points look like cone points, i.e.~the fixed point of an attracting torus action, and one observes that cone points are unibranch. We then deduce the mixed characteristic case by a spreading out argument. The essential point here is that unibranch can be phrased in terms of connectedness of fibers of the normalization map, and normalization is preserved by generic base change. This explains the occurence of the universal polynomial $P$: its vanishing locus is the obstruction to certain properties being preserved under base change.
\subsection{Local models and Emerton--Gee stacks}
Recall that we fixed a finite unramified extension $K/\Qp$.
Let $k$ be the residue field of $K$.
Let $\cJ$ be the set of embeddings $\Hom_{\Qp}(K, \overline{\Q}_p)$ which we identify with $\Hom_{\Qp}(K, E) = \Hom(k, \F)$ using the inclusion $E \subset \overline{\Q}_p$.
To any \emph{tame} inertial type $\tau$ for $I_K$, one can associate a collection $\bf{a}_{\tau} = (\bf{a}_{\tau, j})_{j \in \cJ}$, where $\bf{a}_{\tau, j} \in \cO^n$ records the inertial weights of $\tau$.
In the ``lowest alcove" principal series case, $\bf{a}_{\tau}$ is defined so that $\tau$ is the direct sum \[\bigoplus_{i=1}^n \prod_{j \in \cJ} j \circ \omega_K^{\bf{a}_{\tau,j}^{(i)}}\] where $\omega_K: I_K \ra k^\times$ is the reduction of the Lubin--Tate character $I_K \ra \cO_K^\times$.
Set $\lambda = (\lambda_j)_{j \in \cJ} \in (\Z^n)^{\cJ}$, a Hodge--Tate cocharacter. Define
\[
M_{\cJ}(\lambda, \nabla_{\bf{a}_{\tau}}) = \prod_{j \in \cJ} M(\lambda_j, \nabla_{ \bf{a}_{\tau, j}})
\]
where, for each $j\in\cJ$, the local models $M(\lambda_j, \nabla_{ \bf{a}_{\tau, j}})$ are those appearing in Definition \ref{defn:LMQp}.
The relationship between the local models and the Emerton--Gee stacks is given by the following:
\begin{thm}\emph{(}\cite[Theorem 7.3.2]{LLLM}\emph{)}
\label{thm:intro:LMD} If $\tau$ is sufficiently generic \emph{(}with respect to $\lambda$\emph{)}, then
there exist Zariski open covers $
\underset{\text{\tiny{$\tld{z} $}}}{\text{\Large{$\bigcup$}}}
\cX_{\reg}^{\leq \lambda,\tau}(\tld{z}) $ and $\underset{\text{\tiny{$\tld{z} $}}}{\text{\Large{$\bigcup$}}} U_{\reg}(\tld{z},\leql,\nabla_{\bf{a}_\tau})^{\wedge_p}$ of $\underset{\tiny{\substack{\lambda' \leq \lambda\\ \lambda' \text{\emph{reg. dom.}}}}}{\text{\Large{$\bigcup$}}} \cX^{\lambda',\tau} $ and $ \underset{\tiny{\substack{\lambda' \leq \lambda\\ \lambda' \text{\emph{reg. dom.}}}}}{\text{\Large{$\bigcup$}}}M(\lambda', \nabla_{\bf{a}_\tau})^{\wedge_p}$, respectively, such that for each $\tld{z} $, there exists a local model diagram
\begin{equation}\label{eqn:LMD}
\xymatrix{
& \tld{\cX}_{\reg}^{\leq \lambda,\tau}(\tld{z}) \ar[dl] \ar[dr] & \\
\cX_{\reg}^{\leq \lambda,\tau}(\tld{z}) & &U_{\reg}(\tld{z},\leql,\nabla_{\bf{a}_\tau})^{\wedge_p} \\
}
\end{equation}
\end{thm}
\begin{rmk} \begin{enumerate}
\item In the above diagram, the $T$-fixed points of the local models have a simple Galois theoretic interpretation: they correspond to semisimple $\rbar$.
\item
When $\lambda = \eta$, one has
$
\underset{\tiny{\substack{\lambda' \leq \lambda\\ \lambda' \text{reg. dom.}}}}{\text{{$\bigcup$}}} \cX^{\lambda',\tau}=\cX^{\eta,\tau}.$
Since potentially crystalline deformation rings of type $(\eta,\tau)$ are versal rings to $\cX^{\eta, \tau}$, we see that they appear (up to smooth modifications) as the completion of local rings of $M(\eta, \nabla_{\bf{a}_\tau})$ at closed points. In particular, we deduce the irreducibility of the potentially crystalline deformation rings of Theorem \ref{thm:main_def_ring} from the unibranch property of the local models (at the appropriate points).
This completes the proof of Theorem \ref{thm:main}\eqref{item:genBM} and the first half of Theorem \ref{thm:main}\eqref{item:unitSW} (see Remark \ref{rmk:BM}(\ref{rmk:suffmany})).
\item Combining the theorem with Remark \ref{rmk:explicit_charts}, we get an algorithm to write down explicit presentations of (unions of) $R^{\lambda,\tau}_{\rbar}$ (for regular $\lambda$).
\end{enumerate}
\end{rmk}
We now give a slightly simplified outline of the proof of Theorem \ref{thm:intro:LMD}. The construction of the stacks $\cX^{\lambda,\tau}$ comes in two steps:
\begin{itemize}
\item Using integral $p$-adic Hodge theory, one can attach Breuil--Kisin modules to lattices in (potentially) crystalline representations for $G_K$. Thus the first step is to construct a moduli stack of Breuil--Kisin modules $Y^{\leq \lambda,\tau}$ with tame descent data of type $(\lambda,\tau)$.
\item As not all Breuil--Kisin modules come from lattices in (potentially) crystalline representations, one needs cut down $Y^{\lambda,\tau}$ by appropriate conditions to get $\cX^{\lambda,\tau}$.
\end{itemize}
Accordingly, the proof is divided into two steps:
\begin{itemize}
\item In the first step, we show that $Y^{\leq \lambda,\tau}$ is locally modelled by the Pappas-Zhu model $M(\leql)$. This is not surprising, as Breuil--Kisin modules are a projective $\cO_K[\![u]\!]$-modules with certain semi-linear structures, and thus are closely related to points of $\Gr_{\cG}$. Using the open cover $\Gr_\cG=\bigcup_{\tld{z}} \cU(\tld{z})$ (cf.~Remark \ref{rmk:explicit_charts}), we get an analogue of the local model diagram \eqref{eqn:LMD} for $Y^{\leql,\tau}$ and induced open affine covers on every object in sight.
\item After the first step, we get \emph{two} closed substacks of $Y^{\leql,\tau}(\tld{z})$: the substack $\cX^{\leql,\tau}(\tld{z})$ and the substack $\cX^{\leql,\tau,\star}(\tld{z})$ induced by the $p$-adic completion of $ \bigcup_{\lambda' \leq \lambda} M(\lambda', \nabla_{\bf{a}_\tau})$ along the local model diagram for $Y^{\leql,\tau}$. They are genuinely different substacks, because condition (\ref{eq:monodromy}) is only the ``first order term'' of the condition cutting out $\cX^{\leql,\tau}$ inside $Y^{\leql,\tau}$.
However, the two substacks are $p$-adically close, and using the smoothness of the generic fiber of $M(\lambda, \nabla_{\bf{a}})$, one can produce a non-canonical embedding $\cX^{\leql,\tau}(\tld{z})\into \cX^{\leql,\tau,\star}(\tld{z})$. Since both stacks turn out to have the same dimension, the maximal dimensional part $\cX_\reg^{\leql,\tau}(\tld{z})$ of $\cX^{\leql,\tau}(\tld{z})$ embeds into the maximal dimension part of $\cX^{\leql,\tau,\star}(\tld{z})$. Now, using the results of \cite{LLL} (which ultimately uses Taylor--Wiles patching, and hence automorphic forms), one obtains a lower bound on the number of irreducible components (of the spectrum of the structure sheaf) of the former, while Theorem \ref{thm:intro:unibranchMLM} gives the same upper bound for the number of irreducible components (of the spectrum of the structure sheaf) of the latter. Thus the two maximal dimension parts are (non-canonically) isomorphic to each other, which concludes the proof.
\end{itemize}
As the above outline suggests, the arrows in the local model diagram are produced by Hensel-type lifting arguments, and thus are highly non-canonical. However, modulo $p$ this issue disappears, and the local model diagrams on the open cover produced by Theorem \ref{thm:intro:LMD} glue together. Consequently, the analysis of irreducible components of the special fibers of local models implies the following.
\begin{thm}
\label{thm:intro:irreducible_components_mod_p} For $\tau$ sufficiently generic (with respect to $\lambda$):
\begin{enumerate}
\item $\cX^{\lambda+\eta,\tau}_\red=\cup_\sigma \cC_{\sigma}$, where the union runs over all Serre weights $\sigma\in \JH(\ovl{\sigma(\lambda,\tau)})$.
\item There is a natural bijection between the irreducible components of $\overline{M}(\lambda+\eta, \nabla_{\bf{a}_\tau})$ and the Jordan--H\"older factors of $\ovl{\sigma(\lambda,\tau)}$.
\item For each $\sigma\in \JH(\ovl{\sigma(\lambda,\tau)})$, we have a mod $p$ local model diagram:
\begin{equation}\label{eq:mod_p_local_model}
\xymatrix{
& \tld{\cC}_{\sigma} \ar[dl] \ar[dr] & \\
\cC_{\sigma} & & \overline{M}(\lambda+\eta, \nabla_{\bf{a}_\tau})_{\sigma} \\
}
\end{equation}
where $\overline{M}(\lambda+\eta, \nabla_{\bf{a}_\tau})_{\sigma}$ is the irreducible component of $\overline{M}(\lambda+\eta, \nabla_{\bf{a}_\tau})$ labelled by $\sigma$ and both arrows are torsors for the torus $(T^\vee)^{\cJ}$ \emph{(}with respect to different $(T^\vee)^{\cJ}$-actions\emph{)}.
\end{enumerate}
\end{thm}
\begin{rmk}
\label{rmk:intro:MLM:sp:fib}
\begin{enumerate}
\item The proof of Theorem \ref{thm:intro:irreducible_components_mod_p} does not go through Theorem \ref{thm:intro:LMD}. Because of that it holds under much milder genericity conditions compared to our other theorems: if $\sigma(\tau) = R$, we only require that $R$ is $m$-generic for $m$ sufficiently large depending on $\lambda$ (larger than both $2\langle \lambda,\alpha^\vee\rangle + 2$ and $4n + \langle \lambda,\alpha\rangle$ for all roots $\alpha$).
\item Over $\F$, equation \eqref{eq:monodromy} becomes the equation cutting out a \emph{deformed} affine Springer fiber in $\Fl$, cf.~\cite{Frenkel_Zhu}. Thus the irreducible components $\overline{M}(\lambda+\eta, \nabla_{\bf{a}_\tau})_{\sigma}$ are irreducible components of a (product of) \emph{deformed} affine Springer fiber(s). (It is immediate from the aforementioned \cite[Theorem 4.2.4]{MLM} that these irreducible components are rational varieties.) Theorem \ref{thm:intro:irreducible_components_mod_p} then allows us to get a handle on the irreducible component $\cC_{\sigma}$ of the reduced Emerton--Gee stack for generic $\sigma$.
In particular, one can get a description of the semisimple points on $\cC_\sigma$, and this is the critical ingredient for the verification of Herzig's recipe (Theorems \ref{thm:main}\eqref{item:unitSW} and \ref{thm:main}\eqref{item:GLnSW}).
\item The irreducible components $\overline{M}(\lambda+\eta, \nabla_{\bf{a}_\tau})_{\sigma}$ are fairly easy to implement on computer algebra systems such as Macaulay2. For any given $n$, this allows us to probe the structure of $\cC_\sigma$ in a purely algorithmic manner.
\end{enumerate}
\end{rmk}
\begin{exam}
\begin{enumerate}
\item (Fontaine-Laffaille components) For $\sigma$ in the lowest alcove, the corresponding irreducible component of the deformed affine Springer fiber is isomorphic to a product of flag varieties $(B\backslash \GL_n)^{\cJ}$. We deduce from this that
\[\cC_{\sigma}=[(N\backslash \GL_n)^{\cJ}/T^{\cJ}]\]
where $N$ is the subgroup of unipotent upper triangular matrices, and $T^{\cJ}$ acts via \emph{shifted conjugation}: $(t_j)\cdot (Ng_j)=(Nt_{j}g_j t_{j\circ \phz}^{-1})$, where $\circ \phz$ denotes pre-composition with Frobenius on $K$. In particular, for $n=2$, all components $\cC_{\sigma}$ for generic $\sigma$ are of this form.
\item
When $n=3$, there are two types of irreducible components at each factor $j\in \cJ$: the flag variety $B\backslash \GL_3$ or a more complicated rational and smooth variety (this can be extracted, for example, from the description of minimal primes in \cite[Table 3]{wild}). In particular, there are $2^f$ types of $\cC_\sigma$ (for generic $\sigma$) which correspond to the possible $p$-alcoves containing the highest weight of $\sigma$.
\item For $n=4$, there are generic $\sigma$ for which $\cC_{\sigma}$ is singular (e.g.~for $\sigma$ with highest weight in the highest $p$-restricted alcove). Thus the smoothness of $\cC_{\sigma}$ appears to be a low rank coincidence.
\end{enumerate}
\end{exam}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: cannot resolve symbol 'google' in android studio var origin = new google.maps.LatLng( location.latitude, location.longitude );
The above code shows error as cannot resolve symbol google. which google api i have to use in order to resolve it.
A:
*
*In Eclipse goto File ⇒ Import ⇒ Android ⇒ Existing Android Code Into Workspace
*Click on Browse and select Google Play Services project from your android sdk folder. You can locate play services library project from
android-sdk-windows\extras\google\google_play_services\libproject\google-play-services_lib
*Importantly while importing check Copy projects into workspace option
A: *
*Make sure you have added Google Play Services Library as Library
project.
*if problem persists update your Google Play Service Library by SDK
Manager.
*do Clean and ReBuild project.
*please restart your Eclipse or andorid
studio,
refer :https://developers.google.com/maps/documentation/android-api/
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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The snow is finally falling here in Colorado as we partake in holiday festivities, skiing, and goofing off with Neva.
Peace to you all, on this day and every day.
Merry Christmas and happy new year to you, Jeremy, and Neva. Thank you for another year of lovely photos, delicious recipes, humor, and wisdom. Much love to you!!
And a Happy New Year! Thanks for sharing your Colorado adventures.
Probably Naughty judging from the pictures. Thank you for always sharing your magnificent pictures with us – I showed them to the hubby. My favorite is one of Neva sleeping on the couch (second to last). Happy pagan Saturnalia to those who do celebrate!
Thank you so much, Jen, for sharing your life and delectable recipes with us in such a gorgeous, poetic and spunky fashion. Your blogs feed the spirit as well as our bellies! Wishing you and your family a lovely Holiday Season and, yes, PEACE be with us all!
Spectacular Year of Pics. Your eyes must be filled with happiness with all they get to see.
I hope that you and your family enjoyed the holidays.
Happy Hols, Jen! Thanks for all of your great pics and posts – your blog is always a happy place for me!
Merry Christmas!! Hope you had a lovely holiday! Can't wait t hear about your NYE food plans!
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{
"redpajama_set_name": "RedPajamaC4"
}
| 9,604
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Wilhelm Hedick (* 4. April 1838 in Belecke, Westfalen; † 2. Januar 1897) war ein Lehrer und Pionier in der Tonaufzeichnung.
Ostern 1863 trat er als Probekandidat an der Realschule zu Düsseldorf ein, war von Herbst 1864 kommissarischer und vom 1. Oktober 1865 an ordentlicher Lehrer am Realgymnasium in Köln. Um Ostern 1868 wurde er zum Direktor der höheren Schule in Venlo ernannt und später war er Direktor der Rijks Hogere Burgerschool (HBS, höheren Reichsbürgerschule) in Tilburg-Breda.
Am 6. März 1888 wurde ihm in Deutschland das Patent Nr. 42471 (Brit. pat. 569/88) erteilt, für "... eine Vorrichtung zum Aufzeichnen akustischer und elektrischer Wellen mittels Gas- oder Staubstrahlen, welche ... durch telefonartige Vorrichtung longitudinal vibrierend gegen ein Band geschleudert werden, um dadurch Schrift zu erzeugen, sowie auf Reproduktion ... mit Hilfe der erzeugten Schrift".
Dieses war ähnlich dem von Charles Sumner Tainter (1854–1940) und wurde um 1909 von Ernst Walter Ruhmer verfeinert.
Einzelnachweise
Deutscher
Geboren 1838
Gestorben 1897
Mann
Gymnasiallehrer
Schulleiter (Niederlande)
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{
"redpajama_set_name": "RedPajamaWikipedia"
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\section{INTRODUCTION \& RELATED WORK}
The recent popularity of Virtual Reality (VR) is resulting not only in many new applications available on the market but also in many new improvements and new features of head-mounted displays (HMD). One of the features of virtual experiences is to immerse users into other worlds \cite{zahorik1998presence}. Any contact with the real world such as sensors or controllers can break this feeling \cite{mcgloin2013video}. Therefore, new HMDs are allowing features of interacting via hand tracking to replace interactions with controllers.\footnote{https://www.oculus.com/blog/thumbs-up-hand-tracking-now-available-on-oculus-quest} This new possibility can lead to very interesting use cases in virtual reality in the field of gaming, but also in others fields where virtual reality is used for serious games. Virtual reality is already being widely used in learning \cite{monahan2008virtual}, more particularly in medicine \cite{mcgrath2018using} or manufacturing \cite{nee2013virtual}. Those fields are traditionally associated with precision, as well as hand movements are important. However, it is also important to ensure good quality and user experience to achieve good simulations and results.
Based on the recent release of the build-in hand tracking technology by the company Oculus for their HMD Quest and the fact that there is still little to no scientific studies performed to test for influences on user experience, this paper aims to investigate following questions: 1) How do different interaction types (such as hand tracking or interaction via controllers) in virtual reality influence user experience?, and 2) Do different task types (such as grabbing or typing) for those interaction types have an influence on user experience? The feeling of presence and immersion are critical factors when it comes to simulations in VR, and a lot of research has been done already about it \cite{mania2001effects}, \cite{brooks1999s}. An essential element in the user experience of virtual reality is to accurately represent the user's hand in the virtual environment \cite{manresa2005hand}, \cite{gratch2002creating}.
This not only related to virtual reality but has already been researched in the field of gaming with other types of virtual environments \cite{wang2009real}, putting in focus in particular algorithms on how to achieve precise hand tracking via various sensors \cite{penelle2014multi}, \cite{taylor2016efficient}.
There is a study reporting on hand tracking and visualization in a virtual reality simulation where a design had been developed offering to track the fingers and palm \cite{cameron2011hand}. The study was focusing on experiments where participants had to mimic particular hand positions and reports on performance in terms of completion time.
However, little work has been done with the focus on user experience with hand tracking in virtual reality. To the best of our knowledge, there is none research done with build-in hand tracking in HMD focusing on user experience with different interaction types.
\section{METHODS}
The study was conducted in a separate university's lab room equipped with Oculus Quest head-mounted display (HMD). This device was used in this study as it is supporting interactions using hand tracking as well as interactions with controllers. For this study, two Unity game applications had been created to test two different types of tasks - grabbing and typing.
The first application was a game with a goal to sort five red and five blue balls by grabbing them and putting them to the correct box, as shown in Figure \ref{fig:apps}. The second application was a game where the player had to retype displayed number sequences on a virtual numerical keyboard, as shown in Figure \ref{fig:apps}, and repeat it for five courses.
Both games had a strong focus to be played with hands, depending on the type of the game to grab or to type, and both games had the same four possible types of interactions: 1) Visualized controllers and hands, 2) Visualized controllers, but no visualization of hands, 3) Visualized hands, but no visualization of controllers, 4) Visualized hands while using hand tracking.
All visualizations, for both interaction types with controllers and with hand tracking, are shown in Figure \ref{fig:apps}.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{./figures/Apps.JPG}
\caption{Left: Preview of typing task (upper) and preview of grabbing task (below); Right (top to bottom): interaction with visualization of both controllers and hands, visualization of only controllers, visualization of only hands, and interaction with hand tracking.}
\label{fig:apps}
\vspace{-2em}
\end{figure}
At the beginning of the experiment, participants were welcomed. As part of the introduction to participants, it was explained that the assignment is to play two VR games with different tasks, using controllers and hand tracking as types of interaction. Participants also signed a consent form, and before any VR condition, they were given a questionnaire asking about demographics, their previous experience with virtual reality, and hand tracking. Also, participants were asked about their tendency to actively engage in intensive technology interaction by Affinity for Technology Interaction (ATI) Scale questionnaire \cite{franke2019personal}.
As part of the introduction to virtual reality and interactions, participants were explained how to use controllers and hand tracking for interaction, and they were given a chance to familiarize themselves with virtual environments of both games. After they confirmed to the moderator that they feel comfortable with learned interactions, the moderator could start the experiment. Each participant was playing two games with different task types grabbing or typing four times with each of the interaction types, which means that each participant overall participated in 8 conditions. The moderator asked the participants to observe the visualization and behavior of the interaction at the beginning of each condition. After each condition, participants were asked to fill in the Self-Assessment Manikin (SAM) \cite{bradley1994measuring} as an emotion assessment tool that uses graphic scales. Additionally, after each condition, participants were also rating feeling of presence and realism using a standardized measurement of perceived presence with a subjective scale: the Igroup Presence Questionnaire (IPQ) \cite{Schubert2001}. After all conditions, the participants were asked to rate the usability of reading in virtual reality with the System Usability Scale (SUS) \cite{brooke1996sus} for hand tracking separately for both grabbing and typing task. Finally, at the end, participants were asked to fill in a final questionnaire asking to rank the four different interactions in their experience from best to worse.
In total, there were 20 participants, eight female, and 12 male, with the average age of the participants was 30.20 years (SD = 12.74, min = 19, max = 62). When it comes to previous experience with VR technologies, 4 participants had no experience, while the others tried VR before, but nobody reported to be an expert. Further on, 9 participants reported having no experience at all with hand tracking for interactions, while others had little or some experience, and nobody was an expert with hand tracking.
Data collection was conducted following ethical principles for medical research involving human subjects.
\section{RESULTS}
A repeated measure Analysis of Variance (ANOVA) was performed to determine statistically significant differences to investigate the effects between conditions with different task types and interaction types. The two tasks in the application were grabbing and typing, which had to be fulfilled using different interaction types (using controllers with visualization of only controllers, controllers with visualization of only hands, controllers with visualization of hands and controllers or using hand tracking). Further on, the Friedman's Two-Way Analysis was done to find out significant differences in preference of interaction types. Table \ref{tab:results} and \ref{tab:results2} provide an overview of the statistically significant results of the Friedman and the repeated measure ANOVA tests.
\begin{table}[htbp]
\centering
\caption{Effects of both task types (grabbing and typing), and all interaction types (hand tracking or controllers with different visualizations of hands and controllers) on arousal (SAM\_A), valence (SAM\_V) and dominance (SAM\_D) measured with SAM, and feeling of presence (PRES) and realism (REAL) measured with IPQ.}
\resizebox{\columnwidth}{!}{
\begin{tabular}{llcllrl}
\toprule
\multicolumn{1}{l}{Parameter} & \multicolumn{1}{l}{Effect} & \multicolumn{1}{c}{$df_{\textnormal n}$} & \multicolumn{1}{l}{$df_{\textnormal d}$} & \multicolumn{1}{c}{$F$} & \multicolumn{1}{c}{$p$} & \multicolumn{1}{c}{$\eta_{\textnormal G}^2$}
\\
\midrule
Task & SAM\_A & $1$ & $20$ & $10.35$ & $ .005$ & $0.35$ \\
Task & PRES & $1$ & $20$ & $5.30$ & $ .033$ & $0.22$ \\
Task & REAL & $1$ & $20$ & $4.90$ & $ .039$ & $0.21$ \\
Interaction & SAM\_A & $3$ & $20$ & $5.25$ & $ .003$ & $0.22$ \\
Interaction & SAM\_V & $3$ & $20$ & $20.08$ & $ <.001$ & $0.51$ \\
Interaction & SAM\_D & $3$ & $20$ & $11.43$ & $ <.001$ & $0.38$ \\
\bottomrule
\end{tabular}
}
\label{tab:results}
\vspace{-1.5em}
\end{table}
\begin{table}[htbp]
\centering
\tiny
\caption{Post hoc analysis with Wilcoxon signed-rank tests for significantly different effects of ranking for all interaction types: Controllers with visualization of both hands and controllers (C\_H), controllers with only visualizations of hands (C\_OnlyH), controllers with visualizations of only controllers (C\_noH) and hand tracking (HTrack).}
\resizebox{\columnwidth}{!}{
\begin{tabular}{llcllrl}
\toprule
\multicolumn{1}{l}{Interaction} & \multicolumn{1}{c}{$df_{\textnormal n}$} & \multicolumn{1}{l}{$df_{\textnormal d}$} & \multicolumn{1}{c}{ $Z$ } & \multicolumn{1}{c}{$p$} &
\\
\midrule
C\_H and C\_OnlyH & $3$ & $20$ & $-3.079$ & $ .002$ \\
C\_H and C\_noH & $3$ & $20$ & $-3.162$ & $ .002$ \\
C\_H and HTrack & $3$ & $20$ & $-2.844$ & $ .004$ \\
\bottomrule
\end{tabular}
}
\label{tab:results2}
\vspace{-1.5em}
\end{table}
There were statistically significant differences due to the conditions on perceived arousal, presence, and realism (see Table \ref{tab:results} and Figure \ref{fig:task}).
For the different conditions where participants had different task types, either grabbing or typing, it can be observed that it influenced how the users have rated SAM dimension arousal. For the typing task, arousal was rater as significantly higher (M\,=\,3.83, SE\,=\,0.21) compared to the grabbing task (M\,=\,3.54, SE\,=\,0.22).
Further on, participants have reported feeling significantly higher sense of presence while playing the task where they had to grab the balls (M\,=\,4.00, SE\,=\,0.15) compared to the task where they were typing (M\,=\,3.70, SE\,=\,0.17). Similarly, the feeling of realism was significantly higher for grabbing task (M\,=\,2.88, SE\,=\,0.14) compared to the typing task (M\,=\,2.69, SE\,=\,0.14).
Comparing influences that were caused by different interaction types (with controllers or hand tracking), for all three dimensions of SAM questionnaire, statistically significant differences were found (see Table \ref{tab:results} and Figure \ref{fig:inter}).
SAM dimension of arousal was rated significantly higher for interaction type using only hand tracking (M\,=\,3.38, SE\,=\,0.21) compared to the interaction with the controller where only hands were visualized (M\,=\,3.83, SE\,=\,0.21).
Similarly, the dimension of valence was for hand tracking (M\,=\,2.40, SE\,=\,0.20) significantly higher compared to all other interaction types with controllers visualizes either as only controllers (M\,=\,1.65, SE\,=\,0.14), only hands (M\,=\,1.48, SE\,=\,0.12), or with visualization of both (M\,=\,1.58, SE\,=\,0.16).
Interestingly, the dimension of dominance was reported to be significantly lower for hand tracking (M\,=\,3.13, SE\,=\,0.16) compared to all other conditions with visualization of both controllers and hands (M\,=\,3.85, SE\,=\,0.14), visualization of only controllers (M\,=\,3.75, SE\,=\,0.16), or visualization of only hands (M\,=\,3.70, SE\,=\,0.14).
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{./figures/Task.png}
\caption{Mean values of SAM arousal dimension (SAM\_A), presence (PRES), and realism (REAL) over all participants for both tasks grabbing and typing. Whiskers denote 95 percent confidence intervals.}
\label{fig:task}
\vspace{-2em}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{./figures/Inter.png}
\caption{Mean values of the SAM dimensions: arousal (SAM\_A),valence (SAM\_V), and dominance (SAM\_D) over all participants for all interaction types with controllers (visualised as both controllers and hands, only controller, only hands) and hand tracking. Whiskers denote 95 percent confidence intervals.}
\label{fig:inter}
\vspace{-2em}
\end{figure}
Ranking of the preferred way of interaction types, where participants had to order from best (4) interaction type to worst (1), resulted in statistically significant differences (see Table \ref{tab:results2}).
A side corrected pairwise comparison showed that interaction type with controllers where both where hands and controllers were visualized (M\,=\,3.30, SE\,=\,0.47) were assigned a significantly higher rank compared to all other conditions: controllers with visualizations of only controllers (M\,=\,2.35, SE\,=\,0.93), of only hands (M\,=\,2.25, SE\,=\,1.16), and using hand tracking (M\,=\,2.10, SE\,=\,1.37).
A paired-samples t-test was conducted to compare average SUS scale values in hand tracking for grabbing (M\,=\,56.50, SE\,=\,18.09), and hand tracking for typing (M\,=\,75.62, SE\,=\,20.96). There was a significant difference in the scores; t(19)\,=\,3.01, p\,=\,0.007.
\section{DISCUSSION \& CONCLUSION}
The type of task (grabbing vs. typing) had an impact on how participants experienced the experimental condition. During the grabbing task, participants experienced lower arousal, higher presence, and higher realism. This result can be assigned to the fact that a grabbing task is more natural, therefore resulting in a higher realism and presence experience. The higher arousal of participants during the typing task could be due to higher demand during the more unnatural task.
The type of interaction (controllers visualized as both controllers and hands, the controller only, the controller only hands, and hand tracking) did not influence the experience of presence and realism. Nevertheless, there was an influence of interaction type on all SAM dimensions (valence, arousal, and dominance). It can be seen that participants felt less aroused, have a more positive experience (higher valence), and felt less dominant during the experimental condition hand tracking compared to all interaction types involving controllers. The higher valence for hand tracking can be addressed to the fact that users like the experience better. The lower arousal values could also be due to reduced demand due to the interactions imposed on the participants. The lower dominance experience could be to the fact that interacting with hand tracking leads to a lower feeling of control/precision and, therefore, also to a lower dominance rating. Finally, participants ranked the interaction type, including controllers visualized as both controllers and hands, as being more preferred than all other interaction types. The latter effect could be a combination of being used to interact with controllers and that most tasks are currently rather developed for typing selecting with a controller. Users still seem, when it comes to a visualization, preferred that not only the controller but also the hands are visualized. Finally, hand tracking for typing was rated statistically significantly more usable as hand tracking for the task grabbing on the SUS scale. This effect could be due to the fact that users are still more used to controller interaction and, therefore, also rate this experience a being more usable. This study showed that user experience using pure hand tracking interaction lead to a more positive and less arousing experience while reducing the sense of dominance/control. These results can drive further research and, in the long term, contribute to help selecting the most matching interaction modality for a task.
\bibliographystyle{IEEEtran}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
We consider the problem of testing hypotheses about coefficients in linear models, where the outcome may be non-Gaussian and heteroscedastic, and the number of nuisance coefficients exceeds the sample size.
By the nuisance coefficients we mean the coefficients that are not tested by the particular test at hand, but still need to be dealt with since they lead to confounding effects.
In recent decades, the literature on permutation methods has strongly expanded \citep{tusher2001significance, meinshausen2011asymptotic, hemerik2018false, ganong2018permutation, berrett2018conditional,he2019permutation,albajes2019voxel, hemerik2019permutation, rao2019permutation}.
While the permutation test dates far back \citep{fisher1936coefficient}, most of the permutation tests in the presence of nuisance were published in the last four decades. To our knowledge, the existing methods are limited to low-dimensional nuisance.
For the high-dimensional case, an approach similar to a permutation test is proposed in \citet{dezeure2017high}.
Permutation tests for low-dimensional linear models are valuable for two main reasons. First, they are robust to violations of certain standard assumptions, such as normality and homoscedasticity \citep{winkler2014permutation,hemerik2020robust}.
Second, when the outcome is multidimensional, a permutation-based test can be combined with existing permutation-based multiple testing methods, which tend to be relatively powerful, since they take into account the dependence structure of the outcomes \citep{meinshausen2006false, meinshausen2011asymptotic, hemerik2018false,hemerik2019permutation}.
For example, under strong positive dependence among \emph{p}-values, the Bonferroni-Holm multiple testing method \citep{holm1979simple} is greatly improved by a permutation method \citep{
westfall1993resampling}.
For the low-dimensional general linear model, with identity link but not necessarily Gaussian or homoscedastic residuals, several different permutation tests have been proposed.
The main approach that these methods have in common, is to permute residuals after regressing on the nuisance covariates.
For overviews of the available methods, see \citet{anderson1999empirical}, \citet{anderson2001permutation}, \citet{winkler2016faster} and in particular \citet{winkler2014permutation}.
Among the existing permutation methods, the Freedman-Lane approach \citep{freedman1983nonstochastic} is most commonly used and provides excellent power and type I error control.
Because the existing permutation tests require estimating the nuisance coefficients using maximum likelihood, these methods cannot be used when the number of covariates exceeds the sample size.
In recent years, important theoretical results have been published on testing in such
high-dimensional linear models. Several of these tests have proven asymptotic properties. In particular, the method in \citet{zhang2014confidence} has been shown to be asymptotically optimal under certain assumptions \citep{van2014asymptotically}. \citet{dezeure2017high} propose a bootstrap approach, which is related to the method in \citet{zhang2014confidence}.
Software implementations of tests for high-dimensional models include those described in \citet{dezeure2015high} and \citet{Chernozhukov2016}.
Testing in high-dimensional linear models is very challenging, because a large number of unknown nuisance effects needs to be dealt with, using a relatively small sample size. Consequently, tests tend to sacrifice much power compared to the situation where all nuisance coefficients would be known.
Further, the asymptotic properties of the mentioned methods rely on complex assumptions and sparsity.
The test by \citet{zhang2014confidence} can be rather anti-conservative in settings where a substantial fraction of the coefficients are non-zero. Moreover, these methods are not based on permutations. Hence they do not generally have the above-mentioned advantages, such as robustness against certain violations of the standard linear model. An exception is the bootstrap method in \citet{dezeure2017high}, which tends to be more robust to such violations.
We propose two novel tests, which, to our knowledge, are the first permutation tests in the presence of high-dimensional nuisance.
One is an extension of the low-dimensional method in \citet{freedman1983nonstochastic} and the other is somewhat related to a method by Kennedy \citep{kennedy1995randomization, kennedy1996randomization}. Further, we allow the tested parameter to be multi-dimensional, unlike many existing methods.
Using simulations we show that our methods provide appropriate type I error rate control in a wide range of situations. In particular, we illustrate empirically that our tests have the above-mentioned robustness properties.
The methods in this paper have been implemented in the R package \emph{phd}, available on CRAN.
This paper is built up as follows.
In Section \ref{secldn} we discuss permutation testing in settings with low-dimensional nuisance. This section contains some novel observations that will be used in Section \ref{sechd}.
There, we propose permutation tests for high-dimensional settings. We assess the performance of our methods with simulations in Section \ref{secsims}. An analysis of real data is in Section \ref{secdata}.
\section{Low-dimensional nuisance} \label{secldn}
\subsection{Notation and basic ideas} \label{secnota}
We consider the general linear model
$$\bm{Y}=\bm{X}\bm{\beta}+\bm{Z}\bm{\gamma} + \bm{\epsilon},$$
where $\bm{X}$ is a $n\times d$ matrix of covariates of interest, $\bm{Z}$ an $n\times q$ matrix of nuisance covariates and $\bm{\epsilon}$ an $n$-vector of i.i.d. errors with mean $0$ and non-zero variance, which are independent of the covariates. Here the rows of $\bm{X}$, $\bm{Z}$ and $\bm{Y}$ are i.i.d.. The matrix $\bm{Z}$ is assumed to have full rank with probability $1$.
The parameter $\bm{\beta}\in \mathbb{R}^d$ is of interest and $\bm{\gamma} \in \mathbb{R}^q$ is a nuisance parameter. We want to test the null hypothesis $H_0: \bm{\beta}=\bm{0}\in \mathbb{R}^d$. Here $\bm{0}$ might be replaced by another constant: the extension is straightforward.
Let $w$ be a positive integer, which will denote the number of random permutations or other transformations. In this paper, all permutation \emph{p}-values are of the form
\begin{equation} \label{formulap}
p=w^{-1}\big|\{1\leq j \leq w: T_j\geq T_1\}\big|,
\end{equation}
or, in case of a two-sided test where both small and large values of $T_1$ are evidence against $H_0$,
\begin{equation} \label{formulap2}
p=2 w^{-1} \min\Big\{\big|\{1\leq j \leq w: T_j\geq T_1\}\big|,\big|\{1\leq j \leq w: T_j\leq T_1\}\big|\Big\}.
\end{equation}
Here $T_1,...,T_w\in \mathbb{R}$ are statistics whose definition depends on the particular permutation method. They are specified in the sections below. For every $2\leq j \leq w$, the statistic $T_j$ corresponds to the $j$-th permutation.
The statistic $T_1$ is based on the original, unpermuted data. All existing and novel methods in this paper only differ with respect to how $T_1,...,T_w$ are computed.
Although we will often write `permutation', sign-flipping of residuals can also be used \citep{winkler2014permutation}.
The existing methods, as well as the novel methods in this paper, consist of the following steps.
\vskip3mm
\begin{enumerate}
\item Compute a test statistic $T_1$ based on the original data.
\item Compute a test statistic $T_2$ in a similar way, but after randomly permuting certain residuals. Repeat to obtain $T_3,...,T_w$.
\item The \emph{p}-value equals \eqref{formulap} or \eqref{formulap2}.
\end{enumerate}
\vskip3mm
Most of the existing permutation methods use residualization of $\bm{Y}$ or $\bm{X}$ with respect to the nuisance $\bm{Z}$.
In the low-dimensional situation, the residual forming matrix is
$$\re= \bm{I}-\pro= \bm{I}- \Z(\Z'\Z)^{-1}\Z'.$$
When $d=1$ we will sometimes consider $\re\bm{X}\in \mathbb{R}^n$, which is assumed to be nonzero with probability 1. In Section \ref{secldn} we assume $\Z$ contains a column of $1$'s. This implies that the entries of $\re\X$ and $\re\Y$ sum up to 0.
Note that if we use permutation, we can write the transformed residuals as
$\bm{P}\re\bm{Y},$
where $\bm{P}$ is an $n \times n$ matrix with exactly one $1$ in every row and column and elsewhere 0's. In case of sign-flipping, $\bm{P}$ is instead an $n \times n$ diagonal matrix with diagonal elements in $\{1,-1\}$ \citep{winkler2014permutation}.
We write $\Pe_1,...,\Pe_w$ to distinguish the $w$ random permutation matrices. Here $\Pe_1$ is the identity matrix and $\Pe_2,...,\Pe_w$ are random.
\subsection{Choice of test statistics} \label{fl}
Here we discuss the choice of test statistics within the permutation method of Freedman and Lane \citep{freedman1983nonstochastic,winkler2014permutation}.
The purpose of this section is to discuss some existing and novel results that we will use in Section \ref{sechd}.
The Freedman-Lane permutation method is known to provide excellent type I error control, with both its level and power staying very close to the parametric \emph{F}-test, under the Gaussian model.
The test statistic $T_1$ is based on the unpermuted model $\Y=\X\bm{\beta}+\Z\bm{\gamma}+\epsilon$.
The other statistics are obtained after randomly transforming the residuals. That is,
for $2\leq j \leq w$ the statistic $T_j$ is based on the model $(\Pe_j\re+\pro)\Y=\X\bm{\beta}+\Z\bm{\gamma}+\epsilon$, where the same test statistic, say $T$, is used as for computing $T_1$.
Thus\begin{equation} \label{T1FL}
T_1=T(\X,\Z,\Y),
\end{equation}
\begin{equation} \label{TjFL}
T_j=T\big(\X,\Z,(\Pe_j\re+\pro)\Y\big),
\end{equation}
where $T$ is a suitable test statistic, the choice of which we now discuss.
It is usually important to take $T$ to be an asymptotically pivotal statistic, i.e., a statistic whose asymptotic null distribution does not depend on any unknowns under $H_0$ (\citeauthor{kennedy1996randomization}, \citeyear{kennedy1996randomization}, p.926-927, \citeauthor{winkler2014permutation}, \citeyear{winkler2014permutation}, p.382, \citeauthor{hall1989effect}, \citeyear{hall1989effect}, \citeauthor{hall1991two}, \citeyear{hall1991two}).
A pivotal statistic $T$ will always involve estimation of the nuisance parameters. Thus, after every permutation, the nuisance parameters need to be estimated anew. Examples of pivotal test statistics are the \emph{F}-statistic and Wald statistic. These are equivalent: the resulting permutation \emph{p}-value \eqref{formulap} is the same.
In case $X$ is one-dimensional, the \emph{F}-statistic is also equivalent to the square of the \emph{partial correlation} \citep{fisher1924distribution, agresti2015foundations}, which is used in \citet{anderson2001permutation}. The partial correlation is the sample Pearson correlation of $\re \Y$ and $\re \X$,
\begin{equation} \label{parcor}
\rho\big( \re \Y,\re\X \big)= \frac{ (\re\Y)'\re\X }{ \sqrt{ \sum_i (\re\Y)_i^2 \sum_i (\re\X)_i^2 } }.
\end{equation}
Here we used that the sample means of $\re\Y$ and $\re\X$ are $0$.
If we use the partial correlation in the Freedman-Lane permutation test, this means that we take
$T(\X,\Z,\Y)= \rho\big( \re \Y,\re\X \big),$
so that \eqref{T1FL} and \eqref{TjFL} become
\begin{equation} \label{T1OLS}
T_1= \rho\big( \re \Y,\re\X \big)
\end{equation}
\begin{equation} \label{TjOLS}
\quad T_j= \rho\big( \re (\Pe_j\re+\pro)\Y ,\re\X \big),
\end{equation}
where $\re (\Pe_j\re+\pro)$ could be simplified to $\re \Pe_j\re$, since $\re\pro=\bm{0}$.
The numerator in \eqref{parcor} is
$$(\re\Y)'\re\X=\Y'\re'\re\X=\Y'\re'\X=(\re\Y)'\X,$$
so that \eqref{parcor} equals
\begin{equation} \label{semipc2}
\frac{ (\re\Y)'\X }{ \sqrt{ \sum_i (\re\Y)_i^2 \sum_i (\re\X)_i^2 }}.
\end{equation}
The Freedman-Lane test with $T$ defined by \eqref{semipc2} remains unchanged if in \eqref{semipc2} we replace $ \sum_i (\re\X)_i^2$ by $1$ or by the constant $\sum_i \X_i^2$. Indeed, $T_1,...,T_w$ will just be multiplied by the same constant.
Thus, with respect to the permutation test, the statistic \eqref{parcor} is equivalent to
\begin{equation} \label{semiparcor}
\frac{ (\re\Y)'\X }{ \sqrt{ \sum_i (\re\Y)_i^2 \sum_i \X_i^2 }}.
\end{equation}
If $\X$ has been centered around $0$, then this equals
\begin{equation} \label{T1OLSsemi}
\rho\big( \re \Y,\X \big) = \frac{ (\re\Y)'(\X-\bm{\mu}_x) }{ \sqrt{ \sum_i (\re\Y)_i^2 \sum_i (\X_i-\bm{\mu}_x)^2 }},
\end{equation}
where $\bm{\mu}_x$ denotes the $n$-vector with entries equal to the sample mean of $\X$.
This is the sample correlation of $\re\Y$ and $\X$ and is called the \emph{semi-partial correlation}.
Thus, if $\X$ is centered, using the partial correlation is equivalent to using the semi-partial correlation.
If we take $T$ to be the semi-partial correlation, then \eqref{T1FL} and \eqref{TjFL} become
$T_1= \rho\big( \re \Y,\X \big)$ and
\begin{equation} \label{TjOLSsemi}
\quad T_j= \rho\big( \re (\Pe_j\re+\pro)\Y ,\X \big)=
\frac{ \big(\re (\Pe_j\re+\pro)\Y \big)'(\X-\bm{\mu}_x) }{ \sqrt{ \sum_i \big( \re (\Pe_j\re+\pro)\Y\big)_i^2 \sum_i (\X_i-\bm{\mu}_x)^2 }},
\end{equation} where $\re (\Pe_j\re+\pro)$ could be simplified to $\re \Pe_j\re$.
Note that we could simply leave the constant $\sum_i (\X_i-\bm{\mu}_x)^2$ out without changing the result of the permutation test.
Although for centered $\X$ the statistics \eqref{parcor} and \eqref{T1OLSsemi} are equivalent, their counterparts in the high-dimensional setting are not, as will be discussed in Section \ref{secflhd}.
\section{High-dimensional nuisance} \label{sechd}
When the nuisance parameter $\bm{\gamma}$ has dimension $q\geq n$, the existing permutation methods cannot be used. Here, these approaches are adapted to obtain tests which can account for high-dimensional nuisance. We first consider the case that $X$ is one-dimensional, i.e., $d=1$. The case that $d>1$ is discussed in Section \ref{multidimbeta}.
We assume that the entries of $\Y$, $\X$ and $\Z$ have expected value $0$. Consequently, the intercept is $0$.
All existing tests rely on residualization steps, where $\Y$ or $\X$ is regressed on $\Z$.
A natural way to adapt this step to the high-dimensional setting, is to instead estimate the residuals using some type of elastic net regularization.
We will consider ridge regression. For minimizing prediction error, ridge regression is often preferrable to Lasso, principal components regression, variable subset selection and partial least squares \citep{hastie2009elements,frank1993statistical}.
Compared to the existing methods, including the Freedman-Lane approach discussed in Section \ref{fl}, using ridge regression comes down to replacing the projections $\hat{\Y}=\pro \Y$ and $\hat{\X}=\pro \X$ by ridge estimates
$ \pror_{\lambda}\Y$
and $ \pror_{\lambda_X}\X $, with $\lambda, \lambda_X>0$.
Here, for $\lambda'>0$,
\begin{equation} \label{prorl}
\pror_{\lambda'}=\Z(\Z'\Z+\lambda' \bm{I}_q)^{-1}\Z',
\end{equation}
which satisfies
$$\pror_{\lambda'}\Y= \Z \text{argmin}_{\bm{\gamma}}\Big(\Vert \Y-\Z\bm{\gamma} \Vert_2^2 +\lambda' \Vert \bm{\gamma} \Vert_2^2 \Big)$$ and similarly for $\X$.
The values $\lambda, \lambda_X$ are the regularization parameters, whose selection will be discussed.
Using ridge regression, the residuals become $\rer_{\lambda} \Y$ and $\rer_{\lambda_X} \X$, where $\rer_{\lambda}=(\bm{I}-\pror_{\lambda})$ and $\rer_{\lambda_X} =(\bm{I}-\pror_{\lambda_X})$.
The last two rows of Table \ref{toverviewhd} outline the permutation schemes that we will consider in Sections \ref{secflhd} and \ref{secdres}. The first two rows summarize the Freedman-Lane method discussed in Section \ref{fl} and the Kennedy method \citep{kennedy1995randomization, kennedy1996randomization,winkler2014permutation}.
This table is analogous to Table 2 in \citet{winkler2014permutation} and allows easy comparison of the new methods with the existing methods discussed in \citet{winkler2014permutation}.
Although Table \ref{toverviewhd} outlines the permutation schemes that we will use, several crucial specifics remain to be filled in. For example, several choices of the regularization parameters $\lambda$ and $\lambda_X$ can be considered. Moreover, the computational challenge of performing nuisance estimation in every step needs to be addressed. Finally and importantly, we must determine what test statistics are suitable to use within our permutation tests.
\begin{table}[h!]
\begin{center}
\caption{Permutation schemes for four different methods. The last two methods are novel and can account for high-dimensional nuisance.}
\label{toverviewhd}
\begin{tabular}{ll}
\hline
\textbf{Method} & \qquad\textbf{Model after permutation} \\
\hline
Freedman-Lane & \qquad $(\Pe\re+\pro)\Y=\X\bm{\beta}+\Z\bm{\gamma}+\bm{\epsilon}$\\
Kennedy & \qquad $\Pe\re\Y=\re\X\bm{\beta}+\bm{\epsilon}$\\
Freedman-Lane HD & \qquad $(\Pe\rer_{\lambda}+\pror_{\lambda})\Y=\X\bm{\beta}+\Z\bm{\gamma}+\bm{\epsilon}$\\
Double Residualization & \qquad $(\Pe\rer_{\lambda}+\pror_{\lambda}) \Y=\rer_{\lambda_X}\X\bm{\beta}+\bm{\epsilon}$\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Freedman-Lane HD} \label{secflhd}
As discussed in Section \ref{fl}, the low-dimensional Freedman-Lane method is known to provide excellent type I error control and power. Here we will provide an extension to the case of high-dimensional nuisance. We will refer to this test as \emph{Freedman-Lane HD}. The permutation scheme that we use is analogous to that of Freedman-Lane and is shown in the third row of Table \ref{toverviewhd}.
As in the Freedman-Lane method, after every permutation, we will require nuisance estimation to compute $T_j$.
We will choose ridge regression to do this.
Note however that when many permutations are used, performing a ridge regression after every permutation can be a large computational burden. We will therefore compute $\lambda$ only once, for the unpermuted model.
We take $\lambda$ to be the value that gives the minimal mean cross-validated error; see Section \ref{secsimset} for more details.
After each permutation, we then use the same parameter $\lambda$ in the ridge regression. Thus, after the $j$-th permutation, to compute the new ridge residuals, we will only need to pre-multiply the transformed outcome $(\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y$ by $\rer_{\lambda}$. We only need to compute $\rer_{\lambda}$ once.
Owing to this approach, essentially we need to perform ridge regression only once.
An important consideration is the test statistic $T$ used within the permutation test. The usual \emph{F}-statistic and Wald statistic are only defined when the nuisance is low-dimensional. Extending these definitions to the high-dimensional setting with $q\geq n$ is problematic. For example, a Wald-type statistic would require an unbiased estimate of $\beta$ and a variance estimate.
The partial correlation \eqref{parcor}, however, is more naturally generalized to the $q\geq n$ setting: we can replace the residuals $\re\Y$ and $\re\X$ by the ridge residuals
$\rer_{\lambda}\Y$ and $\rer_{\lambda_X}\X$. Similarly we can generalize the semi-partial correlation \eqref{T1OLSsemi}, by replacing $\re\Y$ by $\rer_{\lambda}\Y$.
This gives the following test statistics, which generalize the partial correlation \eqref{parcor} and the semi-partial correlation \eqref{T1OLSsemi} respectively:
\begin{equation} \label{parcorHD}
\rho\big( \rer_{\lambda} \Y,\rer_{\lambda_X}\X \big)= \frac{ (\rer_{\lambda}\Y-\bm{\mu}_1)' (\rer_{\lambda_X} \X-\bm{\mu}_2 ) }{ \sqrt{\sum_i (\rer_{\lambda}\Y-\bm{\mu}_1)_i^2 \sum_i (\rer_{\lambda_X}\X-\bm{\mu}_2)_i^2 } },
\end{equation}
\begin{equation} \label{semiparcorHD}
\rho\big( \rer_{\lambda} \Y,\X \big)= \frac{ (\rer_{\lambda}\Y-\bm{\mu}_{1})'(\X-\bm{\mu}_{x}) }{ \sqrt{ \sum_i (\rer_{\lambda}\Y-\bm{\mu}_{1})_i^2 \sum_i (\X-\bm{\mu}_x)_i^2 }}.
\end{equation}
Here, $\bm{\mu}_1$, $\bm{\mu}_2$ and $\bm{\mu}_x$ are $n$-vectors whose entries are the sample means of $\rer_{\lambda} \Y$, $\rer_{\lambda_X}\X$ and $\X$ respectively. \citet{zhu2018significance} also use a type of generalized partial correlation as the test statistic.
In Section \ref{fl} we reasoned that if $\X$ has been centered, \eqref{parcor} and \eqref{T1OLSsemi} are equivalent with respect to the permutation test. This does not apply to \eqref{parcorHD} and \eqref{semiparcorHD}. In simulations, using the statistic \eqref{semiparcorHD} tended to result in somewhat higher power than using the statistic \eqref{parcorHD}. In Section \ref{secsims} we consider both methods.
In case the generalization of the partial correlation is used, the test statistics $T_1,...,T_w$ on which Freedman-Lane HD is based are
\begin{equation} \label{eq:flT1par}
T_1 = \rho\big( \rer_{\lambda} \Y,\rer_{\lambda_X} \X \big),
\end{equation}
\begin{equation} \label{eq:flTjpar}
T_j = \rho\big( \rer_{\lambda} \big(\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y,\rer_{\lambda_X}\X \big) =
\end{equation}
$$\frac{ \big(\rer_{\lambda} (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y-\bm{\mu}^j\big)' (\rer_{\lambda_X} \X-\bm{\mu}_2 ) }{ \sqrt{\sum_i \big(\rer_{\lambda} (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y-\bm{\mu}^j\big)_i^2 \sum_i (\rer_{\lambda_X}\X-\bm{\mu}_2)_i^2 } }, $$
where $2\leq j \leq w$. Here $\bm{\mu}^j$ is an $n$-vector whose entries are the sample mean of $\rer_{\lambda}(\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y$.
For the version based on the generalization of the semi-partial correlation, the statistics are
\begin{equation} \label{eq:flT1}
T_1 = \rho\big( \rer_{\lambda} \Y,\X \big),
\end{equation}
\begin{equation} \label{eq:flTj}
T_j = \rho\big( \rer_{\lambda} (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y,\X \big).
\end{equation}
As usual, $T_1$ is just $T_j$ with $\Pe_j=\bm{I}_n$.
The pseudo-code for the version based on semi-partial correlations is in Algorithm \ref{a:FLHD}.
If $q<n$, as $\lambda\downarrow 0$, the test converges to the test for $\lambda=0$, which is the classical Freedman-Lane method.
In the wide range of simulation settings considered in Section \ref{secsims}, the Freedman-Lane HD method stayed on the conservative side, in the sense that the size was less than $\alpha$. This may due to the fact that if $\lambda>0$ and $2\leq j<k\leq w$,
the correlation between $T_1$ and $T_j$ tended to be larger than the correlation between $T_j$ and $T_k$ in simulations. This may be related to the fact that
the correlation between $\Y$ and $\Y^{*j}$ is strictly larger than the correlation between $\Y^{*j}$ and $\Y^{*k}$, where $\Y^{*j} := (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y$. This inequality is proved in the Supplementary Material.
As discussed, to perform the test, $\lambda$ and hence $\rer_{\lambda}$ need to be computed only once.
Thus, like the low-dimensional Freedman-Lane procedure, the test requires nuisance estimation after every permutation, but this is not a large computational burden.
The method is often computationally feasible even when many millions of permutations are used; see Section \ref{secsims}. It is also worth mentioning that there exist approximate methods for reducing the number of permutations while still allowing for very small, accurate \emph{p}-values \citep{knijnenburg2009fewer,winkler2016faster}.
\begin{algorithm}[h!]
\caption{Freedman-Lane HD (version based on semi-partial correlations)}
\begin{algorithmic}[1] \label{a:FLHD}
\STATE Compute $\pror_{\lambda}= \Z(\Z'\Z+\lambda \bm{I}_q)^{-1}\Z'$ and the residual forming matrix $\rer_{\lambda}=\bm{I}-\pror_{\lambda}$. Here $\lambda$ is taken to give the minimal mean cross-validated error (see main text).
\STATE Let $T_1=\rho\big( \rer_{\lambda} \Y,\X \big)$, the sample Pearson correlation of the $\Y$-residuals with $\X$.
\FOR{$2\leq j \leq w$}
\STATE Let $T_j= \rho\big( \rer_{\lambda} (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y,\X \big)$, where the random matrix $\Pe_j$ encodes random permutation or sign-flipping.
\ENDFOR
\STATE The two-sided \emph{p}-value $p$ equals \eqref{formulap2}.
\RETURN $p$
\end{algorithmic}
\end{algorithm}
\subsection{Double residualization} \label{secdres}
Here we propose a test that we refer to as the \emph{Double Residualization} method. The method is somewhat related to the Kennedy procedure \citep{kennedy1995randomization, kennedy1996randomization,winkler2014permutation}, but not analogous.
The Kennedy method residualizes both $\Y$ and $\X$ and proceeds to permute the $\Y$-residuals. Here we replace the least squares regression by ridge regression. Moreover, unlike Kennedy's permutation scheme, we keep $\pror_{\lambda}\Y$ in the model; see Table \ref{toverviewhd}.
The test statistic that we use within the permutation test is the sample correlation. Thus, the test is based on the statistics
$$T_1= \rho \big ( \Y ,\rer_{\lambda_X}\X \big ),$$
\begin{equation} \label{TjDR}
T_j=\rho\big ( (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y ,\rer_{\lambda_X}\X \big ),
\end{equation}
where $2\leq j \leq w$.
The difference between \eqref{TjDR} and \eqref{eq:flTjpar} is that \eqref{eq:flTjpar} contains an additional $\rer_{\lambda}$.
The pseudo-code for the Double Residualization method is in Algorithm \ref{a:DR}.
We take $\lambda$ and $\lambda_X$ to be the values that give the minimal mean cross-validated error; see Section \ref{secsimset} for more details.
For fixed $q$, as $n\rightarrow\infty$, the Double Residualization method becomes equivalent to the Kennedy method and the Freedman-Lane method if the penalty is $o_{\mathbb{P}}(n^{1/2})$, as shown in the Supplementary Material.
The case that $q>n$ is investigated in Section \ref{secsims}.
\begin{algorithm}[h!]
\caption{Double Residualization}
\begin{algorithmic}[1] \label{a:DR}
\STATE Compute $\pror_{\lambda}= \Z(\Z'\Z+\lambda \bm{I}_q)^{-1}\Z'$ and, analogously, $\pror_{\lambda_X}$. Here $\lambda$ and $\lambda_X$ are determined through cross-validation (see main text).
Let $\rer_{\lambda}=\bm{I}-\pror_{\lambda}$ and $\rer_{\lambda_X}=\bm{I}-\pror_{\lambda_X}$.
\STATE Let $T_1= \rho \big ( \Y ,\rer_{\lambda_X}\X \big )$, the sample Pearson correlation of $\Y$ and $\rer_{\lambda_X}\X$.
\FOR{$2\leq j \leq w$}
\STATE Let $T_j=\rho\big ( (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y ,\rer_{\lambda_X}\X \big )$, where the random matrix $\Pe_j$ encodes random permutation or sign-flipping.
\ENDFOR
\STATE The two-sided \emph{p}-value $p$ equals \eqref{formulap2}.
\RETURN $p$
\end{algorithmic}
\end{algorithm}
\subsection{Multi-dimensional parameter of interest} \label{multidimbeta}
In the above we considered the case that the tested parameter $\beta$ has dimension $d=1$. Our tests can be extended to the case $d>1$ by using Pesarin's Non-Parametric Combination (NPC) approach \citep[][ch. 4]{pesarin2010permutation}. This is a general method for combining permutation tests of different hypotheses into a test for the intersection hypothesis.
The NPC principle can be applied in a wide range of scenarios.
In simpler settings with no nuisance, NPC has important proven properties, such as asymptotically optimal power. Here, we will explain how NPC can be applied in our setting. For convenience, we will focus on the application of NPC to our test of Algorithm \ref{a:FLHD}, i.e., Freedman-Lane HD based on the generalized semi-partial correlation. Combining NPC with our other tests can be done similarly,
but can be computationally much less efficient for large $d$, as will be explained below.
Suppose $d>1$. We are interested in $H_0: \bm{\beta}=\bm{\beta}_0$, where we assume $\bm{\beta}_0=\bm{0}$ again for notational convenience. For every $1\leq l \leq d$, let $\beta_l$ be the $l$-th entry of $\bm{\beta}$. The hypothesis of interest $H_0$ is the intersection of $H^1,...,H^d$, where $H^l$ is the hypothesis that $\beta_l$ equals $0$.
To test $H_0=H^1\cap...\cap H^d$, we proceed as follows.
As usual, sample random matrices $\bm{P}_1,...,\bm{P}_w$ that encode permutation (or sign-flipping).
For every $1\leq l \leq d$ and $1\leq j \leq w$, define
$$T_j^l = \rho\big( \rer_{\lambda} (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y,\X_{\cdot l} \big),$$
where $\X_{\cdot l}$ is the $l$-th column of $\X$.
A key point here is that the same permutation matrix $\Pe_j$ is used to compute each of the statistics $T_j^1,....,T_j^d$.
Due to this manner of simultaneous permutation, the dependence structure of $(T_j^1,....,T_j^d)$ mimics that of $(T_1^1,....,T_1^d)$.
Indeed, if $\bm{\gamma}$ were exactly known so that we could replace $\rer_{\lambda}\Y$ and $\pror_{\lambda}\Y$ by
$\bm{\epsilon}$ and $\Z\bm{\gamma}$, then
$(T_j^1,....,T_j^d)$ and $(T_1^1,....,T_1^d)$ would have exactly the same dependence structure under $H_0$.
Consider a function $\Psi:\mathbb{R}^d\rightarrow \mathbb{R}$, which will be used to compute a combination statistic \citep[][ch. 4]{pesarin2010permutation}. For every $1\leq j \leq w$ define $\Psi_j= \Psi(T_j^1,....,T_j^d)$. Note that if $\rer_{\lambda}\Y$ and $\pror_{\lambda}\Y$ would be the exact errors and expected values, then under $H_0$, $\Psi_1,...,\Psi_w$ would be identically distributed and exchangeable.
The \emph{p}-value for testing $H_0$ is now computed as in \eqref{formulap} but with $T_j$ replaced by the combination statistic $\Psi_j$.
The pseudo-code for this test is in Algorithm \ref{a:FLHDmulti}. Note that if $d=1$ and $\Psi$ is the identity and a two-sided \emph{p}-value is computed, then this method reduces to the test of Algorithm \ref{a:FLHD}.
\begin{algorithm}[h!]
\caption{ Extension of the test of Algorithm \ref{a:FLHD} to the case that $d>1$.}
\begin{algorithmic}[1] \label{a:FLHDmulti}
\STATE Compute $\pror_{\lambda}= \Z(\Z'\Z+\lambda \bm{I}_q)^{-1}\Z'$ and the residual forming matrix $\rer_{\lambda}=\bm{I}-\pror_{\lambda}$. Here $\lambda$ is taken to give the minimal mean cross-validated error.
\FOR{$1\leq l \leq d$}
\STATE Let $T_1^l =\rho\big( \rer_{\lambda} \Y,\X_{\cdot l} \big)$, where $\X_{\cdot l}$ is the $l$-th column of $\X$.
\ENDFOR
\FOR{$2\leq j \leq w$}
\STATE Consider a random $n\times n$ matrix $\Pe_j$ encoding random permutation or sign-flipping.
\STATE Compute $\rer_{\lambda} (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y$.
\FOR{$1\leq l \leq d$}
\STATE Let $T_j^l= \rho\big( \rer_{\lambda} (\Pe_j\rer_{\lambda}+\pror_{\lambda})\Y,\X_{\cdot l} \big)$.
\ENDFOR
\ENDFOR
\FOR{$1\leq j \leq w$}
\STATE Compute $\Psi_j= \Psi(T_j^1,....,T_j^d)$, where $\Psi$ is the combining function.
\ENDFOR
\STATE The \emph{p}-value \emph{p} equals $w^{-1}\big|\{1\leq j \leq w: \Psi_j\geq \Psi_1\}\big|$.
\RETURN $p$
\end{algorithmic}
\end{algorithm}
The function $\Psi$ should be chosen such that high values of $\Psi_1$ indicate evidence against $H_0$. The choice of $\Psi$ influences power. Examples of functions $\Psi$ are
$\Psi(t_1,...,t_d)=\max(|t_1|,...,|t_d|)$ and $\Psi(t_1,...,t_d)=d^{-1}\sum_{l=1}^d |t_l|$. The former choice of $\Psi$ if often used when one or few of the coefficients $\beta_1,...,\beta_d$ are expected to be nonzero under the alternative. Otherwise, the latter choice of $\Psi$ is often used. Other examples of combining functions $\Psi$ are in \citet[][ch. 4]{pesarin2010permutation}.
Applying NPC to the other tests of Sections \ref{secflhd} and \ref{secdres} tends to be computationally less efficient than the method of Algorithm \ref{a:FLHDmulti}.
For example, applying NPC to our Double Residualization method would require ridge-regressing each of the $d$ variables of interest (corresponding to $\beta_1,...,\beta_d$) on the nuisance variables.
\section{Simulations} \label{secsims}
We used simulations to gain additional insight into the performance of the new tests, as well as existing tests. The simulations were performed with \emph{R} version 3.6.0 on a server with 40 cores and 1TB RAM. In Section \ref{secsimgaus} we consider scenarios where the outcome $Y$ follows a standard Gaussian high-dimensional linear model. In Section \ref{secsimrob} we consider non-standard settings with non-normality and heteroscedasticity. We consider simulated datasets where the covariates have equal variances. It is well-known that when the data are not standardized, this can affect the accuracy of the model obtained with ridge regression \citep[][p.257]{buhlmann2014high}.
\subsection{Simulation settings and tests} \label{secsimset}
We considered the model in Section \ref{secnota}, where the variable of interest was one-dimensional, i.e., $\beta\in\mathbb{R}$.
The case $d>1$ is considered in Section \ref{secsimmulti}.
In every simulation, the covariates had mean $0$ and variance $1$.
They were sampled from a multivariate normal distribution with homogenous correlation $\rho'$, unless stated otherwise. The errors $\bm{\epsilon}$ had variance 1, unless stated otherwise. The intercept was $\gamma_1=0$, i.e., $Y$ had mean $0$. The tested hypothesis was $H_0:\beta=0$.
The sample size in the reported simulations was $n=30$, unless stated otherwise. We obtained comparable results for other sample sizes.
The estimated probabilities in the tables are based on $10^4$ repeated simulations, unless stated otherwise.
In the power simulations we usually took $|\beta|$ to be relatively large compared to most of the nuisance coefficients. The reason is that testing in high-dimensional models is very challenging. For example, in settings with $|\beta|=|\gamma_1|=...=|\gamma_q| > 0$ the power of all the tests considered (including the competitors) usually barely exceeds the type I error rate.
The penalty $\lambda$ was chosen to give the minimal mean error, based on 10-fold cross validation. The penalty $\lambda_X$ was chosen analogously.
To compute the penalties, we used the \emph{cv.glmnet()} function in the R package \emph{glmnet}.
We used $[10^{-5} ,10^{5}]$ as the range of candidate values for the penalty. The penalty obtained with \emph{cv.glmnet()} is scaled by a factor $n$, so we multiplied this penalty by $n$ to obtain $\lambda$. We included an intercept in the ridge regressions, but excluding the intercept gave very similar results.
All tests used were two-sided. The tests corresponding to the columns of the tables in this section are the following.
``FLH1" is the Freedman-Lane HD test defined in Section \ref{secflhd}, with test statistics $T_1,...,T_w$ based on the generalized partial correlation as in \eqref{eq:flTjpar}.
``FLH2" is the same, except that $T_1,..,T_w$ are based on the generalized \emph{semi}-partial correlation as in \eqref{eq:flTj}.
``DR" is the Double Residualization method of Section \ref{secdres}. Each of these tests used $w=2 \cdot 10^4$ permutations.
``BM" is a high-dimensional test based on ridge projections, proposed in \citet{buhlmann2013statistical}. This test is based on a bias-corrected estimate $|\hat{\beta}_{\text{corr}}|$ of $|\beta|\in \mathbb{R}$ and an asymptotic upper bound of its distribution.
We used the implementation in the R package \emph{hdi} \citep{dezeure2015high}.
``ZZ" is a high-dimensional test based on Lasso projections, proposed in \citet{zhang2014confidence}. This method constructs a different bias-corrected estimate $\hat{b}$ of $\beta$, which has an asymptotically known normal distribution under certain assumptions, such as sparsity.
For this test we also used the \emph{hdi} package. We could not include this test in the simulations with a very high number of nuisance parameters, since it is computationally very time-consuming when $q$ is large, as also noted in \citet{dezeure2015high}. We expect the test to have good power in these settings.
``BO" is the bootstrap approach in \citet{dezeure2017high}, which is also implemented in the \emph{hdi} package. We set the number of bootstrap samples per test to 1000 and considered the robust version of the method. We used the shortcut, which avoids repeated tuning of the penalty. Still, the method was very slow, so that we used $10^3$ instead of $10^4$ repeated simulations of this method per setting. Also, we did not include the test in the simulations with very large $q$.
\subsection{Gaussian, homoscedastic outcome} \label{secsimgaus}
We first consider some settings with a moderately large number of nuisance coefficients, $q=60$.
We first simulated a setting with $\gamma_2=...=\gamma_{60}=0.05$, i.e, $\bm{\gamma}$ was dense.
We took $\rho'=0.5$. The estimated level and power of the tests described above, for different \emph{p}-value cut-offs $\alpha$, are shown in Table \ref{table:hdaspr05}.
The tests rejected $H_0$ if the \emph{p}-value was smaller than $\alpha$. The level of a test should be at most $\alpha$.
Table \ref{table:hdaspr05} shows that the test ZZ by \citet{zhang2014confidence} was rather anti-conservative. Especially for small $\alpha$, its level was many times larger than $\alpha$. This is partly due to the anti-sparsity. Indeed, ZZ only has proven asymptotic properties under a sparsity assumption.
The bootstrap approach BO of \citet{dezeure2017high} was much less liberal, but still seemed to be somewhat anti-conservative for small $\alpha$.
Of the other tests, Freedman-Lane HD 2 (FLH2) often had the most power. The Double Residualization method had relatively low power when $\alpha$ was small, e.g. $0.001$.
\begin{table}[!h] \normalsize
\begin{center}
\caption{Dense setting with $\rho'=0.5$, $n=30$, $q=60$. Power is shown for $\beta= 1.5$. }
\begin{tabular}{ l l l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{5}{l}{\qquad \qquad \qquad \qquad Method} \\ \cline{3-8}
& $\alpha$ & FLH1 & FLH2 & DR & BM & ZZ & BO \\ \hline
& 0.05 \qquad \quad & .0281 & .0333 & .0219 & .0087 & .0666 & .063 \\
level & 0.01 \qquad \quad & .0042 & .0063 & .0021 & .0024 & .0311 & .023 \\
& 0.001 \qquad \quad & .0003 & .0006 & 0001 & .0005 & .0121 & .009 \\ \hline
& 0.05 \qquad \quad & .9062 & .9273 & .9616 & .8901 & .9934 & .982 \\
power & 0.01 \qquad \quad & .8373 & .8819 & .7984 & .7679& .9799 & .939 \\
& 0.001 \qquad \quad & .6716 & .7996 & .3263 & .5795 & .9441 & .857 \\ \hline
\end{tabular}
\label{table:hdaspr05}
\end{center}
\end{table}
We also considered a setting with very high correlation $\rho'=0.9$, see Table \ref{table:hdspr09}. We took $\gamma_2=\gamma_3=1$ and $\gamma_4=....=\gamma_{60}=0$. The first 4 methods provided appropriate type I error control.
For small cut-offs $\alpha$, the method ZZ by \citet{zhang2014confidence} was relatively powerful, but also seemed to be somewhat anti-conservative.
This method seems more suitable for settings where $q$ is many times larger than $n$.
Among our permutation methods, Freedman-Lane HD 2 had the best power, while incurring few type I errors.
The method BM by \citet{buhlmann2013statistical} was relatively conservative.
We repeated the same simulation scenario, but with $n=15$ instead of $n=30$. The results are in Table \ref{table:hdspr09n15}. The methods ZZ of \citet{zhang2014confidence} and BO of \citet{dezeure2017high} were very anti-conservative for $\alpha=0.01$ and $\alpha=0.001$. Our methods provided appropriate type I error control.
\begin{table}[!h] \normalsize
\begin{center}
\caption{ Sparse setting with $\rho'=0.9$, $n=30$, $q=60$. Power is shown for $\beta= 1.5$. }
\begin{tabular}{ l l l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{5}{l}{\qquad \qquad \qquad \qquad Method} \\ \cline{3-8}
& $\alpha$ & FLH1 & FLH2 & DR & BM & ZZ & BO \\ \hline
& 0.05 \qquad \quad & .0302 & .0270 & .0348 & .0106 & .0358 & .051 \\
level & 0.01 \qquad \quad & .0050 & .0035 & .0044 & .0013 & .0104 & .012 \\
& 0.001 \qquad \quad & .0003 & .0001 & .0001 & .0000 & .0022 & .002 \\ \hline
& 0.05 \qquad \quad & .4494 & .5426 & .4804 & .3234 & .6050 & .554 \\
power & 0.01 \qquad \quad & .2283 & .3379 & .2135 & .1506 & .4154 & .346 \\
& 0.001 \qquad \quad & .0685 & .1195 & .0445 & .0501 & .2296 & .206 \\ \hline
\end{tabular}
\label{table:hdspr09}
\end{center}
\end{table}
\begin{table}[!h] \normalsize
\begin{center}
\caption{Sparse setting with $\rho'=0.9$, $n=15$, $q=60$. Power is shown for $\beta= 3$. }
\begin{tabular}{ l l l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{5}{l}{\qquad \qquad \qquad \qquad Method} \\ \cline{3-8}
& $\alpha$ & FLH1 & FLH2 & DR & BM & ZZ & BO \\ \hline
& 0.05 \qquad \quad & .0268 & .0244 & .0294 & .0030 & .0392 & .050 \\
level & 0.01 \qquad \quad & .0048 & .0030 & .0028 & .0004 & .0124 & .026\\
& 0.001 \qquad \quad & .0008 & .0000 & .0000 & .0002 & .0032 & .020 \\ \hline
& 0.05 \qquad \quad & .5020 & .6034 & .5090 & .4038 & .7586 & .692 \\
power & 0.01 \qquad \quad & .2822 & .4558& .2094 & .2384 & .6248 & .552 \\
& 0.001 \qquad \quad & .0730 & .1982 & .0438 & .1244 & .4614 & .386 \\ \hline
\end{tabular}
\label{table:hdspr09n15}
\end{center}
\end{table}
Further, we considered a simulation where there were clusters of correlated covariates. The setting was as before, except that there were three independent clusters of size 20. Each cluster had a multivariate normal distribution with all correlations equal to $0.9$. We took $\gamma_2=...=\gamma_{60}=0.05$. The results are in Table \ref{table:clusters}. As before, the tests ZZ of \citet{zhang2014confidence} and BO of \citet{dezeure2017high} had good power, but were anti-conservative.
\begin{table}[!h] \normalsize
\begin{center}
\caption{ Dense setting with $n=30$, $q=60$ and three clusters of dependent covariates. Power is shown for $\beta= 1.5$. }
\begin{tabular}{ l l l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{5}{l}{\qquad \qquad \qquad \qquad Method} \\ \cline{3-8}
& $\alpha$ & FLH1 & FLH2 & DR & BM & ZZ & BO \\ \hline
& 0.05 \qquad \quad & .0356& .0224& .0344 & .0130 & .0520 & .073 \\
level & 0.01 \qquad \quad & .0059 & .0025& .0048 & .0022 & .0248 & .023 \\
& 0.001 \qquad \quad & .0010 & .0002& .0002 &.0007 & .0087 & .008 \\ \hline
& 0.05 \qquad \quad & .4892 & .5706 & .5043 & .4188 & 7382 & .620 \\
power & 0.01 \qquad \quad & .2672 & .3393& .2226 & .2399 & .6199 & .454 \\
& 0.001 \qquad \quad & .0814 & .1007 & .0382 & .0977 & .4741 & .322 \\ \hline
\end{tabular}
\label{table:clusters}
\end{center}
\end{table}
We also performed simulations with a very large number of nuisance variables ($q=1000$). We first took $\gamma_2=\gamma_3=1$, $\gamma_4=...=\gamma_{10}=0.2$, $\gamma_{11}=...=\gamma_{1000}=0.$ See Table \ref{table:q1000rho05} for simulations with $\rho'=0.5$ and Table \ref{table:q1000rho09} for simulations with $\rho'=0.9$. All permutation methods provided appropriate type I error control.
Double Residualization (DR) had relatively high power for large cut-offs $\alpha$, but not for small cut-offs. The method BM by \citet{buhlmann2013statistical} had relatively good power for $\rho'=0.5$ but low power for $\rho'=0.9$.
We also performed simulations where $\gamma$ was very anti-sparse, e.g. with $\gamma_2=1$, $\gamma_3=...=\gamma_{800}=0.002$ and $\rho'=0.9$.
We also considered negative coefficients and we varied the magnitude of the coefficients and the errors $\bm{\epsilon}$ and the sample size. We also considered more settings where there were multiple independent clusters of correlated covariates.
Also in these settings, the type I error rate was controlled.
\begin{table}[!h] \normalsize
\begin{center}
\caption{ Sparse setting with a large number ($q=1000$) of nuisance variables. Here $\rho'=0.5$, $n=30$. Power is shown for $\beta= 2$. }
\begin{tabular}{ l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{4}{l}{\qquad \qquad \quad Method} \\ \cline{3-6}
& $\alpha$ & FLH1 & FLH2 & DR & BM \\ \hline
& 0.05 \qquad \quad & .0068 & .0065 & .0145 & .0001 \\
level & 0.01 \qquad \quad & .0013 & .0011 & .0011 & .0000 \\
& 0.001 \qquad \quad & .0002 & .0001 & .0000 & .0000 \\ \hline
& 0.05 \qquad \quad & .5577 & .5469 & .9613 & .7820 \\
power & 0.01 \qquad \quad & .5060 & .5043 & .8007 & .6510 \\
& 0.001 \qquad \quad & .3752 & .4049 & .3463 & .4851 \\ \hline
\end{tabular}
\label{table:q1000rho05}
\end{center}
\end{table}
\begin{table}[!h] \normalsize
\begin{center}
\caption{Sparse setting with a large number ($q=1000$) of nuisance variables and high correlation $\rho'=0.9$. Power is shown for $\beta= 2$. }
\begin{tabular}{ l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{4}{l}{\qquad \qquad \quad Method} \\ \cline{3-6}
& $\alpha$ & FLH1 & FLH2 & DR & BM \\ \hline
& 0.05 \qquad \quad & .0236 & .0319 & .0358 & .0006 \\
level & 0.01 \qquad \quad & .0040 & .0074 & .0057 & .0000 \\
& 0.001 \qquad \quad & .0003 & .0006 & .0001 & .0000 \\ \hline
& 0.05 \qquad \quad & .4766 & .5317 & .7127 & .2115 \\
power & 0.01 \qquad \quad & .3106 & .4254 & .4137 & .1042 \\
& 0.001 \qquad \quad & .1303 & .2500 & .1344 & .0407 \\ \hline
\end{tabular}
\label{table:q1000rho09}
\end{center}
\end{table}
\subsection{Violations of the Gaussian model} \label{secsimrob}
Permutation tests can be robust to violations of the standard linear model, such as non-normality and heteroscedasticity. \citep{winkler2014permutation,hemerik2020robust}
The power of parametric methods is often substantially decreased when the residuals have heavy tails. The power of the permutation tests is more robust to such deviations from normality. This is illustrated in Table \ref{table:exp3}. Here, the data distribution was the same as in the setting corresponding to Table \ref{table:hdspr09}, except that the errors $\bm{\epsilon}$ were not standard normally distributed, but had very heavy (cubed exponential) tails, scaled such that the errors had standard deviation 1. Note in Table \ref{table:exp3} that the permutation and bootstrap methods still had roughly the same power as at Table \ref{table:hdspr09}, while the power of BM and ZZ was strongly reduced compared to Table \ref{table:hdspr09}.
\begin{table}[!h] \normalsize
\begin{center}
\caption{Same sparse setting as at Table \ref{table:hdspr09} but with very heavy-tailed errors. }
\begin{tabular}{ l l l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{5}{l}{\qquad \qquad \qquad \qquad Method} \\ \cline{3-8}
& $\alpha$ & FLH1 & FLH2 & DR & BM & ZZ & BO \\ \hline
& 0.05 \qquad \quad & .0345 & .0313 & .0336 & .0034 & .0215 & .022 \\
level & 0.01 \qquad \quad & .0059 & .0051 & .0053 & .0001 & .0043 & .004 \\
& 0.001 \qquad \quad & .0005 & .0002 & .0002 & .0000 & .0006 & .002 \\ \hline
& 0.05 \qquad \quad & .4498 & .5493 & .4593 & .2173 & .5433 & .566 \\
power & 0.01 \qquad \quad & .2295 & .3353 & .2016 & .0730 & .3173 & .390 \\
& 0.001 \qquad \quad & .0780 & .1309 & .0492 & .0151 & .1374 & .215 \\ \hline
\end{tabular}
\label{table:exp3}
\end{center}
\end{table}
As a second type of violation of the standard linear model, we considered heteroscedasticity. We simulated errors $\epsilon_i$ which were normally distributed, but with standard deviation proportional to the absolute value covariate of interest, $|X_i|$. We again took $\gamma_2=\gamma_3=1$, $\gamma_4=...=\gamma_{60}=0$. We took $\rho'=0$ for illustration, since in that case the method ZZ by \citet{zhang2014confidence} turned out to be very anti-conservative under heteroscedasticity. Otherwise, the simulated data were again as those used for Table \ref{table:hdspr09}.
The results are in Table \ref{table:hete}.
Note that despite the heteroscedasticity, the permutation-based tests provided appropriate type I error control.
The bootstrap approach BO of \citet{dezeure2017high} seemed to be anti-conservative for small $\alpha$.
The test BM from \citet{buhlmann2013statistical} had higher power than the permutation methods in this specific setting, but was anti-conservative for small $\alpha$.
In the simulations underlying Table \ref{table:hete}, we did not use sign-flipping, which is known to be robust to heteroscedasticity \citep{winkler2014permutation,hemerik2020robust}. Surprisingly, our tests nevertheless provided appropriate type I control.
We also performed these simulations with sign-flipping instead of permutation (results not shown), which further reduced the level of our tests, but also somewhat reduced the power.
\begin{table}[!h] \normalsize
\begin{center}
\caption{Sparse setting with heteroscedastic errors, $\rho'=0$, $n=30$, $q=60$. Power is shown for $\beta= 1.5$. }
\begin{tabular}{ l l l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{5}{l}{\qquad \qquad \qquad \qquad Method} \\ \cline{3-8}
& $\alpha$ & FLH1 & FLH2 & DR & BM & ZZ & BO \\ \hline
& 0.05 \qquad \quad & .0352 & .0354 & .0271 & .0338 & .1490 & .077 \\
level & 0.01 \qquad \quad & .0065 & .0069 & .0050 & .0109 & .0648 & .028 \\
& 0.001 \qquad \quad & .0010 & .0009 & .0008 & .0029 & .0280 & .011 \\ \hline
& 0.05 \qquad \quad & .7901 & .8060 & .7855 & .9403 & .9902 & .982 \\
power & 0.01 \qquad \quad & .6787 & .6861 & .6454 & .8534 & .9741 & .936 \\
& 0.001 \qquad \quad & .4910 & .4909 & .4498 & .6903 & .9332 & .830 \\ \hline
\end{tabular}
\label{table:hete}
\end{center}
\end{table}
\subsection{Multi-dimensional parameter of interest} \label{secsimmulti}
We simulated the test of Section \ref{multidimbeta} for multi-dimensional $\bm{\beta}$. As the combination statistic we used $\Psi(t_1,...,t_d)=\max(t_1,...,t_d)$. The parameter of interest $\bm{\beta}$ had dimension 10 and there were 490 nuisance variables, i.e., $\dim(\bm{\gamma})=491$, since
$\gamma_1$ is the intercept. The outcome $Y$ followed a Gaussian model, as in Section \ref{secsimgaus}. We considered three simulation settings.
The nuisance parameters were $\gamma_2=3,\gamma_3=2,\gamma_4=1,\gamma_5=...=\gamma_{491}=0$ in the first two settings and $\gamma_2=....=\gamma_{101}=0.03,\gamma_{102}=...=\gamma_{491}=0$ in the third setting. The covariates had a multinormal distribution with homogeneous correlation $\rho'=0.5$ in the first setting and $\rho'=0.9$ in the last two settings.
The results are in Table \ref{table:simmulti}. The test provided appropriate type I error control.
We conclude from the simulations of Section \ref{secsims} that our tests provide good type I error control and are rather robust to several types of model misspecification. The method ZZ from \citet{zhang2014confidence} was often relatively powerful, but was quite anti-conservative in several scenarios.
The bootstrap approach BO of \citet{dezeure2017high} was also anti-conservative in several scenarios, but less so.
The method BM from \citet{buhlmann2013statistical} tended to be relatively conservative.
\begin{table}[!h] \normalsize
\begin{center}
\caption{Multi-dimensional $\bm{\beta}\in\mathbb{R}^{10}$. Power is shown for $\bm{\beta}=(3,2,1,0,...,0)$. }
\begin{tabular}{ l l l l l l }
\hline \\[-0.4cm]
& & \multicolumn{3}{l}{\qquad \quad Simulation setting} \\ \cline{3-5}
& $\alpha$ & Setting 1& Setting 2 & Setting 3 \\ \hline
& 0.05 \qquad \quad & .0174 & .0197 & .0330 \\
level & 0.01 \qquad \quad & .0023 & .0024 & .0055 \\
& 0.001 \qquad \quad & .0004 & .0002 & .0002 \\ \hline
& 0.05 \qquad \quad & .4443 & .5098 & .6286 \\
power & 0.01 \qquad \quad & .3740 & .4552 & .5731 \\
& 0.001 \qquad \quad & .2503 & .3788 & .4736 \\ \hline
\end{tabular}
\label{table:simmulti}
\end{center}
\end{table}
\section{Data analysis} \label{secdata}
We analyze a dataset about riboflavin (vitamin B2) production with \emph{B. subtilis}. This dataset is called \emph{riboflavin} and is publicly available \citep{buhlmann2014high}. It contains normalized measurements of expression rates of 4088 genes from $n=71$ samples. We use these as input variables. Further, for each sample the dataset contains the logarithm of the riboflavin production rate, which is our one-dimensional outcome of interest. We (further) standardized the expression levels by subtracting the means and dividing by the standard deviations. We also shifted the outcome values to have mean zero.
For every $1\leq i \leq 4088$, we tested the hypothesis $H_i$ that the outcome was independent of the expression level of gene $i$, conditional on the other expression levels. We used the same tests as considered in the simulations. This time we used $w=2\cdot10^5$ permutations per test.
The results of the analysis are summarized in Table \ref{table:dataan}. The columns correspond to the same methods as considered in Section \ref{secsims}. For every method, the fraction of rejections is shown for different \emph{p}-value cut-offs $\alpha$. The fraction of rejections is the number of rejected hypotheses divided by 4088, the total number of hypotheses. The hypotheses that were rejected, were those with \emph{p}-values smaller than or equal to the cut-off $\alpha$.
With most methods we obtain many \emph{p}-values smaller than 0.05. This is not the case for the test BM by \citet{buhlmann2013statistical}, which is known to be relatively conservative.
After Bonferroni's multiple testing correction, we reject no hypotheses with any method, suggesting there is no strong signal in the data. \citet{van2014asymptotically} also obtained such a result with this dataset.
\begin{table}[!ht] \normalsize
\caption{Real data analysis. For different \emph{p}-value cut-offs $\alpha$, the fraction of rejected hypotheses is shown. }
\begin{center}
\begin{tabular}{ l l l l l l l }
\hline \\[-0.4cm]
& \multicolumn{5}{l}{\qquad Fraction of rejected hypotheses} \\ \cline{2-7}
$\alpha$ & FLH1 & FLH2 & DR & BM & ZZ & BO \\ \hline
0.05 \qquad \quad & .0005 & .0259 & .0428 & 0 & .0135 & .0272 \\
0.01 \qquad \quad & 0 & .0071 & .0066 & 0 & .0022 & .0051 \\
0.001 \qquad \quad & 0 & .0002 & .0012 & 0 & .0007 & .0024 \\
0.0001 \qquad \quad & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
\end{tabular}
\label{table:dataan}
\end{center}
\end{table}
\section{Discussion}
We have proposed novel permutation methods for testing in linear models, where the number of nuisance variables may be much larger than the sample size.
Advantages of permutation approaches include robustness to certain violations of the standard linear model and compatibility with powerful permutation-based multiple testing methods.
We have proposed two novel permutation approaches, Freedman-Lane HD and Double Residualization. Within these approaches some variations are possible, with respect to how the regularization parameters are chosen and which test statistics are used.
Our methods provided excellent type I error rate control in a wide range of simulation settings.
In particular we considered settings with anti-sparsity, high correlations among the covariates, clustered covariates, fat-tailedness of the outcome variable and heteroscedasticity.
The simulation study was limited to settings with multivariate normal covariates. Future research may address more scenarios.
We compared our methods to the parametric tests in \citet{buhlmann2013statistical} and \citet{zhang2014confidence} and to the bootstrap approach in \citet{dezeure2017high}.
One advantage of our methods compared to those in \citet{buhlmann2013statistical} and \citet{zhang2014confidence}, is that they are defined in the case that the parameter of interest is multi-dimensional.
Further, our tests tended to have higher power than the method by \citet{buhlmann2013statistical}.
The test by \citet{zhang2014confidence} had relatively good power, but was rather anti-conservative in several scenarios, for example under anti-sparsity and heteroscedasticity.
The bootstrap approach of \citet{dezeure2017high} was also anti-conservative in some scenarios, but less so.
Our permutation tests provided appropriate type I error control in all scenarios. Moreover, our permutation tests were computationally much faster than the bootstrap method.
\setlength{\bibsep}{3pt plus 0.3ex}
\def\bibfont{\small}
\bibliographystyle{biblstyle}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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10 Imprescindibles - Campo Grande - 10 Imprescindibles - Campo Grande
Discover Valladolid / Top ten / Detalle 10 Imprescindibles
10 Imprescindibles - Campo Grande
The Campo Grande Park
Right in the city centre lies its most emblematic park in terms of both history and its beauty as a garden. Three of the city's most important avenues border the Campo Grande Park: Acera de Recoletos, Paseo de los Filipinos and Paseo de Zorrilla. The park is surrounded by a fence and has entrance ways on all its sides. Its origin dates back to 1787 when it was first named the Field of Truth, then the Field of Mars and later the Field of the Fair, as herds of animals were kept here next to the entrance to the city. The park occupies a surface area of 115,000 m2 and combines in a unique way its plant life with artistic, aquatic, children's playground and environmentally friendly features.
Maybe you're interested too...
The Cavalry Academy
National Sculpture Museum
The Church of Saint Pau
Church of Saint Mary 'La Antigua'
Old Bullring Square
Church and Monastery of San Benito "El Real"
Museo Patio Herreriano
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 621
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The lower mantle, historically also known as the mesosphere, represents approximately 56% of Earth's total volume, and is the region from 660 to 2900 km below Earth's surface; between the transition zone and the outer core. The preliminary reference Earth model (PREM) separates the lower mantle into three sections, the uppermost (660–770 km), mid-lower mantle (770–2700 km), and the D layer (2700–2900 km). Pressure and temperature in the lower mantle range from 24–127 GPa and 1900–2600 K. It has been proposed that the composition of the lower mantle is pyrolitic, containing three major phases of bridgmanite, ferropericlase, and calcium-silicate perovskite. The high pressure in the lower mantle has been shown to induce a spin transition of iron-bearing bridgmanite and ferropericlase, which may affect both mantle plume dynamics and lower mantle chemistry.
The upper boundary is defined by the sharp increase in seismic wave velocities and density at a depth of . At a depth of 660 km, ringwoodite () decomposes into Mg-Si perovskite and magnesiowüstite. This reaction marks the boundary between the upper mantle and lower mantle. This measurement is estimated from seismic data and high-pressure laboratory experiments. The base of the mesosphere includes the D″ zone which lies just above the mantle–core boundary at approximately . The base of the lower mantle is about 2700 km.
Physical properties
The lower mantle was initially labelled as the D-layer in Bullen's spherically symmetric model of the Earth. The PREM seismic model of the Earth's interior separated the D-layer into three distinctive layers defined by the discontinuity in seismic wave velocities:
660–770 km: A discontinuity in compression wave velocity (6–11%) followed by a steep gradient is indicative of the transformation of the mineral ringwoodite to bridgmanite and ferropericlase and the transition between the transition zone layer to the lower mantle.
770–2700 km: A gradual increase in velocity indicative of the adiabatic compression of the mineral phases in the lower mantle.
2700–2900 km: The D-layer is considered the transition from the lower mantle to the outer core.
The temperature of the lower mantle ranges from at the topmost layer to at a depth of . Models of the temperature of the lower mantle approximate convection as the primary heat transport contribution, while conduction and radiative heat transfer are considered negligible. As a result, the lower mantle's temperature gradient as a function of depth is approximately adiabatic. Calculation of the geothermal gradient observed a decrease from at the uppermost lower mantle to at .
Composition
The lower mantle is mainly composed of three components, bridgmanite, ferropericlase, and calcium-silicate perovskite (CaSiO3-perovskite). The proportion of each component has been a subject of discussion historically where the bulk composition is suggested to be,
Pyrolitic: derived from petrological composition trends from upper mantle peridotite suggesting homogeneity between the upper and lower mantle with a Mg/Si ratio of 1.27. This model implies that the lower mantle is composed of 75% bridgmanite, 17% ferropericlase, and 8% CaSiO3-perovskite by volume.
Chondritic: suggests that the Earth's lower mantle was accreted from the composition of chondritic meteorite suggesting a Mg/Si ratio of approximately 1. This infers that bridgmanite and CaSiO3-perovskites are major components.
Laboratory multi-anvil compression experiments of pyrolite simulated conditions of the adiabatic geotherm and measured the density using in situ X-ray diffraction. It was shown that the density profile along the geotherm is in agreement with the PREM model. The first principle calculation of the density and velocity profile across the lower mantle geotherm of varying bridgmanite and ferropericlase proportion observed a match to the PREM model at an 8:2 proportion. This proportion is consistent with the pyrolitic bulk composition at the lower mantle. Furthermore, shear wave velocity calculations of pyrolitic lower mantle compositions considering minor elements resulted in a match with the PREM shear velocity profile within 1%. On the other hand, Brillouin spectroscopic studies at relevant pressures and temperatures revealed that a lower mantle composed of greater than 93% bridgmanite phase has corresponding shear-wave velocities to measured seismic velocities. The suggested composition is consistent with a chondritic lower mantle. Thus, the bulk composition of the lower mantle is currently a subject of discussion.
Spin transition zone
The electronic environment of two iron-bearing minerals in the lower mantle (bridgmanite, ferropericlase) transitions from a high-spin (HS) to a low-spin (LS) state. Fe2+ in ferropericlase undergoes the transition between 50–90 GPa. Bridgmanite contains both Fe3+ and Fe2+ in the structure, the Fe2+ occupy the A-site and transition to a LS state at 120 GPa. While Fe3+ occupies both A- and B-sites, the B-site Fe3+ undergoes HS to LS transition at 30–70 GPa while the A-site Fe3+ exchanges with the B-site Al3+ cation and becomes LS. This spin transition of the iron cation results in the increase in partition coefficient between ferropericlase and bridgmanite to 10–14 depleting bridgmanite and enriching ferropericlase of Fe2+. The HS to LS transition are reported to affect the physical properties of the iron bearing minerals. For example, the density and incompressibility was reported to increase from HS to LS state in ferropericlase. The effects of the spin transition on the transport properties and rheology of the lower mantle is currently being investigated and discussed using numerical simulations.
History
Mesosphere (not to be confused with mesosphere, a layer of the atmosphere) is derived from "mesospheric shell", coined by Reginald Aldworth Daly, a Harvard University geology professor. In the pre-plate tectonics era, Daly (1940) inferred that the outer Earth consisted of three spherical layers: lithosphere (including the crust), asthenosphere, and mesospheric shell. Daly's hypothetical depths to the lithosphere-asthenosphere boundary ranged from , and the top of the mesospheric shell (base of the asthenosphere) were from . Thus, Daly's asthenosphere was inferred to be thick. According to Daly, the base of the solid Earth mesosphere could extend to the base of the mantle (and, thus, to the top of the core).
A derivative term, mesoplates, was introduced as a heuristic, based on a combination of "mesosphere" and "plate", for postulated reference frames in which mantle hotspots exist.
See also
Large low-shear-velocity provinces
References
Earth's mantle
Structure of the Earth
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,039
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Meet our coworkers
We are rolling-stock professionals with a passion for public transportation solutions and extensive know-how and network in the rolling stock industry. We have chosen this business sector as our playground and we love to work in it. It's a privilege and a passion.
What today is Unstad started as a development project to establish PROSE on the ground in the Norwegian rolling stock market (ie trains, trams, subways, etc.) and an office was started in the fall of 2017. Two years later, the PROSE Group redefined its need to be present and consolidates its resources in its traditional markets. During the same period, we who work here had established a clear presence in the Norwegian industry and decided in December 2019 to buy out the Norwegian part of the company to stand on its own as an independent company.
Unstad started as a breakout from PROSE in December 2019 and we are rolling-stock professionals with a passion for public transport in general and the rolling stock industry in particular. This is a way of life and our livelihood. It is a privilege and a passion. We are like the small community Unstad in Lofoten where the road ends in a wild and beautiful bay and you will find an epicenter for surfers from all over the world with a passion for the perfect wave. The recognition that people with commitment, passion, experience, solid ethical foundations and can-do-attitude will achieve both their own and others' dreams. This is why we chose Unstad as our name. To remind us that it makes personal sense to challenge and strive for our own good - for our customers and for society. Unstad symbolizes for us ingenuity, passion and freedom to deliver the best quality, despite the fact that Unstad is a small place in the world.
Om oss: About
We imagine a future in which most people prefer public transportation solutions.
In 2030, Unstad is the leading professional services company in the Nordic Rolling-Stock sector and at the same time a recognized contributor to an inclusive, environmentally sound and sustainable society.
Our customers trust that we are the most agile, cost-efficient and sustainable partner in the truest sense of these words.
We have arrived at this position by being creative and professional problem solvers with an undisputed focus on our customers' ever-changing business needs as well as the environment we live in.
Our team comprises experts with long-term experience in their fields as well as promising junior professionals we want to be inspired by and seek to boost their skills through a continuous development program. This way we enable young people to develop and define their own professional path.
We are proud to work for Unstad and feel that we are constantly contributing to the success of our customers and to a better society but also to our own personal development. Unstad is regarded by engineering graduates as the best place to build a career in the rolling-stock sector.
Our owners respect us for the ability to focus on a healthy business operation and for our professionalism, both in the day-to-day operations and in the longer term.
The general business society will view Unstad as leader in the professional services industry and will try to replicate our successes to the benefit of other markets.
Unstad is a great place to work!
Our mission is to make public transportation solutions better for the greater good of society.
Our main goal is to provide the best solution for our customers. We have the skills and experience to do this. Our expertise in train technology and our analysis and understanding of each customer's business and needs enable us to perform and deliver the best customized solution. Our proximity to the market allows us to coordinate and execute project tasks effectively.
Integrity in everything we do and what we are. We seek to deliver what we consider to be the best solution for our clients in an objective, in-depth and independent way. We never compromise our values, but act as an ambassador for them.
We respect laws, regulations and our contracts. We maintain our integrity by acting in accordance with them.
For the greater good
We work for the greater good by our contribution to improving public and goods transportation systems. Even though Unstad is a commercially driven company, we support non-profit organizations helping people in need, contributing to a better environment and biodiversity as a company and as individuals. We embrace the UN's Sustainable Development Goals.
We respect everyone as equals, and believe in equality among genders, sexuality and religions. We do not shy away from difficult discussions or decisions but enter them with respect for all parties involved and with the project's best interests in mind.
Welcome to Unstad!
Use this form and your inquiry will be sent to the right person.
To get in touch with our staff directly, use firstname.lastname@unstad.com.
Thank you! We will get back to you as soon as possible.
Unstad AS
Dyvekes vei 2, 0192, Oslo, Norway
Erling Haugnes, General Manager / CEO
T: +47 932 69 597
© 2020 by Unstad AS
Design by www.sisselhovden.com
|
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| 8,231
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Theodorus Marinus Roest van Limburg (Rotterdam, 29 april 1865 – De Bilt, 1 april 1935) was een Nederlands politiefunctionaris.
Militaire carrière
Na enkele jaren gymnasium, ging hij op 14-jarige leeftijd naar een kostschool in Voorburg om opgeleid te worden tot cadet. Hierna studeerde hij van 1881 tot 1885 aan de Koninklijke Militaire Academie (KMA) in Breda in de richting artillerie hier te lande. Vervolgens werd hij aangesteld als tweede luitenant bij de vestingartillerie in Den Helder. In mei 1888 volgde zijn promotie tot eerste luitenant. Een jaar later wordt hij in Leiden ingedeeld bij de veldartillerie en in 1890 werd hij overgeplaatst naar Den Haag waar hij nog steeds ingedeeld blijft bij de veldartillerie. In december 1895 volgde zijn benoeming tot adjudant van de Gouverneur van de KMA zodat hij weer terugkwam in Breda. In 1900 werd hij gepromoveerd tot kapitein en in september 1901 volgde overplaatsing naar het korps Pontonniers te Dordrecht. In 1905 werd hij adjudant van generaal F.G.A. van Ermel Scherer, toenmalig Inspecteur der Artillerie in Den Haag.
Hoofdcommissaris
Op verzoek van de Rotterdamse burgemeester A.R. Zimmerman, die Roest van Limburg kende uit de periode dat ze beiden in Dordrecht werkten (Zimmerman was burgemeester van Dordrecht van 1899 tot 1906), nam hij in 1908 ontslag bij het leger en werd hij hoofdcommissaris van politie in zijn geboorteplaats. Hierbij volgde hij W. Voormolen op die begin 1908 vanwege gezondheidsproblemen ontslag moest nemen en het jaar erop op 53-jarige leeftijd overleed. Voormolen had vanaf 1893 het korps sterk gemoderniseerd en had een aanbod afgeslagen van het bestuur van Amsterdam om daar als hoofdcommissaris hetzelfde te doen.
Onder Roest van Limburg kreeg de Rotterdamse zedenpolitie extra de aandacht. Zo werd in mei 1911 Dina Sanson als eerste vrouwelijke Nederlandse politiebeambte bij het Rotterdamse korps geïnstalleerd waar ze ging werken op de afdeling kinderwetten van de zedenpolitie. Een jaar eerder had hij al een boekje geschreven onder de titel "In den strijd tegen de ontucht" naar aanleiding van een nieuwe Rotterdamse gemeenteverordening waarmee het houden en bezoeken van 'huizen van ontucht' strafbaar werd. In het boekje gaf hij aan dat hij verwachtte dat die verordening onvoldoende was in de strijd tegen de prostitutie en gaf hij aan te verwachten dat die verordening zou leiden tot clandestiene ontucht met als gevolg onder andere gevaren van hygiënische aard. Om een "politie-assistente" aan te nemen, schreef Roest van Limburg diverse vrouwenverenigingen aan zoals de Nederlandsche Vereeniging tot Behartiging van de Belangen van Jonge Meisjes, het Nationaal Comité tegen Handel in Vrouwen en het Nationaal Bureau voor Vrouwenarbeid. Sanson werd voorgedragen door de Vereeniging van Onderlinge Vrouwenbescherming.
Eind 1913 werd Roest van Limburg benaderd om de Amsterdamse hoofdcommissaris H.S. Hordijk op te volgen die vanwege gezondheidsproblemen ontslag had aangevraagd. In tegenstelling tot Voormolen had hij hier wel oren naar en Roest van Limburg wist een veel hoger salaris te bedingen dan Hordijk. Toen dit in Rotterdam bekend werd, hebben ze nog geprobeerd hem te behouden door hetzelfde salaris aan te bieden als wat hij in Amsterdam zou gaan verdienen, maar zonder succes. Op 1 januari 1914 werd Roest van Limburg hoofdcommissaris van politie te Amsterdam. In de hoofdstad pleitte Roest van Limburg voor een afdeling Kinderpolitie met een vrouwelijke beambtes. In 1920 opende het bureau Kinderpolitie met een mannelijke inspecteur aan het hoofd en met C.M van Ooy, de eerste vrouwelijke inspecteur van de Amsterdamse politie.
In 1917 kreeg Amsterdam te maken met het Aardappeloproer waarbij 9 doden en 114 gewonden vielen. Vanwege gezondheidsproblemen nam hij in 1919 ontslag waarna hij ging wonen in de villa 'Rosenegg' in De Bilt. In die plaats overleed hij op 69-jarige leeftijd.
Bibliografie
Feestrede uitgesproken op den 21sten October 1903 in de Groote Kerk te Breda, ter gelegenheid van de viering van het 75-jarig bestaan der Koninklijke Militaire Academie, Naamloze Vennootschap Bredasche Boekhandel en Uitgeversmaatschappij voorheen Broese & Co., 1903
Het kasteel van Breda: aanteekeningen betreffende het voormalig Prinsenhof te Breda, Roelants, 1904
Voormalige Nassausche paleizen in België, 1907
Een Spaansche gravin van Nassau: Mencia de Mendoza, markiezin van Zenete, gravin van Nassau (1508-1554), Sijthoff, 1908
In den strijd tegen de ontucht: enkele opmerkingen en mededeelingen betrekkelijk het prostitutie-vraagstuk, W.J. van Hengel, 1910
Ons volkskarakter: een studie in volkspsychologie, Van Looy, 1917
Externe links
Levensbericht Th. M. Roest van Limburg (Jaarboek van de Maatschappij der Nederlandse Letterkunde, 1935)
Inventaris van het archief van Th. M. Roest van Limburg (Nationaal Archief)
Roest van Limburg, Theodorus Marinus
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,377
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Tail of the Dragon, Deal's Gap
US Route 129, Bryson City, North Carolina 28713 USA
More in Bryson City
"twisting and turning driving road"
Deals Gap is a popular and internationally famous destination for motorcycle and sports car enthusiasts, as it is located along a stretch of two-lane road known since 1981 as "The Dragon". The 11-mile stretch of the Dragon in Tennessee is said to have 318 curves. Some of the Dragon's sharpest curves have names like Copperhead Corner, Hog Pen Bend, Wheelie Hell, Shade Tree Corner, Mud Corner, Sunset Corner, Gravity Cavity, Beginner's End, and Brake or Bust Bend. The road earned its name from its curves being said to resemble a dragon's tail. The stretch bears the street name "Tapoco Road" in North Carolina and "Calderwood Highway" in Tennessee and is signed entirely by U.S. 129. State Route 115 is included on maps, and is the name used by Tennessee Department of Transportation for highway contracts. Since part of the road is also the southwestern border of the Great Smoky Mountains National Park, there is no development along the 11-mile stretch, resulting in no danger of vehicles pulling out in front of those in the right of way. It mostly travels through forested area and there are a few scenic overlooks and pull-off points along the route. The speed limit on the Dragon was 55 mph prior to 1992; it was reduced to 30 mph in 2005. The presence of law enforcement on the Tennessee portion has dramatically increased in 2007, 2008 and 2009.
Tail of the Dragon is great! But this address is totally wrong for it. Please google before making the trek. It's in Robbinsville.
wdrive72
Road most of this on our motorcycles, it was so beautiful!! A great experience until we came upon a horrific accident involving a trike and a pickup truck pulling a boat. The people riding the trike were killed. All I can say is, be careful. As much of a thrill that it is to ride those curves, take your time, lives are precious.
elaine.nall.923
Drove this and loved it! Traffic stopped both ways for mama bear and two little cubs. Saw an 18 wheeler (obviously lost) jackknife his rig trying to make a curve on it.
Be the first to add a review to the Tail of the Dragon, Deal's Gap.
US Route 129
Bryson City, North Carolina
Robbinsville, North Carolina
Blue Waters Mountain Lodge
Whittier, North Carolina
Illuminated Heart Sanctuary
Franklin, North Carolina
Brendle Cabin
Gear Head Inn, In The Great Smoky Mountains
Lakeview At Fontana Boutique Resort & Spa
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,620
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The Ant's Meow
A call for compassion toward animals
Promoting animal rights, plant-based diets and pets looking for forever homes.
Opt to Adopt
Navigation Blog Opt to Adopt Causes Vegan Living About
Image by Giulia Forsythe via Flickr
5 Ways to Get More Engagement for Animal Causes on Social Media
November 15, 2015 by The Ant's Meow in Animal Rights Activism
I often feel apologetic for posting graphic images and depressing stories about the plight of animals. As someone who follows a variety of animal news sources and animal rights organizations on Facebook, I know how overwhelming (and depressing) the stories can be as they flood my newsfeed.
But then I think about the animals. Sure, the graphic images and headlines are rather disturbing to see, but that's just a momentary inconvenience for us. What about the animals that endure the pain and suffering, 24/7—those who can't just simply look away and move onto the next, happier story?
So what's an animal lover to do when your social media audience has grown weary of animal-related postings and petitions?
Here are 5 ways to help you get more engagement for your animal causes
Keep it short and simple. Summarize the article or issue by either quoting a compelling line within the article, or summarizing it in your own words. We know that people have short attention spans, so highlight the most important point when posting or sharing a story. Assume they won't read the whole story and let them know if there's a call-to-action at the end.
Diversify your audience. To help prevent your audience from tuning you out, post on different social media platforms on different days. This way, your Facebook friends will only see that you've posted once every few days, your Twitter followers once every few days, etc. You may be posting every day, but your social media followers won't know that.
Sprinkle in happy stories. No one likes a "Debbie Downer". Make sure to balance sad stories with heartwarming, uplifting or even fun and irreverent stories (think Buzzfeed). By diversifying the stories you share, your content will look fresh and will less likely start sounding the same.
Blog about it! You don't need to have an animal-focused blog to write a post about a worthy animal cause. You've worked hard to build an audience, now share with them a cause that is close to your heart.
Be creative. Present things differently, like a foodie pic turned into a call-to-action. You could post a picture you took of a yummy dessert like crème brule, and then add a line like: "This crème brule is calling my name, but all I hear is the pain and suffering of chickens. Sign this petition to ban cruel battery-cage eggs."
These are things I try to be conscious of and implement. I know it's hard if you're an animal lover and all you want to do is shout to the world about all the cruelty and injustices animals face. It's a struggle, for sure. But hopefully with these tips, we can move people toward compassion for all living beings ... bite-sized posts at a time.
November 15, 2015 /The Ant's Meow
social media engagement, animal causes, animal rights activist, animal rights activism, activism, animal welfare, animal rights, animal petitions
My First Visit to a Farm Animal Sanctuary
October 31, 2015 by The Ant's Meow in Animal Sanctuary
In my effort to move past the pen and paper (or computer) and actively get involved with animal rescue organizations, I set out to find a farm animal sanctuary near Los Angeles. To my surprise, I found several just north of L.A. in and around Santa Clarita. While I knew I'd get an amazing experience out of any of them, I chose The Gentle Barn—particularly after seeing their video of Karma the cow and her miraculous reunion with her calf whom they both rescued. I'll get to that in a bit.
Arriving at the Farm Animal Sanctuary
After a tolerable 45-minute drive, my boyfriend (who has now been vegetarian since June after reading this article) and I pulled up to the crowded parking area bordering the fence of The Gentle Barn. Normally a full parking lot irks me, but in this case, at a $10 donation per person, I was thrilled to see a full parking lot. More cars meant more people and more money toward an incredible cause.
I could tell from the moment I walked up that the sanctuary operates like a well-oiled machine—having been around for 16 years. I fell in line to sign a waiver (basically agreeing not to hold the sanctuary liable for any injuries and for possibly having a picture/video taken of me for their website, etc.). I graciously paid the $10 donation, got my hand stamped, and moved to the waiting area until our tour group was next. To ensure that the smaller animals don't feel overwhelmed or overstressed by large crowds, they limit the size of the groups to about 25 people. Each group is assigned a time to enter the upper barnyard that houses the smaller animals.
A Presentation and Karma's Story
The tour kicked off with about a 15-minute presentation from the founder, Ellie Laks, about how The Gentle Barn came to be, its mission and its vision. She shared the story of Karma the cow whom they rescued from a small farm. For the first day and night since her rescue, Karma would not stop crying. She had passed a veterinary checkup and yet Ellie and her husband, Jay Weiner, could not figure out what was wrong with her. Finally, after hearing Karma's mournful moos throughout the night, Ellie went outside, determined to solve the mystery.
After re-examining Karma, Ellie discovered her swollen udders, which meant that Karma had left behind her baby. Astonished by the realization, Ellie told Jay who immediately called the farmer. The farmer told him that it was too late, as the calf was already sold for slaughter. But luck mercifully stepped in… the truck transporting the calf had broken down. Jay struck a deal with the farmer and told him that if he could fix the truck, the farmer would have to release the calf to him.
Jay rushed to the farm and to his pleasant surprise, the truck's gears were simply stuck and just needed a little push up the hill to get going again. Thankfully, the farmer honored his deal and released the calf to The Gentle Barn. (Grab a box of Kleenex and watch the emotional reunion here.)
Karma's made-for-movie story doesn't end there. Ellie told us about yet another unexpected discovery—Karma was pregnant again. During her labor, they found all the cows at the sanctuary surrounding Karma in a perfect circle. Once the calf was born, the cows orderly formed a hierarchical line—with the eldest in the front—taking their turns to greet the calf. This remarkable ritual is too often made impossible due to the ruthless practices of factory farming and our self-interests.
Ellie finished the presentation stressing that one person can really make a difference. For every person who commits to a plant-based diet, 198 animals' lives are saved. She said that a plant-based diet gives you all the nutrients to set yourself up for success.
It was an inspiring, heartfelt speech without sounding like a plea, which would've rubbed non-vegetarians and non-vegans the wrong way.
The Barnyard Tour
Empowered with that information, we entered the barnyard, which housed animals ranging from chickens and turkeys (even an emu!), to goats, calves, pigs and llamas.
As soon as we got through the gate, Sassy the goat was there to greet us. I didn't think her name was fitting because she seemed more sweet than sassy. I guess she has her moments.
We walked over to the area where most of the goats were hanging out. Many were just standing around, some by the wading pool, others wandering around with the other animals.
The animals were pretty much free to roam within the barnyard, except the pigs had a pen with several barn houses. There was also a row of barn houses outside the pen for the much larger, 1,000-lb. pigs. It was interesting to see that even with multiple sheds, the pigs in the pen were social sleepers, cuddling side by side with each other.
We also met two calves that narrowly missed the slaughterhouse after a girl asked farmers at the Orange County Fair if they'd be willing to release their animals to a sanctuary. Only one farmer said yes, sparing the two calves from brutality and death.
Feeding the Donkeys and Horses
For just $5 for a bag of carrots, I was able to walk through the stable of horses, starting with a few donkeys with the most enviable eyelashes. I felt guilty because by the time I reached the end, I had to skip some of the horses because I ran out of carrots. I was a little too generous at the beginning, spoiling some with more than one serving. Next time I go there, I'll start at the end and finish at the start—I'm sure a lot of the horses at the end feel gipped.
Mooing with the Cows
Our tour wouldn't have been complete without visiting the big cows. They were housed outside of the barnyard and away from the stables. We picked up a couple grooming brushes from a bucket and entered the pen. The cows were wearing eye masks to shield them from pesky flies (I initially thought they were used to keep them calm).
Since it was nearing closing time, most of the visitors had already gone home. We stopped to brush Faith, a cow that was born blind, and another cow (can't remember the name). We brushed their hair for a little bit, but didn't stay long. At that point, I felt like the cows already had enough of people petting them—not that they made any indication that they didn't want us there.
Vegan Kielbasas!
There was a food stand that served vegan hotdogs and vegan kielbasas. Unfortunately they were out of hotdogs by the time we got to it. We topped the kielbasas with relish, mustard, ketchup and onions and enjoyed our vegan meal on one of the picnic tables. I was never a fan of real kielbasas but I thought the vegan version was pretty delicious—maybe it's also because I was starving.
Giving guests the opportunity to sample vegan fare is a great way to help advance The Gentle Barn's mission. People who wouldn't normally consider a plant-based diet might find that they actually like the taste of vegan food and may be open to something like Meatless Mondays. A little bit is better than an all-or-nothing approach.
A Lasting Impression
I think about how many times Ellie must've given that presentation to each tour group, and how much passion she had each time she shared it as though it were for the first time. It's that relentless passion that has helped her create such a magnificent haven for these animals.
What was once a weed-filled land is now a sprawling six acres of lush trees, spacious enclosures, stables and sheds. The love, compassion and empathy grown there have helped the sanctuary and its inhabitants thrive. Ellie and her team's efforts to help animals and people seem to help their mission resonate more strongly than if they were to focus solely on animals. (When I posted pictures of my visit on Facebook, it got a lot of Likes from people who never Like my posts. I think it had a lot to do with me prefacing it with The Gentle Barn's tagline: "A sanctuary for abused and special-needs animals who help heal abused and special-needs children.").
Lessons Learned from The Gentle Barn
The sanctuary is a place of healing to those who live there, and to all those who visit. You can't help but feel a sense of peace. More than that, the positive energy from the good that is done there is contagious. Maybe it's because I love animals so much, but for me, it was an enlightening, uplifting and inspiring experience. It also set me on my gradual journey from vegetarianism to veganism.
Aside from being pecked at a couple times by the emu (whom I was told wasn't normally like that), all the other animals were extremely chill. Despite the horrific treatment many of them endured at the hands of humans, they've learned to forgive and actually enjoy the company of complete strangers. So much could be learned from them, if we give them a chance to teach us.
Compassion and empathy starts with our children, which is why The Gentle Barn is right on point with their mission. If we can raise children to see animals as intelligent, social, caring and feeling beings, we can teach them to love and respect them—not see them as a food source. Only then can we end this cycle of violence, apathy and carelessness toward all living things.
Note: This post was written in August 2015 when I was still building content for my blog.
October 31, 2015 /The Ant's Meow
farm animal sanctuary, farm animals, animal sanctuary, The Gentle Barn, animal welfare
Animal Sanctuary
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,619
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Q: rkhunter - deleted files - whitelist issues I am working to secure a fresh Debian LAMP deployment and decided to include rkhunter (v1.4.2) in my security solution.
I have run it using the following options:
rkhunter -c --enable all --disable none --skip-keypress
All checks completed positive or skipped but for one, the deleted files check.
After a quick look in the .log file I found the culprits to be:
[19:59:10] Info: Starting test name 'deleted_files'
[19:59:11] Checking running processes for deleted files [ Warning ]
[19:59:11] Warning: The following processes are using deleted files:
[19:59:11] Process: /usr/sbin/mysqld PID: 1480 File: /tmp/ib5VMAPQ
[19:59:11] Process: /usr/sbin/apache2 PID: 1792 File: /run/lock/apache2/ssl-cache.1247
[...] # a couple more repetitions here, with different PIDs
[19:59:11] Process: /usr/sbin/apache2 PID: 1813 File: /run/lock/apache2/ssl-cache.1247
I judged these to be harmless/legit and proceeded to whitelist these processes/files.
In /etc/rkhunter.conf I found the line
#ALLOWPROCDELFILE=/usr/sbin/mysqld:/tmp/ib* and un-commented it. I also added ALLOWPROCDELFILE=/usr/sbin/apache2:/run/lock/apache2/ssl-cache.* below the list of commented examples.
Unfortunately, when running rkhunter (with the same options) again I still receive the exact same warnings.
Do I need to enable the whitelisting in general or do something else additionally?
Thank you in advance.
A: ALLOWPROCDELFILE=/usr/sbin/mysqld
ALLOWPROCDELFILE=/usr/sbin/apache2
or
ALLOWPROCDELFILE=/usr/sbin/mysqld:/tmp/ib5VMAPQ
ALLOWPROCDELFILE=/usr/sbin/apache2:/run/lock/apache2/ssl-cache.1247
The PID is different in case by case, so I think the second choice is not realistic. I'm not sure why RKH does not work correctly, but it does not been expanding regexes in $ALLOWPROCDELFILE(S).
or
If you can rewrite RKH script
--- rkhunter 2015-12-07 03:28:53.000000000 +0900
+++ rkhunter.neu 2016-03-30 13:33:19.328416849 +0900
@@ -13395,7 +13395,7 @@
FNAMEGREP=`echo "${RKHTMPVAR3}" | sed -e 's/\([.$*?\\]\)/\\\\\1/g; s/\[/\\\\[/g; s/\]/\\\\]/g'`
- if [ -n "`echo \"${FNAME}\" | grep \"^${FNAMEGREP}$\"`" ]; then
+ if [ -n "`echo \"${FNAME}\" | grep \"^${RKHTMPVAR3}$\"`" ]; then
PROCWHITELISTED=1
fi
else
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,689
|
// Type: Highsoft.Web.Mvc.Stocks.ColumnrangeSeriesData
using System.Collections;
using Newtonsoft.Json;
namespace Highsoft.Web.Mvc.Stocks
{
public class ColumnrangeSeriesData : BaseObject
{
public ColumnrangeSeriesData()
{
this.Color = this.Color_DefaultValue = "undefined";
this.DataLabels = this.DataLabels_DefaultValue = new ColumnrangeSeriesDataLabels();
this.Description = this.Description_DefaultValue = "undefined";
this.Events = this.Events_DefaultValue = new ColumnrangeSeriesDataEvents();
double? nullable1 = new double?();
this.High_DefaultValue = nullable1;
this.High = nullable1;
this.Id = this.Id_DefaultValue = "null";
double? nullable2 = new double?();
this.Labelrank_DefaultValue = nullable2;
this.Labelrank = nullable2;
double? nullable3 = new double?();
this.Low_DefaultValue = nullable3;
this.Low = nullable3;
this.Name = this.Name_DefaultValue = (string) null;
bool? nullable4 = new bool?(false);
this.Selected_DefaultValue = nullable4;
this.Selected = nullable4;
nullable3 = new double?(double.MinValue);
this.X_DefaultValue = nullable3;
this.X = nullable3;
}
public string Color { get; set; }
private string Color_DefaultValue { get; set; }
public ColumnrangeSeriesDataLabels DataLabels { get; set; }
private ColumnrangeSeriesDataLabels DataLabels_DefaultValue { get; set; }
public string Description { get; set; }
private string Description_DefaultValue { get; set; }
public ColumnrangeSeriesDataEvents Events { get; set; }
private ColumnrangeSeriesDataEvents Events_DefaultValue { get; set; }
public double? High { get; set; }
private double? High_DefaultValue { get; set; }
public string Id { get; set; }
private string Id_DefaultValue { get; set; }
public double? Labelrank { get; set; }
private double? Labelrank_DefaultValue { get; set; }
public double? Low { get; set; }
private double? Low_DefaultValue { get; set; }
public string Name { get; set; }
private string Name_DefaultValue { get; set; }
public bool? Selected { get; set; }
private bool? Selected_DefaultValue { get; set; }
public double? X { get; set; }
private double? X_DefaultValue { get; set; }
internal override Hashtable ToHashtable()
{
Hashtable hashtable = new Hashtable();
if (this.Color != this.Color_DefaultValue)
hashtable.Add((object) "color", (object) this.Color);
if (this.DataLabels.IsDirty())
hashtable.Add((object) "dataLabels", (object) this.DataLabels.ToHashtable());
if (this.Description != this.Description_DefaultValue)
hashtable.Add((object) "description", (object) this.Description);
if (this.Events.IsDirty())
hashtable.Add((object) "events", (object) this.Events.ToHashtable());
double? nullable1 = this.High;
double? nullable2 = this.High_DefaultValue;
if (nullable1.GetValueOrDefault() != nullable2.GetValueOrDefault() ||
nullable1.HasValue != nullable2.HasValue)
hashtable.Add((object) "high", (object) this.High);
if (this.Id != this.Id_DefaultValue)
hashtable.Add((object) "id", (object) this.Id);
nullable2 = this.Labelrank;
nullable1 = this.Labelrank_DefaultValue;
if (nullable2.GetValueOrDefault() != nullable1.GetValueOrDefault() ||
nullable2.HasValue != nullable1.HasValue)
hashtable.Add((object) "labelrank", (object) this.Labelrank);
nullable1 = this.Low;
nullable2 = this.Low_DefaultValue;
if (nullable1.GetValueOrDefault() != nullable2.GetValueOrDefault() ||
nullable1.HasValue != nullable2.HasValue)
hashtable.Add((object) "low", (object) this.Low);
if (this.Name != this.Name_DefaultValue)
hashtable.Add((object) "name", (object) this.Name);
bool? selected = this.Selected;
bool? selectedDefaultValue = this.Selected_DefaultValue;
if (selected.GetValueOrDefault() != selectedDefaultValue.GetValueOrDefault() ||
selected.HasValue != selectedDefaultValue.HasValue)
hashtable.Add((object) "selected", (object) this.Selected);
nullable2 = this.X;
nullable1 = this.X_DefaultValue;
if (nullable2.GetValueOrDefault() != nullable1.GetValueOrDefault() ||
nullable2.HasValue != nullable1.HasValue)
hashtable.Add((object) "x", (object) this.X);
return hashtable;
}
internal override string ToJSON()
{
Hashtable hashtable = this.ToHashtable();
if (hashtable.Count > 0)
return JsonConvert.SerializeObject((object) this.ToHashtable());
return "";
}
internal override bool IsDirty()
{
return this.ToHashtable().Count > 0;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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Tag Archives: 广州水伊方88号技师
Posts tagged "广州水伊方88号技师"
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Nimba Budgets US$3M for 2018/2019
Delegates at the County Council Sitting.Nimba County authorities, including the county's 11 lawmakers through the County Council Sitting on Sunday, September 2, budgeted US$3,680,092 for 2018/2019, with 24 percent of the money targeting road works in the county.The 113 delegates, including paramount chiefs, administrative district representatives, city councils, students, the disabled community, constituency representatives and traditional chiefs took the decision through voting, with 108 persons voting in favor of the budget, seven against, and one absent from the voting process.This year's County Council Sitting agenda was one of the heavily debated issues, where the citizens turned out en-masse to witness the occasion and lobbied with some of the delegates to include them or their projects in the budget.Confusion ensued during the Project Management Committee's presentation covering the period from 2016 to June, 2018. Some delegates suspected complications in the report, claiming it was bulky, a situation that was, however, not immediately resolved.At last, the caucus, through the chairman, Senator Prince Y. Johnson, constituted a committee to work along with the county authorities and peruse the documents in order to distribute them among the district commissioners and other stakeholders.Co-Chair Representative Johnson Gwiakolo of District # 9 apologized to the citizens, informing them that the past document/report cannot stop the future plan. He added, "We should go ahead with the meeting, while we wait for a committee to peruse the documents and make their report subsequently."In the new budget, US$804,165 will go to feeder roads rehabilitation and repair of the grounded machines, US$236,000 for affected communities; US$450,000 for electoral districts developments, US$569,505 go into debt owed institutions and other incomplete projects or liabilities.Other allotments include PMC operation of US$210,000; scholarship, US$90,000, Nimba Community Radio, US$30,000; Nimba County Community College, US$50,000; Disabled Community, US$45,000, ABC, US$25,000; LICC, US$30,000, Nimba County Health Team, US$30,000; Traditional Chiefs, US$40,000; County Meeting, US$75,000 and Cities, US$30,000.The amount of US$ 90,000 was allotted for statutory superintendents, district commissioners, township commissioners, paramount chiefs, city mayors, and the renovation of the Superintendent's Compound.The county is expected to generate income, including US$275, 825 from the Government County Development Funds, US$980, 000 from ArcelorMittal Social Development fund, US$186, 190.14 cents from North Star scrap proceeds and US$1,045,665.78 cents expected scraps proceeds from both North Star and Sethi Ferro Fabric Incorporated; but US$510,000 will come from the proceeds of Sethi Ferro Fabric Inc. It should have been US$600,000, but 15 percent was taken off by the government.This County Sitting brought together all the 11 lawmakers, along with chiefs and elders in the county.Each of the speakers, headed by Senator Prince Johnson, called for one Nimba, speaking against disunity and tribalism, even though there were some dissatisfaction raised among those whose entities, such as the Community Watch Forum, were not captured in the budget.David Mahn, head of the Community Watch Forum, expressed frustration, but begged for consideration to allocate money in the county's budget for those regularly assigned with the forum. The forum is an auxiliary security of ordinary residents who devote their time, serving as proxies in places where state security, specifically police officers, are not assigned.During the last County Council Sitting in the year 2016, US$10,000 was allotted for the Community Watch Forum.Other institutions, including the Ganta Christian Community, the Ministry of Education-Nimba, Jackson F. Doe Hospital and women's organizations were not also captured in the budget.Nothing was also allotted for agriculture or any agricultural-related programs.Share this:Click to share on Twitter (Opens in new window)Click to share on Facebook (Opens in new window) read more
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Eriksen-inspired Denmark out to spoil Peru's W. Cup return
However, Denmark have their own goal-machine in Tottenham Hotspur midfielder Eriksen, who netted 11 goals in his team's qualifiers, a record bettered only by Cristiano Ronaldo and Robert Lewandowski on the road to Russia.The 26-year-old struck a stunning hat-trick in the 5-1 demolition away to Ireland last November to secure Denmark's play-off victory, then scored and created another in a 2-0 friendly win over Mexico in a pre-World Cup warm-up.With France installed as Group C favourites, neither team will want to drop points at the first hurdle.The spotlight in Saransk will be Guerrero, 34, who is set to lead Peru's ageing strike force alongside Jefferson Farfan, 33, whose goals against Paraguay and New Zealand, in a play-off victory, carried them to their first World Cup finals for 36 years.(COMBO) This combination of pictures created on June 14, 2018 shows Peru's Jefferson Farfan (L) in Lima, Peru, on November 15, 2017, and Denmark's midfielder Christian Eriksen (R) in Dublin on November 14, 2017. Denmark, in Group C, open their World Cup against Peru in Saransk on June 16 before facing Australia and France. © AFP/File / LUKA GONZALES, Paul FAITHHaving been banned for testing positive after a qualifier against Argentina last October, Guerrero won a last-ditch legal appeal to appear at the finals.He promptly showed his class by scoring twice on his return in a 3-0 friendly win against Saudi Arabia a fortnight ago.However, competition is fierce within the Peru team to win places to face the Danes."We put are creating good pressure in the team to give the coach a problem, it won't be easy for him to choose the players," said defender Anderson Santamaria.Peru winger Edison Flores, scorer of five goals in qualifying, who plies his trade for Aalborg in the Danish league, is confident."I am a small player but with a lot of mental strength. To be in a World Cup is the best thing for a football player. Our coach believes in us and we in him," said the 24-year-old who stands at 1.67 metres (5ft 5ins).The match pits the tallest team at the World Cup, Denmark, who average 1.85m, against the smallest in Peru, who average 1.78m.The Danes' burly centre-back pairing of Simon Kjaer and Andreas Christensen will be tasked with containing Guerrero and Farfan.Denmark have avoided defeat in their last 14 matches, dating back to 2016, and coach Age Hareide has talent up front to choose from with Ajax starlet Kasper Dolberg, 20, and Feyenoord's Nicolai Jorgensen vying to start as striker.0Shares0000(Visited 2 times, 1 visits today) 0Shares0000Peru's forward Paolo Guerrero (C) vies for the ball with teammates during a training session at the Arena Khimki stadium, outside Moscow, on June 11, 2018, ahead of the Russia 2018 World Cup. © AFP / YURI CORTEZSARANSK, Russian Federation, Jun 15 – Paolo Guerrero will look to prove his drug-ban controversy is behind him on Saturday when Peru face Denmark and their star goal-scoring midfielder Christian Eriksen in both teams' World Cup opener.The presence of Flamengo forward Guerrero is a huge relief to Peru boss Ricardo Gareca after his 14-month ban for taking cocaine was overturned just weeks before the World Cup kicked-off. read more
Gabriel praises team effort
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}
In recent years, X-ray spectral imaging has revealed X-ray emitting
ejecta even in a number of very old ($\sim$10,000 yr) remnants where
most of the X-ray emission is from shock-heated interstellar or
circumstellar matter (e.g, Hughes et al. 2003, Park et al. 2003,
Hendrick et al. 2003, 2005, Borkowski et al. 2006). Puppis A is a
middle-aged supernova remnant whose X-ray emission is dominated by
shock-heated interstellar material. It resides in a very complex
interstellar environment including large atomic and molecular clouds
(Dubner \& Arnal 1988, Reynoso et al. 1995) and a large-scale
interstellar density gradient (Petre et al. 1982). It is, however,
one of a handful of remnants in which fast-moving ($\sim 1500$ km/s)
ejecta knots enriched in oxygen have been identified optically
(Winkler \& Kirshner 1985), and is thus a promising target for the
identification of hot, X-ray emitting ejecta. The goal of the Suzaku
observations reported here is to search for and to localize X-ray
emitting ejecta in Puppis A.
The presence of O-enriched ejecta in Puppis A clearly indicates that
the progenitor was a massive star that synthesized the oxygen
hydrostatically, a conclusion that is further confirmed by the
presence of an X-ray emitting compact central source, believed to be a
neutron star (Petre et al. 1996, Winkler \& Petre 2007). The
optically emitting oxygen knots are found in the central and eastern
regions of Puppis (Winkler \& Kirshner 1985). More global oxgyen
ejecta enrichment had been indicated by high-spectral resolution X-ray
observations with the Focal Plane Crystal Spectrometer (FPCS) on the
Einstein Observatory (Canizares \& Winkler 1981). The O enrichment
suggested by the Einstein FPCS could not be localized by it, however,
as the spectra were accumulated over rather large regions of the
remnant ($3'\times 30'$ and $6'$ diameter). The identification and
localization of gas with enriched element abundances requires adequate
spectral imaging. Einstein Solid-State Spectrometer observations with
a 6$'$ aperature did not find evidence for ejecta enrichment
(Szymkowiak 1985 PhD thesis), but a later survey by the ASCA
Observatory (Tamura 1995 PhD thesis) did produce evidence for
localized ejecta enrichment, mostly of Ne in the west and northwest.
Chandra observation of a small $8'$ square field at the eastern edge
of the remnant, where the shock is interacting with molecular and
interstellar clouds, hinted at small pockets of gas enriched with
ejecta (Hwang, Flanagan, \& Petre 2005).
In this paper, we present a partial X-ray imaging and spectral survey
of the Puppis A supernova remnant with the CCD detectors on the Suzaku
Observatory. Five observations cover about two-thirds of this large
50$'$ diameter remnant to the east and center, including the locations
of the optically emitting O ejecta knots. Compared to ASCA, Suzaku
has only slightly improved angular resolution, but much better
sensitivity and spectral resolution, particularly at low energies.
The Suzaku XIS also surpasses the Chandra CCDs for sensitivity and
spectral resolution at low energies, a feature that is particularly
important for studying the X-ray emitting O and Ne ejecta.
\section{Observations and Analysis}
Five observations of Puppis A were performed with the CCD-based X-ray
Imaging Spectrometer (XIS) at the beginning of the AO1 guest observing
phase of the Suzaku Observatory. The XIS features four separate
imagers, one illuminated from the back (XIS1), the other three from
the front (XIS0, XIS2, subsequently defunct, and XIS3). Details can
be found in Mitsuda et al. (2007).
The exposure times were optimized for observation of the oxygen
emission lines (O VII resonance at 574 eV, the dominant component of
the He-like O triplet in Puppis A, and O VIII Ly$\alpha$ at 653 eV;
see Winkler et al. 1981). The observations, which are summarized in
Table 1, cover most of the bright eastern and central portions of the
remnant, but not the west. Given the very soft spectrum of the source
and the overall high count rates of the Bright Eastern Knot (BEK),
Northeast (NE) and Interior (I) fields, data from the Hard X-ray
Detector on Suzaku were not telemetered, in favor of data from the
XIS. Additionally for these three observations, 4s burst mode was
used for the FI chips in order to further reduce the telemetry load
(in this mode, the detector does not collect data for 4 s out of the
normal 8 s data cycle). Shorter exposures also reduce the impact of
pulse pile-up in the detector, which is likely to be an issue at the
10\% level in the bright regions of the remnant. The observations
were carried out without the use of charge injection to moderate the
effects of radiation damage on the CCDs. We use the initial release
version 0.7 of the data, after performing the data screening
recommended in the Suzaku Data Reduction
Guide\footnote{http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc}.
\begin{deluxetable}{lccccc}
\tabletypesize{\footnotesize}
\tablecaption{AO1 Suzaku XIS Observations of Puppis A}
\tablewidth{0pt}
\tablehead{
\colhead{Field}&\colhead{Obs ID}&\colhead{Date}&\colhead{Coordinates}&\colhead{Exposure Time (ks)}&\colhead{FI Burst Mode}}
\startdata
Bright Eastern Knot (BEK) & 501086010 & 2006 Apr 17 & 125.9422, -42.9626 & 16.8 & 4s \\
Northeast (NE) & 501087010 & 2006 Apr 17-18 & 125.7579, -42.7332 & 20.7 & 4s \\
Interior (Int) & 501088010 & 2006 Apr 17 & 125.5915, -42.9172 & 20.3 & 4s \\
Southeast (SE) & 501089010 & 2006 Apr 18 & 126.0121, -43.2023 & 29.8 & 4s, none \\
South (S) & 501090010 & 2006 Apr 18-19 & 125.6826, -43.1664 & 31.2 & none \\
\enddata
\end{deluxetable}
\subsection{Imaging Analysis and Results}
For all five fields, the full-CCD spectra are qualitatively similar.
They fall off rapidly above energies of 3-4 keV, with no evidence for
harder emission, and have numerous strong emission line features.
These are the He$\alpha$ blends of O (dominated by the resonance
line), Ne, Mg, Si, and S, plus weak Fe L features near 827 eV
corresponding to Fe XVII (cf. Winkler et al. 1981). Figure 1 shows
the BI illuminated XIS1 spectrum for the South field as an example,
with the prominent emission lines and blends labelled. In this and all
other spectra considered here, the channels are binned to provide at
least 25 counts per channel.
\begin{figure}
\centerline{\includegraphics[scale=0.40,angle=-90]{f1.eps}}
\figcaption{The XIS1 spectrum for the South field of Puppis A with
the prominent emission lines and blends labelled.}
\end{figure}
\begin{deluxetable}{llll}
\tabletypesize{\footnotesize}
\tablecaption{Image Pulse Height and Energy Cuts}
\tablewidth{0pt}
\tablehead{
\colhead{Image}&\colhead{Pulse Heights}&\colhead{Energies (keV)$^*$}&\colhead{Total Counts ($10^4$)}}
\startdata
Cont 0 & 100-125 & 0.36-0.46 & 18\\
C He$\alpha$ & 125-145 & 0.46-0.53 & 51\\
O VII He$\alpha$ (Res) & 145-164 & 0.53-0.60 & 87\\
O VIII Ly$\alpha$ & 164-192 & 0.60-0.70 & 266\\
Cont 1 & 192-216 & 0.70-0.79 & 259\\
Fe L (Fe XVII) & 216-236 & 0.79-0.86 & 313\\
Ne IX He$\alpha$ & 236-268 & 0.86-0.98 & 573\\
Ne X Ly$\alpha$ & 268-300 & 0.98-1.09 & 322\\
Cont 2 & 312-348 & 1.14-1.27 & 166\\
Mg XI He$\alpha$ & 348-390 & 1.27-1.42 & 175\\
Cont 3 & 390-420 & 1.42-1.53 & 42\\
Si XIII He$\alpha$& 475-538 & 1.74-1.96 & 63\\
Si XIII He$\beta$ & 563-633 & 2.05-2.31 & 15\\
S XV He$\alpha$ & 633-710 & 2.31-2.59 & 12\\
Cont 4 & 710-860 & 2.59-3.13 & 7.4\\
\enddata
\tablenotetext{*}{The energies are computed for a gain of 3.65 eV per pulse height channel.}
\end{deluxetable}
We produced energy-selected images of all the notable line features in
the integrated spectrum, as well as regions of continuum emission, as
summarized in Table 2. We perform a simple correction for the
detector exposure by making the individual line images for each
detector for each observation, then trimming several pixels at all
four edges and dividing by the average exposure time; i.e., we do not
correct for vignetting. No attempt was made to subtract background as
the images are all bright. The final mosaicked images have 8.4$''$
pixels and are scaled to a fiducial exposure time, taken to be the
16.8 ks exposure time for observation 50108600. The Si He$\beta$ and
S He$\alpha$ images have the fewest photons and so were binned by a
further factor of four in each dimension to improve signal-to-noise.
The combined mosaicked line images for all four XIS detectors are
shown in Figure 2; the images mosaicked separately for the BI and FI
detectors are not shown but are similar to those in the Figure. The
continuum images are also not shown, but are similar to the line
images in general morphology. Spatial variations in the line emission
are further highlighted by difference images, five of which are shown
in Figure 3. These difference images are line images with the
broadband image normalized to it and subtracted. While we refer to
the images in Figure 2 and 3 as ``line images'', they are in reality
simply energy-selected images that include any emission in the
relevant energy range, whether from line or continuum. In Figure 2,
the O VII image is overlaid with surface brightness contours of the
ROSAT High Resolution Imager mosaic of the entire remnant (Petre et
al. 1996) and the J2000 coordinates. Coordinates are also given for
the images in Figure 3 to facilitate identification of morphological
features discussed below.
The so-called Bright Eastern Knot (BEK), located just inside the
bottom of the straight eastern shock front, is prominent in all the
images. Its brightness peaks at roughly (J2000) $\alpha$=126.04,
$\delta$=-42.97 (see Figures 2 and 3). It is known to be a
complicated interaction between the supernova remnant shock and
multiple interstellar clouds (Petre et al. 1982, Hwang et al. 2005).
Given that the eastern part of the BEK has lower temperature and
ionization age conducive to stronger O line emission (Hwang et
al. 2005), it is not surprising that O emission is seen to be
relatively prominent at the easternmost edge of the BEK, and Ne and Fe
L emission is strong in a larger region extending towards the remnant
interior (Figure 3). We also point out some of the other
morphological features that we will later discuss in more detail.
These include the elongated region extending southward from the BEK
region (SE Arm, roughly $\alpha$=126.05, $\delta$=-43.15), and the low
surface brightness region that is just west of it (SE Low SB).
Extending north by northwest from the BEK is the northeast shock
front (NE SF); seen by Suzaku, it is bright in the images of Ne, Mg,
Si, S, but not of O.
The relative prominence of lines of higher energy as opposed to oxygen
lines is consistent with the harder spectra seen interior to the NE SF
in ROSAT observations (Aschenbach et al. 1994, Hwang et al. 2005).
The remnant's interior is populated with additional patchy clumps of
emission. The most striking of these is seen most prominently in the
Si images, just inside the northernmost apex of the remnant in the Si
images (it is also seen clearly in Ne, Mg, and S emission; we call
this the ``Si Knot'' ($\alpha$=125.70, $\delta$=-42.76).
Incidentally, the neutron star in Puppis A is visible in Si and S, and
in negative in the Ne IX difference image, at the lower right of the
image mosaics (i.e., in the lower center of the remnant at
$\alpha$=125.48, $\delta$=-43.01).
\begin{figure}
\centerline{\includegraphics[scale=0.8]{f2.eps}}
\vskip -1.5 in
\caption{Energy selected line image mosaics of all four XIS detectors
combined. From left to right: (top) O VII Res, O VIII Ly$\alpha$, Fe
L (Fe XVII) (middle) Ne IX He$\alpha$, Ne X Ly$\alpha$, Mg XI
He$\alpha$ (bottom) Si XIII He$\alpha$, Si XIII He$\beta$, S XV
He$\alpha$. The final two images have been binned further by a factor
of four to improve the signal-to-noise level. The O VII image is
labelled with J2000 coordinates and overlaid with surface brightness
contours of the ROSAT HRI mosaic (Petre et al. 1996) to indicate the
full extent of the remnant. The neutron star can be seen in the
images in the bottom row.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[scale=0.8]{f3.eps}}
\vskip -1.5 in
\caption{Selected line images from Figure 2, with scaled broadband
image subtracted. White indicates excess emission compared to
broadband; black indicates a deficit of emission. For reference, the
spectral regions presented in Figure 6 are shown in the panels for Ne
IX (top: O Knot Center and bottom: Ne Knot South) and Si XIII
(counterclockwise from top: Si Knot North, NE SF, BEK, SE Arm and SE
Low SB)}
\end{figure}
Line-to-continuum ratio images (``line equivalent width'' or EQW) can
be used to make a quick assessment of whether enhancements in the
images are particularly due to enhanced line emission. The continuum
underlying the line emission is first estimated (either by
interpolation or extrapolation from the continuum in nearby energy
regions,
or by extrapolating from a single adjacent continuum region),
then it is subtracted from the ``line'' image before taking the ratio
(the procedure is essentially that used by Hwang et al. 2000). We
examined EQW images for O VIII Ly$\alpha$, Ne He$\alpha$, Mg
He$\alpha$ and Si XIII He$\alpha$ with particular reference to the
emission enhancements at the BEK, the northeastern shock front, and
the various interior knots.
None of the EQW images is bright at the BEK, which is in line with our
understanding that the BEK is associated with shocked interstellar
clouds. The northeast shock front is also absent from the EQW images
for the same reason. Of all the features seen in the EQW images, the
strongest by far is the roughly $3'\times 5'$ Si Knot seen in the Si,
Ne, Mg, and S images. It completely dominates the Si EQW image which
is shown in Figure 4. (This image is binned by a factor of four
relative to the Si He$\alpha$ line image in Figure 2.) We do not show
the O, Ne, or Mg images here, in favor of the abundance maps from
spectral fitting to be discussed below, but note that O and Ne EQW
also appear high in a few regions, particularly in the southern
portion of the Si Knot and in the SE Low SB region adjacent to the SE
Arm.
\begin{figure}
\centerline{\includegraphics[scale=0.8]{f4.eps}}
\vskip -2.5 in
\caption{Left: Line-to-continuum, or ``equivalent width'' image for
Si XIII He$\alpha$.
Right: Line ratio image for O VII Res/O VIII Ly$\alpha$.}
\end{figure}
Element abundances aside, we also briefly consider the O VII Res/O
VIII Ly$\alpha$ ratio image as an example of a single-element
line-ratio image that can be used to diagnose conditions in the X-ray
emitting plasma (Figure 4; the analogous image for Ne is similar and
not shown). Ideally the ratio should be for line
emission only, but we have taken the ratio without subtracting the
continuum; the true continuum level is difficult to determine at this
spectral resolution, and the O VII line in particular is rather weak.
The image highlights the SE Low SB region that extends southward from
the BEK, as well as the eastern edge of the BEK, but cannot tell us to
what extent the O VII enhancement is caused by temperature, ionization
age, or column density differences. Imaging spectral observations
with Chandra indicate that the BEK is composed of at least two
physical components, with the easternmost of these at lower
temperature, lower ionization, and higher column density (Hwang et
al. 2005). Except for the higher column density, these would
contribute to the observed enhancement of O VII/O VIII at the eastern
edge of the BEK. Likewise for the low surface brightness region,
further constraints from spectral analysis are needed to understand
the enhancement observed.
\section{Spectral Analysis}
While the EQW images provide an easily accessible overview of where in
the remnant the line emission is enhanced, a quantitative
interpretation is not possible from these images alone. First, it
must be known how temperature, ionization, and element abundances
affect the line strengths relative to the continuum, and this depends
on the underlying spectral model assumed. Moreover, the parameters of
any model are not expected to be uniform throughout the remnant.
Second, the continuum estimation can be accurate only if the
``continuum'' regions are truly line-free to start with, but this is
not usually a safe assumption for lines such as those of O or Ne that
lie at energies where the spectra are dense with line emission.
Finally, a two-point interpolation is the only feasible method to
estimate and subtract the continuum here, but it lacks sophistication
and probably accuracy. EQW images are thus most likely to be useful
if the line emission is very strong and well-isolated; for example, in
tracing the qualitative large-scale morphology of known, strong ejecta
enhancements as in Cas A (Hwang et al. 2000). They are less reliable
when searching for relatively modest ejecta enhancements. To support
the conclusions of the imaging analysis, it is thus necessary to turn
to the X-ray spectra, as we do next.
We extracted spectra across the portion of the remnant imaged by
Suzaku in a grid following the procedure used by Hwang et al. (2005)
for Chandra observations of the BEK. Spectra were extracted for 259
square and rectangular regions of angular size 100-200$''$, with each
spectrum containing at least 10,000 counts. We focus on the spectral
results for the BI chip.
The spectra were binned to 25 counts per channel and were fitted using
XSPEC with nonequilibrium ionization (NEI) models for a plane parallel
shock, modified by interstellar absorption using cross sections from
Morrison \& McCammon (1983). The models are characterized by the
temperature and the limiting ionization ages; we take the lower limit
for the ionization age to be fixed at $n_et = 0$ cm$^{-3}$~s and have
fitted only for the upper limit. Element abundances were varied for
O, Ne, Mg, Si (with S, Ar, and Ca linked to Si), and Fe (with Ni
linked to Fe).
The spectral models must be converted to pulse-height spectra for
comparison to the data. This conversion is carried out with an
effective area file giving the total efficiency to detect a photon of
a given energy, and a response matrix for the distribution of photon
energy with pulse height. Both of these responses can vary depending
on the region of the detector used and the specific observation
conditions. An important time-dependent effect is the reduction in
XIS low energy efficiency caused by the build-up of contaminants on
the optical blocking filter (Koyama et al. 2007). This effect is
included in the effective area calculation (Ishisaki et al. 2007).
CCDs are also damaged by radiation with the passing of time causing
degradation in the energy resolution and gain. While this effect is
now ameliorated by charge injection, charge injection was not used for
these early observations. We compensate by fitting the spectrum with
an overall scaling factor for the gain (usually 0.993 to 0.996).
The observations were spread out over only two days, and there was no
discernable change in the response matrices computed for the various
observations. We use the matrix appropriate for an observation using
the entire CCD chip because the current matrices do not reproduce
differences in response across the detector.
The effective areas are another story. Ideally, the effective area
should be computed for the specific region on the detector used for a
given spectrum, but this requires time-consuming ray-tracing
simulations for each spectrum. This was impractical for us given
limitations in computer power and the large number of spectral regions
involved. The good signal-to-noise level of our data allowed us to
use relatively small spectral regions, but this makes the effective
area compuations even more time-consuming. We therefore compared the
results of spectral fits using several ray-tracing simulations of the
effective area: five for relatively large regions ($3'$ radius) on the
detector (center, top and bottom to the left and to the right) and
also a smaller 2$'$ grid region. Using the same spectrum and response
matrix, but different effective area files, we find that the
temperatures remain very consistent; abundances and ionization ages
vary from a few to 10 percent, and the column density variation by
less than twenty percent. For all the fits presented in this paper we
used the simulation for a $3'$ radius region on the center of the
detector, which amounts to neglecting the energy- and
spatial-dependent effects of vignetting and the contaminant on the
optical blocking filter. The vignetting removes photons from the edge
of the detector and is more severe at high energies, while the
contamination removes photons from the center of the detector and is
more severe at low energies (see Ishisaki et al. 2007). The two
effects tend to compensate, but the overall effect is
complicated. Since the spectrum of Puppis A peaks at around 1 keV,
neither effect should be too severe. Given that the variation in the
spectral parameters across the remnant is significantly larger than
the spread in spectral parameters that we find from using different
effective area computations, we consider this to be an acceptable
simplification. The biggest limitation imposed by the lack of fully
correct effective area calculations throughout the remnant is that we
are not able to obtain accurate model emission measures.
Given the uncertainties in the spectral response described above, we
do not give numerical values of the spectral parameters obtained for
the entire grid. Rather we present an overview of the fitted
temperatures, ionization ages, column densities, and abundances in
Figure 5, with the smoothed Suzaku broadband mosaic image contours
overlaid for reference. The fitted parameters are represented for
each region by the color scale. Spectra and the fitted model
parameters are shown in Figure 6 and Table 3 for a small sample of
interesting spectral regions.
\begin{figure}
\centerline{\includegraphics[scale=0.8]{f5.eps}}
\vskip -1.5 in
\caption{Maps of the fitted parameters for plane-parallel shock
models, overlaid with smoothed contours of the Suzaku broadband image
mosaic: $\chi^2/d.o.f$, column density $N_H$ ($10^{22}$ cm$^{-2}$),
temperature kT (keV), ionization age $n_et$ (cm$^{-3}$ s), and element
abundances of O, Ne, Mg, Si, S, and Fe by number relative to
solar. The Si abundance map has the explosion center inferred by
Winkler et al. 1985 marked as ``E''.}
\end{figure}
Figure 5 shows that the quality of the fits is generally fair, with
all but a few values of the $\chi^2$ per degree of freedom falling
below 2. While not formally acceptable in a statistical sense, we
consider it acceptable given that the model is simple and many of the
spectra a few tens of thousands of counts or more. Column densities
are distributed mostly between about 0.13 to 0.35 $\times 10^{22}$
cm$^{-2}$, and are at the higher end of this range in the north and
center, away from BEK. The regions of high absorption generally
correspond to the spectrally harder regions in the ROSAT images
(Aschenbach 1994). The upper ionization age is typically
$0.5-2.5\times 10^{11}$ cm$^{-3}$s, but is particularly high along a
relatively narrow ridge behind the straight portion of the NE SF on to
the inside of the BEK. Since the gas near the shock front should have
been shocked relatively recently, these high ionization ages (which
exceed $3\times 10^{11}$ cm$^{-3}$s) are consistent with high gas
densities. The lowest ionization ages occur in the SE Low SB region,
as might be expected since the low surface brightness indicates low
gas densities. The temperature map is somewhat patchy, but more
uniform than the others, with temperatures distributed fairly
uniformly between 0.5-0.8 keV.
The abundance maps all show a prominent double peak at the Si Knot in
the north central region. Two strong peaks are seen in Mg, Si and Fe,
but the southernmost of these is more prominent in O and Ne. All the
fitted element abundances have their peak values in this region; these
are all above solar, except for Fe. Ne and Si have higher peak
abundance values than the other elements, but Ne abundances are skewed
higher than the other elements throughout the remnant. In most of the
remnant, O, Mg, Si, and Fe abundances are generally between 0.3 to
0.6-0.7 solar, but the Ne abundances fall mostly between 0.55-0.85
solar. Ne and Mg lines are blended with Fe L emission, so their
abundances are somewhat more model-dependent than those of elements
with cleaner line emission, such as Si. It should also be noted here
that sub-solar abundances are the norm in Puppis A when the spectra
are fitted with single-component models. For example, Tamura (1995)
typically required abundances from 0.1 to 0.6 solar for the ASCA
spectra.
The O and Ne abundance maps are patchier than the others. The fitted
abundances are lowest in the SE Low SB region, and highest in the
southern peak of the Si Knot; we refer to this as the O Knot Center in
Table 3 ($\alpha$=125.67 $\delta$=-42.84). The absence of this
feature in the O EQW image (not shown here) underlines the difficulty
of identifying O ejecta from the X-ray images alone; the imaging
analysis is particularly difficult for lines like those of O that lie
in crowded spectral regions. Only by examining the spectra could we
learn that the O abundance is high at the Si Knot. Ne shows another
relative enhancement in an extended region just south of the broadband
contours that is also echoed more faintly by O. This region is
clearly visible in the Ne difference image in Figure 3, so we refer to
it as Ne Knot South in Table 3 ($\alpha$=125.66, $\delta$=-43.12).
For Mg, Si and Fe, the Si Knot is the only region with a definite
element enhancement.
We take a closer look at a few spectra selected for their interest, in
Figure 6. The top panel compares the spectrum of a region in the
northern peak of the Si Knot with one in the NE SF. Both have strong
Si line emission in the Si line image in Figure 2, but only the Si
Knot knot has enhanced EQW (Figure 4) and enhanced Si abundance from
spectral fitting. The Si He-like blend is indeed stronger at the
knot. From Table 3, we see that the column densities and temperatures
are similar for the two fits, whereas the NE SF has a somewhat higher
ionization age. All the fitted element abundances are higher by at
least a factor of two at the Si Knot compared to the NE SF, with Si
being higher by a factor of four. Given that the statistical errors
are quite tight, the Si Knot is clearly enriched with ejecta.
The second panel shows two spectra with O and Ne enhancements---those
of the O Knot Center (the southern peak of the Si Knot) and the Ne
Knot South. While the O Knot Center also has higher-than-average
abundances for all the elements except perhaps Fe, the Ne Knot South
does not really show evidence for significant enhancement of elements
other than Ne, which is especially prominent; O is only slightly
higher than average.
In the bottom panel of Figure 6, we compare regions without
significant abundance enhancements---the SE Arm extending south from
the BEK, the SE Low SB region, and the eastern edge of the BEK. The
latter two are prominent in the O VII/O VIII ratio image, and the SE
Low SB region also appears in the O VIII Ly$\alpha$ EQW image.
Enhanced O VII/O VIII in the SE Low SB region and the BEK is affirmed
by the spectra, as the O VII line is clearly stronger in these regions
than in the SE arm, whereas O VIII emission is comparably strong in
all three regions. Enhanced O EQW in the SE Low SB region is not
affirmed by the fitted O abundance, however. The explanation would
appear to be that the ``continuum band'' used to subtract the
continuum contribution (``Cont 1'' at 0.70-0.79 keV from Table 2) is
relatively weaker here (due in part to lower Fe abundance), which
artificially enhances the O EQW. The spectral parameters given in
Table 3 show that these three regions have similar columns,
temperatures, and abundances. The primary difference is a lower
ionization age for the SE Low SB region, which is enough to shift the
ion balance in favor of He-like O and to enhance the O VII/O VIII line
intensity ratio. Compared to other regions in the remnant, the SE Low
SB region also has very low column density and relatively low
temperature, which combines with the low ionization age to enhance O
VII relative to O VIII.
\begin{figure}
\centerline{\includegraphics[scale=0.8]{f6.eps}}
\vskip -2.0 in
\caption{(Top) Spectra comparing strong Si line emission at the NE SF
(left) and the northern peak of the Si Knot (right). (Middle)
Spectrum of the O Knot Center (the southern peak of the Si Knot) (left) and the Ne Knot South (right).
(Bottom) Spectra from SE Arm with low O VII/O VIII and O VIII EQW
(left) and the SE Low SB region with high O VII/O VIII and O VIII EQW
(middle) and the eastern edge of the BEK (right). Fitted parameters
are given in Table 3.}
\end{figure}
\begin{deluxetable}{lllllllll}
\tabletypesize{\scriptsize}
\rotate
\tablecaption{Spectral Fits}
\tablewidth{0pt}
\tablehead{\colhead{Region}&\colhead{$\chi^2, \chi^2/dof$}&\colhead{N$_{\rm H}^*$}&\colhead{kT}&\colhead{$n_et$}&\colhead{O}&\colhead{Ne}&\colhead{Si}&\colhead{Fe}\\
\colhead{J2000 RA, DEC}&\colhead{}&\colhead{$10^{22}$ cm$^{-2}$}&\colhead{(keV)}&\colhead{$10^{11}$cm$^{-3}$s}&\colhead{$\odot$}&\colhead{$\odot$}&\colhead{$\odot$}&\colhead{$\odot$}}
\startdata
NE SF & 587.4, 1.41 & 0.345$\pm0.005$ & 0.708 (0.707-0.727) & 2.63e11 (2.56-2.72e11) & 0.53 (0.51-0.56) & 0.64 (0.63-0.67) & 0.41 (0.38-0.43) & 0.45 (0.44-0.47) \\
125.89, -42.79 &&&&&&&&\\
Si Knot & 558.6, 1.56 & 0.33$\pm0.01$ & 0.88 (0.87-0.91) & 6.5e10 (6.3-6.9e10) & 0.86 (0.84-0.89) & 1.27 (1.24-1.33) & 1.5 (1.4-1.6) & 0.94 (0.90-0.97) \\
125.70, -42.76 &&&&&&&&\\
O Knot Center & 510.7, 1.49 & 0.35 (0.33-0.36) & 0.77 (0.73-0.86) &
7.1e10 (6.2-7.2e10) & 1.3 (1.1-1.4) & 1.8 (1.7-2.0) & 1.3 (1.2-1.3)
& 0.57 (0.54-0.65) \\
125.67, -42.84 &&&&&&&&\\
Ne Knot South & 271.0, 1.14 & 0.30$\pm 0.01$ & 0.66 (0.64-0.68) & 5.30e10 (5.28-5.30e10) & 0.73 (0.70-0.75) & 1.41 (1.35-1.46) & 0.48 (0.39-0.54) & 0.41 (0.38-0.44) \\
125.66, -43.12 &&&&&&&&\\
SE Arm & 479.2, 1.45 & 0.22$\pm0.01$ & 0.50 (0.48-0.52) & 1.95e11 (1.62-2.03e11) & 0.42 (0.41-0.45) & 0.67 (0.63-0.70) & 0.47 (0.43-0.51) & 0.50 (0.45-0.54) \\
126.05, -43.15 &&&&&&&&\\
SE Low SB & 489.8, 1.42 & 0.173 (0.166-0.183) & 0.64 (0.61-0.68) & 8.2e10 (7.4-9.0e10) & 0.35 (0.32-0.37) & 0.61 (0.59-0.63) & 0.43 ( 0.37-0.47) & 0.39 (0.36-0.42) \\
125.91, -43.14 &&&&&&&&\\
BEK, east edge & 363.7, 1.23 & 0.15 (0.14-0.17) & 0.58 (0.52-0.59) & 1.1e11 (1.0-1.3e11) & 0.38 (0.35-0.40) & 0.69 (0.64-0.74) & 0.40 (0.34-0.49) & 0.43 (0.38-0.47) \\
126.08, -42.98 &&&&&&&&\\
\enddata
\tablenotetext{*}{Error ranges given are 90\% confidence for a single interesting parameter.}
\end{deluxetable}
\section{Discussion}
To summarize the observational results, Suzaku clearly detects X-ray
emitting ejecta in Puppis A, and shows where it is
located. The most striking feature of the spectral maps is the
spatial correlation of the element abundances, especially at the
location of the Si Knot. All the elements are found to be
particularly enriched this region, with Si appearing to show the most
enrichment relative to other regions. Ne, and to a lesser extent O (
but not other elements), is also enriched in the south center of the
remnant.
The two peaks of the Si Knot are both regions with somewhat
higher-than-average temperatures, but even a forced temperature of 0.6
keV closer to the remnant average requires clearly enhanced element
abundances (Si=1.3 and O=0.65 for the weaker northern peak of the
knot). We consider it more likely that the temperature difference is
real and is associated with the different composition of the knot
compared to its surroundings.
The spectral parameters also show strong patterns. The high column
density to the north (combined with relatively uniform temperature)
echoes the patterns seen in spectral hardness by ROSAT. There is some
hint that the fitted column densities may be correlated with detector
position in the two northernmost fields (covering the northern peak
and the interior southwest of it), with the center of the detector
giving higher values. The other fields do not give the same
impression, however, and the large-scale differences in column density
are certainly robust. The ridge of high ionization age that follows
the boundary of the northeast shock front is qualitatively consistent
with the strong interaction of the blast wave seen in that region.
Radio HI observations show a very large molecular cloud whose straight
edge fits against the X-ray boundary of the remnant (Reynoso et
al. 1995). The blast wave will be strongly decelerated as it
propagates through the dense cloud, but the reflected shock will be
associated with higher temperature gas (cf. Levenson et al. 2002).
The difference in ionization age is a factor of several on the ridge
compared to the region behind it, and implies a comparable density
difference assuming that the shock times are roughly the same. Higher
angular resolution observations would be useful to make more reliable
quantitative estimates, but in any case, the relative narrowness of
the ridge seems to suggest that there was a sudden increase in the
ambient gas density, which presumably occured at the boundary of the
molecular cloud.
The Chandra study of the BEK region in Puppis A showed that there are
significant spectral differences even within the BEK. The eastern
0.5$'$ of the region is a compact knot having different spectral
parameters from the portion of the BEK to the interior. These
differences are unfortunately not accessible to this study, being on
too small an angular scale compared to the angular resolution of the
spectral grid. Nevertheless, it is encouraging to see that other,
somewhat larger scale spectral features seen in the BEK region by
Chandra are also seen by Suzaku. For example, there is a clump of
high ionization age gas inside the main part of the BEK which
corresponds well with the bottom end of the ``ridge'' of high
ionization age in the Suzaku map. The increase in temperature just
north of the BEK is also reproduced by the Suzaku analysis. The only
sufficiently large angular-scale spectral feature that we did not see
reproduced in the Suzaku spectral images is the higher column density
gas, not only at the eastern edge of the BEK, but also at the eastern
boundary above and below it. We do not have a complete explanation
for this, but it may be related in part to the contaminants that build
up on CCDs, since these reduce low energy sensitivity and can be
masked as increased absorption column densities; this is relevant for
both the Suzaku and Chandra CCDs.
The overall subsolar element abundances also require some comment. As
noted earlier, similarly low abundances had been obtained by Tamura
(1995) in his analysis of ASCA X-ray spectra, but there are otherwise
no indications that the interstellar medium around Puppis A has
depleted element abundances. Dust depletion is a possibility, given
the brightness of the remnant at infrared wavelengths, and the
excellent correlation of infrared and X-ray morphologies (Arendt et
al. 1990). Dust is clearly present in the remnant and is apparently
collisionally heated. The signature of dust depletion in the X-ray
emission would be higher Ne abundance to that of O, Mg, Si, and Fe,
since Ne is not depleted onto dust grains as are these other elements.
The Ne abundance does have a tendency to be slightly higher than that
of O in Puppis A, but so is the Si abundance. Moreover, the Ne
abundance in CCD-resolution spectra is subject to additional
uncertainty because the Ne lines are blended with the complicated Fe L
line emission. In the south-central Ne knot of Puppis A, the fitted
Ne abundance is near the solar value and significantly enhanced
relative to the other elements; perhaps this is a region where dust
depletion is important. The Ne abundance is nevertheless generally
fitted at subsolar values, however, so it would appear that the
overall abundances are indeed subsolar, independent of any depletion
of some of the elements onto dust grains.
Element abundances obtained from an overly simple spectral model may
be inaccurate, but there are no definite indications of additional
components in Puppis A that would, if present and unaccounted for,
result in artifically lower element abundances. Neither is Puppis A
an isolated incidence of low abundances inferred for the ISM-phase of
a supernova remnant. The Cygnus Loop in particular has yielded
extremely low ISM-phase abundances of 0.1-0.2 solar to analysis using
data from ASCA (Miyata et al. 1994), Chandra (Levenson et al. 2002,
Leahy et al. 2004), XMM-Newton (Tsunemi et al. 2007) and Suzaku
(Miyata et al. 2007). Finally, the interstellar medium in the
Magellanic Clouds is known to have subsolar abundances that are in
fact reproduced by Sedov models fitted to the entire remnant spectrum
(Hughes et al. 1998)\footnote{Plane-parallel shocks should provide a
good approximation to Sedov models (Borkowski et al. 2001).}.
\subsection{Lack of Widespread O Enrichment}
The modest O ejecta enrichment seen with Suzaku is at odds with the
significant enrichment of O and Ne relative to Fe that was reported by
Canizares \& Winkler (1981; hereafter CW81). In most regions of
Puppis A, we do not see O significantly enhanced relative to Fe, even
where O is enhanced above its average value. This echoes earlier
spectral studies of Puppis A with the Einstein Solid-State
Spectrometer (Szymkowiak 1985) and the ASCA Solid-State Imaging
Spectrometer (Tamura 1995), neither of which found a significant O/Fe
enrichment anywhere in Puppis A. The FPCS has also reported
enrichments of O relative to Fe for other sources, including
M87 and the supernova remnant N132D. For M87, XMM-Newton Reflection
Grating Spectrometer observations indicated a O/Fe abundance ratio of
0.5 (Sakelliou et al. 2002), compared to the factor of 3-5 enrichment
reported by the FPCS (Canizares et al. 1982). For N132D, ASCA finds that
O/Fe is roughly solar (Hughes, Hayashi, \& Koyama 1998), in contrast
to values of twice solar or higher as deduced from Einstein
observations (Hwang et al. 1993).
The Einstein FPCS observations were the first high-resolution X-ray
spectra ever obtained of non-solar cosmic sources---and the FPCS
observation of Puppis A was the very first of these. The FPCS used a
number of different crystals to scan limited energy ranges around a
feature of interest. The calibration was thus challenging, and not
fully optimal because the calibration data files taken just prior to
launch were unusable. The final calibration was based on extensive,
earlier laboratory measurements\footnote{Mark Schattenburg, private
communication}. For M87, the relative abundance estimate was based on
the O VIII Ly $\alpha$ line at 654 eV and the Fe XVII line at 826 eV,
for which two different crystals were required to scan the separate
lines, adding to the calibration uncertainties. This uncertainty is
eliminated for the observations of Puppis A and N132D since a single
crystal was used to scan the O VIII Ly $\gamma$ (816 eV) and Fe XVII
lines for the relative abundance determination. There is added
uncertainty, however, in that this is a crowded spectral region, and
the relatively weak Ly $\gamma$ line had to be modelled carefully.
For Puppis A, the assumptions made by CW81 to interpret their spectra
were reasonable enough at the time, but updated X-ray observations and
atomic physics show some possible pitfalls. These can be summarized
as arising from (1) the assumption of a single-component (i.e., single
temperature, single ionization age) plasma within the large FPCS
apertures (2) comparison to a solar coronal spectrum with a
temperature distribution that turns out not to be appropriate for most
of Puppis A (3) the assumption that Fe XVII is always the dominant Fe
ion species when O VII and O VIII ions coexist. (1) Spectral maps of
Puppis A, including the one in this work, clearly show a wide variety
of temperatures, column densities, and ionization timescales
throughout the remnant. (2) From the same maps, we see that
temperatures in Puppis A are relatively high, averaging near 0.6 keV.
By contrast the solar coronal spectrum used to calibrate the FPCS
observations peaked at roughly 0.2 keV (3) Evaluation of the ion
populations using the NEI models in XSPEC show that O VII, O VIII, and
Fe XVII do not necessarily coexist as assumed, particularly at
temperatures below kT=0.2 keV and, more pertinently for this
discussion, at temperatures above kT=0.6-0.7 keV. The same evaluation
of ion populations show that they can vary by factors of at least 3-5
in the optically thin NEI models, particularly when ionization effects
are also taken into account.
Given the foregoing, we believe that the current Suzaku abundance
results are more accurate, the high spectral resolution of the FPCS
notwithstanding.
\subsection{Spatial Correlations}
Although the peak abundances for the different elements vary from
subsolar (Fe) to nearly twice solar (Ne and Si), the abundance maps
all have the same general appearance, with the double-peaked Si Knot
being the most prominent feature throughout. At the southern peak,
all the elements with line emission have element abundances that are
significantly higher than average, and reach their highest values (in
this survey). The northern peak of this knot is relatively weaker for
O, Ne, and Mg compared to Si, but when the errors on the fitted
abundances are considered (Table 3), there is no significant
difference in the Si/O abundance ratio at the northern vs the southern
peak. The presence of Si ejecta at a radius comparable to that of O
ejecta is certainly interesting, because supernova ejecta may be
either hydrostatic or explosive in origin; O is synthesized completely
hydrostatically by the progenitor before the explosion, and Si in the
inner layers of the exploding star. Thus Si originates close to the
center of the explosion, but has been propelled outward in the
northern region of Puppis A and mixed with O and Ne ejecta.
Most of the fast optically emitting ejecta [O III] knots reported by
Winkler et al. (1985) are in the eastern half of Puppis A observed
with Suzaku. The bulk of them lie 25 to 90 degrees east of north, at
5$'$ to 9$'$ from the explosion center determined from their proper
motions (this center is indicated in the Si map subpanel of Figure 5).
Many of the optical knots are part of a large (roughly $8'\times 4'$)
tangle of [O III] filaments lying to the south and east of the
southern peak of the X-ray Si Knot. A smaller clump of optical knots
touches the X-ray knot at its southeastern boundary. Thus, while the
optically emitting O ejecta are in the general vinicity of the O
element abundances identified through the X-ray spectral fits, there
is no close physical correspondence between the optical and X-ray
emitting ejecta. On the whole, the X-ray and optically emitting O
ejecta appear to be spatially disjoint.
It is clear from the Suzaku observations that the Si knot is the only
example of significant Si ejecta enhancement in the eastern half of
Puppis A. In a result simultaneous with the preparation of this
paper, Katsuda et al. 2007 (in preparation) independently confirm the
Si enhancement here using XMM-Newton data, which provides
complementary coverage to Suzaku of the western half of the remnant.
They survey a larger fraction of the remnant, but do not find other
regions with substantial Si ejecta enrichment. If we allow that the
average interstellar element abundances near Puppis A are below the
solar value, it would appear that Fe is enhanced at the Si Knot as
well, since the fitted Fe abundances are higher here than elsewhere in
the remnant. The Si Knot is thus the only recognizable structure that
contains significant amounts of any explosively synthesized ejecta,
whether Si or Fe. Unlike the case in Cas A, where there is evidence
for physical separation of the Si and Fe ejecta (Hughes et al. 2000),
Puppis shows Si and Fe ejecta that appear to be coincident with each
other, as well as being mixed with lower Z elements.
\subsection{Knot Ejecta Mass Ratios}
The north central knot being the cleanest sample of ejecta in the
remnant, we consider how its element abundances compare to the
predictions of nucleosynthesis models. We convert the fitted
element abundances in Table 3, which are by number relative to the
solar values of Anders \& Grevesse (1989), to the ratio of the element
mass relative to that of Si. The observed mass ratios are compared to
various calculations of core-collapse supernova nucleosynthesis by
Woosley \& Weaver (1994), Thielemann et al. (1996), Rauscher et
al. (2002) and Limongi \& Chieffi (2003).
Nucleosynthesis calculations are complex, and are sensitive to the
treatment of the stellar and nuclear physics, some of which is not yet
fully understood. For these reasons, models have yet to converge in
detail on what the element mass ratios should be for a progenitor of
given mass. The models we consider predict that a Type II explosion
for a 25 M$_\odot$ progenitor should produce 0.1-0.3 M$_\odot$ Si
ejecta, with the models of Thielemann et al. giving Si masses at the
low end of this range. Considering the various model calculations for
abundances of O, Ne, Mg, and Fe relative to Si, we do not find perfect
agreement with the observed mass ratios, but progenitors between 15
and 25 M$_\odot$ appear to be generally indicated. In no case do we
find a plausible model for a progenitor more massive than 30
M$_\odot$, whereas models are seldom computed for masses below about
15 M$_\odot$.
Of course, this particular knot need not even be a representative
sample of the global ejecta abundances, but at the least it is
reassuring that the high abundances of O, Ne and Mg relative to Si
require a core-collapse rather than a thermonuclear explosion.
It is challenging to put tight constraints on the progenitor of Puppis
A from the X-ray observations because the remnant is relatively old
and most of the X-ray emission is dominated by interstellar material,
some of which is highly structured as dense clouds. Even where ejecta
are present, they are mixed with ISM, but complex models cannot always
be reliably constrained using moderate-resolution X-ray spectra. We
have only considered simple one-component models. Technically, the
presence of a second component could skew the derived ejecta, but this
should tend to increase the element abundances. We found that if we
did attempt to add a second thermal component representing emission
behind the blast wave (hence with the blast wave abundances fixed at
0.4 solar), the fitted ejecta abundances increased overall by a factor
of two or so; the temperature and ionization age for the Si Knot did
not change much, increasing by no more than 10-20\%. We chose not to
use two-component fits throughout because the spectra generally cut
off above 2-3 keV, and we considered it difficult to put believable
constraints on a second component. In any case, this is a
conservative assumption for identifying ejecta enhancements, as we
have pointed out.
In spite of the remnant's advanced age, the ejecta in this mature
remnant are tantalizing to pursue for the hints it may give to the
explosion that formed it. Ideally, better angular resolution is
desirable since Chandra observations (Hwang et al. 2005) have hinted
at relatively small localized regions with abundance enhancements.
These may contribute to the patchy, low-level enhancements seen with
Suzaku. Aside from the X-ray emitting Si Knot, the remaining Si may
either already be shocked and cooled, shocked and diluated, or still
unshocked. If the former, we might hope to see it in optical
emission, but optical [S II] emission in Puppis A resembles H$\alpha$
emission rather than the X-ray Si emission, and appears more likely to
be associated with blast wave emission (P. F. Winkler, private
communication). Shocked and diluted Si will be difficult to detect
with the significant dilution by the ISM expected for Puppis A after
4000 years. Roughly 0.1 M$_\odot$ of Si is present in a few hundred
solar masses of interstellar material at 0.4 solar abundances.
Unshocked ejecta might be revealed by infrared observations, but the
available data are unfortunately insufficient for identifying ejecta
(Arendt et al. 1990). We hope that future observations will shed
interesting light on the ejecta in Puppis A.
\acknowledgments
We have benefitted from scientific discussions with Satoru Katsuda and
other collaborators on the XMM-Newton observation, and from a careful
reading by an anonymous referee. We are grateful to Hideyuki Mori for
expert help in configuring the observations, Mark Schattenburg, Dale
Graessle, and Claude Canizares for discussion of the FPCS observations
and calibration, Takashi Okajima for discussion of the Suzaku
mirrors, and Frank Winkler for sharing his optical images.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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\section{Introduction}
In this paper we consider nonlocal cell-cell adhesion models on a multidimensional bounded domain. Cellular adhesion is a fundamental feature of multicellular organisms with relevance for e.g., embryogenesis, wound healing and self-organization. The regulation of cellular adhesion is critical, too, for cancer growth and spread. \\
\indent
Mathematical models of cancer invasion have been widely studied for decades; the proliferation and movement of tumour cells are governed by random diffusion, aggregations and reactions which are often of logistic type \cite{ACN,BLM,CA,CL,PB}. In most cancer migration models, aggregations arise from the cell-cell and cell-ECM (extracellular matrix) adhesions, movement of cells in response to stimulii of diffusing chemicals (chemotaxis), and movement of cells in response to non-diffusing environmental factor such as ECM (haptotaxis). There have been numerous mathematical analysis studies of parabolic or parabolic-ellipic-ODE system consisting of diffusion, aggregation and reaction for describing tumour dynamics; they addressed local and global well-posedness, blow-up, asymptotic behavior etc. on the whole space or space with boundary. {See \cite{LMR10,MR08,SSW14,TW11,TM201,TM20,TW08} and references therein}.
\\
\indent
In this paper we study a two-populations cancer model \eqref{MODEL01} set in a multidimensional bounded domain and focused on nonlocal cell-cell adhesion. The first successful continuum model of cellular adhesion was proposed by Armstrong et al. in \cite{APS06}, where
they introduced a nonlocal integral term to describe the adhesive forces between cells. The basic single population model on the real line, for instance, is derived to be
\begin{align}\label{Single} \partial_t u = u_{xx}- \al \left( u(x,t) \int_{-R} ^{R} g(u(x+r)) { \omega }(r) dr \right)_x
\end{align}
where $\al$ is the strength of cell adhesion, $g(u)$ describes the nature of the adhesion force, ${ \omega }(r)$ is a function describing the direction and magnitude of the adhesion force, and $R$ sensing radius of a cell.
In the same paper a two-dimensional version of \eqref{Single} involving two cell populations was also proposed with numerical simulations supporting the active cell-sorting process from the randomly distributed mixture. The model was the first to reproduce Steinberg's cell segregating experiment \cite{Stein}, which is a classical result in developmental biology.
\\
\indent
The regulation of cellular adhesion is critical in cancer formations as well. Naturally \eqref{Single} has been extensively used to model cancer cell invasion and developmental processes \cite{ANCH,APS09,BCE17,DTGC,GC08,GP10,PBSG,PAS10,SGAP}. Cauchy problems were studied in \cite{ANCH,EPSZ,HB19,HPW17,SGAP} {\it e.g.}, among which \cite{HPW17} obtained general results on local and global well-posedness of classical solutions to a version of \eqref{Single} in $\mathbb R^n$.
\\
\indent
On the other hand, there are only a few studies of the adhesion model posed on the bounded domain \cite{BCE17,EPSZ, HB19}. Related to our result, Hillen and Buttensch\"on {were} the first to consider the well-posedness of the initial-boundary problem of \eqref{Single} with the {explicit} sensing domain under the independent boundary condition in \cite{HB19}, where they constructed a bounded solution globally in time. Recently, Eckardt, Painter, Surulescu and Zhigun considered more general nonlocal models involving adhesion or nonlocal chemotaxis under the nonlocal Robin boundary condition in \cite{EPSZ} (For more complete review of the recent advancements of the nonlocal models, please see \cite{CPSZ}). It seems more realistic to consider the cell adhesion in a bounded domain, since cells interact with the boundaries of the regions in which they are contained.
In this paper we extend the work from \cite{HB19} to a multidimensional bounded domain with two kinds of boundary conditions satisfying a no-flux requirement.\\
\indent
As mentioned in \cite{HB19}, there is another class of non-local models so called {\it aggregation equation}, where the non-local term has a singular interaction kernel in general.
The aggregation equation arises as a gradient flow of a potential, and the well-posedness and blow-up features of the equations are extensively studied {\it{e.g.}} in \cite{BLR, BL}. Recent studies of such equation on a bounded domain can be found in \cite{FK,WS}. \\
\indent
Let $u(x,t)$ and $v(x,t)$ denote the two types cancer cell population densities at spatial location $x\in \Omega$ and time $t$, where $\Omega \subset{ \mathbb{R} }^{n}$, $n\ge2$, is a bounded domain. Following the simplifying assumptions in \cite{APS06}, we assume that the cell adhesion force is linearly proportional to the population density and that both cell types have the same sampling radius $R$; we set the velocities of nonlocal adhesions between two population as
\begin{align}\label{MODEL01}
\begin{aligned}
\mathcal{K}[u,v](x,t)& =\int_{E(x)}\bkt{ M_{11}u(x+y,t)+ M_{12}v(x+y,t) }\omega(y)dy,
\\
\mathcal{S}[u,v](x,t)& =\int_{E(x)}\bkt{ M_{21}u(x+y,t)+ M_{22}v(x+y,t) }\omega(y)dy,
\\
E(x)&: \mbox{ sensing domain depending on $x$, to be specified later, }
\\
\omega&: =(\omega_{1},\cdots, \omega_{n}) \mbox{ for } \, \omega_{i} \,\mbox{ bounded}.
\end{aligned} \end{align}
The positive constants $M_{11}$ and $M_{22}$ represent the self-adhesive strength of populations $u$ and $v$, respectively, and the positive constants $M_{12}$ and $M_{21}$ represent the cross-adhesive strengths between the populations. \\
\indent
Below, we set the two initial-boundary problems on $u(x, t)$ and $v(x,t)$ according to boundary conditions.\\
{\bf I.} Nonlocal Robin boundary condition\\
Let $\Omega$ be a $C^2$ smooth bounded domain and ${\mathcal K}[u, v]$, ${ \mathcal S }[u,v]$ be defined as in \eqref{MODEL01}. We consider the system
\begin{equation}\label{MODEL00}
\left\{
\begin{array}{ll}
\vspace{1.5mm}
\partial_t u - \Delta u= -\nabla \cdot (u \,\mathcal{K}[u,v] )-mu +\displaystyle\frac{\lambda}{k} u(k-(u+v)), \qquad& x\in \Omega,\,\, t>0, \\
\vspace{1.5mm}
\partial_t v - \Delta v= -\nabla \cdot (v \,\mathcal{S}[u,v] )+mu +\displaystyle\frac{\mu}{k} v(k-(u+v)), \qquad& x\in \Omega,\,\, t>0, \\
\partial_{\nu} u =u\mathcal{K}[u,v]\cdot \nu,\quad \partial_{\nu} v=v\mathcal{S}[u,v]\cdot \nu, \qquad& x\in\partial \Omega,\,\, t>0, \\
u(x,0)=u_{0}(x),\,\, v(x,0)=v_{0}(x),\quad &x\in \Omega,
\end{array}
\right.
\end{equation}
where $\nu(\cdot)$ is the $C^1$-smooth unit outward normal vector field to $\partial\Omega$.
The positive constants $m,k,\lambda$, and $\mu$ denote the mutation rate, the carrying capacity, the growth rate of $u$, and the growth rate of $v$. The sensing domain $E(x)$ is given by
\begin{equation}\label{sensing1} E(x) = \bket{ y\in { \mathbb{R} }^{n} \,|\, x+y\in\Omega,\,\,|y|<R }, \end{equation}where $R$ is a fixed sensing radius. Moreover we assume
\begin{equation}\label{omega1}
\omega=(\omega_{1},\cdots, \omega_{n}) \mbox{ for } \omega_{i} \mbox{ bounded and smooth. }
\end{equation}
\indent
The equations \eqref{MODEL00} can be seen as a similar but simplified version of the model considered in \cite{BCE17},
describing dynamics of two type {of} cancer cells on a bounded domain in $\mathbb R^n$. In \cite{BCE17} the terms $\mp m u$ refer that healthy or early stage cancer cells mutate into later stage cells at a constant rate $m$. See \eqref{GENMODEL} in Remark \ref{Rmk1} for a full equations considered in \cite{BCE17}.
\indent
On the other hand the similar reaction terms in \eqref{MODEL00} can be found in two-phenotypes cancer model undergoing
{\it epithelial -mesenchymal transition }(EMT). In this case we interpret $u$ and $v$ as the density of epithelial tumor cells and the density of mesenchymal tumor cells respectively. The terms $\mp m u$ stand for EMT that epithelial phenotype cells undergo to acquire the ability to migrate into surroundings \cite{YY}.
Following this kind of view, the equation \eqref{MODEL00} should be understood to consider a local cancer cell invasion step according to terms in \cite{FLBC} and assumes the EMT takes place only. The logistic competition terms $u(k-(u+v))$ and $v(k-(u+v))$ describe the production of $u$ and $v$ are according to logistic law and they compete for free space. Two-species cancer model with haptotactic invasion undergoing EMT is analyzed in \cite{DL20, GI}. For more details on the model we refer to references therein.
Note that we require the total flux zero on the boundary in \eqref{MODEL00}, which leads to a nonlocal boundary condition of Robin type,
\begin{equation}\label{totalzero} \partial_{\nu} u -u\mathcal{K}[u,v]\cdot \nu=0,\quad \partial_{\nu} v-v\mathcal{S}[u,v]\cdot \nu=0, \qquad x\in\partial \Omega,\,\, t>0.
\end{equation}
One way to attain the total flux zero condition is to assume for the solution of \eqref{MODEL00} to satisfy
\begin{align}\label{ind1}
&\partial_{\nu} u = 0, \quad \partial_{\nu} v = 0 \quad x\in \partial{ \Omega }, \\ \label{ind2}
&{\mathcal K}[u,v] =0, \quad { \mathcal S }[u,v] =0 \quad x\in \partial{ \Omega }.
\end{align}
on the boundary, which is the boundary condition used in \cite{BCE17, HB19}. The first condition is the usual Neumann zero condition given on the solutions $u$ and $v$, however, \eqref{ind2} is the condition imposed to the nonlocal operators ${\mathcal K}$ and ${ \mathcal S }$.
Following \cite{HB19}, we call \eqref{ind1}, \eqref{ind2} {\it independent} or {\it zero-zero flux condition}.
An independent case allows the constant solution
\begin{equation}\label{constant}
u\equiv 0, \quad v\equiv k.
\end{equation}
which is one of two non-negative constant solutions. The other one is $(0, 0)$.\footnote{ There is the other steady state,
$u^{*}=-\frac{\mu}{\la}v^{*}, v^{*}=\frac{k(1-\frac{m}{\la})}{1-\frac{\mu}{\la}}$, which
are of different signs, hence unrealistic.} The $\mp m u$ terms in \eqref{MODEL00} suggest that the later stage population dominates the total cell population if the mutation rate $m$ is large. The linear asymptotic stability around $(0, k)$ is considered in Appendix. The linear stability analysis for the related one dimensional model is performed in \cite{BE}.\\
\indent
We shall study the {\it independent } case with an explicit example of the sensing domain $E(x)$ satisfying $ |E(x)| = 0 $
as $x$ approaches to $\partial { \Omega }$ by which the condition \eqref{ind2} is assured. Note that the volume of \eqref{sensing1} cannot vanish on the boundary. We formulate the second initial-boundary problems under \eqref{ind1}--\eqref{ind2} as follows.
\\
{\bf II.} Zero-zero flux condition\\
\indent Let $\Omega $ be the open ball of radius $L$, $B_L(0)$, and ${\mathcal K}[u, v]$, ${ \mathcal S }[u,v]$ be defined as in \eqref{MODEL01}. We have \begin{equation}\label{MODEL11}
\left\{
\begin{array}{ll}
\vspace{1.5mm}
\partial_t u - \Delta u= -\nabla \cdot (u \,\mathcal{K}[u,v] )-mu +\displaystyle\frac{\lambda}{k} u(k-(u+v)), \qquad& x\in \Omega,\,\, t>0, \\
\vspace{1.5mm}
\partial_t v - \Delta v= -\nabla \cdot (v \,\mathcal{S}[u,v] )+mu +\displaystyle\frac{\mu}{k} v(k-(u+v)), \qquad& x\in \Omega,\,\, t>0, \\
\partial_{\nu} u = 0, \quad \partial_{\nu} v = 0, \quad
{\mathcal K}[u,v] =0, \quad { \mathcal S }[u,v] =0, \qquad& x\in \partial { \Omega }\,\, t>0,
\\
u(x,0)=u_{0}(x),\,\, v(x,0)=v_{0}(x),\quad &x\in \Omega,
\end{array}
\right.
\end{equation}
where $m, k, \mu,\la$ are same as before in case {I}, and $\nu(\cdot)$ denotes the unit outward normal vector to $\partial\Omega$. The sensing domain $E(x)$ is given for $0<R<L$ by
\begin{equation}\label{sensing2}
E(x) = B_R(0) \quad \mbox{ for } |x| < L-R, \quad E(x)= B_{L-|x|}(0) \mbox{ for } L-R \le |x| < L,
\end{equation}
and we assume
\begin{equation}\label{omega2}
\omega(x):=\displaystyle\frac{x}{|x|}w(|x|)
\end{equation}
for $ w\in \mathcal{C}^{\infty}_{0}(\Omega)$ non-negative.
In \eqref{sensing2} we choose $E(x)$ as simple as possible to satisfy the following property ; when $x$ is away from the boundary of the domain, $E(x)$ is $B_R(0)$ as was set for two dimensional model in \cite{APS06} . When $x$ is close to the boundary, it shrinks to a smaller region such that $ x + {\mathbf r }\in { \Omega }$ for ${\mathbf r} \in E(x)$ and $|E(x)| = 0$ as $x$ reaches to the boundary.
The choice of varying integration domain $E(x)$ affects the extent of regularity of the adhesion terms ${\mathcal K}[u,v]$ and ${ \mathcal S }[u,v]$. In Lemma \ref{LEMKS} and Lemma \ref{LEMKS1}, we only show ${\mathcal K}[u,v]$ and ${ \mathcal S }[u,v]$ to be Lipschitz continuous however smooth $u$, $v$, $\omega$ are.
\footnote{ For case {II}, we could change the shrinking rate and shape of $E(x)$ as $x$ approaches the boundary so that the regularity of adhesion terms is possibly worse.}\\
\indent In case {I} the nonlocal nonlinear Robin type boundary condition as well as the restricted regularity of ${\mathcal K}[u,v]$ and ${ \mathcal S }[u,v]$ cause difficulty in constructing local-in-time strong solutions directly by iterations. For the local well-posedness we take two steps; we first construct the generalized solution
relying on the semi-group theory for parabolic boundary value problem found in Amann's seminar works \cite{A86,AM93}.
In particular we introduce a certain extension of the unit outward normal vector field, and employ the {\it generalized variation-of-constants formula} for this case. See Section~\ref{SEC21}, and Section~\ref{SEC22} for details. A similar construction can also be found in \cite{DGM-RS10,DM-RST10}. We can show the generalized solutions are indeed strong and satisfy the maximal regularity estimates employing the result of Denk-Hieber-Pr\"uss \cite{DHP07}. The global well-posedness follows from several {\it a priori } estimates and Moser-Alikakos type estimate.
\\
\indent
{In the zero-zero flux case the diffusive flux and the adhesion flux are independently zero on the boundary.
The global well-posedness is obtained in a standard way though ${\mathcal K}[u,v]$ and ${ \mathcal S }[u,v]$ are less regular than the adhesion terms in the nonlocal Robin type boundary case due to the shrinking sensing domain $E(x)$. As mentioned earlier, this case allows the constant steady state $(0,0)$ and $(0,k)$. We provide a linear stability analysis in the Appendix, where $(0,0)$ is found linearly unstable, and $(0,k)$ is linearly asymptotically stable if $m> \mu$. Also we
find that a Lyapunov type inequality holds when $v$ has no adhesion term and $\mu> \la$, see Remark $1$.}
\\
\indent
We are ready to state the main results in this paper. We first provide the global strong solvability for case {I}.
\begin{theorem}\label{GETHM}
Let $\Omega \subset{ \mathbb{R} }^{n}$, $n\ge2$, be an open bounded domain with $C^2$ boundary $\partial \Omega$. Suppose that the non-negative initial conditions $u_{0}$ and $v_{0}$ belong to $W^{2,p}(\Omega)$, $n<p <\infty$ and satisfy the compatibility condition
\[
\partial_{\nu}u_{0}-u_0\mathcal{K}[u_0,v_0]\cdot \nu=0,\qquad \partial_{\nu}v_{0}-v_0\mathcal{S}[u_0,v_0]\cdot \nu=0,\qquad\quad x\in\partial{\Omega}.
\]
Then, \eqref{MODEL00}--\eqref{omega1} admits a unique non-negative strong solution $(u,v)$ such that
\[
u,v\in C([0,t);W^{1,p}(\Omega))\cap W^{1,p}(0,t; L^{p}(\Omega))\cap L^{p}(0,t; W^{2,p}(\Omega)), \qquad t>0.
\]
Moreover, the solution $(u,v)$ has a boundedness property
\[
\sup_{t>0}( \| u(\cdot,t) \|_{L^{\infty}(\Omega)}+ \| v(\cdot,t) \|_{L^{\infty}(\Omega)} )\le C.
\]
\end{theorem}
We next state the global strong solvability for case {II}.
\begin{theorem}\label{GETHM1}
Let $\Omega\subset{ \mathbb{R} }^{n}$, $n\ge2$ be an open ball of radius $L$, $B_L(0)$. Suppose that the non-negative initial data $u_{0}$ and $v_{0}$ belong to $W^{2,p}(\Omega)$, $n<p<\infty$, and satisfy the compatibility condition
\[
\partial_{\nu}u_{0}=\partial_{\nu}v_{0}=0,\qquad x\in\partial{\Omega}.
\]
Then, \eqref{MODEL11}--\eqref{omega2} admits a unique non-negative strong solution $(u,v)$ such that
\[
u,v\in C([0,t);W^{1,p}(\Omega))\cap W^{1,p}(0,t; L^{p}(\Omega))\cap L^{p}(0,t; W^{2,p}(\Omega)), \qquad t>0.
\]
Moreover, the solution $(u,v)$ has a boundedness property
\[
\sup_{t>0}( \| u(\cdot,t) \|_{L^{\infty}(\Omega)}+ \| v(\cdot,t) \|_{L^{\infty}(\Omega)} )\le C.
\]
\end{theorem}
\indent The paper is organized as follows.
We prove Theorem \ref{GETHM} in Section \ref{SEC2} and Theorem \ref{GETHM1} in Section \ref{SEC3}. Both sections start from proving preliminary results about the adhesion velocities
${\mathcal K}[u,v]$, and ${ \mathcal S }[u,v]$. The local well-posedness of \eqref{MODEL00}-\eqref{omega1} is proved in Lemma \ref{LEM1}.
The global well-posedness follows from the blow-up criteria (Lemma \ref{BUCLEM} and Lemma \ref{LEMLINFUV}), where the solutions are found to be uniformly bounded in the $\| \cdot \|_{{L^{\infty}(\Om)}}$ norm. Section \ref{SEC3} is organized analogously; the local well-posedness of \eqref{MODEL11}-\eqref{omega2}
in Lemma \ref{LEM4} and the uniform boundedness in Lemma \ref{LEMLINFUV1}. Finally we provide a linear stability analysis in the Appendix for the steady state $(0,0)$, $(0, k)$ of the zero-zero case.
Before closing this section we write down the related nonlocal $($local$)$ adhesion models arising in cancer cell invasion \cite{BCE17, DL20, EPSZ} and explain simplifications and differences of our model compared to those studied in \cite{BCE17}.
\begin{remark}\label{Rmk1}
A class of nonlocal(local) adhesion models proposed in \cite{BCE17}$($\cite{DL20}$)$ respectively
can be written in an integrated manner as
\begin{equation}\label{GENMODEL}
\left\{
\begin{array}{ll}
\vspace{1.5mm}
\partial_t u_{1} = D_{1}\Delta u_{1} -\nabla \cdot (u_{1} \, F_{1}[u_{1},u_{2},f,c,b] )-mu_{1} +G_{1}(u_{1}, u_{2},f,b), \\
\vspace{1.5mm}
\partial_t u_{2} =D_{2}\Delta u_{2} -\nabla \cdot (u_{2} \,F_{2}[u_{1},u_{2},f,c,b] )+mu_{1} +G_{2}(u_{1}, u_{2},f,b),\\
\vspace{1.5mm}
\partial_t f = -\alpha_{1}u_{1}f-\alpha_{2}u_{2}f \pm \alpha_{3}bf+\alpha_{4}f(\alpha_{5}-f),
\\
\vspace{1.5mm}
\partial_t c = \beta_{1}u_{1}+\beta_{2}u_{2}+ \beta_{3}bc- \beta_{4}c,
\\
\vspace{1.5mm}
\partial_t b = D_{3}\Delta b+\mu_{1}u_{1}+\mu_{2}u_{2}-\mu_{3}b.
\end{array}
\right.
\end{equation}
Bitsouni et al. \cite{BCE17} considered the system \eqref{GENMODEL} with nonlocal adhesion forces $F_{i}$, $i=1,2$. More precisely
\begin{equation}\label{remarkNA} F_i[u_1, u_2, f, c, b]:= \int_{{ \Omega }} K(|y-x|)g_i(u_1, u_2, f, c, b)(y) dy \end{equation}
for a kernel $K_i \in (W^{1, \infty}({ \Omega }))^5$, where $g_i$ describes the nature of cell-cell and cell-matrix adhesive forces, chosen to be a linear function in $u_i$ and $f$ with bounded coefficients. The system describes the dynamics of two cancer cell populations coupled with ordinary differential equations describing the ECM (extracellular matrix) degradation, the production and decay of integrins $c$, and with an equation governing the evolution of the TGF-$\beta$ concentration. We denote
by $u_1, u_2, f, c, b$ the density of healthy or early stage cancer cell population, the density of late stage cancer cell population,
the ECM density, the level of integrins, and TGF-$\beta$ concentration. $\al_i, \be_i, \mu_i$ are given parameters. The first two equations also refer to homogeneous and heterogeneous cancer invasions. The movement of early stage cell is due to random diffusion and directed motility in response to adhesive forces, and the movement of late stage cell due to directed motility only $(D_2=0)$, exhibiting a heterogeneous type invasion. Note that $u_1$ mutates into $u_2$ at a constant rate $m$ in the model. On the boundary of the domain ${ \Omega }$, they assumed
\begin{align}\label{bce17bd}
\nabla u_i\cdot \nu = F_i\cdot \nu = \nabla b \cdot \nu = 0, \quad i=1, 2.
\end{align}
On the other hand,
Dai and Liu \cite{DL20} considered the system \eqref{GENMODEL} with local adhesion forces $F_{i}=\chi_{i}\nabla f$, $i=1,2$ and $c\equiv0$, where they modeled two species cancer invasion of surrounding healthy tissue. More precisely, their model involves the epithelial-mesenchymal transition $($EMT$)$ from epithelial phenotype cancer cells $(u_1)$ to mesenchimal phenotype cells $(u_2)$ at a constant rate $m$, the migration of two cancer cell populations due to random diffusion $D_{i}>0$, $i=1,2$, and haptotaxis $F_{i}$, $i=1,2$, the proliferation of cancer cells $G_{i}$, $i=1,2$, and their competition for space with ECM $(f)$, the production and decay of matrix metalloproteinases $(b)$, the degradation of ECM by matrix metalloproteinases upon contact, $-\alpha_{3}bf$, etc. On the boundary of the domain, they assumed $D_{i}\nabla u_{i}\cdot\nu-\chi_{i}u_{i}\nabla f\cdot\nu=\nabla b\cdot \nu=0$, $i=1,2$.
As is mentioned earlier, Eckardt et al. proposed the nonlocal models \footnote
{Eckardt et al. considered averaging nonlocal operators $\mathcal A_r$ and $\mathring{\nabla_r}$. Following notations in this paper, we have
\[\mathcal A_Ru(x) = \frac{1}{R}\frac{1}{|B_R(0)|} \int_{E(x)} u(x+y) \frac{y}{|y|} w(|y|) dy= \frac{1}{M_{11}R}\frac{1}{|B_R(0)|}\mathcal K [u,0] \]
by choosing ${ \omega }(x)=\frac{y}{|y|} w(|y|)$ in \eqref{MODEL01}.
}
describing
cell-cell and cell-tissue adhesions or nonlocal taxis (for a single cell phenotype) in \cite{EPSZ}. They proved global existence of weak-strong solution and investigated solution properties for such complex systems, including non-linear motility coefficients. Moreover they verified a rigorous limit procedure linking nonlocal adhesion models and their local counterparts.
\end{remark}
\begin{remark}\label{Rmk2}
A global in time existence of a weak solution of the parabolic-hyperbolic system \eqref{GENMODEL} was shown in
\cite{BCE17} under the zero-zero flux condition \eqref{bce17bd}. In this paper we simplify the system into \eqref{MODEL00} or
\eqref{MODEL11} by ignoring the cell-matrix adhesion force as well as effects of integrins and TGF-$\beta$ protein. Instead, we focus on mathematical analysis on nonlocal cell-cell adhesion under either the nonlocal Robin boundary condition,
which was not addressed in \cite{BCE17} or \cite{HB19}, or the zero-zero flux condition \eqref{bce17bd}.
The same analysis works as well without difficulty if we include the ECM equation and consider the nonlocal adhesion forces in the form of
\eqref{remarkNA} with $c=b=0$. In the zero-zero conditions, $F_i\cdot \nu=0$
is not usual since it is imposed on the nonlocal operator $F_i$, not on the solutions. In Section $3$ we work with a concrete
sensing domain to find that such boundary condition indeed makes sense. However, the choice of $E(x)$ seems made arbitrary and we have not yet found a more meaningful choice connected to the biological context considered here.
Apart from the above simplifications, we assume $D_2$ strictly positive in the $u_2$-equation. Compared with \cite{BCE17}, we are interested in the boundary value problems with nonlocal Robin boundary condition or zero-zero flux condition and strong solutions for the boundary value problems. It is not clear to us whether the similar methods of our research can be applied for the case $D_2=0$. When $D_2=0$, Lagrangian coordinates $($\cite{DA, FU}$)$ seem to be more useful to construct a bounded solution along the characteristic curve if $u_1, u_2$ have same directed motility or velocity, $F_1, F_2$, which corresponds to
$M_{11}=M_{22}$, $M_{12}=M_{21}$ in \eqref{MODEL01}.
\end{remark}
\section{Nonlocal Robin boundary case}\label{SEC2}
\begin{subsection}{Preliminary}\label{SEC21}
We first consider a lemma on the extension $\mathcal{N}$ of the normal vector field $\nu$ into the domain $\Omega$. The extension $\mathcal{N}$ is used in Section~\ref{SEC22} to interpret our nonlocal Robin boundary value problems as inhomogeneous Neumann boundary value problems.
The lemma holds for a bounded domain with $C^k$-smooth boundary for any integer $k\ge 2$ without difficulty. Before stating the lemma, we remind the definition of a $C^k$-boundary in \cite[Appendix C.1]{EVANS}.
\begin{defn}\label{defbound}
Let $U \subset { \mathbb{R} }^n$ be an open bounded domain, and $k\in \{1, 2, \dots\}$. We call $\partial U$ is $C^k$ if
for each point $x^0 \in \partial U$ there exist $r >0$ and a $C^k$ function $\xi: \mathbb R^{n-1} \to \mathbb R$
such that upon an orthogonal change of coordinates we have
\[
U\cap B_r(x^0)= \{ x\in B_r(x^0) \, |\, x_n > \xi(x_1, \dots, x_{n-1})\}.
\]
\end{defn}
\begin{lemma}\label{LEMNU}
Let $\Omega \subset{ \mathbb{R} }^{n}$, $n\ge2$, be an open bounded domain with $C^2$ boundary $\partial \Omega$. Then, there exists at least one vector field $ \mathcal{N}\in C^{1}(\overline{\Omega})$, a continuous extension of the unit outward normal vector field $\nu$, such that
$\mathcal{N}=\nu$ on $\partial\Omega$.
\end{lemma}
\begin{proof}
For $x^0 \in \partial { \Omega }$ there is $r, \xi$, and a relabeled tuple $(x_1, \dots, x_n)$ as in the definition such that $\partial U \cap B_r(x^0) = \{ x\in B_r(x^0) \,|\, \xi(x_1, \dots, x_{n-1}) -x_n = 0\}$.
The outward normal vector field $\nu(x)$ is well defined by
\[ \nu(x) = \frac{ (\nabla \xi(\bar x), -1)}{ \sqrt{ 1 + |\nabla \xi(\bar x)|^2}}\]
where $x= (\bar x, x_n)$.
\\
\indent Let us consider the function $E: \partial { \Omega } \times \mathbb R \to \mathbb R^n$ given by
\[ E(x, t) = x - t\nu(x). \]
Applying the tubular neighborhood theorem in Cahpter $10$ of \cite{Lee} there exist $\rho>0$ such that $E$ is the $C^1$-diffeomorphism \footnote {We follow a definition of $C^1$-functions on a submanifold $\partial { \Omega } \times \mathbb R$ embedded in $\mathbb R^n \times \mathbb R$. The open set $E(V)$is called a tubular neighborhood of $\partial { \Omega }$,
} on
$ V: =\{(x, t) \in \partial { \Omega } \times \mathbb R \,|\, |t| < \rho\}$ and
$T_{\rho}:=\{ E(x, t) | x\in \partial { \Omega }, 0 < t <\rho \}$ is in ${ \Omega }$.
We find a smooth extension of $\nu$ on $\overline { \Omega }$ as follows. For $y\in T_{\rho}$
there is a unique $(x, t) \in \partial { \Omega } \times \mathbb R$ such that $ y = x -t \nu(x)$, since $E$ is one to one. Note that the mapping $y \to (x, t) $ is $E^{-1}$, hence $C^1$. We define a continuous extension of $\nu$ by
\[ { N} (y) =
\begin{cases} \frac{\rho -t}{\rho} \nu(x)\quad \mbox{ for } y\in T_{\rho} \\
0 \quad \quad \quad \quad \mbox{ for } y \in { \Omega } \setminus T_{\rho}.
\end{cases} \]
$N$ being $C^1$ on $T_{\rho}$ follows from the construction; Define $\pi_1 : \partial { \Omega } \times \mathbb R \to \partial { \Omega } $ to be the projection onto the first slot, and $\pi_2$ onto the second slot. Then we can write
\[ N(y) = \frac{\rho - \pi_2 \circ E^{-1}(y)}{\rho } \nu ( \pi_1\circ E^{-1}(y))\]
with $\nu(x) = \frac{ (\nabla \xi(\bar x), -1)}{ \sqrt{ 1 + |\nabla \xi(\bar x)|^2}}$ upon an orthogonal change of coordinates.
Finally smoothing out $N$ in $T_{\rho} \setminus T_{\frac{\rho}{2}}$ we obtain a smooth extension ${\mathcal N}$ of $\nu$ on $\overline { \Omega }$.
\end{proof}
Before closing the section we prepare a $W^{1,\infty}(\Omega)$ estimate for the non-local terms $\mathcal{K}$ and $\mathcal{S}$.
\begin{lemma}\label{LEMKS}
Let $\Omega \subset{ \mathbb{R} }^{n}$, $n\ge2$, be an open bounded domain with $C^2$ boundary $\partial \Omega$. Suppose that $f$, $g\in L^{1}(\Omega)$.
Assume further that $\mathcal{K}[f,g]$, $\mathcal{S}[f,g]$, $E$, and $\omega$ are given by \eqref{MODEL01}, and
\eqref{sensing1}--\eqref{omega1}.
Then, there exists a constant $C>0$ satisfying
\[\norm{\mathcal{K}[f,g]}_{W^{1,\infty}(\Omega)}+\norm{\mathcal{S}[f,g]}_{W^{1,\infty}(\Omega)}\le C(\norm{f}_{L^{1}(\Omega)}+ \norm{g}_{L^{1}(\Omega)}).
\]
\end{lemma}
\begin{proof}
It suffices to prove the Lipschitz continuity of
\[
\mathcal{I}[f](x)=\displaystyle\int_{E(x)} f(x+y) \omega(y)dy,
\]
where $E(x)$ and $\omega$ are introduced in \eqref{sensing1}--\eqref{omega1}. By change of variable, $\mathcal{I}[f](x)$ can be written as
\[
\mathcal{I}[f](x)=\int_{V_{x}}f(z) \omega(z-x)dz,\qquad V_{x}=\bket{ z\in\Omega \,|\,|z-x|<R }.
\]
Choose any two points $x, y \in { \Omega }$ and let $h= y-x$. We have
\[
\mathcal{I}[f](x+h)-\mathcal{I}[f](x)=\int_{V_{x+h}} f(z)\omega(z-x-h) dz-\int_{V_{x}} f(z)\omega(z-x) dz
\]
\[
=\int_{V_{x+h}} f(z)\bkt{\omega(z-x-h)-\omega(z-x)} dz+\int_{V_{x+h}} f(z)\omega(z-x) dz-\int_{V_{x}} f(z)\omega(z-x) dz.
\]
We estimate
\[
\biggr{|}\int_{V_{x+h}} f(z)\bkt{\omega(z-x-h)-\omega(z-x)} dz\biggr{|}\le C|h|\norm{f}_{L^{1}(\Omega)}\norm{\omega}_{W^{1,\infty}(\Omega)}.
\]
For the the other terms, we note that
$ (V_{x+h}\cup V_{x}) \setminus (V_{x+h}\cap V_{x})$ is a subset of $B_R(x+h) \cup B_R(x) \setminus B_R(x+h) \cap B_R(x)$ in $\mathbb R^n$ and the volume of the latter is bounded by $C |h|$ with a uniform constant $C$ if $y= x+h$ are in $B_R(x)$. We have
\[
\biggr{|}\int_{V_{x+h}} f(z)\omega(z-x) dz-\int_{V_{x}} f(z)\omega(z-x) dz\biggr{|}\le \norm{f}_{L^{1}(\Omega)}\norm{\omega}_{L^{\infty}({ \Omega }) }\int_{(V_{x+h}\cup V_{x}) \setminus (V_{x+h}\cap V_{x})}1dz
\]
\[
\le C|h|\norm{f}_{L^{1}(\Omega)}\norm{\omega}_{L^{\infty} (\Omega)},
\]
hence
it follows that
\[
\bigr{|}\mathcal{I}[f](x+h)-\mathcal{I}[f](x)\bigr{|}\le C|h|\norm{f}_{L^{1}(\Omega)}\norm{\omega}_{W^{1,\infty}(\Omega)}.
\]
This completes the proof.
\end{proof}
\end{subsection}
\begin{subsection}{Local well-posedness and blow-up criteria}\label{SEC22}
To prove a local well-posedness of \eqref{MODEL00}--\eqref{omega1}, we shall employ {\it a generalized variation-of-constants formula} \eqref{INTF} for Neumann type parabolic boundary value problem established by Amann (\cite{A86,AM93} {\it e.g.}).
Below, we introduce the interpolation scale of spaces $E_{d}$ and the set of linear operators $A_{\delta}$ needed for writing down the formula. We work with $E_{d} $ and $A_{\delta}$ for $-1< d, \delta <0$, while they can be defined for $d, \delta>-1$. \\
\indent
We denote the boundary trace operator by $\gamma$. Note that
\begin{equation}\label{TRNOT}
\gamma\in\mathcal{L}( W^{1,p}(\Omega), W^{1-\frac{1}{p},p}(\partial\Omega)) \mbox{ for } 1< p<\infty.
\end{equation}
We also denote the boundary trace of normal derivative by $B$;
\[
Bf:= \ga \nabla f\cdot \nu \, \mbox{ for }\, f\in W^{s, p}({ \Omega }),\,1+\frac{1}{p}<s\le 2,\, 1<p<\infty.
\]
Consider the sectorial operator $A:= {(I-\Delta)}|_{D(A)}$ with its domain
\[
D(A):=\bket{f\in W^{2,p}(\Omega) \, | \, Bf=0\,\mbox{ on }\, \partial\Omega }\, \mbox{ for }\, 1<p<\infty.
\]
Note that
$ \inf Re\, \Sigma(A)>0$, where $\Sigma(A)$ is the spectrum of $A$.
In \cite[Section 4]{AM93} the pair $(A, B)$ is found to satisfy the condition of {\it normally elliptic problem}, which enables one to construct an interpolation scale of spaces.
Let
\[
W^{s,p}_{B}:=\left\{
\begin{array}{lll}
\bket{f\in W^{s,p}(\Omega) \, | \, B f=0 }, \quad &\,\,\,\,1+\frac{1}{p}<s\le 2, \\
W^{s,p}(\Omega),\quad &-1+\frac{1}{p}<s<1+\frac{1}{p}, \\
(W^{-s,p'}(\Omega))', \quad &-2+\frac{1}{p}<s\le -1+\frac{1}{p},
\end{array}
\right.
\]
and take $E_{0}=L^{p}(\Omega)=W^{0,p}_{B}$, $E_{1}=W^{2,p}_{B}$. Following \cite[Section 6]{AM93},
we construct an interpolation scale of spaces
\[
E_{\theta}:=(E_{0},E_{1})_{\theta,p}=W^{2\theta,p}_{B}\] for $2\theta\in(0,2)\setminus\{1,1+\frac{1}{p}\}$, where $(\cdot,\cdot)_{\theta,p}$ denotes the real interpolation functor.
Introducing a completion of the normed space $(E_{0},\|A^{-1}\cdot\|_{E_{0}})$, which is denoted by $E_{-1}$,
we can inductively extend the definition of $E_{k+\theta}$ and $A_{k+\theta}$ for $-1< k+\theta<\infty$, $k=-1,0,1,\cdots$ (see \cite[(6.4)]{AM93}). Then, we have a family of operators
\[
A_{k+\theta}\in\mathcal{L}(E_{k+1+\theta},E_{k+\theta})
\]
such that $-A_{k+\theta}$ is the infinitesimal generator of an analytic semigroup
\[
\bket{e^{-tA_{k+\theta}} \,|\, t\ge0}\, \mbox{ on }\, E_{k+\theta},
\]
and $A_{k+\theta}$ is a $W^{2(k+\theta), p}$- realization
\footnote{ $A_{\beta}$ is a $W^{s,p}$- realization of $A$ if
$ A= A_{\beta}$ in $D(A)$ and the range of $A_{\beta}$ is in $W^{s,p}$. }of $A$ for {$ -1<k+\theta<\infty$, $k=-1,0,1,\cdots$, $0<\theta<1$, and $1<p<\infty$.} Let us specify
\begin{equation}\label{aal}
A_{\alpha-1}=W^{2(\alpha-1),p}_{B}\mbox{- realization of }\, A \,\mbox{ for }\, 2\alpha\in \Bigr{(}1, 1+\frac{1}{p}\Bigr{)} \,\mbox{ with }\, n<p <\infty.
\end{equation}
The semigroup $e^{-t A_{\alpha-1}}$ satisfies the smoothing estimate (\cite[Lemma 3.1]{CM-R13}):\\
If $1<p<\infty$, $1<\beta<2\alpha<1+\frac{1}{p}$, $f\in W^{2\alpha-2,p}_{B}$, then there exist positive constants $\sigma=\sigma(\beta)<1$, $\kappa<1$ and $C(\alpha,\beta,p)$ such that
\begin{equation}\label{NONEST}
\norm{ e^{-tA_{\alpha-1}}f }_{W^{\beta,p}(\Omega)}\le Ct^{-\sigma}e^{-\kappa t}\norm{ f }_{W^{2\alpha-2,p}_{B}},\qquad t>0.
\end{equation}
We will use \eqref{NONEST} in the proof of Lemma~\ref{LEM1} to control the nonlinear terms.
Lastly, we define $B^c$ by the continuous extension of $(B|_{Ker (I-\Delta)} )^{-1}$ to $W^{2\alpha-1-\frac{1}{p}}(\partial\Omega)$. Note that \begin{equation}\label{bc}
B^c\in\mathcal{L}(W^{2\alpha-1-\frac{1}{p}, \frac{1}{p}}(\partial\Omega), W^{2\al, p}({ \Omega }))
\end{equation} for $\al, p$ of same range in \eqref{aal}.
\\
\indent
Let $\mathcal{N}\in C^{1}(\overline{\Omega})$ be a fixed vector field satisfying $\mathcal{N}=\nu$ on $\partial\Omega$ constructed in Lemma~\ref{LEMNU}. Now
we consider the inhomogeneous Neumann boundary value problems for \eqref{MODEL00}--\eqref{omega1}:
\begin{equation}\label{NPBV0}
\begin{aligned}
&\partial_{t}u+(I-\Delta)u=g_{1},& &\partial_{t}v+(I-\Delta)v=g_{2}, \quad\qquad&x\in\Omega, \,\,t\le T,\,\,\,&\\
&\partial_{\nu}u= h_1, &\qquad\quad &\partial_{\nu} v=h_2, &x\in\partial\Omega, \,\,t\le T,&\\
&u(x,0) =u_{0}(x),&\qquad &v(x,0) =v_{0}(x), &x\in\partial\Omega,\quad\qquad\,&
\end{aligned}
\end{equation}
and its {\it generalized variation-of-constants formulas:}
\begin{equation}\label{INTF}
\begin{array}{ll}
u=e^{-tA_{\alpha-1}}u_{0}+\int_{0}^{t}e^{-(t-\tau)A_{\alpha-1}}(g_{1}(\tau)+ A_{\alpha-1} B^{c} \gamma h_{1}(\tau))d\tau,\vspace{1mm}
\\
v=e^{-tA_{\alpha-1}}v_{0}+\int_{0}^{t}e^{-(t-\tau)A_{\alpha-1}}(g_{2}(\tau)+ A_{\alpha-1} B^{c} \gamma h_{2}(\tau))d\tau,
\end{array}
\end{equation}
where
\begin{equation}\label{INTF1}
\begin{array}{ll}
g_{1}:=-\nabla \cdot (u \,\mathcal{K}[u,v] )+(1-m)u +\displaystyle\frac{\lambda}{k} u(k-(u+v)),
\\
g_{2}:=-\nabla \cdot (v \,\mathcal{S}[u,v] )+mu+v +\displaystyle\frac{\mu}{k} v(k-(u+v)),
\\
h_{1}:=u\mathcal{K}[u,v]\cdot\mathcal{\mathcal{N}},\qquad h_{2}:=v\mathcal{S}[u,v]\cdot\mathcal{\mathcal{N}}.
\end{array}
\end{equation}
The formal argument to write \eqref{NPBV0} into \eqref{INTF} is presented in \cite[(11.16)--(11.20)]{AM93}.
{\begin{defn}
Let ${ \Omega }, {\mathcal K}, { \mathcal S }$ and ${ \omega }$ be given as for Theorem \ref{GETHM}. Assume $u_0$ and $v_0$ are functions belonging to $W^{2,p}({ \Omega }), p>n$. We call $(u, v)$ in $(L^{\infty}(0, T; W^{1,p}({ \Omega })))^2$ satisfying the integral equation \eqref{INTF}, \eqref{INTF1} a \it{generalized solution} of
\eqref{NPBV0} for $T>0$.
\end{defn}}
In what follows we show that \eqref{NPBV0} has a unique local-in-time generalized
solution, which coincides with a strong solution satisfying maximal regularity estimates.
\begin{lemma}\label{LEM1}
Let ${ \Omega }, {\mathcal K}, { \mathcal S }$ and ${ \omega }$ be given as for Theorem \ref{GETHM}.
Assume that $u_{0}$ and $v_{0}$ are non-negative functions belonging to $W^{2,p}(\Omega)$, $p>n$, and satisfying
the compatibility condition
\[
\partial_{\nu}u_{0}=u_0\mathcal{K}[u_0,v_0]\cdot \nu,\qquad \partial_{\nu}v_{0}=v_0\mathcal{S}[u_0,v_0]\cdot \nu,\qquad x\in\partial{\Omega}.
\]
Then, there is a maximal time, $T_{\rm max}\le\infty$, such that a unique non-negative strong solution $(u,v)$ of \eqref{NPBV0} exists and satisfies
\[
u,v\in C([0,t]; W^{1,p}(\Omega))\cap W^{1,p}(0,t;L^{p}(\Omega))\cap L^{p}(0,t;W^{2,p}(\Omega)),\qquad t<T_{\rm max}.
\]
Moreover, if $T_{\rm max}<\infty$, then
\begin{equation}\label{BUC}
\lim_{t\rightarrow T_{\rm max}}(\norm{u(\cdot,t)}_{W^{1,p}(\Omega)}+\norm{v(\cdot,t)}_{W^{1,p}(\Omega)})=\infty.
\end{equation}
\end{lemma}
\begin{proof}
We divide the proof into three parts. We first obtain a unique generalized solution $(u,v)$, and then show that it is indeed a strong solution. In the last step, the non-negativity of solution components is shown.\\
{\bf Step 1} (Generalized solution)
Let $n<p <\infty$. With positive constants $T<1$ and $R_0$ to be specified below, we introduce the Banach space $X_{T}:=C([0,T];W^{1,p}(\Omega))$ and its closed convex subset $S_{T}\subset X_{T}$,
\[
S_{T}:=\bigr{\{}\,f\in X_{T}\,|\, \norm{f}_{L^{\infty}(0,T; W^{1,p}(\Omega))}\le R_0 \bigr{\}}.
\]
Let $u,v \in S_{T}$, $2\alpha\in (1, 1+\frac{1}{p})$ and $0<t<T$. As in \eqref{INTF}--\eqref{INTF1}, we consider \begin{equation}\label{PFPHI1}
\Phi_{1}(u,v):=e^{-tA_{\alpha-1}}u_{0}+\int_{0}^{t}e^{-(t-\tau)A_{\alpha-1}}(g_{1}(\tau)+ A_{\alpha-1} B^{c} \gamma h_{1}(\tau))d\tau,
\end{equation}
\begin{equation}\label{PFPHI2}
\Phi_{2}(u,v):=e^{-tA_{\alpha-1}}v_{0}+\int_{0}^{t}e^{-(t-\tau)A_{\alpha-1}}(g_{2}(\tau)+ A_{\alpha-1} B^{c} \gamma h_{2}(\tau))d\tau,
\end{equation}
where $\gamma$, $A_{\al-1}$ , and $B^{c}$ are as previously defined with \eqref{TRNOT}, \eqref{aal} and \eqref{bc}.
We now show $\Phi_{1}(u,v),\Phi_{2}(u,v)\in S_{T}$. Let $1<\beta<2\alpha$.
Using \eqref{NONEST} and $W^{\beta,p}(\Omega)\hookrightarrow W^{1,p}(\Omega)$, we compute
\[
\begin{aligned}
\|&\Phi_{1}(u,v)(t)\|_{W^{1,p}(\Omega)}\\
&\le \norm{e^{-tA_{\alpha-1}}u_{0}}_{W^{1,p}(\Omega)}+\int_{0}^{t}\norm{e^{-(t-\tau)A_{\alpha-1}}(g_{1}(\tau)+ A_{\alpha-1} B^{c} \gamma h_{1}(\tau))}_{W^{1,p}(\Omega)}d\tau\\
&\le K_{1}\norm{u_{0}}_{W^{1,p}(\Omega)}+C\int_{0}^{t}\norm{e^{-(t-\tau)A_{\alpha-1}}(g_{1}(\tau)+ A_{\alpha-1} B^{c} \gamma h_{1}(\tau))}_{W^{\beta,p}(\Omega)}d\tau\\
&\le K_{1}\norm{u_{0}}_{W^{1,p}(\Omega)}+C\int_{0}^{t}e^{-\kappa(t-\tau)}(t-\tau)^{-\sigma}(\norm{g_{1}(\tau)}_{W^{2\alpha-2,p}_{B}}+ \norm{A_{\alpha-1} B^{c} \gamma h_{1}(\tau)}_{W^{2\alpha-2,p}_{B}})d\tau.
\end{aligned}
\]
Note that $A_{\alpha-1} B^{c}\gamma $ is well defined due to Lemma~\ref{LEMKS} and
\begin{equation}\label{BDRY}
W^{1-\frac{1}{p},p}(\partial\Omega)\hookrightarrow W^{2\alpha-1-\frac{1}{p},p}(\partial\Omega),\qquad A_{\alpha-1}B^{c}\in\mathcal{L}(W^{2\alpha-1-\frac{1}{p},p}(\partial\Omega), W^{2\alpha-2,p}_{B}).
\end{equation}
\
Using $L^{p}(\Omega)=W^{0,p}_{B}\hookrightarrow W^{2\alpha-2,p}_{B}$, Lemma~\ref{LEMKS}, and $W^{1,p}(\Omega)\hookrightarrow L^{\infty}(\Omega)$, we can estimate
\[
\norm{g_{1}(\tau)}_{W^{2\alpha-2,p}_{B}}\le C\norm{g_{1}(\tau)}_{L^{p}(\Omega)}\le C(R_0+R_0^{2}).
\]
Using Lemma~\ref{LEMKS} and \eqref{BDRY}, we also compute that
\[
\norm{A_{\alpha-1} B^{c} \gamma h_{1}(\tau))}_{W^{2\alpha-2,p}_{B}} \le C\norm{u\mathcal{K}[u,v](\tau)\cdot\mathcal{N}}_{W^{1,p}(\Omega)}\le CR_0^{2}.
\]
Then, combining the above computations leads to
\[
\norm{\Phi_{1}(u,v)(t)}_{W^{1,p}(\Omega)}\le K_{1}\norm{u_{0}}_{W^{1,p}(\Omega)}+C_{1}(R_0+R_0^{2})t^{1-\sigma},
\]
where $C_{1}$ is a positive constant independent of $R_0$.
Analogously to above, we can see that
\[
\norm{\Phi_{2}(u,v)(t)}_{W^{1,p}(\Omega)}\le K_{2}\norm{v_{0}}_{W^{1,p}(\Omega)}+C_{2}(R_0+R_0^{2})t^{1-\sigma},
\]
where $C_{2}$ is a positive constant independent of $R_0$.
Choosing
\[
R_0:=K_{1}\norm{u_{0}}_{W^{1,p}(\Omega)}+K_{2}\norm{v_{0}}_{W^{1,p}(\Omega)}+1,
\]
\[
T<T_{1}:=\min\bket{ 1, \frac{1}{(2C_{1}(R_0+R_0^{2}))^{\frac{1}{1-\sigma}}}, \frac{1}{(2C_{2}(R_0+R_0^{2}))^{\frac{1}{ 1-\sigma}}} },
\]
and taking supremum over $0<t\le T$, we have
$
\Phi_{1}(u,v),\Phi_{2}(u,v)\in S_{T}$.
We next show the mapping $(u,v)\mapsto (\Phi_{1},\Phi_{2})$ is a contraction. Note from \eqref{PFPHI1}--\eqref{PFPHI2} that
\[
\Phi_{1}(u,v)(t)-\Phi_{1}(\tilde{u},\tilde{v})(t)=\int_{0}^{t}e^{-(t-\tau)A_{\alpha-1}}((g_{1}-\tilde{g}_{1})(\tau)+ A_{\alpha-1} B^{c} \gamma (h_{1}-\tilde{h}_{1})(\tau))d\tau,
\]
\[
\Phi_{2}(u,v)(t)-\Phi_{2}(\tilde{u},\tilde{v})(t)=\int_{0}^{t}e^{-(t-\tau)A_{\alpha-1}}((g_{2}-\tilde{g}_{2})(\tau)+ A_{\alpha-1} B^{c} \gamma (h_{2}-\tilde{h}_{2})(\tau))d\tau,
\]
where
\[
\tilde{g}_{1}=-\nabla \cdot (\tilde{u} \,\mathcal{K}[\tilde{u},\tilde{v}] )+(1-m)\tilde{u} +\displaystyle\frac{\lambda}{k} \tilde{u}(k-(\tilde{u}+\tilde{v})),\qquad \tilde{h}_{1}:=\tilde{u}\mathcal{K}[\tilde{u},\tilde{v}]\cdot\mathcal{\mathcal{N}},
\]
\[
\tilde{g}_{2}=-\nabla \cdot (\tilde{v} \,\mathcal{S}[\tilde{u},\tilde{v}] )+ m \tilde{u}+\tilde{v} +\displaystyle\frac{\mu}{k} \tilde{v}(k-(\tilde{u}+\tilde{v})),\qquad \tilde{h}_{2}:=\tilde{v}\mathcal{S}[\tilde{u},\tilde{v}]\cdot\mathcal{\mathcal{N}}.
\]
Then, by similar computations as above, we have
\[
\sup_{t\le T}\norm{(\Phi_{1},\Phi_{2})(u,v)(t)-(\Phi_{1},\Phi_{2})(\tilde{u},\tilde{v})(t)}_{W^{1,p}(\Omega)}
\]
\[
\le C_{3}(R_0+1)T^{1-\sigma}\sup_{t\le T}\norm{ (u,v)(t)- (\tilde{u},\tilde{v})(t)}_{W^{1,p}(\Omega)},
\]
where $C_{3}>0$ is a constant independent of $R_0$.
Taking
\[
T<T_{2}:=\min \bket{ T_{1}, \frac{1}{(2C_{3}(R_0+1))^{\frac{1}{1-\sigma}}} },
\]
we obtain
\[
\sup_{t\le T}\norm{(\Phi_{1},\Phi_{2})(u,v)(t)-(\Phi_{1},\Phi_{2})(\tilde{u},\tilde{v})(t)}_{W^{1,p}(\Omega)}\le \frac{1}{2} \sup_{t\le T}\norm{ (u,v)(t)- (\tilde{u},\tilde{v})(t)}_{W^{1,p}(\Omega)},
\]
i.e., the mapping is a contraction. According to the Banach fixed point theorem, this mapping has a fixed point in $S_{T}$, denoted again $(u,v)$. Thus, the generalized solution $(u,v)$ for \eqref{NPBV0} is obtained.
By the standard extension argument and the fact that the above choice of $T$ depends only on $\|u_0\|_{W^{1,p}(\Omega)}$, and $\|v_0\|_{W^{1,p}(\Omega)}$, it should be noted that there exists $T_{\rm M}\le \infty$ such that $ u,v\in C([0,T_{M}); W^{1,p}(\Omega))$, and
\begin{equation}\label{TMP}
\mbox{either }\, T_{\rm M}=\infty,\,\mbox{ or }\, \lim_{t\rightarrow T_{\rm M}}(\norm{u(\cdot,t)}_{W^{1,p}(\Omega)}+\norm{v(\cdot,t)}_{W^{1,p}(\Omega)})=\infty.
\end{equation}
We next show the uniqueness. Let $T<T_{M}$, and let $(u,v)$ and $(\tilde{u}, \tilde{v})$ be two constructed solutions for $t\le T$.
Analogously to above, we can estimate
\[
\begin{aligned}
\|( u&-\tilde{u})(t)\|_{W^{1,p}(\Omega)}\\
&\le \int_{0}^{t} e^{-\kappa(t-\tau)}(t-\tau)^{-\sigma}(\norm{( g_{1}-\tilde{g}_{1})(\tau)}_{L^{p}(\Omega)}+\|A_{\alpha-1}B^{c}\gamma( h_{1}-\tilde{h}_{1})(\tau)\|_{L^{p}(\Omega)} ) d\tau,
\end{aligned}
\]
\[
\begin{aligned}
\|( g_{1}&-\tilde{g}_{1})(\tau)\|_{L^{p}(\Omega) }
\\
&\le C\sup_{t\le T}(\norm{u(t)}_{W^{1,p}(\Omega)}+ \norm{v(t)}_{W^{1,p}(\Omega)}+ \norm{\tilde{u}(t)}_{W^{1,p}(\Omega)}+\norm{\tilde{v}(t)}_{W^{1,p}(\Omega)})
\\
&\qquad\times( \|( u-\tilde{u})(\tau)\|_{W^{1,p}(\Omega)}+ \|( v-\tilde{v})(\tau)\|_{W^{1,p}(\Omega)})
\\
&\le C( \|( u-\tilde{u})(\tau)\|_{W^{1,p}(\Omega)}+ \|( v-\tilde{v})(\tau)\|_{W^{1,p}(\Omega)}),
\end{aligned}
\]
and
\[
\begin{aligned}
\| A_{\alpha-1}B^{c}\gamma( h_{1}&-\tilde{h}_{1})(\tau)\|_{L^{p}(\Omega)}\\
&
\le \|(h_1-\tilde{h}_1)(\tau)\|_{W^{1,p}}(\Omega)\\
&\le C\sup_{t\le T}(\norm{u(t)}_{W^{1,p}(\Omega)}+ \norm{v(t)}_{W^{1,p}(\Omega)}+ \norm{\tilde{u}(t)}_{W^{1,p}(\Omega)}+\norm{\tilde{v}(t)}_{W^{1,p}(\Omega)})
\\
&\qquad\times( \|( u-\tilde{u})(\tau)\|_{W^{1,p}(\Omega)}+ \|( v-\tilde{v})(\tau)\|_{W^{1,p}(\Omega)})
\\
&\le C( \|( u-\tilde{u})(\tau)\|_{W^{1,p}(\Omega)}+ \|( v-\tilde{v})(\tau)\|_{W^{1,p}(\Omega)}).
\end{aligned}
\]
Thus, we have
\[
\|( u-\tilde{u})(t)\|_{W^{1,p}(\Omega)}\le C \int_{0}^{t} e^{-\kappa(t-\tau)}(t-\tau)^{-\sigma}( \|( u-\tilde{u})(\tau)\|_{W^{1,p}(\Omega)}+ \|( v-\tilde{v})(\tau)\|_{W^{1,p}(\Omega)})d\tau.
\]
Similarly, we also have
\[
\|( v-\tilde{v})(t)\|_{W^{1,p}(\Omega)}\le C \int_{0}^{t} e^{-\kappa(t-\tau)}(t-\tau)^{-\sigma}( \|( u-\tilde{u})(\tau)\|_{W^{1,p}(\Omega)}+ \|( v-\tilde{v})(\tau)\|_{W^{1,p}(\Omega)})d\tau.
\]
Adding the above two estimates and
using a Gr\"onwall type inequality, $(u,v)=(\tilde{u},\tilde{v})$ is obtained for $t\le T$. Since $T<T_{\rm M}$ is arbitrary, we have the uniqueness of solutions.\\
{\bf Step 2} (Strong solution)
We next consider the regularity of the constructed solution $(u,v)$. Let $t\le T<T_{M}$.
As $1<\beta<2\alpha<1+\frac{1}{p}$, we first note that
\begin{equation}\label{INITWBP}
\| e^{-tA_{\alpha-1}}u_{0}\|_{W^{\beta,p}(\Omega)}\le C\|u_{0}\|_{W^{2,p}(\Omega)}, \qquad \| e^{-tA_{\alpha-1}}v_{0}\|_{W^{\beta,p}(\Omega)}\le C\|v_{0}\|_{W^{2,p}(\Omega)}.
\end{equation}
If we replace the computations for the initial counterparts in the previous step by \eqref{INITWBP}, then we have
$
u,v\in C ([0,T]; W^{\beta,p }_{B})$.
Thus, as in \cite[(3.5)]{A86}, it can be shown that
\[
u,v\in C^{1}([0,T]; W^{\beta-2,p }_{B})\quad\mbox{ as well as}\quad u,v\in C^{\frac{1}{2}-\frac{1}{2p}}([0,T]; W^{\beta-1+\frac{1}{p}, p }_{B}).
\] Therefore, we have $u,v\in W^{\frac{1}{2}-\frac{1}{2p},p}(0,T; W^{\beta-1, p }(\partial\Omega))$ and, in particular,
\[
h_{1},h_{2}\in W^{\frac{1}{2}-\frac{1}{2p},p}(0,T; W^{\beta-1, p }(\partial\Omega)).
\]
Due to these facts along with $h_{1}, h_{2}\in L^{\infty}(0,T; W^{1-\frac{1}{p},p}(\partial\Omega))$ and $g_{1}, g_{2}\in L^{\infty}(0,T; L^{p}(\Omega))$, by
applying the maximal regularity theorem \cite[Theorem 2.1]{DHP07} to
\begin{equation}\label{NPBV00}
\begin{aligned}
&\partial_{t}f_{1}+(I-\Delta)f_{1}=g_{1},& &\partial_{t}f_{2}+(I-\Delta)f_{2}=g_{2}, \quad\qquad&x\in\Omega, \,\,t\le T,\,\,\,&\\
&\partial_{\nu}f_{1}= h_1, &\qquad\quad &\partial_{\nu} f_{2}=h_2, &x\in\partial\Omega, \,\,t\le T,&\\
&f_{1}(x,0) =u_{0}(x),&\qquad &f_{2}(x,0) =v_{0}(x), &x\in\partial\Omega,\quad\qquad\,&
\end{aligned}
\end{equation}
we have the unique strong solution $(f_1,f_2)$ for \eqref{NPBV00} in the class
\[
f_1,f_2\in W^{1,p}(0,T;L^{p}(\Omega))\cap L^{p}(0,T;W^{2,p}(\Omega)),\qquad T<T_{\rm M}.
\]
Since the strong solution is the generalized solution (\cite{A86}, \cite{LaTri}), we put \eqref{NPBV00} into the generalized variation-of-constants formulas with respect to $f_1$, $f_2$ with the same right hand side terms as in \eqref{INTF}, \eqref{INTF1}.
Then we have $(f_1,f_2)=(u,v)$ due to the uniqueness of the generalized solution and thus,
\[
u,v\in W^{1,p}(0,T;L^{p}(\Omega))\cap L^{p}(0,T;W^{2,p}(\Omega)),\qquad T<T_{\rm M}.
\]
{\bf Step 3} (Non-negativity)
It remains to show the non-negativity of the constructed solution $(u,v)$. Let $t\le T<T_{M}$. Define $u_{-}:=-\min\bket{u,0}$.
Multiplying the $u$-equation in \eqref{NPBV0} by $u_{-}$ and integrating over $\Omega$, using the direct computation and Young's inequality, we can compute
\begin{equation}\label{UNU}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}&\int_{\Omega}|u_{-}|^{2}+\int_{\Omega}|\nabla u_{-}|^{2}\\
&
\le \frac{1}{2}\int_{\Omega}|\nabla u_{-}|^{2}+C(1+ \|\mathcal{K}[u,v]\|_{L^{\infty}(\Omega)}^{2}+\|u\|_{L^{\infty}(\Omega)}+\|v\|_{L^{\infty}(\Omega)})\int_{\Omega}|u_{-}|^{2},\qquad t\le T,
\end{aligned}
\end{equation}
where we used Lemma~\ref{LEMKS} and $W^{1,p}(\Omega)\hookrightarrow L^{\infty}(\Omega)$.
Then, $u\ge0$ can be obtained by using Gr\"onwall's lemma. Similarly, testing the $v$-equation in \eqref{NPBV0} by $v_-:=- \min\{v,0\}$ and using Young's inequality, we observe that
\begin{equation}\label{UNV}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}&\int_{\Omega}|v_{-}|^{2}+\int_{\Omega}|\nabla v_{-}|^{2} \\
&\le \frac{1}{2}\int_{\Omega}|\nabla v_{-}|^{2}+C(1+ \|\mathcal{S}[u,v]\|_{L^{\infty}(\Omega)}^{2}+\|u\|_{L^{\infty}(\Omega)}+\|v\|_{L^{\infty}(\Omega)})\int_{\Omega}|v_{-}|^{2}-m\int_{\Omega}uv_{-}
\\
&\le \frac{1}{2}\int_{\Omega}|\nabla v_{-}|^{2}+C\int_{\Omega}|v_{-}|^{2},\quad t\le T.
\end{aligned}
\end{equation}
Again by Gr\"onwall's lemma, we have $v\ge0$. Since $T<T_{\rm M}$ is arbitrary, we obtain the non-negativity of solutions.
Finally taking $T_{\rm max}$ as $T_{\rm M}$ in \eqref{TMP} completes the proof.
\end{proof}
We next prove a refined blow-up criteria.
\begin{lemma}\label{BUCLEM}
Let the same assumptions as in Lemma~\ref{LEM1} be satisfied. The solution $(u,v)$ of \eqref{NPBV0} given by Lemma~\ref{LEM1} satisfies
\[
\mbox{either }\, T_{\rm max}=\infty,\,\mbox{ or }\, \lim_{t\rightarrow T_{\rm max}}(\norm{u(\cdot,t)}_{L^{\infty}(\Omega)}+\norm{v(\cdot,t)}_{L^{\infty}(\Omega)})=\infty.
\]
\end{lemma}
\begin{proof}
By \eqref{BUC}, it suffices to show that if
\begin{equation}\label{BUCPF0}
\norm{u}_{L^{\infty}(0,T_{\rm max};L^{\infty}(\Omega))}+\norm{v}_{L^{\infty}(0,T_{\rm max};L^{\infty}(\Omega))}\le C,\qquad T_{\rm max}<\infty,
\end{equation}
then
\begin{equation}\label{BUCPF1}
\norm{u}_{L^{\infty}(0,T_{\rm max};W^{1,p}(\Omega))}+\norm{v}_{L^{\infty}(0,T_{\rm max};W^{1,p}(\Omega))}\le C.
\end{equation}
Let $p>n$, $2\alpha\in (1, 1+\frac{1}{p})$, $1<\beta<2\alpha$, and let $\varepsilon>0$ be a number such that $0<T_{\rm max}-\varepsilon<t<T_{\rm max}$.
Using \eqref{INTF}, \eqref{NONEST} and $W^{\beta,p}(\Omega)\hookrightarrow W^{1,p}(\Omega)$, we compute
\[
\begin{aligned}
\|u&(t)\|_{W^{1,p}(\Omega)} \\
&\le K_{1}\norm{u(T_{\rm max}-\varepsilon)}_{W^{1,p}(\Omega)}\\
&\quad+C\int_{T_{\rm max}-\varepsilon}^{t}\|e^{-(t-\tau)A_{\alpha-1}}(g_{1}(\tau)+ A_{\alpha-1} B^{c} \gamma h_{1}(\tau))\|_{W^{\beta,p}(\Omega)}d\tau\\
&\le K_{1}\norm{u(T_{\rm max}-\varepsilon)}_{W^{1,p}(\Omega)}\\
&\quad+C\int_{T_{\rm max}-\varepsilon}^{t}e^{-\kappa(t-\tau)}(t-\tau)^{-\sigma}(\norm{g_{1}(\tau)}_{W^{2\alpha-2,p}_{B}}+ \norm{A_{\alpha-1} B^{c} \gamma h_{1}(\tau))}_{W^{2\alpha-2,p}_{B}})d\tau,
\end{aligned}
\]
where $\gamma\in\mathcal{L}( W^{1,p}(\Omega), W^{1-\frac{1}{p},p}(\partial\Omega))$ is the boundary trace operator, $B^{c}$ is the continuous extension of $(B|_{Ker(I-\Delta)})^{-1}$ to $W^{2\alpha-1-\frac{1}{p}}(\partial\Omega)$, and $h_{1}$, $g_{1}$ are given in \eqref{INTF1}.
Using $L^{p}(\Omega)=W^{0,p}_{B}\hookrightarrow W^{2\alpha-2,p}_{B}$, Lemma~\ref{LEMKS}, \eqref{BDRY}, and \eqref{BUCPF0}, we compute
\[
\begin{aligned}
\|g_{1}&(\tau)\|_{W^{2\alpha-2,p}_{B}}\\
&\le C\norm{g_{1}(\tau)}_{ L^{p}({ \Omega })}\\
&\le C(\norm{u(\tau)}_{L^{\infty}(\Omega)}+\norm{u(\tau)}_{L^{\infty}(\Omega)}^{2}+\norm{u(\tau)}_{L^{\infty}(\Omega)}\norm{v(\tau)}_{L^{\infty}(\Omega)})\\
&\quad+C(\norm{u(\tau)}_{W^{1,p}(\Omega)}\norm{\mathcal{K}[u,v](\tau)}_{L^{\infty}(\Omega)}+\norm{u(\tau)}_{L^{\infty}(\Omega)}\norm{\mathcal{K}[u,v](\tau)}_{W^{1,p}(\Omega)})\\
&\le C
+C(\norm{u(\tau)}_{W^{1,p}(\Omega)}+\norm{v(\tau)}_{W^{1,p}(\Omega)}),
\end{aligned}
\]
and
\[
\begin{aligned}
\|A_{\alpha-1}& B^{c} \gamma h_{1}(\tau))\|_{W^{2\alpha-2,p}_{B}}\\
&
\le C\norm{u\mathcal{K}[u,v](\tau)\cdot\mathcal{N}}_{W^{1,p}(\Omega)}\\
&
\le C(\norm{u(\tau)}_{W^{1,p}(\Omega)}\norm{\mathcal{K}[u,v](\tau)}_{L^{\infty}(\Omega)}+\norm{u(\tau)}_{L^{\infty}(\Omega)}\norm{\mathcal{K}[u,v](\tau)}_{W^{1,p}(\Omega)})\\
&\le C
+C(\norm{u(\tau)}_{W^{1,p}(\Omega)}+\norm{v(\tau)}_{W^{1,p}(\Omega)}).
\end{aligned}
\]
It then follows that
\[
\begin{aligned}
\|u&(t)\|_{W^{1,p}(\Omega)} \\
&\le K_{1}\norm{u(T_{\rm max}-\varepsilon)}_{W^{1,p}(\Omega)}
+C_{4}\varepsilon^{1-\sigma}\sup_{T_{\rm max}-\varepsilon<t<T_{\rm max}}(\norm{u(t)}_{W^{1,p}(\Omega)}+\norm{v(t)}_{W^{1,p}(\Omega)}),
\end{aligned}
\]
where $C_{4}$ is a positive constant independent of $\varepsilon$.
Analogously to above, we can compute
\[
\begin{aligned}
\|v&(t)\|_{W^{1,p}(\Omega)} \\
&\le K_{2}\norm{v(T_{\rm max}-\varepsilon)}_{W^{1,p}(\Omega)}
+C_{5}\varepsilon^{1-\sigma}\sup_{T_{\rm max}-\varepsilon<t<T_{\rm max}}(\norm{u(t)}_{W^{1,p}(\Omega)}+ \norm{v(t)}_{W^{1,p}(\Omega)}),
\end{aligned}
\]
where $C_{5}$ is a positive constant independent of $\varepsilon$. Adding above two inequalities and taking supremum over $T_{\rm max}-\varepsilon<t<T_{\rm max}$, we have
\[
\begin{aligned}
\sup_{T_{\rm max}-\varepsilon<t<T_{\rm max}}\bigr{(}\| &u(t)\|_{W^{1,p}(\Omega)} +\|v(t)\|_{W^{1,p}(\Omega)} \bigr{)}\\
&\le K_{1}\norm{u(T_{\rm max}-\varepsilon)}_{W^{1,p}(\Omega)}+K_{2}\norm{v(T_{\rm max}-\varepsilon)}_{W^{1,p}(\Omega)}\\
&\quad
+(C_{4}+C_{5})\varepsilon^{1-\sigma}\sup_{T_{\rm max}-\varepsilon<t<T_{\rm max}}\bigr{(}\norm{u(t)}_{W^{1,p}(\Omega)} +\norm{v(t)}_{W^{1,p}(\Omega)} \bigr{)}.
\end{aligned}
\]
Therefore, taking sufficiently small $\varepsilon$, \eqref{BUCPF1} is obtained. This completes the proof.
\end{proof}
\end{subsection}
\begin{subsection}{ A priori estimates}\label{SEC23}
Next, we provide some {\it a priori} estimates (Lemma~\ref{LEML1} and Lemma~\ref{LEMLINFUV}).
\begin{lemma}\label{LEML1}
Let the same assumptions as in Lemma~\ref{LEM1} be satisfied. The solution $(u,v)$ of \eqref{NPBV0} given by Lemma~\ref{LEM1} for $T<T_{\rm max}$ satisfies
\begin{equation}\label{UL1}
\sup_{t\le T}\int_{\Omega}u(\cdot,t)\le C(\norm{u_{0}}_{L^{1}(\Omega)}),
\end{equation}
\begin{equation}\label{VL1}
\sup_{t\le T}\int_{\Omega}v(\cdot,t)\le C(\norm{u_{0}}_{L^{1}(\Omega)}, \norm{v_{0}}_{L^{1}(\Omega)}),
\end{equation}
\begin{equation}\label{KLINF}
\sup_{t\le T} \norm{\mathcal{K}[u,v](\cdot, t)}_{L^{\infty}(\Omega)}\le C(\norm{u_{0}}_{L^{1}(\Omega)}, \norm{v_{0}}_{L^{1}(\Omega)}),
\end{equation}
and
\begin{equation}\label{SLINF}
\sup_{t\le T} \norm{\mathcal{S}[u,v](\cdot, t)}_{L^{\infty}(\Omega)}\le C(\norm{u_{0}}_{L^{1}(\Omega)}, \norm{v_{0}}_{L^{1}(\Omega)}).
\end{equation}
\end{lemma}
\begin{proof}
Integrating $\eqref{MODEL00}_{1}$ and $\eqref{MODEL00}_{2}$ over $\Omega$, we obtain
\begin{equation}\label{LEML1_1}
\frac{d}{dt}\int_{\Omega}u+\frac{\lambda}{k}\int_{\Omega}u(u+v)=(\lambda-m)\int_{\Omega}u,
\end{equation}
and
\begin{equation}\label{LEML1_2}
\frac{d}{dt}\int_{\Omega}v+\frac{\mu}{k}\int_{\Omega}v(u+v)=m\int_{\Omega}u+ \mu\int_{\Omega}v.
\end{equation}
As we have
\[
(|\lambda-m|+1)\int_{\Omega}u\le \frac{\lambda}{k}\int_{\Omega}u^{2}+C,
\]
it follows from \eqref{LEML1_1} that
\[
y'+ y\le C,\qquad y(t):=\int_{\Omega}u(\cdot,t).
\]
Therefore, \eqref{UL1} is obtained by standard ODE argument.
Similarly, as Young's inequality gives
\[
(\mu+1)\int_{\Omega}v\le \frac{\mu}{k}\int_{\Omega}v^{2}+C,
\]
it follows from \eqref{UL1} and \eqref{LEML1_2} that
\[
\frac{d}{dt}\int_{\Omega}v+\int_{\Omega}v\le C(\norm{u_{0}}_{L^{1}(\Omega)}).
\]
Thus, we can also obtain \eqref{VL1}. Then, \eqref{KLINF} and \eqref{SLINF} are direct consequences of \eqref{UL1}--\eqref{VL1} and \eqref{omega1}.
This completes the proof.
\end{proof}
\begin{lemma}\label{LEMLINFUV}
Let the same assumptions as in Lemma~\ref{LEM1} be satisfied. The solution $(u,v)$ of \eqref{NPBV0} given by Lemma~\ref{LEM1} for $T<T_{\rm max}$ satisfies
\begin{equation}\label{ULINFTY}
\sup_{t\le T}\norm{u(\cdot,t)}_{L^{\infty}(\Omega)}\le C(\norm{u_{0}}_{(L^{1}\cap L^{\infty})(\Omega)}, \norm{v_{0}}_{L^{1}(\Omega)}),
\end{equation}
and
\begin{equation}\label{VLINFTY}
\sup_{t\le T}\norm{v(\cdot,t)}_{L^{\infty}(\Omega)}\le C(\norm{u_{0}}_{(L^{1}\cap L^{\infty})(\Omega)}, \norm{v_{0}}_{(L^{1}\cap L^{\infty})(\Omega)}).
\end{equation}
\end{lemma}
\begin{proof}
Let $p>1$ and $t\le T<T_{\rm max}$.
Multiplying $\eqref{MODEL00}_{1}$ by $u^{p-1}$, integrating over $\Omega$, and using integrating by parts, we have
\[
\frac{1}{p}\frac{d}{dt}\int_{\Omega}u^{p}+\frac{4(p-1)}{p^{2}}\int_{\Omega}|\nabla u^{\frac{p}{2}}|^{2}+\frac{\lambda}{k}\int_{\Omega} u^{p}(u+v)
=(\lambda-m)\int_{\Omega}u^{p}+(p-1)\int_{\Omega}u^{p-1}\nabla u \,\mathcal{K}[u,v].
\]
Using Young's inequality and \eqref{KLINF}, we can compute the rightmost term as
\[
(p-1)\int_{\Omega}u^{p-1}\nabla u \,\mathcal{K}[u,v]\le \frac{p-1}{p^{2}}\int_{\Omega}|\nabla u^{\frac{p}{2}}|^{2}+C(p-1)\int_{\Omega}u^{p},
\]
and thus, it follows that
\[
\frac{1}{p}\frac{d}{dt}\int_{\Omega}u^{p}+\frac{3(p-1)}{p^{2}}\int_{\Omega}|\nabla u^{\frac{p}{2}}|^{2}
\le C_{6} (p+1)\int_{\Omega}u^{p},
\]
where $C_{6}>0$ is a constant independent of $p$. Then, \eqref{ULINFTY} can be deduced by Moser-Alikakos iteration argument \cite{A79}. Indeed, for $p=p_{k}:=2^{k}$, $k=1,2,3,\cdots$, the last inequality becomes
\begin{equation}\label{MA_1}
\frac{d}{dt}\int_{\Omega}u^{p_k}+\frac{3(p_k-1)}{p_k}\int_{\Omega}|\nabla u^{ p_{k-1}} |^2\le C_{6}p_k(p_k+1)\int_{\Omega}u^{p_k}.
\end{equation}
Using the Gagliardo-Nirenberg interpolation inequality and Young's inequality, we note that
\[
\norm{u^{ p_{k-1}}}_{L^{2}(\Omega)}^{2}\le C\norm{ u^{ p_{k-1}} }_{L^{1}(\Omega)}^{\frac{4}{n+2}}\norm{\nabla u^{ p_{k-1}} }_{L^{2}(\Omega)}^{\frac{2n}{n+2}}+C\norm{ u^{ p_{k-1}} }_{L^{1}(\Omega)}^{2}
\]
\[
\le \frac{1}{2C_{6}p_{k}(p_k+1)}\norm{\nabla u^{ p_{k-1}} }_{L^{2}(\Omega)}^{2}+Cp_{k}^{\frac{n(n+2)}{n+1}}\norm{ u^{ p_{k-1}} }_{L^{1}(\Omega)}^{2}
\]
for some constant $C>0$ independent of $p_{k}$. Plugging it into \eqref{MA_1}, due to $\frac{3(p_k-1)}{p_k}\ge \frac{3}{2}$, we have
\begin{equation}\label{MA_2}
\frac{d}{dt}\int_{\Omega}u^{p_k}+2C_{6}p_k(p_k+1)\int_{\Omega}u^{p_k}\le C_{7}p_{k}^{\frac{n(n+2)}{n+1}+2}\bke{ \int_{\Omega} u^{ p_{k-1}} }^{2},
\end{equation}
where $C_{7}>0$ is a constant independent of $p_k$. We take a sufficiently large constant $C_{8}\ge1$ independent of $p_{k}$ satisfying
$C_{8}p_{k}^{\frac{n(n+2) }{n+1}} \ge C_{7}p_{k}^{\frac{n(n+2)}{n+1}+2}/(2C_{6}p_k(p_k+1))$ and
define
\[
M:=\max\{ \norm{u_{0}}_{L^{1}(\Omega)}, \norm{u_{0}}_{L^{\infty}(\Omega)}, 1\},\qquad\delta_{k}:=C_{8}p_{k}^{\frac{n(n+2) }{n+1}}.
\]
Then, it follows from \eqref{MA_2} that
\[
\sup_{t\le T}\int_{\Omega}u^{p_k}(\cdot,t)\le \max\bket{ M^{p_k},\quad \delta_{k}\biggr{(} \sup_{t\le T}\int_{\Omega}u^{p_{k-1}}(\cdot,t)\biggr{)}^{2}}.
\]
Note that $\delta_k\ge1$ for all $k=1,2,3,\cdots$.
By an inductive computation, we have
\[
\sup_{t\le T}\biggr{(}\int_{\Omega}u^{p_k}(\cdot,t)\biggr{)}^{\frac{1}{p_{k}}}\le \bigr{[}\delta_{k}\delta_{k-1}^{p_{1}}\delta_{k-2}^{p_{2}}\cdots\delta_{1}^{p_{k-1}} (1+\sup_{t\le T}\norm{u(\cdot,t)}_{L^{1}(\Omega)})^{p_{k}} \bigr{]}^{\frac{1}{p_{k}}}M
\]
\begin{equation}\label{MA_3}
\le C_{8}^{\sum_{i=1}^{k}\frac{1}{2^{i}}}2^{\frac{n(n+2)}{n+1}\sum_{i=1}^{k}\frac{i}{2^{i}}}(1+\sup_{t\le T}\norm{u(\cdot,t)}_{L^{1}(\Omega)}) M.
\end{equation}
Thus, by \eqref{UL1} and $\sum_{i=1}^{k}\frac{i}{2^{i}}<\infty$, taking the limit $k\rightarrow\infty$, we can obtain \eqref{ULINFTY}.
Similarly, we can see from $\eqref{MODEL00}_{2}$ that
\[
\begin{aligned}
\frac{1}{p}\frac{d}{dt}\int_{\Omega}v^{p}&+\frac{4(p-1)}{p^{2}}\int_{\Omega}|\nabla v^{\frac{p}{2}}|^{2}+\frac{\mu}{k}\int_{\Omega} v^{p}(u+v)\\
&=m\int_{\Omega}uv^{p-1}+\mu\int_{\Omega}v^{p}+(p-1)\int_{\Omega}v^{p-1}\nabla v \,\mathcal{S}[u,v].
\end{aligned}
\]
Using Young's inequality and \eqref{SLINF}, we note that
\[
(p-1)\int_{\Omega}v^{p-1}\nabla v \,\mathcal{S}[u,v]\le \frac{p-1}{p^{2}}\int_{\Omega}|\nabla v^{\frac{p}{2}}|^{2}+C(p-1)\int_{\Omega}v^{p}.
\]
Analogously as above, using \eqref{ULINFTY}, we have
\begin{align}\label{MA_33}
\frac{1}{p}\frac{d}{dt}\int_{\Omega}v^{p}+\frac{3(p-1)}{p^{2}}\int_{\Omega}|\nabla v^{\frac{p}{2}}|^{2}\le C(p+1)\int_{\Omega}v^{p}+C,
\end{align}
where $C>0$ is a constant independent of $p$. Now, likewise \eqref{MA_2}, we can find positive constants $C_{9}$, $C_{10}$, and $C_{11}$ independent of $p=p_{k}=2^{k}$, $k=1,2,3,\cdots$ such that
\begin{equation}\label{MA_4}
\frac{d}{dt}\bke{\int_{\Omega}v^{p_k}+C_{9}}+2C_{10}p_k(p_k+1)\bke{\int_{\Omega}v^{p_k}+C_{9}}\le C_{11}p_{k}^{\frac{n(n+2)}{n+1}+2}\bke{ \int_{\Omega} v^{ p_{k-1}} +C_{9} }^{2}.
\end{equation}
We take a sufficiently large constant $C_{12}\ge1$ independent of $p_{k}$ satisfying
\[C_{12}p_{k}^{\frac{n(n+2) }{n+1}} \ge C_{11}p_{k}^{\frac{n(n+2)}{n+1}+2}/(2C_{10}p_k(p_k+1))\] and
define
\[
M:=\max\{ \norm{v_{0}}_{L^{1}(\Omega)}, \norm{v_{0}}_{L^{\infty}(\Omega)}, C_{9}+1 \},\qquad\delta_{k}:=C_{12}p_{k}^{\frac{n(n+2) }{n+1}}.
\]Then, we obtain from \eqref{MA_4} that
\[
\sup_{t\le T}\bke{\int_{\Omega}v^{p_k}(\cdot,t)+C_{9}}\le \max\bket{ (2M)^{p_k},\quad \delta_{k}\sup_{t\le T}\biggr{(} \int_{\Omega}v^{p_{k-1}}(\cdot,t)+C_{9}\biggr{)}^{2}}.
\]
Note that $\delta_k\ge1$ for all $k=1,2,3,\cdots$.
By an inductive computation, we have
\[
\sup_{t\le T}\biggr{(}\int_{\Omega}v^{p_k}(\cdot,t)+C_{9}\biggr{)}^{\frac{1}{p_{k}}}
\]
\[
\le \bigr{[}\delta_{k}\delta_{k-1}^{p_{1}}\delta_{k-2}^{p_{2}}\cdots\delta_{1}^{p_{k-1}} (\sup_{t\le T}\norm{v(\cdot,t)}_{L^{1}(\Omega)}+C_{9}+1)^{p_{k}} \bigr{]}^{\frac{1}{p_{k}}} 2M
\]
\[
\le C_{12}^{\sum_{i=1}^{k}\frac{1}{2^{i}}}2^{\frac{n(n+2)}{n+1}\sum_{i=1}^{k}\frac{i}{2^{i}}}(\sup_{t\le T}\norm{v(\cdot,t)}_{L^{1}(\Omega)}+C_{9}+1) 2M.
\]
Analogously to \eqref{MA_3}, we can see that
\[
\sup_{t\le T}\biggr{(}\int_{\Omega}v^{p_k}(\cdot,t)\biggr{)}^{\frac{1}{p_{k}}} \le \sup_{t\le T}\biggr{(}\int_{\Omega}v^{p_k}(\cdot,t)+C_{9}\biggr{)}^{\frac{1}{p_{k}}}
\]
\[
\le C_{12}^{\sum_{i=1}^{k}\frac{1}{2^{i}}}2^{\frac{n(n+2)}{n+1}\sum_{i=1}^{k}\frac{i}{2^{i}}}(\sup_{t\le T}\norm{v(\cdot,t)}_{L^{1}(\Omega)}+C_{9}+1) 2M.
\]
Due to \eqref{VL1} and $\sum_{i=1}^{k}\frac{i}{2^{i}}<\infty$, we have \eqref{VLINFTY} by taking the limit $k\rightarrow\infty$. This completes the proof.
\end{proof}
\end{subsection}
\begin{pfthm1}
It is a direct consequence of local-in-time existence, uniqueness, non-negativity (Lemma~\ref{LEM1}), blow-up criterion (Lemma~\ref{BUCLEM}), and {\it a priori} estimates (Lemma~\ref{LEMLINFUV}). This completes the proof.
\end{pfthm1}
\section{Independent case}\label{SEC3}
In this section, we prove Theorem~\ref{GETHM1}.
\begin{subsection}{Preliminary}
In what it follows we take such an example of ${ \Omega }$ and $E(x)$ and show a similar lemma as Lemma \ref{LEMKS} holds for ${{\mathcal K}}$ and ${{ \mathcal S }}$.
\begin{lemma}\label{LEMKS1}
Let $\Omega$ be the open ball of radius $L$, $B_{L}(0)$, and let $0<R<L$. Suppose that $f,g \in W^{1,p}(\Omega)$, $p>n$. Assume further that $\mathcal{K}[f,g]$, $\mathcal{S}[f,g]$, $E$, and $\omega$ are given by \eqref{MODEL01}, and \eqref{sensing2}--\eqref{omega2}.
Then, there exists a constant $C>0$ satisfying
\[ \norm{\mathcal{K}[f,g]}_{{L^{\infty}(\Om)}}+\norm{\mathcal{S}[f,g]}_{{L^{\infty}(\Om)}}\le C(R,\Omega)(\norm{f}_{L^1(\Omega)}+ \norm{g}_{L^1(\Omega)}),\]
\[\norm{\mathcal{K}[f,g]}_{W^{1,\infty}(\Omega)}+\norm{\mathcal{S}[f,g]}_{W^{1, \infty}(\Omega)}\le C(R,\Omega)(\norm{f}_{W^{1,p}(\Omega)}+ \norm{g}_{W^{1,p}(\Omega)}).
\]
\end{lemma}
\begin{proof}
It is sufficient to show
\begin{align}
\label{lemks1eq}
\| \mathcal{I}[f]\|_{L^{\infty}(\Omega)} &\le C(R,{ \Omega })\| f\|_{L^1(\Omega)}, \\
\label{lemks1eq0}
\| \mathcal{I}[f]\|_{W^{1,\infty}(\Omega)}
&\le C(R,{ \Omega })\| f\|_{W^{1,p}(\Omega)},
\end{align} where
\[
\mathcal{I}[f](x)=\displaystyle\int_{E(x)} f(x+\mathbf r) \omega(\mathbf r)d\mathbf r.
\]
\eqref{lemks1eq} is obvious.
We prove \eqref{lemks1eq0} when $B_L(0) $ is the two dimensional disc in $\mathbb R^2$ for a computational simplicity.
Let us denote the radial coordinate of $B_L(0)$ by $(s, \varphi)$, thus consider $ x=(x_1, x_2)= (s\cos \varphi, s\sin \varphi)$. We may assume $f$ is in $C^1(B_L(0))$ since $ f\in W^{1.p}({ \Omega })$ has an approximating sequences from $C^{\infty}({ \Omega })$ in $W^{1. p}({ \Omega })$.\\
\indent
When $ |x| <L-R$, we have
\begin{equation}\label{lemks1eq1}
\partial_{x_1} { \mathcal I }[f](x) = \int_{E(x)} f_{x_1}(x+\mathbf r) { \omega }(\mathbf r) d\mathbf r:= I^{in}(x).
\end{equation}
When $ L-R<|x| <L$, we use polar coordinates $(r, \theta)$ on $ B_{L-s}(0)$ to write
\begin{align*}
{ \mathcal I }[f](x)= \int_0^{L-s} \int_0^{2\pi} f(x+\mathbf r) { \omega }(\mathbf r) r d\theta dr.
\end{align*}
In this region we have
\begin{equation}\label{lemkseq2}
\begin{aligned}
\partial_{x_1} { \mathcal I }[f](x)=& \frac{\partial s}{\partial x_1} \frac{\pa}{\pa s} \int_0^{L-s} \int_0^{2\pi} f(x+\mathbf r) { \omega }(\mathbf r) r d\theta dr +
\frac{\partial \varphi}{\partial x_1}\frac{\pa }{\pa \vp} \int_0^{L-s} \int_0^{2\pi} f(x+\mathbf r) { \omega }(\mathbf r) r d\theta dr\\
= &- \frac{\partial s}{\partial x_1}\int_0^{2\pi} f(x + (L-s)(\cos \theta, \sin \theta) ){ \omega } ((L-s)(\cos\theta, \sin\theta)) d\theta \\
&+ \int_{E(x)} f_{x_1}(x+\mathbf r){ \omega }(\mathbf r) d\mathbf r\\
:=& I^o(x)
\end{aligned}
\end{equation}
with $ \frac{\partial s}{\partial x_1}= \cos \varphi$.
From the above we have
\begin{align*}
\| \partial_{x_1} { \mathcal I }[f]\|_{L^{\infty}({ \Omega })} \le C\|f\|_{L^{\infty}({ \Omega })} \| { \omega }\|_{L^{\infty}({ \Omega })}
+C \|{ \omega } \|_{L^{p'}({ \Omega })} \| f_{x_1}\|_{L^p({ \Omega })}
\end{align*} for $\frac{1}{ p'} + \frac{1}{p} =1$.
The same bound holds for $\| \partial_{x_2} { \mathcal I }[f]\|_{L^{\infty}({ \Omega })}$.
Hence we prove
\begin{equation}\label{lemks1eq3}
\| \partial_{x} { \mathcal I }[f]\|_{L^{\infty}({ \Omega })} \le C(R, { \Omega }) \|f\|_{W^{1,p}({ \Omega })},
\end{equation}
where $\partial_x{ \mathcal I }[f]$ denotes the pointwise differentiation as above.\\
\indent
For $\psi \in C_0^{\infty}({ \Omega })$ we have
\begin{align*}
\int_{{ \Omega }} { \mathcal I }[f](x) \partial_{x_1} \psi(x) dx
&=\lim_{\epsilon\to 0}\int_{|x|< L-R-\epsilon} { \mathcal I }[f](x) \partial_{x_1} \psi(x) dx\\
&\quad +
\lim_{\epsilon\to 0}\int_{L-R + \epsilon <|x|< L} { \mathcal I }[f](x) \partial_{x_1} \psi(x) dx\\
& = -\int_{|x|< L-R }I^{in}(x)\psi(x) dx
- \int_{L-R <|x|< L } I^o(x)\psi(x) dx\\
&\quad + \lim_{\epsilon\to 0} \int_{|x|= L-R-\epsilon} { \mathcal I }[f](x) \psi(x) \frac{x_1}{L-R-\epsilon} dx\\
&\quad -\lim_{\epsilon\to 0} \int_{|x|= L-R+\epsilon} { \mathcal I }[f](x) \psi(x) \frac{x_1}{L-R+\epsilon} dx\\
& =
-\int_{|x|< L-R }I^{in}(x)\psi(x) dx
- \int_{L-R <|x|< L } I^o(x)\psi(x) dx
\end{align*}
since $ { \mathcal I }[f]$ is continuous on $ |x|= L-R$. Hence
\[ D_{x_1} { \mathcal I }[f](x): = \begin{cases}
I^{in}(x) \quad |x| < L-R\\
I^o(x) \quad L-R< |x| < L
\end{cases}\]
is the weak derivative of ${ \mathcal I }[f]$ in ${x_1}$ and \eqref{lemks1eq0} follows due to \eqref{lemks1eq3}. This completes the proof.
\end{proof}
\end{subsection}
\begin{subsection}{Local well-posedness, blow-up criteria, and a priori estimates}
To prove local well-posedness of \eqref{MODEL11}--\eqref{omega2}, we introduce the sectorial operator $A:=(I-\Delta)|_{D(A)}$ with its domain
\[
D(A):=\{ f\in W^{2,p}(\Omega)\,|\,Bf:=\partial_{\nu}f=0\,\mbox{ on }\,\partial\Omega \},\quad 1<p<\infty.
\]
Since $(A,B)$ is a sectorial operator, it generates an analytic semigroup $\{ e^{-tA}\,|\,t\ge0 \}$ on $L^{p}(\Omega)$.
Note that the fractional powers of $A$ are well-defined (see \cite[Section 1.4]{H81}). We denote the domains of fractional powers by
\[
X^{\eta}_{p}:=D(A^{\eta}),\quad \eta\in(0,1).
\]
We also note from \cite[Theorem 1.6.1]{H81} that
\begin{equation}\label{FRACEM}
X^{\eta}_{p}\hookrightarrow W^{\kappa,q}(\Omega)\,\mbox{ for }\,\kappa-\frac{n}{q}<2\eta-\frac{n}{p}, \,q\ge p.
\end{equation}
Now we consider the homogenous Neumann boundary value problems for \eqref{MODEL11}--\eqref{omega2}:
\begin{equation}\label{NPBV1}
\begin{aligned}
&\partial_{t}u+(I-\Delta)u=g_{1},& &\partial_{t}v+(I-\Delta)v=g_{2}, \quad\qquad&x\in\Omega, \,\,t\le T,\,\,\,&\\
&\partial_{\nu}u= 0, &\qquad\quad &\partial_{\nu} v=0, &x\in\partial\Omega, \,\,t\le T,&\\
&u(x,0) =u_{0}(x),&\qquad &v(x,0) =v_{0}(x), &x\in\partial\Omega,\quad\qquad\,&
\end{aligned}
\end{equation}
and their integral representation formulas:
\begin{equation}\label{INTF11}
u=e^{-tA }u_{0}+\int_{0}^{t}e^{-(t-\tau)A } g_{1}(\tau) d\tau,\qquad v=e^{-tA }v_{0}+\int_{0}^{t}e^{-(t-\tau)A} g_{2}(\tau) d\tau,
\end{equation}
where
\begin{equation}\label{INTF12}
\begin{array}{ll}
g_{1}:=-\nabla \cdot (u \,\mathcal{K}[u,v] )+(1-m)u +\displaystyle\frac{\lambda}{k} u(k-(u+v)),
\\
g_{2}:=-\nabla \cdot (v \,\mathcal{S}[u,v] )+mu+v +\displaystyle\frac{\mu}{k} v(k-(u+v)).
\end{array}
\end{equation}
Before stating the local-in-time result, let us note the smoothing estimates for $e^{-t A} $ from \cite[Theorem 1.4.3]{H81}, which will be used in the proof of the next lemma:\\
If $p>1$, $0<\eta<1$, $f\in L^{p}(\Omega)$, then there exist positive constants $\kappa$, and $C=C(\eta)$ such that
\begin{equation}\label{NMNEMB}
\norm{ e^{-tA}f}_{X^{\eta}_{p}}\le Ce^{-\kappa t}t^{-\eta}\norm{f}_{L^{p}(\Omega)},\quad t>0.
\end{equation}
\begin{lemma}\label{LEM4}
Let $\Omega \subset{ \mathbb{R} }^{n}$, $n\ge2$, be an open ball of radius $L$, $B_{L}(0)$. Assume that $u_{0}$ and $v_{0}$ are non-negative functions belong to $W^{2,p}(\Omega)$, $p>n$, and satisfy compatibility condition
\[
\partial_{\nu}u_0=0,\quad \partial_{\nu}v_0=0,\qquad x\in\partial\Omega.
\]
Then, there is a maximal time, $T_{\rm max}\le\infty$, such that a pair of unique non-negative strong solution $(u,v)$ of \eqref{NPBV1} exists and satisfies
\[
u,v\in C([0,T_{\rm max}); W^{1,p}(\Omega))\cap W^{1,p}(0,t; L^{p}(\Omega))\cap L^{p}(0,t;W^{2,p}(\Omega)),\qquad t<T_{\rm max}.
\]
Moreover, if $T_{\rm max}<\infty$, then
\begin{equation}\label{BUC1}
\lim_{t\rightarrow T_{\rm max}} (\norm{u(\cdot,t)}_{W^{1,p}(\Omega)}+\norm{v(\cdot,t)}_{W^{1,p}(\Omega)} )=\infty.
\end{equation}
\end{lemma}
\begin{proof}
Let $n<p<\infty$. With positive constants $T<1$ and $R_0$ to be specified below, we introduce the Banach space $X_{T}:=C([0,T];W^{1,p}(\Omega))$ and its closed convex subset $S_{T}\subset X_{T}$,
\[
S_{T}:=\bigr{\{}\,f\in X_{T}\,|\, \norm{f}_{L^{\infty}(0,T; W^{1,p}(\Omega))}\le R_0 \bigr{\}}.
\]
Let $u,v \in S_{T}$. As in \eqref{INTF11}--\eqref{INTF12}, we consider \begin{equation}\label{PFPHI11}
\Phi_{1}(u,v):=e^{-tA }u_{0}+\int_{0}^{t}e^{-(t-\tau)A }g_{1}(\tau) d\tau,
\end{equation}
\begin{equation}\label{PFPHI22}
\Phi_{2}(u,v):=e^{-tA }v_{0}+\int_{0}^{t}e^{-(t-\tau)A } g_{2}(\tau) d\tau,
\end{equation}
where $g_{i}$ for $i=1,2$ are functions given in \eqref{INTF12}.
First, we show $\Phi_{1}(u,v),\Phi_{2}(u,v)\in S_{T}$. Let $\frac{1}{2}<\eta<1$ and note from \eqref{FRACEM} that
\begin{equation}\label{FRACEM1}
X^{\eta}_{p}\hookrightarrow W^{1,p}(\Omega).
\end{equation}
Using \eqref{NMNEMB} and \eqref{FRACEM1},
we can compute
\[
\begin{aligned}
\norm{\Phi_{1}(u,v)}_{W^{1,p}(\Omega)}&\le \norm{e^{-tA}u_{0}}_{W^{1,p}(\Omega)}+\int_{0}^{t}\|e^{-(t-\tau)A}g_{1}(\tau)\|_{W^{1,p}(\Omega)}d\tau\\
&
\le K_{3}\norm{u_{0}}_{W^{1,p}(\Omega)}+C\int_{0}^{t}\|e^{-(t-\tau)A}g_{1}(\tau)\|_{X^{\eta}_{p}}d\tau\\
&
\le K_{3}\norm{u_{0}}_{W^{1,p}(\Omega)}+C\int_{0}^{t}e^{-\kappa(t-\tau)}(t-\tau)^{-\eta}\norm{g_{1}(\tau)}_{L^{p}(\Omega)} d\tau.
\end{aligned}
\]
By Lemma~\ref{LEMKS1} and $W^{1,p}(\Omega)\hookrightarrow L^{\infty}(\Omega)$, we have
\[
\norm{g_{1}(\tau)}_{L^{p}(\Omega)}=\bigr{\|}-\nabla \cdot (u \,\mathcal{K}[u,v] )+(1-m)u +\displaystyle\frac{\lambda}{k} u(k-(u+v))\bigr{\|}_{L^{p}(\Omega)}\le C (R_0+R_0^{2}),
\]
and thus,
\[
\norm{\Phi_{1}(u,v)}_{W^{1,p}(\Omega)}\le K_{3}\norm{u_{0}}_{W^{1,p}(\Omega)}+ C_{13}(R_0+R_0^{2})t^{1-\eta},
\]
where $C_{13}$ is a positive constant independent of $R_0$.
Similarly, we also have
\[
\norm{\Phi_{2}(u,v)}_{W^{1,p}(\Omega)}\le K_{4}\norm{v_{0}}_{W^{1,p}(\Omega)}+ C_{14}(R_0+R_0^{2})t^{1-\eta},
\]
where $C_{14}$ is a positive constant independent of $R_0$.
Taking
\[
R_0:=K_{3}\norm{u_{0}}_{W^{1,p}(\Omega)}+K_{4}\norm{v_{0}}_{W^{1,p}(\Omega)}+1,
\]
and
\[
T<T_{3}:=\min \bket{ 1, \frac{1}{(2 C_{13}(R_0+R_0^{2}))^{\frac{1}{ 1-\eta}}}, \frac{1}{(2 C_{14} (R_0+R_0^{2}))^{\frac{1}{1-\eta}}} },
\]
we obtain
$
\Phi_{1}(u,v),\Phi_{2}(u,v)\in S_{T}$.
We next show the mapping $(u,v)\mapsto (\Phi_{1},\Phi_{2})$ is a contraction. Using \eqref{INTF11}, we note that
\[
\Phi_{1}(u,v)-\Phi_{1}(\tilde{u},\tilde{v})=\int_{0}^{t}e^{-(t-\tau)A}(g_{1}-\tilde{g}_{1})(\tau)d\tau,
\]
\[
\Phi_{2}(u,v)-\Phi_{2}(\tilde{u},\tilde{v})=\int_{0}^{t}e^{-(t-\tau)A}(g_{2}-\tilde{g}_{2})(\tau) d\tau,
\]
where
\[
\tilde{g}_{1}=-\nabla \cdot (\tilde{u} \,\mathcal{K}[\tilde{u},\tilde{v}] )+(1-m)\tilde{u} +\displaystyle\frac{\lambda}{k} \tilde{u}(k-(\tilde{u}+\tilde{v})),
\]
\[
\tilde{g}_{2}=-\nabla \cdot (\tilde{v} \,\mathcal{S}[\tilde{u},\tilde{v}] )+ m\tilde{u} +\tilde{v}+\displaystyle\frac{\mu}{k} \tilde{v}(k-(\tilde{u}+\tilde{v})).
\]
Then, analogously to the above computations, we can estimate
\[
\norm{(\Phi_{1},\Phi_{2})(u,v)-(\Phi_{1},\Phi_{2})(\tilde{u},\tilde{v})}_{W^{1,p}(\Omega)}\le C_{15} (R_0+1)T^{1-\eta}\norm{ (u,v)- (\tilde{u},\tilde{v})}_{W^{1,p}(\Omega)}.
\]
Taking
\[
T<T_{4}:=\min \bket{T_{3}, \frac{1}{(2 C_{15} (R_0+1))^{\frac{1}{ 1-\eta}}} } ,
\] we obtain
\[
\norm{(\Phi_{1},\Phi_{2})(u,v)-(\Phi_{1},\Phi_{2})(\tilde{u},\tilde{v})}_{W^{1,p}(\Omega)}\le \frac{1}{2}\norm{ (u,v)- (\tilde{u},\tilde{v})}_{W^{1,p}(\Omega)},
\]
i.e., the mapping is a contraction. According to the Banach fixed point theorem, this mapping has a fixed point in $S_{T}$, denoted again as $(u,v)$.
The uniqueness of $(u,v)$, and the blow-up criterion \eqref{BUC1} can be obtained as in the proof of Lemma~\ref{LEM1}. By the maximal regularity theorem (see, e.g., Ladyzhenskaya-Solonnikov-Ural\'ceva \cite[Section 4. Theorem 9.1]{LSU88}), we have
\[
u,v\in L^{p}(0,T; W^{2,p}(\Omega))\cap W^{1,p}(0,T; L^{p}(\Omega)).
\]
Then, the non-negativity of solutions is followed as \eqref{UNU}--\eqref{UNV}. This completes the proof.
\end{proof}
Next, we provide a refined blow-up criterion (Lemma~\ref{BUCLEM1}), and a priori estimates (Lemma~\ref{LEML11} and Lemma~\ref{LEMLINFUV1}). In order to reduce the redundancies, we refer to see the proofs of Lemma~\ref{BUCLEM}, Lemma~\ref{LEML1}, and Lemma~\ref{LEMLINFUV}.
\begin{lemma}\label{BUCLEM1}
Let the same assumptions as in Lemma~\ref{LEM4} be satisfied. The solution $(u,v)$ of \eqref{NPBV1} given by Lemma~\ref{LEM4} satisfies
\[
\mbox{either }\, T_{\rm max}=\infty,\,\mbox{ or }\, \lim_{t\rightarrow T_{\rm max}}(\norm{u(\cdot,t)}_{L^{\infty}(\Omega)}+\norm{v(\cdot,t)}_{L^{\infty}(\Omega)})=\infty.
\]
\end{lemma}
\begin{lemma}\label{LEML11}
Let the same assumptions as in Lemma~\ref{LEM4} be satisfied. The solution $(u,v)$ of \eqref{NPBV1} given by Lemma~\ref{LEM4} for $T<T_{\rm max}$ satisfies
\begin{equation}\label{UL11}
\sup_{t\le T}\int_{\Omega}u(\cdot,t)\le C(\norm{u_{0}}_{L^{1}(\Omega)}),
\end{equation}
\begin{equation}\label{VL11}
\sup_{t\le T}\int_{\Omega}v(\cdot,t)\le C(\norm{u_{0}}_{L^{1}(\Omega)}, \norm{v_{0}}_{L^{1}(\Omega)}),
\end{equation}
\begin{equation}\label{KLINF1}
\sup_{t\le T} \norm{\mathcal{K}[u,v](\cdot, t)}_{L^{\infty}(\Omega)}\le C(\norm{u_{0}}_{L^{1}(\Omega)}, \norm{v_{0}}_{L^{1}(\Omega)}),
\end{equation}
and
\begin{equation}\label{SLINF1}
\sup_{t\le T} \norm{\mathcal{S}[u,v](\cdot, t)}_{L^{\infty}(\Omega)}\le C(\norm{u_{0}}_{L^{1}(\Omega)}, \norm{v_{0}}_{L^{1}(\Omega)}).
\end{equation}
\end{lemma}
\begin{lemma}\label{LEMLINFUV1}
Let the same assumptions as in Lemma~\ref{LEM4} be satisfied. The solution $(u,v)$ of \eqref{NPBV1} given by Lemma~\ref{LEM4} for $T<T_{\rm max}$ satisfies
\begin{equation}\label{ULINFTY1}
\sup_{t\le T}\norm{u(\cdot,t)}_{L^{\infty}(\Omega)}\le C(\norm{u_{0}}_{(L^{1}\cap L^{\infty})(\Omega)}, \norm{v_{0}}_{L^{1}(\Omega)}),
\end{equation}
and
\begin{equation}\label{VLINFTY1}
\sup_{t\le T}\norm{v(\cdot,t)}_{L^{\infty}(\Omega)}\le C(\norm{u_{0}}_{(L^{1}\cap L^{\infty})(\Omega)}, \norm{v_{0}}_{(L^{1}\cap L^{\infty})(\Omega)}).
\end{equation}
\end{lemma}
\end{subsection}
\begin{pfthm2}
It is a direct consequence of local-in-time existence, uniqueness, non-negativity (Lemma~\ref{LEM4}), the blow-up criterion (Lemma~\ref{BUCLEM1}), and a priori estimates (Lemma~\ref{LEMLINFUV1}). This completes the proof.
\end{pfthm2}
\begin{remark}
If the adhesive strength of $v$ is negligible, $\mathcal{S}[u,v]=0$, and the growth rate of $u$ is smaller than that of $v$, $\lambda<\mu$,
then the solution asymptotically tends to constant equilibrium. Indeed, the solution $(u,v)$ of \eqref{MODEL11}--\eqref{omega2} with $\inf_{\overline{\Omega}}v_0>0$ given by Theorem~\ref{GETHM1} satisfy
\[
\frac{d}{dt}\bkt{ \frac{1}{\lambda}\int_{\Omega}u +\frac{1}{\mu}\int_{\Omega}v -\frac{k}{\mu}\int_{\Omega}\log v }
+\frac{k}{\mu}\int_{\Omega} \biggr{|}\frac{\nabla v}{v} \biggr{|}^{2}+\frac{mk}{\mu}\int_{\Omega}\frac{u}{v}+m\bke{ \frac{1}{\lambda}-\frac{1}{\mu} }\int_{\Omega}u+\frac{1}{k}\int_{\Omega}|u-(k-v)|^{2}
\]
\[
=\frac{k}{\mu}\int_{\Omega}\mathcal{S}[u,v]\frac{\nabla v}{v}.
\]
Thus, one can verify that $\int_{\Omega}u(\cdot,t)\rightarrow 0$ and $\int_{\Omega}|u-(k-v)|^2(\cdot,t)\rightarrow 0$ as time tends to infinity whenever $\mathcal{S}[u,v]=0$ and $\lambda<\mu$. Then, by $u\ge0$, we have $(u,v)\rightarrow(0,k)$.\end{remark}
\section*{Appendix}
Let ${ \Omega }, {\mathcal K}, { \mathcal S }$ and ${ \omega }$ be given as in Section $3$. We define the operator
$ F: W^{2,p}_B \times W^{2,p}_B \to L^2\times L^2$ by
\begin{align*}
F(u,v)&= (F_1(u,v), F_2(u,v)),\\
F_1(u,v)&= \Delta u-\nabla \cdot (u \,\mathcal{K}[u,v] )+(1-m)u +\displaystyle\frac{\lambda}{k} u(k-(u+v)) \\
F_2(u,v)&= \Delta v-\nabla \cdot (v \,\mathcal{S}[u,v] )+mu+v +\displaystyle\frac{\mu}{k} v(k-(u+v)).
\end{align*}
We denote the G\^ateaux derivative of $F$ at $U=(u,v)$ by $T_{U}$;\\
\[ T_{U}(W) = \lim_{t\to 0} \frac{F(U+tW)- F(U)}{t} = (\delta_W F_1(U), \delta_W F_2(U))\] where
$ W= (w, z)
\in W^{2,p}_B \times W^{2,p}_B$.
By computation we have
\begin{align*}
\delta_WF_1(0,k)& = \Delta w-mw \\
\delta_WF_2(0,k)& = \Delta z - \nabla \cdot( k{ \mathcal S }[w,0]+kS[w,z]+z{ \mathcal S }[0,k]) + (m-\mu)w - \mu z\\
\delta_WF_1(0,0)& = \Delta w - mw\\
\delta_WF_2(0,0)& = \Delta z + mw + \mu z.
\end{align*}
We consider the two linearized equations at $(0,k)$ and $(0,0)$ respectively with initial data
$(w_0, z_0)\in W^{2,p}_B \times W^{2,p}_B$;
\begin{align}\label{L1}
\begin{aligned}
\partial_t w &= \Delta w-mw\\
\partial_t z &= \Delta z - \nabla \cdot( k{ \mathcal S }[w,0]+kS[w,z]+z{ \mathcal S }[0,k]) + (m-\mu)w - \mu z,
\end{aligned}
\end{align}
and
\begin{align}\label{L2}
\begin{aligned}
\partial_t w &= \Delta w-mw \\
\partial_t z &= \Delta z + mw + \mu z. \phantom{ \nabla \cdot( k{ \mathcal S }[w,0]+kS[w,z]+z{ \mathcal S }[0,k]) + m+m}
\end{aligned}
\end{align}
The equations \eqref{L1}, \eqref{L2} are decoupled and it is immediate that
\begin{align}\label{apeq1}
\| w\|_{W^{1,p}({ \Omega })} \le e^{-mt}\|w_0\|_{W^{1,p}}, \quad p\ge 1
\end{align}
from
\[ \partial_t(e^{m t} w ) = \Delta (e^{m t} w ).\]
Let $ \tilde z$ denote $e^{\mu t} z$.
Multiplying to the $z$- equation of \eqref{L1} by ${e^{\mu t}}$, we have
\begin{align}\label{apeq3}
\partial_t \tilde z - \Delta \tilde z = -\nabla \cdot( 2k{ \mathcal S }[e^{\mu t}w,0]+kS[0,\tilde z]+ \tilde z{ \mathcal S }[0,k]) + (m-\mu) e^{\mu t}w.
\end{align}
It holds that
\[ \frac{d}{dt} \int_{\Om} |\tilde z |\le |m-\mu| e^{\mu t} \int |w| \le |m-\mu| e^{-(m-\mu)t} \| w_0\|_{L^1({ \Omega })},\]
which implies
\begin{align*}
\int_{\Om} |\tilde z |& \le \int_{\Om} |z_0| - \frac{|m-\mu|}{m-\mu}(e^{-(m-\mu)t}-1)\int_{\Om} |w_0|
\end{align*}
and
\begin{align}
\int_{\Om} |z |& \le e^{-\mu t} \int_{\Om} |z_0| + (e^{-mt}- e^{-\mu t})\int_{\Om} |w_0| \label{apeq111},
\end{align}
where we abuse the notation by
$ | m-\mu|/(m-\mu) =0$ if $ \mu=m$. When $m > \mu$, it holds that
\begin{align}\label{apeq1111}
\int | \tilde z| \le \| z_0\|_{L^1({ \Omega })} + \| w_0\|_{L^1({ \Omega })}.
\end{align}
\indent
In what follows we find that
the different signs of $\mp \mu z$ in \eqref{L1} and \eqref{L2} imply that $(0, k)$ is linearly stable and $(0,0)$ is linearly unstable as expected.
\begin{proposition}\label{stab}
Each of the linearized equations \eqref{L1}, \eqref{L2} have a unique global solution $(w,z)$ for each in
\[C([0,t); W^{1,p}(\Omega))\cap W^{1,p}(0,t; L^{p}(\Omega))\cap L^{p}(0,t;W^{2,p}(\Omega))\]
for any $t>0$. When $m> \mu$, the solution $(w,z)$ for \eqref{L1} is asymptotically stable such that
\begin{align}\label{appeq2}
\|z\|_{L^p({ \Omega }) }\le e^{-\mu t}\| z_0\|_{L^p({ \Omega })} \quad \mbox{ for } p\ge 1.
\end{align}
The solution $(w,z)$ for \eqref{L2} grows exponentially in its $L^1$-norm if the initial data is non-negative;
\begin{align}\label{appeq3}
\int_{ \Omega } |z| \ge e^{\mu t}\int_{ \Omega } |z_0|.
\end{align}
\end{proposition}
\begin{proof}
Due to the a priori estimates \eqref{apeq1} and \eqref{apeq111}, the global well-posedness part for \eqref{L1} follows from the same argument in the subsection $2.3$ or the subsection $3.2$. Repeating the argument of Lemma \ref{LEMLINFUV} to \eqref{L1} it holds that
\begin{equation}\label{MA-z}
\| z\|_{L^{\infty}(\Om)} \le C(\|w_0\|_{L^1\cap L^{\infty}({ \Omega })}, \|z_0\|_{L^1\cap L^{\infty}({ \Omega })}).
\end{equation}
For details see \eqref{tz}--\eqref{tzinf} for $\tilde z$, where the similar estimates are given.
By Lemma \ref{BUCLEM1} it also holds that
\begin{equation}\label{bw}
\| z\|_{W^{1,p}} \le C(\|w_0\|_{L^1\cap L^{\infty}({ \Omega })}, \|z_0\|_{L^1\cap L^{\infty}({ \Omega })} )
\end{equation}
for any $p\ge 1$.
Let us prove \eqref{appeq3} first. The solution $(w,z)$ remains non-negative and we have
\begin{align*}
\int_{\Om} w& = e^{-mt}\int_{\Om} w_0,\\
\frac{d}{dt}(e^{-\mu t} \int_{\Om} z) & = m e^{-(\mu+ m)t} \int_{\Om} w_0.
\end{align*}
Integrating the second equation, we have \eqref{appeq3}.
\\
\indent
For \eqref{appeq2} we proceed as in Lemma \ref{LEMLINFUV}. Multiplying $|\tilde z|^{p-2}\tilde z$
into \eqref{apeq3} for $p \ge2$, we have
\begin{align}\label{tz}
\frac 1p \frac{d}{dt} \int_{\Om} |\tilde z|^{p} + \frac{4(p-1)}{p^2}\int_{\Om} |\nabla \tilde z^{\frac p2}|^2 = &
\int_{\Om} |\tilde z|^{p-2}\tilde z \nabla \cdot( 2k{ \mathcal S }[e^{\mu t}w,0]+kS[0,\tilde z]+ \tilde z{ \mathcal S }[0,k]) \\
& + (m-\mu) \int_{\Om} e^{\mu t}w |\tilde z|^{p-2}\tilde z.
\end{align}
By Lemma \ref{LEMKS1}, \eqref{apeq1}, \eqref{bw} and using $m >\mu$, we have
\begin{align*}
& \| \nabla \cdot { \mathcal S } [ e^{\mu t} w, 0]\|_{L^{\infty}(\Om)} \le C\| e^{\mu t} w\|_{W^{1,q}({ \Omega })} \le C \|w_0\|_{W^{1,q}({ \Omega })} (q >n)\\
& \| { \mathcal S }[ 0, \tilde z] \|_{L^{\infty}(\Om)} \le C\| \tilde z\|_{L^1({ \Omega })}
\le
C(\|w_0\|_{L^1}, \|z_0\|_{L^1})\\
& \| \nabla { \mathcal S }[0, k]\|_{L^{\infty}(\Om)} \le C
\end{align*}
and estimate the right hand side of \eqref{tz} as follows,
\begin{align*}
&\int_{\Om} |\tilde z|^{p-2}\tilde z \nabla \cdot( 2k{ \mathcal S }[e^{\mu t}w,0]) + (m-\mu) \int_{\Om} e^{\mu t}w |\tilde z|^{p-2}\tilde z \le C\int_{\Om} |\tilde z|^{p-1} \\
&\int_{\Om} |\tilde z|^{p-2}\tilde z \nabla \cdot kS[0,\tilde z] \le \frac{p-1}{p^2} \int_{\Om} | \nabla \tilde z^{\frac p2|}|^2 + C(p-1) \int \tilde z^{p-2}
\| { \mathcal S }[0, \tilde z\|_{{L^{\infty}(\Om)}}^2 \\
& \phantom{\int_{\Om} |\tilde z|^{p-2}\tilde z \nabla \cdot kS[0,\tilde z] } \le \frac{p-1}{p^2} \int_{\Om} | \nabla \tilde z^{\frac p2}|^2 + C \frac{p-1}{p} \left(|{ \Omega }| +(p-2)\int_{\Om} |\tilde z|^{p} \right),\\
& \int_{\Om} |\tilde z|^{p-2}\tilde z \nabla \cdot( \tilde z{ \mathcal S }[0,k]) \le \frac{1}{p^2} \int_{\Om} | \nabla \tilde z^{\frac p2}|^2 + C\int_{\Om} |\tilde z|^{p} \| \nabla { \mathcal S }[0, k]\|_{{L^{\infty}(\Om)}}.
\end{align*}
Summing up, we have
\begin{align*}
\frac 1p \frac{d}{dt} \int_{\Om} |\tilde z|^{p} + \frac{3(p-1)}{p^2}\int_{\Om} |\nabla \tilde z^{\frac p2}|^2 \le C+ C\int_{\Om} |\tilde z|^{p}, \quad p\ge 2,
\end{align*}
where $C$ is a uniform constant depending on $\| w_0\|_{L^1({ \Omega })}$, $\|z_0\|_{L^1({ \Omega })}$, and given constants $\mu, m, k$
etc.. As was derived from \eqref{MA_33} for $v$ in Lemma \ref{LEMLINFUV} it holds that
\[\sup_ {0 <t \le T} \| \tilde z\|_{L^{p_k}({ \Omega })} \le C(\|z_0\|_{L^1({ \Omega })}, \|z_0\|_{{L^{\infty}(\Om)}}) \sup_ {0 <t \le T}\left(\| \tilde z \|_{L^1({ \Omega })}
+C \right)\quad p_k= 2^k, k=0, 1\dots.\]
and
\begin{align}\label{tzinf} \sup_ {0 <t \le T} \| \tilde z\|_{L^{\infty}(\Om)} \le C(\|w_0\|_{L^1\cap L^{\infty}({ \Omega })}, \|z_0\|_{L^1\cap L^{\infty}({ \Omega })}).
\end {align}
That implies \eqref{appeq2}.
\end{proof}
\section*{Acknowledgement}
Jaewook Ahn's work is supported by NRF-2018R1D1A1B07047465. Jihoon Lee's work is supported by SSTF-BA1701-05.
Myeongju Chae's work is supported by NRF-2018R1A1A 3A04079376.
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Q: mapping a 2d delay vector from a 1d vector in numpy I am trying to generate a 2D vector from a 1D vector where the element is shifted along the row by an increment each row.
i would like my input to look like this:
input:
t = [t1, t2, t3, t4, t5]
out =
[t5, 0, 0, 0, 0]
[t4, t5, 0, 0, 0]
[t3, t4, t5, 0, 0]
[t2, t3, t4, t5, 0]
[t1, t2, t3, t4, t5]
[ 0, t1, t2, t3, t4]
[ 0, 0, t1, t2, t3]
[ 0, 0, 0, t1, t2]
[ 0, 0, 0, 0, t1]
im unaware of a way to do this without using a for loop, and computational efficieny is important for the task im using this for. Is there a way to do this without a for loop?
this is my code using a for loop:
import numpy as np
t = np.linspace(-3, 3, 7)
z = np.zeros((2*len(t) - 1, len(t)))
diag = np.arange(len(t))
for index, val in enumerate(np.flip(t, 0)):
z[diag + index, diag] = val
print(z)
A: What you're asking for here is known as a Toeplitz Matrix, which is:
a matrix in which each descending diagonal from left to right is constant
The one difference is that you want the lower triangle of your matrix.
You happen to be in luck, you can use scipy.linalg.toeplitz to construct your matrix, and then np.tril to access the lower triangle.
import numpy as np
from scipy.linalg import toeplitz
v = np.array([1, 2, 3, 4, 5])
t = np.pad(v[::-1], (0, 4), mode='constant')
Solve the matrix and access the lower triangle:
np.tril(toeplitz(t, v))
And our output!
array([[5, 0, 0, 0, 0],
[4, 5, 0, 0, 0],
[3, 4, 5, 0, 0],
[2, 3, 4, 5, 0],
[1, 2, 3, 4, 5],
[0, 1, 2, 3, 4],
[0, 0, 1, 2, 3],
[0, 0, 0, 1, 2],
[0, 0, 0, 0, 1]])
To generalize this method, simply calculate the necessary padding for t from the shape of v:
v = # any one dimension array
t = np.pad(v[::-1], (0, v.shape[0]-1), mode='constant')
A: Not aware of a non-loopy method, but you could probably speed this up a bit with roll and column_stack.
v = np.array([1, 2, 3, 4, 5])
t = np.pad(v[::-1], (0, 4), mode='constant')
np.column_stack([np.roll(t, i) for i in range(len(v))])
array([[5, 0, 0, 0, 0],
[4, 5, 0, 0, 0],
[3, 4, 5, 0, 0],
[2, 3, 4, 5, 0],
[1, 2, 3, 4, 5],
[0, 1, 2, 3, 4],
[0, 0, 1, 2, 3],
[0, 0, 0, 1, 2],
[0, 0, 0, 0, 1]])
A: Here's one vectorized way leveraging strides-tricks that simply uses a view into a zeros-padded array and being a view its memory efficient and hence performant -
def map2D(a):
n = len(a)
p = np.zeros(n-1,dtype=a.dtype)
a_ext = np.r_[p,a,p]
s0 = a_ext.strides[0]
strided = np.lib.stride_tricks.as_strided
return strided(a_ext[-n:], (len(a_ext)-n+1,n), (-s0,s0), writeable=False)
Sample run -
In [81]: a = np.array([2,5,6,7,9])
In [82]: map2D(a)
Out[82]:
array([[9, 0, 0, 0, 0],
[7, 9, 0, 0, 0],
[6, 7, 9, 0, 0],
[5, 6, 7, 9, 0],
[2, 5, 6, 7, 9],
[0, 2, 5, 6, 7],
[0, 0, 2, 5, 6],
[0, 0, 0, 2, 5],
[0, 0, 0, 0, 2]])
If you need the output to have its own memory space, use .copy()
Timings on 5k elements array -
In [83]: a = np.random.randint(0,9,(5000))
# From this post's soln
In [84]: %timeit map2D(a)
10000 loops, best of 3: 26.3 µs per loop
# If you need output with its own memory space
In [97]: %timeit map2D(a).copy()
10 loops, best of 3: 43.6 ms per loop
# @user3483203's soln
In [87]: %%timeit
...: t = np.pad(a[::-1], (0, len(a)-1), mode='constant')
...: out = np.tril(toeplitz(t, a))
1 loop, best of 3: 264 ms per loop
# @coldspeed's soln
In [89]: %%timeit
...: t = np.pad(a[::-1], (0, len(a)-1), mode='constant')
...: out = np.column_stack([np.roll(t, i) for i in range(len(a))])
1 loop, best of 3: 336 ms per loop
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{"url":"https:\/\/byjus.com\/chemistry\/amines-identification\/","text":"Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :\n\n# Identification of Primary Amines, Secondary Amines and Tertiary Amines\n\n## What are Amines?\n\nWhen an organic compound with one or more hydrogen atoms of the ammonia molecule is replaced with an alkyl or even an aryl group, these compounds are classified as amines.\n\nThese are carbon atoms which are bonded to the nitrogen atom and their bond is quite strong.\n\n## Primary Amines, Secondary Amines and Tertiary Amines\n\nAmines are similarly divided into 3 different types; which are:\n\n1. Primary amine\n2. Secondary amine\n3. Tertiary amine\n\nOnce it bonds with 2 carbon atoms it is called as a secondary amine and when it bonds with 3, it is called as a tertiary amine. Primary, Secondary and Tertiary amines all show different chemical properties and also have physical, observable changes. They are mainly used in industrial and commercial applications.\n\nAmines usually have particularly distinct properties about them, such as their characteristic odours. These odours usually that of rotting eggs or fishes. Aliphatic amines are amines that are less dense than water and are usually stronger bonds of ammonia than aromatic amines. The major industrial applications are in making rubber, dyes, pharmaceuticals and synthetic resins and fibres. Certain tests are carried out for the identification of primary amines, secondary amines, and tertiary amines. One of the most popular tests is Hinsberg test and the reaction produced from this test is called the Hinsberg reaction.\n\n## Hinsberg test\n\nA chemical test that is most commonly used for the identification of primary, secondary and tertiary amines is called the Hinsberg test. An amine in the presence of an aqueous alkali interacts with a Hinsberg reagent. Thus, this is what is meant as the Hinsberg test. After the reaction, the following observations are observed:\n\nGetting a primary amine to react with a Hinsberg reagent usually leads to an amide being formed, called N-ethylbenzenesulphonyl amide, whereas the reagent used is called benzene sulfonyl chloride. This is strongly acidic as the hydrogen that is attached as a part of the nitrogen compound. Thus, this solution is also soluble in alkali.\n\nThus, Hinsberg test is effective for the identification of primary amines, secondary amines, and tertiary amines. Amines are usually used for a variety of different purposes such as a variety of different medicines and photographs. Apart from this, the amine is also used for the synthesis of rocket propellants and insecticides. Thus, amines have a whole host of different uses in the industry along with their traditional chemical uses. It is also used in heavy-duty military functions such as the creation of synthetic fibres, used in the production of Kevlar, which is the key component in creating helmets and bulletproof vests for protection of soldiers in warfare.\n\n## Hoffman Mustard Oil Reaction\n\nIn the Hofmann mustard oil reaction of primary amines, the black precipitate is due to HgS. It is a test of primary amine.\n\nPrimary amine gives alkyl isothiocyanate having mustard oil like smell.\n\n$$\\begin{array}{l}RNH_{2} + CS_{2}\\rightarrow S=C(SH)-NHR \\overset{HgCl_{2}}{\\rightarrow} RNCS + HgS + 2HCl\\end{array}$$\n\nSecondary amine doesn\u2019t show Hofmann\u2019s mustard oil reaction.\n\n## Frequently Asked Questions \u2013 FAQs\n\n### What is a primary amine group?\n\nAn amine in which the amino group is directly bonded to one carbon of any hybridization which cannot be a carbonyl group carbon. X = any atom but carbon; usually hydrogen.\n\n### Is NH2 a primary amine?\n\nAmines in the IUPAC system: the \u201ce\u201d ending of the alkane name for the longest chain is replaced with amine. More complex primary amines are named with -NH2 as the amino substituent.\n\n### What is an amine used for?\n\nAmines are used in making azo-dyes and nylon apart from medicines and drugs. They are widely used in developing chemicals for crop protection, medication and water purification.\n\n### Why is amine so important?\n\nAmines play an important role in the survival of life \u2013 they are involved in the creation of amino acids, the building blocks of proteins in living beings. Many vitamins are also built from amino acids. Serotonin is an important amine that functions as one of the primary neurotransmitters for the brain.\n\n### What is a basic amine? Is amine flammable?\n\nAmines, phosphines, and pyridines are generally high-boiling liquids or solids at room temperature and are combustible, but not highly flammable. The combustion of amines yields noxious NOx gases.\n\nFor the detailed discussion on identification of primary amines, secondary amines, and tertiary amines, please download BYJU\u2019S \u2013 The Learning App.\n\nTest your knowledge on amines identification!","date":"2022-07-06 16:30:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3164653778076172, \"perplexity\": 5129.281795475651}, \"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-2022-27\/segments\/1656104675818.94\/warc\/CC-MAIN-20220706151618-20220706181618-00537.warc.gz\"}"}
| null | null |
(14 de octubre de 1926-13 de febrero de 2011) fue un ingeniero mecánico y electrónico japonés. Trabajó en la corporación Sony desde 1947 hasta 2006. Fue reconocido, principalmente, por su trabajo en el desarrollo del primer modelo de walkman en la década de 1970.
Además ha trabajado en: radio transistores, magnetoscopios, magnetófonos portátiles de casete, sistemas de sonido estéreo, el formato Betamax, la cámara digital, etc.
Se le conocía como «Mr. Walkman» en la prensa, y dentro de la corporación Sony como «El tesoro de Sony».
Biografía
Nobutoshi Kihara nació en Tokio el 14 de octubre de 1926. Estudió Ingeniería Mecánica en la Universidad de Waseda. Para pagarse sus estudios universitarios Kihara buscaba radios estropeadas, que reparaba y vendía.
Kihara se graduó en 1947, y ese mismo año comenzó a trabajar en la compañía electrónica Sony. Estuvo bajo la supervisión personal de Masaru Ibuka, fundador de Sony, desde la misma entrevista de acceso a Sony; y pasó rápidamente de su trabajo inicial (fabricación de productos) a ser ingeniero de diseño.
En la década de 1950 colaboró activamente en el desarrollo tecnológico del primer radio transistor con éxito comercial de los años 50 y también en una amplia variedad de tecnologías de grabación. En 1964, Kihara lideró el equipo de desarrollo del CV-2000, el primer magnetoscopio destinado al mercado doméstico que utilizaba cintas de casete.
Kihara, siendo Ingeniero Mecánico, también aprendió de forma autodidacta Ingeniería Electrónica, llegando a ser un experto en ambas ramas tecnológicas. Esto le permitió desarrollar investigaciones que se han protegido mediante cientos de patentes en Japón y de otras cientos internacionales.
En 2006, con 80 años de edad, decidió retirarse de Sony. Cinco años más tarde, el 13 de febrero de 2011 falleció en Tokio a los 84 años debido a un ataque cardiaco.
Trayectoria profesional
1947 abril: Empieza a trabajar en Tokio Tsushin Kogyo K. K. (Corporación de Ingeniería de Telecomunicaciones Tokio)
1949: Desarrolla la primera grabadora de cinta magnética japonesa.
1950 enero: Desarrolla la Type G, la primera grabadora de cintas japonesa y el Type H, la primera grabadora de cintas destinada al mercado doméstico.
1951: Desarrolla la Type P, una grabadora de cintas portátil.
1951 abril: Desarrolla la Type M, también denominado como Densuke, una grabadora de cintas para radiodifusión.
1955 julio: Desarrolla el TR-55, el primer radio transistor de bolsillo japonés.
1956: Desarrolla la televisión en blanco y negro por transistores.
1958 enero: La empresa Tokyo Tsushin Kogyo K.K cambia su nombre a Sony Corporation.
1962: Desarrolla la PV-100, la primera grabadora de cintas de vídeo destinada a teledifusión y entornos industriales.
1965: Desarrolla la CV-2000, la primera grabadora de cintas por transistores destinada al mercado doméstico.
1966: Desarrolla la AV-3400, una grabadora de cintas de vídeo portátil.
1967 enero: Es nombrado segundo director de la sección de desarrollo.
1969 noviembre: Desarrolla un prototipo del U-matic, el primer magnetoscopio que utiliza cintas de casete y graba a color.
1970 junio: Es nombrado director de Sony.
1971: Desarrolla la versión definitiva de U-matic junto a 7 empresas asociadas.
1975 abril: Desarrolla el magnetoscopio Betamax, el primer magnetoscopio destinado al mercado doméstico.
1975 junio: Es nombrado director ejecutivo de Sony.
1980 julio: Desarrolla el Video Movie, un sistema compacto de vídeo de 8mm que integra cámara y reproductor de vídeo.
1981 agosto: Desarrolla la Mavica, una cámara magnética de vídeo estático.
1982 marzo: Desarrolla la Mavigraph, una impresora de color para vídeo estático.
1982 mayo: Es nombrado consejero delegado de Sony y director del centro de investigación.
1988 octubre: Es nombrado presidente del centro de investigación Sony-Kihara.
1989 junio: Es nombrado asesor superior de Sony.
1996 octubre: Es nombrado asesor de Sony.
2000 abril: Es nombrado presidente del consejo del centro de investigación Sony-Kihara.
2006: Abandona la empresa Sony
Reconocimientos y premios
1961 mayo: Premio Especial Sony.
1965: Premio Nikkan Kogyo Shinbun.
1967 abril: Reconocimiento al mérito en el campo de la Ingeniería, otorgado por el Ministerio de Ciencia y Tecnología de Japón.
1968 agosto: Décimo Premio Anual de Diseño Industrial WESCON.
1976 mayo: Primer premio Ibuka de IEEE (Instituto de Ingenieros Eléctricos y Electrónicos).
1977 junio: Premio Outstanding Papers de IEEE.
1979 agosto: Premio Eduard Rhein de 1978.
1981 mayo: Cuarto premio Ibuka de IEEE.
1982 junio: Premio Outstanding Papers de IEEE de 1981.
1982 junio: Medalla David Sarnoff de IEEE por su contribución a la grabación de vídeo en cintas magnéticas.
1983 enero: Grado Fellow en IEEE.
1983 junio: Premio Transactions Paper Award de IEEE de 1982.
1989 abril: Premio de Honor Internacional Interkamera '89 (Checoslovaquia)
1989 septiembre: Premio por la contribución a la tecnología fotográfica en su 150 aniversario (Checoslovaquia)
1990 abril: Medalla de honor con cinta púrpura del Emperador de Japón (紫綬褒章).
1992 mayo: Doctorado Honorífico de Investigación por la Universidad de Oklahoma.
Referencias
Enlaces externos
Entrevista realizada en mayo de 1994 para el Center for the History of Electrical Engineering (en inglés)
Artículo de EETimes sobre Sony centrado en Kihara, publicado en julio de 2000 (en inglés)
Entrevista realizada en junio de 2005 para la revista de la Sociedad Japonesa de Ingenieros Mecánicos (en japonés)
Entrevista realizada en julio de 2005 en el Centro de Investigación Sony-Kihara (en japonés)
Personas relacionadas con la electrónica
Fallecidos por infarto agudo de miocardio
Ingenieros de Japón
Personajes destacados en sonido
Sony
Japoneses del siglo XX
Ingenieros mecánicos
Ingenieros electrónicos
|
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Q: Populate A DataGridViewComboBox Looked at some of the suggested solutions couldn't find one that got me rolling.
I have a DataGrid that gets filled with the Following Code
DataGridViewBranches.AutoGenerateColumns = False
DataGridViewBranches.DataSource = FillBranchesDGV()
Private Function FillBranchesDGV()
Dim InstitutionsDT As New DataTable
Dim cmdText As String
cmdText = "SELECT examiners.last_name, examiners.examiner_ID, BranchInstitutions.* FROM BranchInstitutions INNER JOIN examiners ON BranchInstitutions.FldExaminer_ID = examiners.Examiner_ID where Company_Parent_ID=" & TextBoxCompanyParentID.Text
Using conn As New OleDbConnection(PubConnectionStrng)
Using cmd As New OleDbCommand(cmdText, conn)
conn.Open()
Dim reader As OleDbDataReader = cmd.ExecuteReader()
InstitutionsDT.Load(reader)
End Using
End Using
End Function
I have added a DgVComboBox and want to populate it with the Examiner Name. So that a user can change the examiners from the DGV.
I want to get rid on the 'Field Examiner' column and have the CBO there instead and have it changeable.
Being a ShadeTree VB coder, I am stumped.....
Any Help or push in the right direction appreciated.
A
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{"url":"https:\/\/quant.stackexchange.com\/questions\/16141\/gamma-and-delta-pl-example-question","text":"# Gamma and delta P&L example question\n\nI'm trying to get a basic understanding of this example delta ladder\n\nPrice Delta\n80 43\n90 31\n100 25\n110 11\n120 -5\n130 -20\n140 -12\n150 10\n160 15\n170 30\n\n\nwith spot at 110\n\nSo it's long gamma on the upside above 130 but short initially, and short gamma downside.\n\nI want to know in a easy to understand way, what is the delta and gamma P&L if the price were to end up at each of the above price points.\n\nMy understanding for say 130 is\n\nDelta = 11*(130-110) = 220\nGamma = ((-20-11) * (130-110)) \/2 = -310\nTotal P&L = delta + gamma = -90\n\n\nThus if the price were to go from 110 to 170 then it's\n\nP&L from 110-130 = -90\nP&L from 130-170 =\nDelta = -20 * (170-130) = -800\nGamma = ((30-20) * (170-130))\/2 = 200\nTotal = -600\nTotal P&L = -90-600 = -690\n\n\nIf this is correct, then is it not the same as the area under the curve ? ie, I can just do a discrete integral?\n\nFirst, I think you made a mistake in your computations above. Where you wrote $(30-20)$, I think you really meant $(30-(-20))$ i.e. $30+20$, yielding a gamma P&L of $1000$ instead of $200$. Your total P&L over $[90,170]$ would then be $110$ instead of $-690$. It doesn't matter for my answer either way, just thought I'd point it out for confused readers.\n\nBy definition, $\\Delta = \\frac{\\partial V}{\\partial S}$, where $V$ is the price of a financial derivative and $S$ is the price of its underlying.\n\nSo if $S$ experiences a move from $S_0$ to $S_1$, it follows logically that\n\n$$\\Delta V = V(S=S_1) - V(S=S_0) = \\int_{S_0}^{S_1}{\\Delta \\mathrm{d}S}$$\n\nSo, your P&L is indeed the area under the curve. Now, unfortunately, what you are computing here is not that. Indeed, by writing your P&L as the sum of these simplistic delta and gamma terms, what you really are saying, mathematically, is:\n\n$$\\Delta V = \\int_{S_0}^{S_1}{\\Delta \\mathrm{d}S} = \\Delta(S=S_0)\\cdot(S_1-S_0) + \\frac{\\Delta(S=S_1)-\\Delta(S=S_0)}{2}\\cdot(S_1-S_0)$$ i.e. $$\\Delta V = \\frac{\\Delta(S=S_0)+\\Delta(S=S_1)}{2}\\cdot(S_1-S_0)$$\n\nThis would hold if for example $\\Delta$ is assumed linear over $[S_0,S_1]$, but unfortunately, isn't true in the general case. For example, over $[110, 130]$, your computed $\\Delta$ is slightly concave (it would have to be equal to $4.5$ at $120$ to be linear), so your estimation of the P&L is slightly off. Over $[130,170]$, as the interval is larger and the shape more complex, the error is obviously worse.\n\nA better estimation when you know some values of $\\Delta$ over a discrete interval would be to assume that it is piecewise linear in between observations. It would be equivalent to using your method, but over the smallest possible intervals, which is indeed as you suggest the same as doing a discrete integral. In this case\n\n$$\\int_{S_i}^{S_j}{\\Delta \\mathrm{d}S} = \\sum_{k=i}^{k=j-1}\\left(\\frac{\\Delta(S=S_k)+\\Delta(S=S_{k+1})}{2}\\cdot(S_{k+1}-S_k)\\right)$$\n\nIn your specific case, the computation would yield\n\n$$\\Delta V = \\left(\\frac{11-5}{2}+\\frac{-20-5}2+\\frac{-12-20}2+\\frac{-12+10}2+\\frac{15+10}2+\\frac{30+15}2\\right)\\cdot10=85$$\n\nSo as you can see, it gives a quite different result ;-)\n\n\u2022 Thank you greatly. This is a really useful explanation that I've been searching for. Jan 9 '15 at 11:55","date":"2022-01-16 23:35:11","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\": 0.9033316373825073, \"perplexity\": 561.1959430305632}, \"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-2022-05\/segments\/1642320300244.42\/warc\/CC-MAIN-20220116210734-20220117000734-00309.warc.gz\"}"}
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{"url":"http:\/\/www.ntg.nl\/pipermail\/ntg-context\/2012\/065998.html","text":"# [NTG-context] howto execute loaded xml?\n\nMeer, H. van der H.vanderMeer at uva.nl\nThu Mar 22 13:04:24 CET 2012\n\n```It took hours (some far in the night) to solve. Finally. I am posting this message just in case someone starts delving into it.\n\nStill standing is the problem of \\xmlcommand not obeying the attribute selection but generating an error: \\xmlcommand{#1}{path[@attrib==whatever]}{command}\n\nHans van der Meer\n\nOn 21 mrt. 2012, at 10:29, Meer, H. van der wrote:\n\nThe output was missing in my previous post. Sorry. Here it is.\n\nHans van der Meer\n\nOn 21 mrt. 2012, at 10:27, H. van der Meer wrote:\n\nI made a minimal example showing that load'ed xml is not typeset. In contrast to the first <text>-node the second comes out as xml and not as typeset text.\n\nWhat has to be done to change that?","date":"2014-04-18 08:05:07","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.862494170665741, \"perplexity\": 6797.639390490922}, \"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-2014-15\/segments\/1397609533121.28\/warc\/CC-MAIN-20140416005213-00036-ip-10-147-4-33.ec2.internal.warc.gz\"}"}
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VOLTAIRE
Le philosophe ignorant
PRÉSENTATION
NOTES
GLOSSAIRE
DOSSIER
BIBLIOGRAPHIE
par Véronique Le Ru
GF-Flammarion
Voltaire
Le Philosophe ignorant
Flammarion
Collection : GF Flammarion
Maison d'édition : Flammarion
Présentation, notes et dossier par Véronique Le Ru
© Éditions Flammarion, Paris, 2009
ISBN numérique : 978-2-0812-4309-5
N° d'édition numérique : N.01EHPN000192.N001
Le livre a été imprimé sous les références :
ISBN : 978-2-0812-2290-8
N° d'édition : L.01EHPN000250.N001
Ouvrage composé et converti par Nord Compo
**Présentation de l'éditeur :
** Voltaire est âgé de 72 ans en 1766 lorsque paraît Le Philosophe ignorant, malicieuse invitation à un voyage autour du monde de la philosophie. Raillant Descartes, Spinoza et Leibniz - la volonté n'est pas plus libre qu'elle n'est bleue ou carrée, oppose-t-il au premier -, louant les analyses de Pierre Bayle et de John Locke, Voltaire critique avant tout l'esprit de système des philosophes, que guettent les travers de son Pangloss. Contrairement à eux, le philosophe ignorant qu'est Voltaire ne dissimule pas ses contradictions : oui, on peut être à la fois déiste et profondément sceptique ; oui, on peut soutenir que les principes de la morale, comme toutes les idées, s'acquièrent par les sens, et néanmoins affirmer qu'il existe une morale universelle et naturelle fondée en Dieu. Car le philosophe ignorant ne cesse de rechercher la vérité. Tel est l'autoportrait que nous livre ici Voltaire. |
Virginie Berthemet © Flammarion
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_Œuvres philosophiques de Voltaire dans la même collection_
_Dictionnaire philosophique._
_Lettres philosophiques. Derniers écrits sur Dieu._
_Lettres philosophiques._
_Traité sur la tolérance._
Présentation
L'autoportrait de Voltaire
en philosophe ignorant
Voilà bien un titre provocateur : _Le Philosophe ignorant_ , qui paraît en 1766, révèle toute la charge subversive que Voltaire entend donner à son essai. Le philosophe, n'est-ce pas traditionnellement le sage, celui qui est doté de sapience, c'est-à-dire de sagesse et de science ? En joignant au terme « philosophe » le qualificatif d'« ignorant », Voltaire s'insurge contre la manière de penser des philosophes prétendument savants, contre l'esprit de système de ceux qui s'arrogent le droit de résumer l'univers dans leur construction philosophique ou métaphysique. Voltaire ne condamne pas l'esprit philosophique, sans doute ce qu'il y a de plus précieux pour lui, mais les faiseurs de système et leurs sectateurs, les professeurs de philosophie. Il ne se moque pas de l'homme, il ne se moque pas de l'animal qui se nourrit de transcendentaux, il sait parfaitement qu'il appartient à cette espèce, mais il tourne en dérision les philosophes qui les mettent en système et qui s'en éprennent à tel point qu'ils le pensent comme la seule réalité qui vaille. Les philosophes à système et leurs sectateurs, voilà la cible sur laquelle Voltaire tire à boulets rouges. L'essai de 1766 est une arme destinée à écraser l'infâme, décliné non pas dans la version triviale de la superstition religieuse et du fanatisme des prêtres, mais sous la forme instruite des systèmes philosophiques. _Le Philosophe ignorant_ combat l'intolérance des faiseurs et défenseurs des systèmes et le dogmatisme de ceux qui les professent. Sans doute la figure la plus emblématique du professeur qui ne pense plus par lui-même mais seulement par le système qu'il a fait sien est-elle celle de Pangloss, celui qui enseigne la « métaphysico-théologo-cosmolonigologie1 »... Nigologie : la science des nigauds enserrée en un système de métaphysique, de théologie et de cosmologie. À la logorrhée de Pangloss, Voltaire oppose le rire, le rire de Démocrite. À la différence de ce dernier, Voltaire ne descend pas tous les jours au port d'Abdère pour rire des frénétiques activités des hommes, des choses graves et légères, de l'exercice d'un métier, du désir de reconnaissance et de promotion sociale, des discours des orateurs donnés devant la foule, du mariage d'un tel, des mésaventures de tel autre, mais il écrit _Le Philosophe ignorant_ en confiant à chacun de ses doutes un éclat du rire du philosophe grec. Quand, à la demande des Abdéritains, inquiets pour la santé mentale du philosophe, Hippocrate rend visite à Démocrite, il le trouve assis à l'ombre d'un platane en train d'écrire un traité sur la folie. À la question du médecin : « pourquoi ris-tu ? », il répond : « je ris d'un unique objet, l'homme plein de déraison qui transgresse les lois de la vérité pour des biens dont nul en mourant ne demeure le maître ». Après quoi Hippocrate considérera Démocrite comme l'homme le plus sain d'esprit et le plus sensé qui soit. Après avoir lu _Le Philosophe ignorant_ , on est conduit au même jugement. L'apparente contradiction du titre s'efface : seul un préjugé nous faisait y voir un oxymore. Après avoir ri avec Voltaire de tous les systèmes, on comprend qu'il nous faut renouer avec la racine grecque du mot « philosophe » : le philosophe n'est pas celui qui est sage et savant mais celui qui aime la sagesse et la vérité et qui ne cesse de les rechercher, tout en sachant que sa quête n'aura pas de fin puisque la quête est la fin même de sa vie. Voltaire nous invite à mettre nos pas, comme il le fait lui-même, dans ceux de Socrate pour qui le premier geste est de prendre conscience de son ignorance : on croyait savoir mais en réalité on ignorait qu'on ignorait. Quand on sait qu'on ignore, alors on peut commencer à philosopher, c'est pourquoi le philosophe doit être ignorant.
Dans _Le Philosophe ignorant_ , Voltaire fait le point sur ses lectures philosophiques. Mais ce point est un coup de poing dans la figure des bâtisseurs de systèmes en papier, dans la figure des bâtisseurs d'univers en carton pâte. Au terme de ce voyage au bout de l'extravagance dogmatique, une chose est claire : on ne sait rien mais on le sait, et cela est le plus merveilleux aiguillon qui soit pour se mettre à la recherche de la vérité.
LE TESTAMENT PHILOSOPHIQUE DE VOLTAIRE : DU _T RAITÉ DE MÉTAPHYSIQUE_ AU _P HILOSOPHE IGNORANT,_ OU COMMENT BOUCLER LA BOUCLE
Trente-deux ans séparent _Le Philosophe ignorant_ publié en 1766 du _Traité de métaphysique_ que Voltaire a rédigé en 1734 et qui sera publié dans l'édition posthume des _Œuvres complètes_ de Kehl (1785-1789). En 1734, Voltaire a 40 ans, il vient de rencontrer Gabrielle-Émilie de Breteuil, marquise du Châtelet, avec qui il vivra à Cirey jusqu'à la mort de la marquise en 1749. Il lui dédicace ainsi son traité :
L'auteur de la métaphysique
Que l'on apporte à vos genoux
Mérita d'être cuit dans la place publique,
Mais il ne brûla que pour vous.
En 1766, Voltaire a 72 ans, il vient d'obtenir la réhabilitation (en 1765) de Jean Calas, négociant calviniste qui avait été injustement accusé d'avoir tué son fils, retrouvé pendu. Jean Calas fut condamné à mort et exécuté en 1762. Voltaire, pour le réhabiliter, publia le _Traité sur la tolérance, à l'occasion de la mort de Jean Calas_ en 1763.
Et pourtant, malgré le grand laps de temps qui sépare la rédaction des deux ouvrages, on peut toutefois lire, à plus d'un titre, _Le Philosophe ignorant_ comme une reprise du _Traité de métaphysique_. D'une part, il faut noter la concision de ces deux textes (une cinquantaine de pages pour le _Traité de métaphysique_ , une ou deux pages de plus pour _Le Philosophe ignorant_ ). D'autre part, remarquons aussi l'homogénéité des questions abordées : qu'est-ce que l'homme ? Dieu existe-t-il ? Que valent les opinions matérialistes ? D'où viennent les idées ? Les objets extérieurs existent-ils ? Qu'est-ce que l'âme de l'homme ? L'homme est-il libre ? L'homme est-il un être sociable ? Les vertus et les vices sont-ils relatifs aux conventions ?
En outre, malgré les trente-deux ans écoulés, dans la construction de la problématique, Voltaire choisit la même entrée en matière pour ses deux ouvrages : le doute. En 1734, il ouvre son traité par une introduction qu'il intitule « Doutes sur l'homme ». En 1766, il présente un sommaire intitulé « Table des doutes » comportant cinquante-six titres dont plus du tiers sous forme de questions telles que _L'homme est-il libre ? Y a-t-il une morale ? Consentement universel est-il preuve de vérité ? La nature est-elle toujours la même ? La philosophie est-elle une vertu ?_
Cependant, le doute voltairien est loin d'être cartésien : il n'est ni hyperbolique ni provisoire, il est modeste, modéré et il dure puisque trente-deux ans ne suffisent pas à le lever. La permanence de ce doute fait planer un soupçon sur le portrait que l'on fait habituellement de Voltaire en habit de déiste : Que cache cet habit ? Un croyant ou un penseur sceptique ? Ce qui est certain, c'est que les incrédules ont exercé une grande influence sur la pensée de Voltaire. Dans l'introduction du _Traité de métaphysique_ , il mentionne Hobbes, Locke, Descartes et Bayle parmi les philosophes et les esprits sages et il termine son traité en faisant référence aux incrédules et aux libertins que la philosophie n'empêche pas d'être d'une vertu rigide. Pour illustrer ce point, il cite La Mothe Le Vayer, Bayle de nouveau, Locke de nouveau, Spinoza et Shaftesbury. Ne répond-il pas ainsi à Bayle qui se demandait s'il peut exister une société d'athées vertueux2 ? Si l'on tient compte de la mention privilégiée faite à ces penseurs (dans l'introduction et dans la conclusion), on peut avancer que Voltaire rédige le _Traité de métaphysique_ sous l'égide de deux références principales : Locke et Bayle.
Or ces deux philosophes constituent aussi deux références extrêmement importantes dans _Le Philosophe ignorant_ : le nom de Locke apparaît dans trois titres de doutes (XXIX, XXXIV et XXXV) et si le nom de Bayle n'apparaît qu'une fois aux côtés de celui de Spinoza (dans la Table des Doutes, on trouve en effet : XXIV. _Justice rendue à Spinoza et à Bayle_ , alors que, dans le corps du texte le Doute XXIV porte juste le titre _Spinoza_ ), il est pourtant très présent dans l'ouvrage de Voltaire, ne serait-ce que par le choix d'intituler « Doute » chaque question philosophique qu'il discute, et aussi par la lecture très baylienne qu'il fait de Spinoza (voir le Doute XXIV). Le scepticisme de Bayle que l'on peut résumer par le double mot d'ordre – savoir attendre et douter – se diffuse dans les doutes voltairiens animés par la même modestie teintée d'ironie devant les faiseurs de système. Quant à Locke, c'est son empirisme qui convainc Voltaire : toutes nos idées viennent des sens. Cette origine empirique des idées que Locke expose dans le Livre II de son _Essai philosophique concernant l'entendement humain_ va de pair avec le démantèlement du système cartésien des idées innées (qu'il a opéré dans le Livre I), ce qui n'est pas pour déplaire à Voltaire dont l'œuvre est marquée par un anticartésianisme assez fort, corrélé à sa défense et à sa diffusion de la pensée de Newton en France3.
Ces deux philosophies que Voltaire fait siennes le conduisent à des acrobaties intellectuelles : comment concilier son déisme et son scepticisme (Doutes X, XI, XV et XXIV) ? Comment soutenir, d'une part, que la morale est naturelle et universelle car fondée en Dieu et, d'autre part, que les principes de la morale s'acquièrent mais que cependant ils ne varient pas d'une civilisation à l'autre, d'un continent à l'autre, d'une langue à l'autre (Doutes XXXV, XXXVI et XXXVIII) ? Voltaire n'hésite pas à tenir dans ses mains les rênes de la contradiction en disant : je ne sais qui je suis ni qui sont les autres (Doutes I à V, XII) mais je crois en cet axiome : tout ouvrage démontre un ouvrier (Doute XV), donc Dieu existe même si je ne le peux connaître (Doutes XVII à XXIII). De même pour la morale, Voltaire préfère la contradiction qui consiste à affirmer à la fois le caractère universel et naturel de la morale (Doutes XXXI, XXXII) et l'acquisition empirique des principes de la morale plutôt que de sombrer dans le relativisme des valeurs (Doutes XXXIII à XXXVIII).
Cependant, à propos de la question de la liberté de l'homme, Voltaire est obligé de reconnaître que sa pensée a évolué depuis le _Traité de métaphysique_. Ainsi, à la fin du Doute XIII, Voltaire écrit : « L'ignorant qui pense ainsi, n'a pas toujours pensé de même, mais il est enfin contraint de se rendre », il est enfin contraint de se rendre, c'est-à-dire de réviser sa conception positive de la liberté de l'homme développée dans le chapitre VII du _Traité de métaphysique_ dans lequel Voltaire affirmait que certes il y a des hommes plus libres que d'autres mais que la liberté est la santé de l'âme, même si elle est faible et bornée comme notre raison et comme toutes nos facultés. À l'âge de 72 ans, Voltaire remise son habit joyeux d'homme libre au placard : l'heure n'est plus au déisme optimiste de 1734 mais au constat que « tout ce qui arrive est nécessaire » (Doute XIII). Que s'est-il passé entre 1734 et 1766 ? De l'eau puis des torrents de boue ont coulé sous les ponts, et les ponts eux-mêmes se sont effondrés à la suite du terrible tremblement de terre du 1er novembre 1755 qui a secoué une capitale européenne, Lisbonne, causant soixante mille morts (plus du tiers des habitants). Après cette catastrophe, Voltaire ne traduit plus de la même manière la formule « Tout est bien » de Pope4 qu'il avait adoptée lors de son séjour en Angleterre en 1726-1728.
DE « TOUT EST BIEN » À « AINSI SOIT-IL »
En même temps que la terre a tremblé à Lisbonne, la confiance et l'admiration qu'il éprouve envers Pope, l'inventeur de l'expression « Tout est bien », se sont fissurées : Voltaire devient de plus en plus critique vis-à-vis de cette formule, qu'il met en scène, avec un malin plaisir, d'abord dans ses contes au milieu du siècle ( _Candide ou l'optimisme_ , 1759), puis en 1766 dans _Le Philosophe ignorant_ (Doute XXVI : « Du meilleur des mondes »), enfin dans les _Questions sur l'Encyclopédie_ vers 1770 (article « Bien, tout est bien »). Voltaire ne croit plus comme jadis que le paradis est là où il vit5, mais il se gausse de ceux qui, comme Pangloss, défendent en faisant preuve de mauvaise foi la thèse leibnizienne de l'optimisme : « Pangloss avouait qu'il avait toujours horriblement souffert ; mais ayant soutenu une fois que tout allait à merveille, il le soutenait toujours, et n'en croyait rien6. » Quand Pangloss cherche, dans la conclusion de _Candide_7, à discuter avec le derviche du sens de la vie humaine, et lui demande : « Que faut-il donc faire ? », celui-ci lui répond « Te taire » et quand Pangloss insiste encore : « Je me flattais [...] de raisonner un peu avec vous des effets et des causes, du meilleur des mondes possibles, de l'origine du mal, de la nature de l'âme, de l'harmonie préétablie », le derviche lui ferme la porte au nez.
Pour suivre l'évolution de la pensée de Voltaire entre le _Traité de métaphysique_ de 1734 où il affiche un déisme souriant compatible avec la liberté de l'homme, et _Le Philosophe ignorant_ de 1766 où son déisme est mis en tension par le constat qu'il y a du mal sur la terre et que tout arrive nécessairement, appuyons-nous sur les trois traductions successives que propose Voltaire de l'expression « Tout est bien ».
1) Entre 1734 et 1746, ce qui correspond à peu près à la période où il vit à Cirey avec la célèbre mathématicienne newtonienne, Gabrielle-Émilie de Breteuil, « Tout est bien » signifie pour Voltaire à la fois « tout est ce qu'il doit être » et « tout est relatif ». Face à cette reconnaissance de l'ordre du monde et de la relativité générale du bien et du mal, les hommes sont d'égale condition, tout est égal.
2) Dans un deuxième temps, entre 1747 et 1755, Voltaire traduit « Tout est bien » par « tout est passable ». L'argument du dessein qui consiste à inférer de l'ordre et de la nature, le principe d'ordre qu'est le Dieu créateur, bat de l'aile et le Dieu des déistes est aussi menacé que celui de la Providence chrétienne. Mais Voltaire a peur de l'athéisme : il juge Jean Meslier (1664-1729), prêtre et philosophe français qui dans son _Testament_ se déclare ouvertement athée, comme un penseur dangereux. Il ne peut ni ne veut renoncer au « Tout est bien ». Dieu existe mais l'humanité souffre, la philosophie est impuissante à résoudre le conflit qui naît de ces deux certitudes et mieux vaut écrire des contes que faire un système vain par définition8.
3) Enfin, à partir de 1756 et de son _Poème sur le désastre de Lisbonne_ , Voltaire nie que tout soit bien aujourd'hui mais laisse une place à l'espérance : « un jour (peut-être ?), tout sera bien ». Dans les _Questions sur l'Encyclopédie_ rédigées en 1770, il revient une dernière fois sur la question du « Bien, Tout est bien » et son dernier mot est de renoncer à lire Spinoza : « Des raisonneurs ont prétendu qu'il n'est pas dans la nature de l'Être des êtres que les choses soient autrement qu'elles ne sont. C'est un rude système ; je n'en sais pas assez pour oser seulement l'examiner9. » Au « Tout est ainsi », Voltaire préfère finalement le « Ainsi soit-il ». Même si la condition humaine est régie par les lois de la nature, le regard de Voltaire demeure celui d'un déiste sceptique qui préfère se désoler de la souffrance humaine que de la renvoyer au réalisme indifférent du « Tout est nécessaire ». Pour cela, il maintient Dieu à bout de bras et il conclut que tout est une question de point de vue : ce qui est ordre au regard de Dieu est pure nécessité pour l'humanité souffrante.
L'idée forte qui transparaît, à travers ces variations sur le thème du « Tout est bien », est que Voltaire est demeuré fidèle au scepticisme de Bayle sans que cette fidélité l'ait jamais conduit à rejeter radicalement le déisme ni à laisser toute espérance.
Mais l'infléchissement de ses traductions du « Tout est bien » témoigne du scepticisme grandissant de Voltaire à qui l'on pourrait peut-être reprocher ce que lui-même critique chez Pangloss : soutenir avec une certaine mauvaise foi la thèse déiste alors qu'il est persuadé que tout arrive nécessairement. Malgré qu'il en ait, son Dieu ne serait-il pas au fond celui de Spinoza ?
LA RECHERCHE DE LA VÉRITÉ
Les deux procédés narratifs récurrents qui caractérisent les ouvrages de Voltaire, qu'il s'agisse des contes ou des ouvrages dits sérieux, sont le regard du dessus et le voyage autour du monde. Or il se sert aussi de ces deux procédés dans les deux traités que nous avons mis en résonance, à savoir le _Traité de métaphysique_ et _Le Philosophe ignorant_. Dans l'introduction du _Traité de métaphysique_ , Voltaire suppose qu'il descend du globe de Mars ou de Jupiter et qu'il peut ainsi porter une vue rapide sur tous les siècles, sur tous les pays et par conséquent sur « toutes les sottises de ce petit globe10 ». Dans _Le Philosophe ignorant_ , les Doutes XXVI à XXIX peuvent être lus comme un micro-conte, une variation de _Candide_ sur le thème du meilleur des mondes (Doute XXVI), où Pangloss et sa logorrhée se laissent entendre dans les trois points de suspension à la fin du Doute XXVII sur les monades, où le narrateur reprend son baluchon et rencontre un Anglais nommé Cudworth (Doute XXVIII) et où, harassé, il finit son voyage dans les bras de son père Locke (retour de l'enfant prodigue dans le Doute XXIX). C'est cette perméabilité entre l'univers des contes et _Le Philosophe ignorant_ qui rend la lecture du traité de 1766 si singulière : Voltaire ne s'invente-t-il pas dans le Doute LIII un lieu de naissance fictif : « Ma mère m'a dit que j'étais né au bord du Rhin » ? Ne s'invente-t-il pas dans ce même Doute un double : « J'ai demandé à mon ami le savant Apédeutès » ? Apédeutès : celui qui est _a-paideion_ , ce qui signifie en grec sans éducation, sans instruction, ignorant. Le savant Apédeutès est bien un double, mot pour mot, du philosophe ignorant, mais cette fois « natif de Courlande » (Doute LIII). En 1767, Voltaire réutilise ce terme comme nom commun dans _L'Ingénu_ où il parle des apédeutes de Constantinople qui dénoncent l'hérésie en très mauvais grec11.
Cependant, dans cet itinéraire de Voltaire, _Le Philosophe ignorant_ a une place insigne en ce qu'il revêt une valeur testamentaire : ce qu'il dit ensuite dans les contes ou dans le _Dictionnaire philosophique_ ou dans les _Questions sur l'Encyclopédie_ n'ajoute rien au _modus vivendi_ du _Philosophe ignorant_ qui doit vivre avec ses contradictions : Dieu existe, mais il y a du mal sur terre ; la morale, fondée en Dieu, est naturelle et universelle, mais ses principes comme toutes les idées s'acquièrent par les sens et pourtant ne varient pas d'une civilisation à l'autre ; l'homme n'est pas libre, tout arrive nécessairement, mais il vaut mieux déformer Spinoza pour s'en moquer que le lire.
Enfin, ce qui est remarquable dans ce petit testament philosophique, c'est la vitalité qui s'en dégage : certes l'auteur n'est plus le jeune homme amoureux de la marquise du Châtelet de 1734, mais la présence de son esprit et de son ironie perce sous chaque Doute jusqu'à atteindre la profession de foi finale du philosophe qui aime par-dessus tout la recherche de la vérité, véritable point d'orgue de tout l'essai : « Pour moi, je crois que la vérité ne doit pas plus se cacher devant ces monstres [l'hydre du fanatisme, l'envie et la calomnie], que l'on ne doit s'abstenir de prendre de la nourriture dans la crainte d'être empoisonné » (Doute LVI). Ce que nous apprend ainsi Voltaire, c'est que rechercher la vérité, c'est prendre des risques mais qu'il vaut mieux oser penser par soi-même que de « rester oisif dans les ténèbres » (Doute LVI).
Cette leçon, Voltaire la rend visible tout au long de son essai par la lutte du philosophe ignorant avec ses contradictions qu'il préfère assumer plutôt qu'effacer. Du reste, François Arouet n'a-t-il pas choisi son pseudonyme Voltaire en raison de l'esprit virevoltant qui l'anime mais aussi de l'esprit de révolte qu'il veut insuffler : Voltaire ou l'ironie d'un anagramme ou d'un mot de verlan avant l'heure Volt-air/Volt–èr pour mettre de la révolte dans l'air. Car ce qui est impressionnant, c'est de voir que Voltaire, à l'âge de 72 ans, ne lâche pas prise : il faut chercher la vérité, il faut inlassablement chercher à s'instruire : « Mais malgré ce désespoir [dû à l'impossibilité de résoudre les grandes questions métaphysiques], je ne laisse de désirer d'être instruit, et ma curiosité trompée est toujours insatiable » (Doute IV).
La soixantaine passée, Voltaire montre dans sa vie même que sa recherche de la vérité s'est muée en un véritable combat pour la vérité, comme l'attestent les deux affaires qui le préoccupent dans les années 60, à savoir l'affaire Calas et celle du chevalier de La Barre.
L'affaire Calas est l'histoire d'une erreur judiciaire due à l'intolérance religieuse : le 13 octobre 1761, l'aîné des six enfants de Jean Calas, Marc-Antoine, se pend. Les parents commettent l'imprudence, pour éviter l'opprobre, de cacher le suicide. La justice accuse alors le père Jean, de religion calviniste, d'avoir tué son fils parce que celui-ci aurait manifesté l'intention de se convertir au catholicisme. En vertu d'un arrêt du parlement de Toulouse, Calas fut rompu vif sur la roue en 1762. Voltaire recueillit à Ferney sa veuve et deux de ses enfants, les trois autres ayant été enfermés dans des couvents. Il obtint un arrêt du Conseil du Roi qui cassa celui de Toulouse et prononça la réhabilitation de Jean Calas en 1765.
La deuxième affaire est celle du chevalier de La Barre. Ce gentilhomme, né en 1745, fut accusé d'avoir mutilé un crucifix et d'avoir proféré des blasphèmes, et arrêté en compagnie de trois jeunes gens, tous suspectés de ne pas s'être découverts au passage du saint sacrement en 1765. Le chevalier de La Barre fut condamné par le présidial d'Abbeville à avoir le poing coupé, la langue arrachée, puis à être brûlé. Il en appela au parlement de Paris, qui ordonna qu'il serait décapité avant d'être livré aux flammes. Il mourut avec courage. Réclamée vainement par Voltaire, sa réhabilitation fut décrétée par la Convention en 1793.
Or l'injustice qui frappe le chevalier de La Barre et son exécution atroce se produisent au moment où _Le Philosophe ignorant_ sort des presses genevoises, ce qui oblige Voltaire à différer l'envoi de son ouvrage à Paris. De fait, la livraison de l'ouvrage est retardée de quelques mois, il n'est diffusé en France qu'au début de 1767. Voltaire est contraint de faire profil bas car cette affaire l'implique comme responsable moral des écarts du jeune contestataire (on a en effet retrouvé chez le chevalier un exemplaire du _Dictionnaire philosophique_ qu'on a brûlé sur le corps du supplicié). François Marin, censeur royal de la Librairie, de la Police et des Théâtres, conseille à Voltaire la plus grande prudence, ce qui ne l'empêche pas de recueillir à Ferney un compagnon en fuite du chevalier de La Barre.
Le cheminement du _Philosophe ignorant_ peut se lire ainsi comme un dialogue socratique où la docte ignorance n'est jamais synonyme de scepticisme radical ni de renoncement, mais est une incitation à chercher encore et toujours la vérité. Le cheminement du philosophe ignorant, comme le dialogue socratique, se déroule comme une danse à trois temps : premier temps, on ignore qu'on ignore, deuxième temps, on sait qu'on ignore, troisième temps, on désire savoir. Voltaire dansant, pourquoi pas ? D'après Sainte-Beuve ( _Causeries du Lundi_ , 17 juin 1850), Voltaire ne se peint-il pas lui-même en ces termes ?
Toujours un pied dans le cercueil
De l'autre faisant des gambades
Le dernier Doute du _Philosophe ignorant_ nous conduit au troisième temps de la danse : au désir de savoir, au désir de rechercher la vérité comme un soleil à l'horizon qui recule au fur et à mesure qu'on avance. Peu importe que l'on sache que l'on ne pourra jamais rejoindre le soleil, l'essentiel est de courir après...Et cette silhouette recroquevillée et malingre qui gesticule au loin dans le couchant, n'est-ce pas le philosophe ignorant, n'est-ce pas Voltaire lui-même ?
Véronique LE RU.
1\- _Candide ou l'optimisme_ , in _Romans et contes_ , Paris, GF-Flammarion, 1966, p.180, édition de René Pomeau.
2\- Voir Pierre Bayle, _Pensées diverses sur la comète_ , Rotterdam, 1683, § CLXXII à CLXXVI, p. 525-540 ; Paris, GF-Flammarion, 2007, édition de Joyce et Hubert Bost.
3\- Sur ce point, nous nous permettons de renvoyer le lecteur à V. Le Ru, _Voltaire newtonien_ , Paris, Vuibert-Adapt, 2005.
4\- Voltaire lit son ouvrage _Essay on man_ en Angleterre vers 1727, et la première traduction française de l' _Essai sur l'homme_ faite par Silhouette paraît en 1736.
5\- En 1736, Voltaire n'hésitait pas à écrire : « Le Paradis terrestre est où je suis », c'est le dernier vers du _Mondain_ , _in_ tome X, p. 83 des _Œuvres complètes_ , établies par Louis Moland, Paris, Garnier frères, 1877-1883, 52 vol.
6\- _Candide ou l'optimisme_ in _Romans et contes_ , éd. citée, p. 257.
7\- _Ibid._ , p. 257-258.
8\- René Pomeau remarque à ce propos que le conte voltairien naît de la crise de 1748 : « De frêles humanités y courent, dont l'existence même est un reproche à Dieu [...] » in _La Religion de Voltaire_ , p. 243.
9\- Voir « Bien, Tout est bien », in tome XVII des _Œuvres complètes_ , éd. Moland.
10\- Voir l'introduction du _Traité de métaphysique_ , in tome XVII des _Œuvres complètes_.
11\- _L'Ingénu_ , in _Romans et contes_ , éd. citée, p. 353-354.
Note sur cette édition
La présente édition est fondée sur le texte originellement établi et présenté par Roland Mortier en 1987 dans les _Œuvres complètes_ de Voltaire (volume 62) publiées par Voltaire Foundation Ltd, University of Oxford. Pour toute précision sur l'établissement du texte, les éditions et les particularités de graphie, nous renvoyons le lecteur aux pages 11 à 24 du volume 62 des _Œuvres complètes_ de Voltaire (1987), qui constitue l'édition de référence du _Philosophe ignorant_.
Comme le note Roland Mortier, on connaît sept éditions séparées du _Philosophe ignorant_ datées des années 1766 et 1767. L'édition choisie pour texte de base est l'édition originale sortie des presses de Cramer à Genève en 1766. Roland Mortier y a supprimé les trois derniers Doutes LVII, LVIII et LIX, par souci de conformité avec la pratique de Voltaire à partir de 1767.
Le philosophe ignorant
Table des Doutes
Premier Doute1
1\- La subdivision du texte en « Doutes » exprime l'esprit sceptique qui domine tout l'ouvrage. Voltaire, en philosophe ignorant, prend l'habit du sceptique et se veut l'héritier de Bayle : savoir attendre et douter. Du reste, l'édition princeps (1766) qui sert ici de référence comporte un sommaire intitulé « Table des Doutes » (voir ci-dessus p. 25).
Qui es-tu ? d'où viens-tu ? que fais-tu ? que deviendras-tu ? C'est une question qu'on doit faire à tous les êtres de l'univers, mais à laquelle nul ne nous répond. Je demande aux plantes quelle vertu les fait croître, et comment le même terrain produit des fruits si divers ? Ces êtres insensibles et muets, quoique enrichis d'une faculté divine, me laissent à mon ignorance et à mes vaines conjectures.
J'interroge cette foule d'animaux différents, qui tous ont le mouvement et le communiquent, qui jouissent des mêmes sensations que moi, qui ont une mesure d'idées et de mémoire avec toutes les passions1. Ils savent encore moins que moi ce qu'ils sont, pourquoi ils sont, et ce qu'ils deviennent.
Je soupçonne, j'ai même lieu de croire que les planètes, les soleils innombrables qui remplissent l'espace, sont peuplées d'êtres sensibles et pensants ; mais une barrière éternelle nous sépare, et aucun de ces habitants des autres globes ne s'est communiqué à nous2.
Monsieur le prieur3, dans le _Spectacle de la nature,_ a dit à monsieur le chevalier4, que les astres étaient faits pour la terre, et la terre, ainsi que les animaux, pour l'homme. Mais comme le petit globe de la terre roule avec les autres planètes autour du soleil, comme les mouvements réguliers et proportionnels des astres peuvent éternellement subsister sans qu'il y ait des hommes comme il y a sur notre petite planète infiniment plus d'animaux que de mes semblables ; j'ai pensé que monsieur le prieur avait un peu trop d'amour-propre en se flattant que tout avait été fait pour lui5. J'ai vu que l'homme pendant sa vie est dévoré par tous les animaux, s'il est sans défense, et que tous le dévorent encore après sa mort. Ainsi j'ai eu de la peine à concevoir que monsieur le prieur et monsieur le chevalier fussent les rois de la nature. Esclave de tout ce qui m'environne, au lieu d'être roi, resserré dans un point, et entouré de l'immensité, je commence par me chercher moi-même.
1\- Voltaire opte pour la thèse selon laquelle les animaux ont une âme, mais plus limitée que celle des hommes. La question de l'âme des bêtes agite les esprits depuis l'affirmation par Descartes que les animaux sont des machines et n'ont pas d'âme. Voir l'article ÂME DES BÊTES de l'abbé Yvon, p. 343-353 du tome I, 1751, de l' _Encyclopédie_ de Diderot et de d'Alembert, Paris, Briasson, David, Le Breton et Durand en 35 vol., 1751-1780, rééd. Fromann, 1966-1967, voir aussi l'article _Bêtes_ de Voltaire dans le _Dictionnaire philosophique_ , présentation, notes et annexes par Béatrice Didier, d'après l'édition de 1769, Paris, Imprimerie nationale, 1994, p. 111-113.
2\- L'idée qu'il existe d'autres mondes habités avait été mise à la mode par Fontenelle dans ses _Entretiens sur la pluralité des mondes_ (1686) qui lui-même la reprenait de Cyrano (voir _L'Autre Monde_ , 1657-1662, divisé en _Histoire comique des États et Empires de la Lune_ et en _Histoire comique des États et Empires du Soleil_ ). Voltaire s'en inspirera à son tour dans _Micromégas_ (1752) et la développe ci-dessous dans le Doute XXII.
3\- Pluche, plus connu sous l'appellation de l'abbé Pluche (1688-1761), est l'auteur du _Spectacle de la nature_ , véritable best-seller d'histoire naturelle, paru en 9 volumes entre 1732 et 1750.
4\- Newton (1642-1727) a découvert la loi de l'attraction universelle (voir ses _Principes mathématiques de la philosophie naturelle_ , 1687), c'est en 1705 que la reine d'Angleterre le fit chevalier.
5\- Voltaire est souvent critique envers le finalisme naïf de Pluche qui met effectivement l'homme au centre de la création. On lit, par exemple dans le tome IV du _Spectacle de la nature_ , p. 13 : « L'histoire de la physique est vraiment le récit de nos besoins, et des riches secours que Dieu a mis à notre portée pour y pourvoir » (Paris, Estienne, 1739).
II. Notre faiblesse
Je suis un faible animal ; je n'ai en naissant ni force ni connaissance, ni instinct ; je ne peux même me traîner à la mamelle de ma mère, comme font tous les quadrupèdes ; je n'acquiers quelques idées que comme j'acquiers un peu de force quand mes organes commencent à se développer. Cette force augmente en moi jusqu'au temps où ne pouvant plus s'accroître, elle diminue chaque jour. Ce pouvoir de concevoir des idées s'augmente de même jusqu'à son terme, et ensuite s'évanouit insensiblement par degrés1.
Quelle est cette mécanique2 qui accroît de moment en moment les forces de mes membres jusqu'à la borne prescrite ? Je l'ignore ; et ceux qui ont passé leur vie à rechercher cette cause n'en savent pas plus que moi.
Quel est cet autre pouvoir qui fait entrer des images dans mon cerveau, qui les conserve dans ma mémoire ? Ceux qui sont payés pour le savoir l'ont inutilement cherché ; nous sommes tous dans la même ignorance des premiers principes où nous étions dans notre berceau3.
1\- Voltaire affirme l'origine empirique des idées dans le sillage de Locke (voir le livre II de son _Essai philosophique concernant l'entendement humain_ , trad. Coste en 1700, reprod. par Vrin, Paris, 1983, ouvrage de référence pour Voltaire – voir _Bibliothèque de Voltaire : catalogue des livres_ , Moscou et Leningrad, 1961, n° 2149, 2150). Il soutient que toutes nos idées viennent des sens, qu'elles augmentent avec le développement des sens et qu'elles diminuent avec leur affaiblissement ; ce faisant, il s'inscrit, à l'instar de Locke, contre la théorie des idées innées de Descartes.
2\- Le terme « mécanique » a été mis à la mode au XVIIIe siècle par le succès et la diffusion du mécanisme cartésien. Ici le terme a le sens général de dispositif.
3\- Voltaire, fidèle à Bayle mais aussi à Newton, renonce à rechercher les premiers principes parmi lesquels il compte l'origine de la force vitale, la formation des images dans le cerveau à partir des impressions sensibles et leur conservation dans la mémoire.
III. Comment puis-je penser ?
Les livres faits depuis deux mille ans, m'ont-ils appris quelque chose ? Il nous vient quelquefois des envies de savoir comment nous pensons, quoiqu'il nous prenne rarement l'envie de savoir comment nous digérons, comment nous marchons. J'ai interrogé ma raison ; je lui ai demandé ce qu'elle est ? Cette question l'a toujours confondue.
J'ai essayé de découvrir par elle, si les mêmes ressorts qui me font digérer, qui me font marcher, sont ceux par lesquels j'ai des idées. Je n'ai jamais pu concevoir comment et pourquoi ces idées s'enfuyaient quand la faim faisait languir mon corps, et comment elles renaissaient quand j'avais mangé.
J'ai vu une si grande différence entre des pensées et la nourriture, sans laquelle je ne penserais point, que j'ai cru qu'il y avait en moi une substance qui raisonnait, et une autre substance qui digérait. Cependant, en cherchant toujours à me prouver que nous sommes deux, j'ai senti grossièrement que je suis un seul ; et cette contradiction m'a toujours fait une extrême peine1.
J'ai demandé à quelques-uns de mes semblables qui cultivent la terre notre mère commune, avec beaucoup d'industrie, s'ils sentaient qu'ils étaient deux, s'ils avaient découvert par leur philosophie qu'ils possédaient en eux une substance immortelle, et cependant formée de rien, existante sans étendue, agissant sur leurs nerfs sans y toucher, envoyée expressément dans le ventre de leur mère six semaines après leur conception2 ; ils ont cru que je voulais rire, et ont continué à labourer leurs champs sans me répondre.
1\- Peine au sens de difficulté : Voltaire discute en effet de l'épineux problème de l'union et de l'interaction de l'âme et du corps. Sommes-nous composés de deux substances, pensante et corporelle, ou sommes-nous un seul être ? Descartes résolvait cette contradiction en disant que l'homme est l'union substantielle de l'âme et du corps, deux substances réellement distinctes.
2\- Voltaire résume ici, non sans ironie, la conception cartésienne de l'âme comme substance pensante réellement distincte du corps et qui pourtant agit sur lui sans le toucher car l'esprit n'est pas étendu ; il y ajoute la conception traditionnelle, depuis Thomas d'Aquin, de l'animation médiate : l'âme spirituelle s'unit à l'embryon au bout de six semaines dans le ventre de la mère, conception que reprend Descartes.
IV. M'est-il nécessaire de savoir ?
Voyant donc qu'un nombre prodigieux d'hommes n'avait pas seulement la moindre idée des difficultés qui m'inquiètent, et ne se doutait pas de ce qu'on dit dans les écoles, de l'être en général, de la matière et de l'esprit etc., voyant même qu'ils se moquaient souvent de ce que je voulais le savoir ; j'ai soupçonné qu'il n'était point du tout nécessaire que nous le sussions. J'ai pensé que la nature a donné à chaque être la portion qui lui convient ; et j'ai cru que les choses auxquelles nous ne pouvions atteindre ne sont pas notre partage. Mais, malgré ce désespoir, je ne laisse pas de désirer d'être instruit, et ma curiosité trompée est toujours insatiable1.
1\- Savoir qu'on ne peut connaître les premiers principes ne détruit pas pour autant le désir de savoir. La pensée sceptique est une pensée inquiète, qui ne se résigne pas à l'ignorance.
V. Aristote, Descartes et Gassendi
Aristote commence par dire que l'incrédulité est la source de la sagesse1 ; Descartes a délayé cette pensée, et tous deux m'ont appris à ne rien croire de ce qu'ils me disent. Ce Descartes surtout, après avoir fait semblant de douter, parle d'un ton si affirmatif de ce qu'il n'entend point ; il est si sûr de son fait quand il se trompe grossièrement en physique ; il a bâti un monde si imaginaire ; ses tourbillons et ses trois éléments sont d'un si prodigieux ridicule, que je dois me défier de tout ce qu'il me dit sur l'âme, après qu'il m'a tant trompé sur les corps2.
Il croit, ou il feint de croire que nous naissons avec des pensées métaphysiques3. J'aimerais autant dire qu'Homère naquit avec l' _Iliade_ dans la tête. Il est bien vrai qu'Homère en naissant avait un cerveau tellement construit, qu'ayant ensuite acquis des idées poétiques, tantôt belles, tantôt incohérentes, tantôt exagérées, il en composa enfin l' _Iliade_. Nous apportons en naissant le germe de tout ce qui se développe en nous ; mais nous n'avons pas réellement plus d'idées innées, que Raphaël et Michel-Ange n'apportèrent en naissant de pinceaux et de couleurs.
Descartes pour tâcher d'accorder les parties éparses de ses chimères, supposa que l'homme pense toujours4 ; j'aimerais autant imaginer que les oiseaux ne cessent jamais de voler, ni les chiens de courir, parce que ceux-ci ont la faculté de courir, et ceux-là de voler.
Pour peu que l'on consulte son expérience et celle du genre humain, on est bien convaincu du contraire. Il n'y a personne d'assez fou pour croire fermement qu'il ait pensé toute sa vie, le jour et la nuit, sans interruption, depuis qu'il était fœtus jusqu'à sa dernière maladie. La ressource de ceux qui ont voulu défendre ce roman, a été de dire qu'on pensait toujours, mais qu'on ne s'en apercevait pas5. Il vaudrait autant dire qu'on boit, qu'on mange, et qu'on court à cheval sans le savoir. Si vous ne vous apercevez pas que vous avez des idées, comment pouvez-vous affirmer que vous en avez ? Gassendi se moqua comme il le devait de ce système extravagant6. Savez-vous ce qui en arriva ? On prit Gassendi et Descartes pour des athées7.
1\- On ne trouve pas explicitement chez Aristote une telle conception mais au XVIIIe siècle se développe une lecture d'Aristote qui le tire vers l'athéisme. Sylvain Maréchal, dans son _Dictionnaire des Athées anciens et modernes_ (Bruxelles, 1833, p. 12), attribue cette fausse citation d'Aristote au marquis d'Argens et à ses _Mémoires secrets de la République des Lettres_ parus au milieu du XVIIIe siècle.
2\- Voltaire s'oppose à la théorie cartésienne des tourbillons et lui préfère la théorie newtonienne de l'attraction, il rejette les trois éléments du Feu, de l'Air et de la Terre qui composent, selon Descartes, les grands corps (voir _Le Monde ou le Traité de la lumière_ ), Voltaire est newtonien et anticartésien comme l'attestent ses _Éléments de la philosophie de Newton_ , 1738 ; 1741 (in _Œuvres complètes_ , t. XV, critical edition by R. L. Walters and W. H. Barber, Voltaire Foundation, Oxford, 1992).
3\- Voltaire entérine son rejet de la conception cartésienne des idées innées, qui se laissait déjà deviner dans le Doute II.
4\- Cette thèse cartésienne avait été discutée par les cartésiens eux-mêmes, voir Malebranche, livre III, chapitre 2 du tome I de la _Recherche de la vérité_ (Paris, 1675, repris par Vrin en deux tomes, Paris, 1965 et 1967). Elle est réfutée par Locke dans l' _Essai philosophique concernant l'entendement humain_ , livre II, chapitre I, § 9 et 10. D'Alembert, en 1759, dans le chapitre VI des _Éléments de philosophie_ (Paris, Fayard, 1986, p. 47-48) juge la question indécidable.
5\- C'est effectivement la manière dont Malebranche défend la thèse cartésienne, voir livre III, chapitre 2 du tome I de la _Recherche de la vérité_ , p. 220 : « ainsi il arrive quelquefois qu'on a un si grand nombre de pensées différentes qu'on s'imagine que l'on ne pense à rien ».
6\- Gassendi (1592-1655) est l'auteur des cinquièmes Objections aux _Méditations métaphysiques_ de Descartes : une dispute aigre et longue les opposa, Gassendi soutenant qu'il n'y a pas d'idées innées et que toutes nos idées viennent des sens.
7\- Pirouette de Voltaire sur fond de vérité : Gassendi fut accusé d'athéisme à cause de son atomisme et Descartes à cause de sa conception du monde indéfini, sans bornes assignables.
VI. Les bêtes
De ce que les hommes étaient supposés avoir continuellement des idées, des perceptions, des conceptions, il suivait naturellement que les bêtes en avaient toujours aussi ; car il est incontestable qu'un chien de chasse a l'idée de son maître auquel il obéit, et du gibier qu'il lui rapporte1. Il est évident qu'il a de la mémoire et qu'il combine quelques idées. Ainsi donc si la pensée de l'homme était aussi l'essence de son âme, la pensée du chien était aussi l'essence de la sienne ; et si l'homme avait toujours des idées, il fallait bien que les animaux en eussent toujours. Pour trancher cette difficulté, le fabricateur des tourbillons et de la matière cannelée2, osa dire que les bêtes étaient de pures machines, qui cherchaient à manger sans avoir appétit, qui avaient toujours les organes du sentiment pour n'éprouver jamais la moindre sensation, qui criaient sans douleur, qui témoignaient leur plaisir sans joie, qui possédaient un cerveau pour n'y pas recevoir l'idée la plus légère, et qui étaient ainsi une contradiction perpétuelle3.
Ce système était aussi ridicule que l'autre ; mais au lieu d'en faire voir l'extravagance, on le traita d'impie ; on prétendit que ce système répugnait à l'Écriture sainte, qui dit, dans la Genèse, _que Dieu a fait un pacte avec les animaux, et qu'il leur redemandera le sang des hommes qu'ils auront mordus et mangés_4 ; ce qui suppose manifestement dans les bêtes l'intelligence, la connaissance du bien et du mal.
1\- Voltaire prend un malin plaisir à contredire Descartes qui affirmait, d'une part, que l'âme pense toujours et, d'autre part, que les animaux n'ont pas d'âme et sont des machines.
2\- Voltaire fait ici référence à la physique cartésienne que l'on trouve par exemple dans _Le Monde ou le Traité de la lumière_ ou dans les _Principes de la philosophie_ , partie III.
3\- Pour mieux critiquer Descartes, Voltaire le caricature. En effet, Descartes, contrairement à ce qui est affirmé ici, reconnaît que les animaux sentent corporellement, et expriment par des mouvements et des cris, les trois passions primitives de la crainte, de l'espérance et de la joie, dont on se sert dans le dressage : « ainsi toutes les choses qu'on fait faire aux chiens, aux chevaux et aux singes, ne sont que des mouvements de leur crainte, de leur espérance, ou de leur joie, en sorte qu'ils les peuvent faire sans aucune pensée » (lettre à Newcastle du 23 novembre 1646, p. 694-695 du tome III des _Œuvres philosophiques_ , en trois tomes, Paris, Garnier, 1963-1973).
4\- Voir Genèse, IX, 5, _in_ La Bible de Jérusalem, Paris, Éd. du Cerf, 1986.
VII. L'expérience
Ne mêlons jamais l'Écriture sainte dans nos disputes philosophiques ; ce sont des choses trop hétérogènes, et qui n'ont aucun rapport. Il ne s'agit ici que d'examiner ce que nous pouvons savoir par nous-mêmes, et cela se réduit à bien peu de chose. Il faut avoir renoncé au sens commun pour ne pas convenir que nous ne savons rien au monde que par l'expérience ; et certainement si nous ne parvenons que par l'expérience, et par une suite de tâtonnements et de longues réflexions, à nous donner quelques idées faibles et légères du corps, de l'espace, du temps, de l'infini, de Dieu même, ce n'est pas la peine que l'Auteur de la nature mette ces idées dans la cervelle de tous les fœtus, afin qu'il n'y ait ensuite qu'un très petit nombre d'hommes qui en fassent usage1.
Nous sommes tous sur les objets de notre science, comme les amants ignorants Daphnis et Chloé, dont Longus nous a dépeint les amours et les vaines tentatives2. Il leur fallut beaucoup de temps pour deviner comment ils pouvaient satisfaire leurs désirs, parce que l'expérience leur manquait. La même chose arriva à l'empereur Léopold3 et au fils de Louis XIV4 ; il fallut les instruire. S'ils avaient eu des idées innées, il est à croire que la nature ne leur eût pas refusé la principale et la seule nécessaire à la conservation de l'espèce humaine.
1\- Voltaire réaffirme ici sa position empiriste dans le sillage de Locke et son rejet des idées innées.
2\- Longus ou Longos (fin IIe \- déb. IIIe siècle) est l'auteur de _Daphnis et Chloé_ , une pastorale traduite à l'origine par Jacques Amyot (1513-1593), humaniste et prélat français.
3\- Léopold Ier, empereur d'Allemagne, né en 1640 et mort en 1705, élu empereur en 1658, connu pour sa ferveur religieuse, sa vie monacale et sa chasteté (voir Jean Bérenger, _Léopold I er (1640-1705), fondateur de la puissance autrichienne_, Paris, PUF, 2004).
4\- Louis de France, dit le Grand Dauphin ou Monseigneur (1661-1711), était réputé pour son tempérament doux et placide et son peu d'intelligence (d'après son précepteur Bossuet). Il eut trois enfants.
VIII. Substance
Ne pouvant avoir aucune notion que par expérience, il est impossible que nous puissions jamais savoir ce que c'est que la matière. Nous touchons, nous voyons les propriétés de cette substance ; mais ce mot même _substance_ , _ce qui est dessous_ , nous avertit assez que ce dessous nous sera inconnu à jamais : quelque chose que nous découvrions de ses apparences, il restera toujours ce dessous à découvrir. Par la même raison, nous ne saurons jamais par nous-mêmes ce que c'est qu'esprit. C'est un mot qui originairement signifie _souffle_ , et dont nous nous sommes servis pour tâcher d'exprimer vaguement et grossièrement ce qui nous donne des pensées. Mais quand même, par un prodige qui n'est pas à supposer, nous aurions quelque légère idée de la substance de cet esprit, nous ne serions pas plus avancés ; et nous ne pourrions jamais deviner comment cette substance reçoit des sentiments et des pensées. Nous savons bien que nous avons un peu d'intelligence, mais comment l'avons-nous ? C'est le secret de la nature, elle ne l'a dit à nul mortel1.
1\- Tout le paragraphe s'inspire de la critique lockienne de l'idée de substance qui concluait aussi que nous n'avons pas d'idée claire de la substance corporelle ni de la substance spirituelle (voir _l'Essai philosophique concernant l'entendement humain_ , livre II, chapitre XXIII).
IX. Bornes étroites
Notre intelligence est très bornée, ainsi que la force de notre corps. Il y a des hommes beaucoup plus robustes que les autres ; il y a aussi des Hercules en fait de pensées ; mais au fond cette supériorité est fort peu de chose. L'un soulèvera dix fois plus de matière que moi, l'autre pourra faire de tête et sans papier une division de quinze chiffres, tandis que je ne pourrai en diviser que trois ou quatre avec une extrême peine1 ; c'est à quoi se réduira cette force tant vantée ; mais elle trouvera bien vite sa borne ; et c'est pourquoi dans les jeux de combinaison, nul homme après s'y être formé par toute son application et par un long usage, ne parvient jamais, quelque effort qu'il fasse, au-delà du degré qu'il a pu atteindre ; il a frappé à la borne de son intelligence. Il faut même absolument que cela soit ainsi ; sans quoi nous irions de degré en degré jusqu'à l'infini2.
1\- Petit trait autobiographique quand on sait que Gabrielle-Émilie de Breteuil, avec qui Voltaire vécut de 1735 jusqu'à la mort de la marquise en 1749, était capable de faire de tête de telles divisions à quinze chiffres (voir R. Vaillot, _Voltaire en son temps. Avec Mme Du Châtelet_ , Oxford, Voltaire Foundation, 1988 ; E. Badinter, _Émilie ou l'ambition féminine au XVIIIe siècle_, Flammarion, Paris, 1983 et 2006).
2\- Voltaire, une fois n'est pas coutume, est cartésien sur ce point : notre esprit fini ne peut comprendre l'infini (voir Descartes, _Principes de la philosophie_ , I, 26, in _Œuvres philosophiques_ , III, p. 107-108).
X. Découvertes impossibles
Dans ce cercle étroit où nous sommes renfermés, voyons donc ce que nous sommes condamnés à ignorer, et ce que nous pouvons un peu connaître. Nous avons déjà vu qu'aucun premier ressort, aucun premier principe ne peut être saisi par nous1.
Pourquoi mon bras obéit-il à ma volonté ? Nous sommes si accoutumés à ce phénomène incompréhensible que très peu y font attention ; et quand nous voulons rechercher la cause d'un effet si commun, nous trouvons qu'il y a réellement l'infini entre notre volonté et l'obéissance de notre membre : c'est-à-dire qu'il n'y a nulle proportion de l'un à l'autre, nulle raison, nulle apparence de cause ; et nous sentons que nous y penserions une éternité, sans pouvoir imaginer la moindre lueur de vraisemblance2.
1\- Voltaire rappelle ici que nous ignorons les premiers principes (voir Doute II) et enchaîne, comme dans le Doute III, sur le problème de l'interaction de l'âme et du corps.
2\- Voltaire hérite de la crise de la causalité ouverte par Descartes qui affirme à la fois la distinction réelle de l'esprit et du corps et l'interaction de l'esprit et du corps dans l'homme. Malebranche proposait de résoudre le problème de l'interaction de l'âme et du corps par l'occasionnalisme : ma volonté de lever le bras n'est que l'occasion pour Dieu d'agir sur mon corps et de me faire lever le bras. Voltaire critique à plusieurs reprises l'interprétation occasionnaliste de Malebranche : voir, par exemple, _Tout en Dieu_ , _in_ vol. XXVIII des _Œuvres complètes_ de Voltaire, éditées par Moland en 52 vol., Paris, Garnier frères, 1877-1883.
XI. Désespoir fondé
Ainsi arrêtés dès le premier pas, et nous repliant vainement sur nous-mêmes, nous sommes effrayés de nous chercher toujours, et de ne nous trouver jamais. Nul de nos sens n'est explicable.
Nous savons bien à peu près, avec le secours des triangles1, qu'il y a environ trente millions de nos grandes lieues géométriques2 de la terre au soleil ; mais qu'est-ce que le soleil ? et pourquoi tourne-t-il sur son axe ? et pourquoi en un sens plutôt qu'en un autre ? et pourquoi Saturne et nous tournons-nous autour de cet astre plutôt d'occident en orient que d'orient en occident ? Non seulement nous ne satisferons jamais à cette question ; mais nous n'entreverrons jamais la moindre possibilité d'en imaginer seulement une cause physique. Pourquoi ? c'est que le nœud de cette difficulté est dans le premier principe des choses3.
Il en est de ce qui agit au-dedans de nous comme de ce qui agit dans les espaces immenses de la nature. Il y a dans l'arrangement des astres, et dans la conformation d'un ciron et de l'homme, un premier principe dont l'accès doit nécessairement nous être interdit4. Car si nous pouvions connaître notre premier ressort, nous en serions les maîtres, nous serions des dieux. Éclaircissons cette idée, et voyons si elle est vraie.
Supposons que nous trouvions en effet la cause de nos sensations, de nos pensées, de nos mouvements, comme nous avons seulement découvert dans les astres la raison des éclipses et des différentes phases de la lune et de Vénus, il est clair que nous prédirions alors nos sensations, nos pensées et nos désirs, résultant de ces sensations, comme nous prédisons les phases et les éclipses5. Connaissant donc ce qui devrait se passer demain dans notre intérieur, nous verrions clairement par le jeu de cette machine, de quelle manière ou agréable ou funeste nous devrions être affectés. Nous avons une volonté qui dirige, ainsi qu'on en convient, nos mouvements intérieurs en plusieurs circonstances. Par exemple, je me sens disposé à la colère, ma réflexion et ma volonté en répriment les accès naissants. Je verrais, si je connaissais mes premiers principes, toutes les affections auxquelles je suis disposé pour demain, toute la suite des idées qui m'attendent ; je pourrais avoir sur cette suite d'idées et de sentiments la même puissance que j'exerce quelquefois sur les sentiments et sur les pensées actuelles, que je détourne et que je réprime. Je me trouverais précisément dans le cas de tout homme qui peut retarder et accélérer à son gré le mouvement d'une horloge, celui d'un vaisseau, celui de toute machine connue.
Étant le maître des idées qui me sont destinées demain, je le serais pour le jour suivant, je le serais pour le reste de ma vie ; je pourrais donc être toujours tout-puissant sur moi-même, je serais le dieu de moi-même. Je sens assez que cet état est incompatible avec ma nature ; il est donc impossible que je puisse rien connaître du premier principe qui me fait penser et agir6.
1\- Voltaire fait ici référence à la méthode de triangulation utilisée pour mesurer le degré d'un arc du méridien, ce qui a permis de connaître la figure de la Terre et son diamètre puis de calculer la distance de la Terre au Soleil (voir les expéditions scientifiques en Laponie et au Pérou, voir aussi la mesure de la méridienne par Picard que Voltaire présente comme décisive pour l'élaboration par Newton de la théorie de l'attraction dans la lettre XV des _Lettres philosophiques_ , Paris, GF-Flammarion, 1964, p. 99). Mais, en réalité, la méthode de triangulation était trop peu précise pour connaître la distance exacte de la Terre au Soleil car le triangle est trop aplati. Il faut attendre 1769 pour avoir une valeur correcte de la distance Terre-Soleil (149,5 millions de km), par l'utilisation de la troisième loi de Kepler.
2\- La lieue que Voltaire appelle géométrique est sans doute la lieue métrique qui vaut exactement 4 km. Selon la méthode de triangulation donc, la distance de la Terre au Soleil ne serait que de 120 millions de km.
3\- Effectivement, la théorie newtonienne de l'attraction laisse ce problème en suspens ; Voltaire en profite pour renvoyer la réponse à cette question à Dieu et ainsi affirmer son déisme. À la fin de la lettre XV des _Lettres philosophiques_ , il disait déjà à propos de l'attraction : « La cause de cette cause est dans le sein de Dieu. »
4\- L'accent est cette fois pascalien : devant la double infinité, l'infiniment grand de l'univers et l'infiniment petit du ciron, l'homme est condamné à ne pas comprendre l'infini (voir Pascal, _Pensées_ , fr. 199, « Disproportion de l'homme » in _Œuvres complètes_ , Paris, Seuil, 1963, p. 526 : « Qu'est-ce qu'un homme dans l'infini ? [...] Un néant à l'égard de l'infini, un tout à l'égard du néant, un milieu entre rien et tout, infiniment éloigné de comprendre les extrêmes »).
5\- Si l'on connaissait le premier principe qui nous fait agir, notre pouvoir de prédiction serait le même que celui de la théorie newtonienne de l'attraction qui permet de calculer les phases de la Lune et les éclipses du Soleil.
6\- Voltaire développe la formule qui termine le troisième alinéa de ce Doute : si nous connaissions notre premier ressort, nous en serions les maîtres, nous serions des dieux. Mais il faut noter que son raisonnement ne vaut que si l'on suppose que la volonté puisse agir et modifier ce que l'on connaît par anticipation, c'est-à-dire qu'elle puisse changer ce qui est déterminé, mais alors à quoi sert notre pouvoir de prédiction puisque ce qui doit se passer demain – je dois me mettre en colère – n'arrive pas. En outre, le raisonnement ne vaut que pour mon moi isolé mais, en réalité, mon moi ne cesse d'être affecté par autrui, et si chacun est dieu, quelle toute-puissance sur moi-même me reste-t-il ?
XII. Doute
Ce qui est impossible à ma nature si faible, si bornée, et qui est d'une durée si courte, est-il impossible dans d'autres globes, dans d'autres espèces d'êtres ? Y a-t-il des intelligences supérieures, maîtresses de toutes leurs idées, qui pensent et qui sentent tout ce qu'elles veulent1 ? Je n'en sais rien ; je ne connais que ma faiblesse, je n'ai aucune notion de la force des autres.
1\- Ces êtres supérieurs pourraient être de purs esprits, comme les anges, par exemple.
XIII. Suis-je libre ?
Ne sortons point encore du cercle de notre existence ; continuons à nous examiner nous-mêmes autant que nous le pouvons. Je me souviens qu'un jour, avant que j'eusse fait toutes les questions précédentes, un raisonneur voulut me faire raisonner. Il me demanda si j'étais libre ; je lui répondis que je n'étais point en prison, que j'avais la clef de ma chambre, que j'étais parfaitement libre. Ce n'est pas cela que je vous demande, me répondit-il, croyez-vous que votre volonté ait la liberté de vouloir ou de ne vouloir pas vous jeter par la fenêtre ? pensez-vous, avec l'Ange de l'École1 que le libre arbitre soit une puissance appétitive, et que le libre arbitre se perde par le péché2 ? Je regardai mon homme fixement, pour tâcher de lire dans ses yeux s'il n'avait pas l'esprit égaré ; et je lui répondis que je n'entendais rien à son galimatias3.
Cependant, cette question sur la liberté de l'homme m'intéressa vivement ; je lus des scolastiques, je fus comme eux dans les ténèbres4 ; je lus Locke, et j'aperçus des traits de lumière5 ; je lus le traité de Colins, qui me parut Locke perfectionné6 ; et je n'ai jamais rien lu depuis qui m'ait donné un nouveau degré de connaissance. Voici ce que ma faible raison a conçu, aidée de ces deux grands hommes, les seuls, à mon avis, qui se soient entendus eux-mêmes en écrivant sur cette matière, et les seuls qui se soient fait entendre aux autres.
Il n'y a rien sans cause7. Un effet sans cause n'est qu'une parole absurde. Toutes les fois que je veux, ce ne peut être qu'en vertu de mon jugement bon ou mauvais ; ce jugement est nécessaire, donc ma volonté l'est aussi. En effet, il serait bien singulier que toute la nature, tous les astres obéissent à des lois éternelles, et qu'il y eût un petit animal haut de cinq pieds, qui au mépris de ces lois pût agir toujours comme il lui plairait au seul gré de son caprice. Il agirait au hasard ; et on sait que le hasard n'est rien. Nous avons inventé ce mot pour exprimer l'effet connu de toute cause inconnue8.
Mes idées entrent nécessairement dans mon cerveau, comment ma volonté qui en dépend serait-elle libre ? Je sens en mille occasions que cette volonté n'est pas libre ; ainsi quand la maladie m'accable, quand la passion me transporte, quand mon jugement ne peut atteindre aux objets qu'on me présente, etc. je dois donc penser que les lois de la nature étant toujours les mêmes, ma volonté n'est pas plus libre dans les choses qui me paraissent les plus indifférentes que dans celles où je me sens soumis à une force invincible9.
Être véritablement libre, c'est pouvoir. Quand je peux faire ce que je veux, voilà ma liberté ; mais je veux nécessairement ce que je veux ; autrement je voudrais sans raison, sans cause, ce qui est impossible. Ma liberté consiste à marcher quand je veux marcher et que je n'ai point la goutte10.
Ma liberté consiste à ne point faire une mauvaise action quand mon esprit se la représente nécessairement mauvaise ; à subjuguer une passion quand mon esprit m'en fait sentir le danger, et que l'horreur de cette action combat puissamment mon désir. Nous pouvons réprimer nos passions (comme je l'ai déjà annoncé nombre IV11) mais alors nous ne sommes pas plus libres en réprimant nos désirs qu'en nous laissant entraîner à nos penchants ; car dans l'un et l'autre cas, nous suivons irrésistiblement notre dernière idée ; et cette dernière idée est nécessaire ; donc je fais nécessairement ce qu'elle me dicte. Il est étrange que les hommes ne soient pas contents de cette mesure de liberté, c'est-à-dire du pouvoir qu'ils ont reçu de la nature de faire en plusieurs cas ce qu'ils veulent ; les astres ne l'ont pas ; nous la possédons, et notre orgueil nous fait croire quelquefois que nous en possédons encore plus. Nous nous figurons que nous avons le don incompréhensible et absurde de vouloir sans autre raison, sans autre motif que celui de vouloir. Voyez le nombre XXIX12.
Non, je ne puis pardonner au docteur Clarke13 d'avoir combattu avec mauvaise foi ces vérités dont il sentait la force, et qui semblaient s'accommoder mal avec ses systèmes. Non, il n'est pas permis à un philosophe tel que lui d'avoir attaqué Colins en sophiste, et d'avoir détourné l'état de la question en reprochant à Colins d'appeler l'homme _un agent nécessaire._ Agent, ou patient, qu'importe ! agent quand il se meut volontairement, patient quand il reçoit des idées. Qu'est-ce que le nom fait à la chose ? L'homme est en tout un être dépendant, comme la nature entière est dépendante, et il ne peut être excepté des autres êtres.
Le prédicateur, dans Samuel Clarke, a étouffé le philosophe14 ; il distingue la nécessité physique et la nécessité morale. Et qu'est-ce qu'une nécessité morale ? Il vous paraît vraisemblable qu'une reine d'Angleterre qu'on couronne et que l'on sacre dans une église, ne se dépouillera pas de ses habits royaux pour s'étendre toute nue sur l'autel, quoiqu'on raconte une pareille aventure d'une reine de Congo. Vous appelez cela une nécessité morale dans une reine de nos climats ; mais c'est au fond une nécessité physique, éternelle, liée à la constitution des choses. Il est aussi sûr que cette reine ne fera pas cette folie, qu'il est sûr qu'elle mourra un jour. La nécessité morale n'est qu'un mot ; tout ce qui se fait est absolument nécessaire. Il n'y a point de milieu entre la nécessité et le hasard : et vous savez qu'il n'y a point de hasard : donc tout ce qui arrive est nécessaire15.
Pour embarrasser la chose davantage, on a imaginé de distinguer encore entre nécessité et contrainte ; mais au fond la contrainte n'est autre chose qu'une nécessité dont on s'aperçoit ; et la nécessité est une contrainte dont on ne s'aperçoit pas16. Archimède est également nécessité à rester dans sa chambre quand on l'y enferme, et quand il est si fortement occupé d'un problème qu'il ne reçoit pas l'idée de sortir17.
_Ducunt volentem fata, nolentem trahunt_18.
L'ignorant qui pense ainsi, n'a pas toujours pensé de même, mais il est enfin contraint de se rendre19.
1\- Thomas d'Aquin (1225-1274), auteur de la _Somme théologique_ ( _Bibliothèque de Voltaire_ n° 3292), est le père de la philosophie scolastique qui interprète Aristote dans le contexte de la religion chrétienne.
2\- Thomas d'Aquin discute du libre arbitre et de la puissance appétitive, c'est-à-dire du désir, dans le tome I de la _Somme théologique_ , en deux tomes, Paris, Éd. du Cerf, 1984, question 83.
3\- Autrement dit, aux questions discutées par le philosophe scolastique.
4\- Thomas d'Aquin, dans la question 83, se réfère à l'Ecclésiastique, XV, 14, pour affirmer que Dieu est la cause du libre arbitre. Et comme le libre arbitre est la cause du péché, on pourrait déduire que Dieu est la cause du péché mais il n'en est rien car le péché est dû à une mauvaise délibération et à un mauvaix choix de la volonté qui aspire pourtant au bien en général, le problème étant que le libre arbitre se détermine et fixe son choix sur un bien particulier.
5\- Locke discute de la liberté dans le chapitre XXI du livre II de _l'Essai philosophique concernant l'entendement humain ;_ au § 24 il en livre une conception toute négative : l'homme est libre quand il n'est pas empêché ni contraint d'agir, quand il n'est pas prisonnier dans des chaînes.
6\- Anthony Collins (1676-1729), élève et ami de Locke, développe lui aussi une conception négative de la liberté comme exemption de la contrainte physique ; voir ses _Recherches sur la liberté de l'homme_ (1717 ; trad. fr. 1720 ; _Bibliothèque de Voltaire_ n° 2889).
7\- Formulation du principe de raison.
8\- Cette idée que le hasard est un nom mis sur notre ignorance vient de Collins et de Locke aussi bien que de Spinoza : nous croyons agir librement et parfois au hasard parce que nous ignorons les causes qui nous déterminent.
9\- Le libre arbitre est donc une illusion.
10\- Voltaire reprend exactement la conception négative de la liberté défendue par Locke et par Collins : je suis libre quand je ne suis pas empêché d'agir.
11\- Ce que Voltaire appelle « nombre » désigne la numérotation des Doutes mais il fait ici une erreur de référence interne car le Doute IV ne parle nullement de la régulation des passions, il en parle en revanche dans le Doute XI à propos de la colère : « Par exemple, je me sens disposé à la colère, ma réflexion et ma volonté en répriment les accès naissants ».
12\- Voir le Doute XXIX intitulé « De Locke » où Voltaire reprend le problème de la liberté.
13\- Samuel Clarke est le chef de file du cercle des théologiens qui gravitent autour de Newton. Il publia en 1705-1706 son _Traité de l'existence et des attributs de Dieu_ (trad. fr. 1727-1728) et fut le porte-parole de Newton, notamment dans ses discussions avec Leibniz. Il critiqua aussi la conception négative de la liberté de Collins dans ses _Remarques sur un ouvrage intitulé « Recherches sur la liberté de l'homme »_ , publiées en annexe de la correspondance de Clarke avec Leibniz, en 1717. C'est à cet ouvrage que Voltaire fait référence ensuite.
14\- Dans les _Éléments de la philosophie de Newton_ , Voltaire écrit à propos de Clarke : « Je me souviens que dans plusieurs conférences que j'eus en 1726 avec le docteur Clarke, jamais ce philosophe ne prononçait le nom de Dieu qu'avec un air de recueillement et de respect très remarquable. Je lui avouai l'impression que cela faisait sur moi, et il me dit que c'était de Neuton qu'il avait pris insensiblement cette coutume, laquelle doit être en effet celle de tous les hommes » (p. 195-196).
15\- Dans cet alinéa, Voltaire combat la distinction traditionnelle entre nécessité morale (ou conditionnelle) et nécessité physique défendue notamment par Leibniz dans le § 13 de son _Discours de métaphysique_ , il rabat la première sur la seconde et développe ici les idées de Locke et de Collins, pas si éloignées du nécessitarisme de Spinoza.
16\- Voltaire reprend fidèlement Locke et Collins.
17\- Même exemple dans le Doute XXIX.
18\- « Les destins guident ceux qui acquiescent, ils entraînent ceux qui résistent », dit Sénèque dans la lettre 107 des _Lettres à Lucilius_ , en 5 vol. établis par F. Préchac et trad. H. Noblot, Paris, Les Belles Lettres, 1956-1964.
19\- Voltaire souligne ici qu'il n'a pas toujours pensé ainsi et que c'est malgré lui qu'il se range à l'opinion de Locke et de Collins. Effectivement, en 1734, dans le chapitre VII de son _Traité de métaphysique_ , il défendait une tout autre conception : l'homme est libre, la liberté est la santé de l'âme, même si elle est faible et bornée, comme toutes nos autres facultés.
XIV. Tout est-il éternel ?
Asservi à des lois éternelles comme tous les globes qui remplissent l'espace, comme les éléments, les animaux, les plantes ; je jette des regards étonnés sur tout ce qui m'environne, je cherche quel est mon auteur, et celui de cette machine immense dont je suis à peine une roue imperceptible1.
Je ne suis pas venu de rien : car la substance de mon père et de ma mère qui m'a porté neuf mois dans sa matrice est quelque chose. Il m'est évident que le germe2 qui m'a produit n'a pu être produit de rien ; car comment le néant produirait-il l'existence ? Je me sens subjugué par cette maxime de toute l'Antiquité : _Rien ne vient du néant, rien ne peut retourner au néant_3. Cet axiome porte en lui une force si terrible, qu'il enchaîne tout mon entendement, sans que je puisse me débattre contre lui. Aucun philosophe ne s'en est écarté ; aucun législateur, quel qu'il soit, ne l'a contesté. Le _Cahut_ des Phéniciens4, le _Chaos_ des Grecs, le _Tohu bohu_ des Chaldéens5 et des Hébreux6, tout nous atteste qu'on a toujours cru l'éternité de la matière. Ma raison, trompée par cette idée si ancienne et si générale, me dit : Il faut bien que la matière soit éternelle, puisqu'elle existe ; si elle était hier, elle était auparavant. Je n'aperçois aucune vraisemblance qu'elle ait commencé à être, aucune cause pour laquelle elle n'ait pas été, aucune cause pour laquelle elle ait reçu l'existence dans un temps plutôt que dans un autre. Je cède donc à cette conviction, soit fondée, soit erronée ; et je me range du parti du monde entier, jusqu'à ce qu'ayant avancé dans mes recherches je trouve une lumière supérieure au jugement de tous les hommes, qui me force à me rétracter malgré moi7.
Mais si, comme tant de philosophes de l'Antiquité l'ont pensé, l'Être éternel a toujours agi, que deviendront le _Cahut_ et l' _Ereb_8 des Phéniciens, le _Tohu bohu_ des Chaldéens, le _Chaos_ d'Hésiode9 ? Il restera dans les fables. Le _Chaos_ est impossible aux yeux de la raison ; car il est impossible que, l'intelligence étant éternelle, il y ait jamais eu quelque chose d'opposé aux lois de l'intelligence ; or le _Chaos_ est précisément l'opposé de toutes les lois de la nature. Entrez dans la caverne la plus horrible des Alpes10, sous ces débris de rochers, de glace, de sable, d'eaux, de cristaux, de minéraux informes, tout y obéit à la gravitation. Le _Chaos_ n'a jamais été que dans nos têtes, et n'a servi qu'à faire composer de beaux vers à Hésiode et à Ovide11.
Si notre sainte Écriture a dit que le _Chaos_ existait, si le _Tohu bohu_ a été adopté par elle, nous le croyons sans doute, et avec la foi la plus vive. Nous ne parlons ici que suivant les lueurs trompeuses de notre raison12. Nous nous sommes bornés, comme nous l'avons dit, à voir ce que nous pouvons soupçonner par nous-mêmes. Nous sommes des enfants qui essayons de faire quelques pas sans lisières13.
1\- La conception voltairienne de l'univers s'inscrit dans un cadre mécaniste général même si Voltaire n'est pas cartésien : il paye ici son tribut au paradigme dominant du XVIIIe siècle.
2\- Voltaire défend la thèse de la préexistence des germes à l'encontre de celle de la génération spontanée, voir ci-dessous Doute XX.
3\- Perse, _Satires_ , III, 84 : _de nihilo nihil, in nihilum nil posse reverti_. Perse reprend ici l'axiome fondamental de la physique ancienne – rien ne se fait de rien – que Diderot, dans l'article CHAOS ( _Encyclopédie_ , III, 1753, 156), considère comme excellent et que l'article anonyme MATÉRIALISTES ( _Encyclopédie_ , X, 1765, 188) présente comme la définition même de la pensée matérialiste.
4\- Le territoire de la Phénicie correspond au Liban auquel il faudrait ajouter certaines portions de la Syrie, d'Israël et de la Palestine. Les Phéniciens étaient un peuple antique d'habiles navigateurs et commerçants. Après avoir supporté les assauts des Athéniens, des Assyriens, de Nabuchodonosor puis de Darius III, la Phénicie disparut finalement avec sa conquête par Alexandre le Grand en 332 av. J.-C.
5\- La Chaldée est une ancienne région située entre les cours inférieurs de l'Euphrate et du Tigre. Les Chaldéens habitaient au sud-ouest de Babylone. Du IXe siècle au VIe siècle av. J.-C., les Chaldéens jouèrent un rôle important dans l'histoire de l'Asie et contribuèrent à la destruction de l'empire assyrien. Pour une courte période, ils firent de la Babylonie, qui progressivement s'appela la Chaldée, la puissance dominante de la Mésopotamie.
6\- Cahut ou Chaos ont une même racine indo-européenne qui signifie abîme, désordre complet, ou confusion. Diderot, dans l'article CHAOS ( _Encyclopédie_ , III, 1753, 156), définit ainsi le terme : « Les anciens philosophes ont entendu par ce mot, un mélange confus de particules de toute espèce, sans forme ni régularité, auquel ils supposent le mouvement essentiel, lui attribuant en conséquence la formation de l'univers. Ce système est chez eux un corollaire d'un axiome excellent en lui-même [...] savoir, que _rien_ ne se fait sans _rien_. » Quant à Tohu Bohu, il vient de l'hébreu _tohou oubohou_ qui signifie aussi le chaos, c'est-à-dire le ciel du ciel et la terre de la terre, encore informes (voir Genèse, I, 2).
7\- Voltaire se range à la thèse de l'éternité de la matière et s'oppose à la création _ex nihilo_ de la tradition judéo-chrétienne. Pourtant, l'idée de Tohu Bohu n'est pas incompatible avec la création _ex nihilo_ car c'est Dieu qui crée le ciel et la terre dans leur premier état informe (Genèse, I, 1).
8\- Terme phénicien qui signifie sombre, soir, couchant.
9\- Hésiode est un poète grec du VIIe siècle av. J.-C., auteur de la _Théogonie_ , le plus ancien poème religieux grec, qui livre une généalogie des dieux à laquelle s'ajoute une cosmogonie qui retrace la formation du monde à partir du Chaos.
10\- Longtemps, on évita la montagne. Obstacle redoutable, elle compliquait les déplacements, se montrait hostile à l'habitat humain, à l'exploitation. Les Alpes notamment suscitaient la crainte et l'effroi. Au XVIIIe siècle, cependant, la perception des montagnes commence à changer : Albrecht von Haller, Jean-Jacques Rousseau et William Windham présentent la montagne sous un jour meilleur. En 1786, les Français Paccard et Balmat réussissent à gravir le mont Blanc, exploit réitéré l'année suivante par le Suisse Horace-Bénédict de Saussure. Puis le XIXe siècle romantique donne une vision idyllique des Alpes. Les Anglais de l'ère victorienne découvrent la montagne.
11\- Ovide, poète latin du Ier siècle apr. J.-C., auteur des _Métamorphoses_ , poème mythologique en quinze livres.
12\- Tergiversations de Voltaire : s'il semblait tendre au début du Doute vers la thèse de l'éternité de la matière, c'était par une raison trompée par l'idée ancienne du chaos. La raison lui fait plutôt concevoir que tout est soumis à la gravitation et que l'idée de chaos est une fiction. Mais la Bible dit que le chaos existe, donc Voltaire dit, non sans ironie, qu'il parle selon « les lueurs trompeuses de la raison ».
13\- Cette image de l'enfant qui essaie de marcher sans lisières, c'est-à-dire sans soutien, s'accompagne de l'injonction de penser par soi-même chère à Voltaire et aux encyclopédistes (Diderot dit dans l'article AUTORITÉ dans les discours et dans les écrits – _Encyclopédie_ , I, 1751, 901 – que celui qui ne se délivre pas de l'argument d'autorité et ne pense pas par lui-même est comme un enfant dont les jambes ne s'affermissent point ; Voltaire écrit dans l'article _Liberté de penser_ du _Dictionnaire philosophique_ , Paris, Imprimerie nationale, 1994, p. 331 : « il est honteux de mettre son âme dans les mains de ceux à qui vous ne confierez pas votre argent : osez penser par vous-même »). Cette image de l'enfant et l'injonction d'oser penser par soi-même auront une postérité importante notamment chez Kant : dans sa _Réponse à la question : Qu'est-ce les Lumières ?_ (trad. J.-F. Poirier et F. Proust, Paris, GF-Flammarion, 1991), il en fera la devise des Lumières qui ont osé faire un pas sans la roulette d'enfant que l'état de tutelle impose aux hommes pour marcher.
XV. Intelligence
Mais en apercevant l'ordre, l'artifice prodigieux, les lois mécaniques et géométriques qui règnent dans l'univers, les moyens, les fins innombrables de toutes choses, je suis saisi d'admiration et de respect1. Je juge incontinent que si les ouvrages des hommes, les miens même, me forcent à reconnaître en nous une intelligence, je dois en reconnaître une bien supérieurement agissante dans la multitude de tant d'ouvrages. J'admets cette intelligence suprême, sans craindre que jamais on puisse me faire changer d'opinion. Rien n'ébranle en moi cet axiome, tout ouvrage démontre un ouvrier2.
1\- La philosophie naturelle de Newton, sa théorie de l'attraction, conduit naturellement à une théologie naturelle ; du constat de l'ordre du monde, on infère le principe d'ordre : Dieu, qui a eu le dessein de faire un monde ordonné. Tel est l'argument du dessein cher à Voltaire et à tous les tenants de la théologie naturelle (les newtoniens, les leibniziens ; et les déistes, nombreux au XVIIIe siècle, qui affirment que Dieu existe mais rejettent la Révélation).
2\- Sur ce point-là, la pensée de Voltaire n'a effectivement pas changé depuis 1745 et sa _Courte réponse aux longs discours d'un docteur allemand_ (in _Éléments de la philosophie de Newton_ , p. 755), où il utilise la fameuse formule : « Je serai toujours persuadé qu'une horloge prouve un horloger et que l'univers prouve un Dieu. » Malgré les désastres et tremblements de terre (celui de Lisbonne notamment en novembre 1755 qui a fait 60 000 morts), il reste un déiste invétéré.
XVI. Éternité
Cette intelligence est-elle éternelle ? Sans doute, car soit que j'aie admis ou rejeté l'éternité de la matière, je ne peux rejeter l'existence éternelle de son artisan suprême1 ; et il est évident que s'il existe aujourd'hui, il a existé toujours.
1\- Le déisme de Voltaire se pose en s'opposant au matérialisme représenté en France par La Mettrie, Diderot, Helvétius et le baron d'Holbach : en effet, à l'encontre du raisonnement de Voltaire qui infère de l'éternité de la matière l'éternité de Dieu, les matérialistes soutiennent que la matière est éternelle, qu'elle s'auto-organise et que ses lois suffisent à expliquer l'ordre du monde sans recourir à un quelconque Dieu (voir, sur cette question, _Le Matérialisme du XVIII e siècle_, dir. O. Bloch, Paris, Vrin, 1982, et J.-C. Bourdin, _Les Matérialistes au XVIII e siècle_, textes choisis, Paris, Payot, 1996).
XVII. Incompréhensibilité
Je n'ai fait encore que deux ou trois pas dans cette vaste carrière ; je veux savoir si cette intelligence divine est quelque chose d'absolument distinct de l'univers, à peu près comme le sculpteur est distingué de la statue ; ou si cette âme du monde1 est unie au monde, et le pénètre à peu près encore comme ce que j'appelle mon âme est unie à moi, et selon cette idée de l'Antiquité si bien exprimée dans Virgile et dans Lucain :
_Mens agitat molem et magno se corpore miscet_2 _._
_Juppiter est quodcumque vides quocumque moveris_3 _._
Je me vois arrêté tout à coup dans ma vaine curiosité. Misérable mortel, si je ne puis sonder ma propre intelligence, si je ne puis savoir ce qui m'anime, comment connaîtrai-je l'intelligence ineffable qui préside visiblement à la matière entière ? Il y en a une, tout me le démontre ; mais où est la boussole qui me conduira vers sa demeure éternelle et ignorée4 ?
1\- Introduite par Platon dans le _Timée_ , la notion d'âme du monde coéternelle au monde fut développée par les stoïciens, voir aussi Doutes XX et XXI.
2\- Virgile (poète latin né en 70 av. J.-C. et mort en 19 av. J.-C.), _Énéide_ , VI, 727. Dans l'article _Causes finales_ des _Questions sur l'Encyclopédie_ (1770), Voltaire choisit comme exorde le même vers latin qu'il traduit ainsi : « L'esprit régit le monde, il s'y mêle, il l'anime », _in_ vol. XIX des _Œuvres complètes_ de Voltaire, éd. Moland.
3\- Lucain (poète latin né en 39 apr. J.-C. et mort en 65 apr. J.-C.), _La Pharsale_ , IX, 580 : « Jupiter est tout ce que tu vois, tout ce que tu sens en toi-même » (trad. Marmontel, Paris, Garnier frères, 1865).
4\- Le déisme de Voltaire est tout sauf dogmatique, c'est un déisme sceptique en quelque sorte.
XVIII. Infini
Cette intelligence est-elle infinie en puissance et en immensité, comme elle est incontestablement infinie en durée ? Je n'en puis rien savoir par moi-même. Elle existe, donc elle a toujours existé, cela est clair. Mais quelle idée puis-je avoir d'une puissance infinie ? Comment puis-je concevoir un infini actuellement existant ? Comment puis-je imaginer que l'intelligence suprême est dans le vide ? Il n'en est pas de l'infini en étendue comme de l'infini en durée. Une durée infinie s'est écoulée au moment que je parle, cela est sûr1 ; je ne peux rien ajouter à cette durée passée, mais je peux toujours ajouter à l'espace que je conçois, comme je peux ajouter aux nombres que je conçois. L'infini en nombres et en étendue est hors de la sphère de mon entendement2. Quelque chose qu'on me dise, rien ne m'éclaire dans cet abîme. Je sens heureusement que mes difficultés et mon ignorance ne peuvent préjudicier à la morale ; on aura beau ne pas concevoir ni l'immensité de l'espace remplie, ni la puissance infinie qui a tout fait, et qui cependant peut encore faire ; cela ne servira qu'à prouver de plus en plus la faiblesse de notre entendement ; et cette faiblesse ne nous rendra que plus soumis à l'Être éternel dont nous sommes l'ouvrage3.
1\- Voltaire, en soutenant que la durée ou le temps est infini, s'oppose ici à la conception augustinienne selon laquelle il n'y a point de temps avant la création (voir Augustin, _Les Confessions_ , livre XI, § X à XIII, Paris, Desclée de Brouwer, 1962).
2\- Comme à la fin du Doute IX, ou dans le Doute XI, Voltaire réitère l'idée que mon esprit fini ne peut comprendre l'infini en puissance ni en acte, ni la double infinité.
3\- À la manière de Pascal, Voltaire infère de l'impuissance de la raison une soumission de la raison, voir _Pensées_ , section XIII, fr. 167 à 188 et fr. 199, in _Œuvres complètes_ , Paris, Seuil, 1963.
XIX. Ma dépendance
Nous sommes son ouvrage. Voilà une vérité intéressante pour nous ; car de savoir par la philosophie en quel temps il fit l'homme, ce qu'il faisait auparavant, s'il est dans la matière, s'il est dans le vide, s'il est dans un point, s'il agit toujours ou non, s'il agit partout, s'il agit hors de lui ou dans lui ; ce sont des recherches qui redoublent en moi le sentiment de mon ignorance profonde1.
Je vois même qu'à peine il y a eu une douzaine d'hommes en Europe qui aient écrit sur ces choses abstraites avec un peu de méthode ; et quand je supposerais qu'ils ont parlé d'une manière intelligible, qu'en résulterait-il ? Nous avons déjà reconnu (nomb. 42) que les choses que si peu de personnes peuvent se flatter d'entendre, sont inutiles au reste du genre humain. Nous sommes certainement l'ouvrage de Dieu, c'est là ce qu'il m'est utile de savoir ; aussi la preuve en est-elle palpable3. Tout est moyen et fins dans mon corps, tout est ressort, poulie, force mouvante, machine hydraulique, équilibre de liqueurs, laboratoire de chimie. Il est donc arrangé par une intelligence (nomb. 154). Ce n'est pas l'intelligence de mes parents à qui je dois cet arrangement, car assurément ils ne savaient ce qu'ils faisaient quand ils m'ont mis au monde ; ils n'étaient que les aveugles instruments de cet éternel fabricateur, qui anime le ver de terre, et qui fait tourner le soleil sur son axe.
1\- Voltaire est à l'unisson de d'Alembert sur ce point : les questions métaphysiques et théologiques traditionnelles sont indécidables (voir le chapitre VI sur la métaphysique des _Éléments de philosophie_ , Paris, Fayard, 1986).
2\- C'est-à-dire Doute IV.
3\- La preuve qui suit, tirée du parfait arrangement mécanique du corps humain, prend appui sur la conception cartésienne du corps vivant comme machine hydraulique (voir la deuxième partie intitulée « L'Homme » du _Monde ou Traité de la lumière_ ). Cependant, Descartes renierait la preuve de Voltaire car elle relève de l'utilisation des causes finales qu'il proscrit dans la philosophie (voir _Principes de la philosophie_ , I, 28). Voltaire, même s'il se moque de l'abbé Pluche, n'hésite pas à verser dans le finalisme et l'anthropocentrisme quand cela l'arrange. Rien d'étonnant alors à ce qu'Helvétius l'appelle un cause-finalier, ce que Voltaire commente en ces termes dans son article _Causes finales_ des _Questions sur l'Encyclopédie_ : « Si une horloge n'est pas faite pour montrer l'heure, j'avouerai alors que les causes finales sont des chimères ; et je trouverai fort bon qu'on m'appelle _cause-finalier,_ c'est-à-dire un imbécile », _in_ vol. XIX des _Œuvres complètes_ de Voltaire, éd. Moland.
4\- C'est-à-dire Doute XV.
XX. Éternité encore
Né d'un germe venu d'un autre germe, y a-t-il eu une succession continuelle, un développement sans fin de ces germes, et toute la nature a-t-elle toujours existé par une suite nécessaire de cet Être suprême qui existait de lui-même1 ? Si je n'en croyais que mon faible entendement, je dirais, Il me paraît que la nature a toujours été animée2. Je ne puis concevoir que la cause qui agit continuellement et visiblement sur elle, pouvant agir dans tous les temps, n'ait pas agi toujours. Une éternité d'oisiveté dans l'être agissant et nécessaire, me semble incompatible. Je suis porté à croire que le monde a toujours émané de cette cause primitive et nécessaire, comme la lumière émane du soleil. Par quel enchaînement d'idées me vois-je toujours entraîné à croire éternelles les œuvres de l'Être éternel ? Ma conception, toute pusillanime qu'elle est, a la force d'atteindre à l'Être nécessaire existant par lui-même, et n'a pas la force de concevoir le néant. L'existence d'un seul atome, me prouve l'éternité de l'existence ; mais rien ne me prouve le néant. Quoi ! il y aurait eu le _rien_ dans l'espace où est aujourd'hui quelque chose ? Cela paraît absurde et contradictoire. Je ne puis admettre ce _rien,_ à moins que la révélation ne vienne fixer mes idées qui s'emportent au-delà des temps3.
Je sais bien qu'une succession infinie d'êtres qui n'auraient point d'origine, est aussi absurde ; Samuel Clarke le démontre assez4 ; mais il n'entreprend pas seulement d'affirmer que Dieu n'ait pas tenu cette chaîne de toute éternité ; il n'ose pas dire qu'il ait été si longtemps impossible à l'Être éternellement actif de déployer son action. Il est évident qu'il l'a pu ; et s'il l'a pu, qui sera assez hardi pour me dire qu'il ne l'a pas fait ? La révélation seule, encore une fois, peut m'apprendre le contraire. Mais nous n'en sommes pas encore à cette révélation qui écrase toute philosophie, à cette lumière devant qui toute lumière s'évanouit5.
1\- Voltaire pose ici la question de la préexistence des germes : leur développement est-il immanent à la nature ou est-elle prévue de toute éternité par Dieu ? S'il ne tranche pas cette question, en revanche il se déclare clairement, comme il le faisait déjà dans le Doute XIV, partisan de la thèse selon laquelle le vivant naît du vivant, il s'inscrit donc contre la thèse adverse de la génération spontanée soutenue par Needham, puis Diderot (qui changera d'avis à la toute fin de sa vie), puis au XIXe siècle par Lamarck. C'est Pasteur qui mettra un terme au débat en montrant que le vivant ne peut naître que d'un germe.
2\- Voltaire, de la question de la préexistence des germes, passe à celle de l'origine de la vie qu'il conçoit comme éternelle. Comme il le notait à la fin du Doute IV, le fait que ces questions soient indécidables (voir le début du Doute XIX) ne l'empêche pas de les soulever.
3\- La révélation est le seul rempart contre la propension de Voltaire à croire que tout est éternel et même coéternel à Dieu. Autrement dit, selon le déisme de Voltaire, il n'y a pas de création _ex nihilo_ ni un temps d'avant la création où Dieu ne ferait rien, mais le Dieu voltairien est plutôt à rapprocher de l'âme du monde des stoïciens coéternelle au monde (voir ci-dessus le Doute XVII et ci-dessous le Doute XXI).
4\- Voir Clarke, _Traité de l'existence et des attributs de Dieu_ (publié en 1705-1706 et trad. fr. en 1727-1728).
5\- Voltaire ici attaque frontalement la révélation : elle écrase toute philosophie, et sa lumière produit de l'obscurité en faisant passer pour trompeuses les lueurs de la raison (voir la fin du Doute XIV).
XXI. Ma dépendance encore
Cet Être éternel, cette cause universelle, me donne mes idées ; car ce ne sont pas les objets qui me les donnent. Une matière brute ne peut envoyer des pensées dans ma tête ; mes pensées ne viennent pas de moi, car elles arrivent malgré moi, et souvent s'enfuient de même. On sait assez qu'il n'y a nulle ressemblance, nul rapport entre les objets et nos idées et nos sensations1. Certes il y avait quelque chose de sublime dans ce Malebranche, qui osait prétendre que nous voyons tout dans Dieu même2. Mais n'y avait-il rien de sublime dans les Stoïciens, qui pensaient que c'est Dieu qui agit en nous, et que nous possédons un rayon de sa substance3 ? Entre le rêve de Malebranche et le rêve des Stoïciens, où est la réalité ? Je retombe (nomb. 24) dans l'ignorance, qui est l'apanage de ma nature, et j'adore le Dieu par qui je pense, sans savoir comment je pense.
1\- Voir aussi Doute X, 2e alinéa. Depuis Descartes, l'hétérogénéité entre la cause (l'objet extérieur) et l'effet (la sensation de l'objet) n'a cessé de poser problème. Pour Malebranche, la volonté de l'âme d'agir sur le corps est l'occasion pour Dieu d'exercer sa puissance, c'est-à-dire de produire le mouvement du corps. De même, le mouvement du corps est l'occasion pour Dieu d'exercer sa puissance, c'est-à-dire de produire l'affection correspondante au mouvement dans l'âme. C'est Dieu la cause réelle de l'interaction de l'esprit et du corps, et de toute action en général.
2\- Malebranche, _De la recherche de la vérité_ , livre III, chap. VI « Que nous voyons toutes choses en Dieu ».
3\- En effet, pour les stoïciens, l'homme, parce qu'il est une parcelle de la raison divine peut comprendre, par sa raison, la nécessité des choses et se disposer favorablement à son égard ; par son âme, il participe à l'âme du monde (voir _Les Stoïciens_ , choix de textes établi par J. Brun, Paris, PUF, 1966).
4\- C'est-à-dire Doute II.
XXII. Nouveau Doute
Convaincu par mon peu de raison qu'il y a un Être nécessaire, éternel, intelligent, de qui je reçois mes idées, sans pouvoir deviner ni le comment, ni le pourquoi, je demande ce que c'est que cet Être ? s'il a la forme des espèces intelligentes et agissantes supérieures à la mienne dans d'autres globes ? J'ai déjà dit que je n'en savais rien (nomb. 11). Néanmoins je ne puis affirmer que cela soit impossible ; car j'aperçois des planètes très supérieures à la mienne en étendue, entourées de plus de satellites que la terre2. Il n'est point du tout contre la vraisemblance qu'elles soient peuplées d'intelligences très supérieures à moi, et de corps plus robustes, plus agiles et plus durables3. Mais leur existence n'ayant nul rapport à la mienne, je laisse aux poètes de l'Antiquité le soin de faire descendre Vénus de son prétendu troisième ciel, et Mars du cinquième4 ; je ne dois rechercher que l'action de l'Être nécessaire sur moi-même.
1\- C'est-à-dire Premier Doute dans lequel Voltaire a soulevé la question des habitants des autres mondes.
2\- Comme, par exemple, Jupiter dont Galilée a observé les quatre satellites.
3\- Dans le Premier Doute, Voltaire avait déjà développé cette hypothèse d'autres mondes habités.
4\- L'auteur de cette conception n'est pas un poète de l'Antiquité mais le poète florentin Dante (1265-1321). Dans le Paradis de _La Divine Comédie_ , aux Chants VIII et IX, il décrit le troisième ciel de Vénus et dans les Chants XIV à XVIII le cinquième ciel de Mars.
XXIII. Un seul artisan suprême
Une grande partie des hommes voyant le mal physique et le mal moral répandus sur ce globe, imagina deux êtres puissants, dont l'un produisait tout le bien, et l'autre tout le mal1. S'ils existaient, ils étaient nécessaires ; ils existaient donc nécessairement dans le même lieu ; car il n'y a point de raison pourquoi ce qui existe par sa propre nature serait exclu d'un lieu ; ils se pénétreraient donc l'un l'autre, cela est absurde. L'idée de ces deux puissances ennemies ne peut tirer son origine que des exemples qui nous frappent sur la terre ; nous y voyons des hommes doux et des hommes féroces, des animaux utiles et des animaux nuisibles, de bons maîtres et des tyrans. On imagina ainsi deux pouvoirs contraires qui présidaient à la nature ; ce n'est qu'un roman asiatique2. Il y a dans toute la nature une unité de dessein manifeste ; les lois du mouvement et de la pesanteur sont invariables ; il est impossible que deux artisans suprêmes, entièrement contraires l'un à l'autre, aient suivi les mêmes lois. Cela seul, à mon avis, renverse le système manichéen3, et l'on n'a pas besoin de gros volumes pour le combattre.
Il est donc une puissance unique, éternelle, à qui tout est lié, de qui tout dépend, mais dont la nature m'est incompréhensible4. St Thomas5 nous dit, _que Dieu est un pur acte, une forme, qui n'a ni genre, ni prédicat, qu'il est la nature et le suppôt, qu'il existe essentiellement, participativement, et noncupativement_6. Lorsque les dominicains furent les maîtres de l'Inquisition7, ils auraient fait brûler un homme qui aurait nié ces belles choses ; je ne les aurais pas niées, mais je ne les aurais pas entendues8.
On me dit que Dieu est simple ; j'avoue humblement que je n'entends pas la valeur de ce mot davantage. Il est vrai que je ne lui attribuerai pas des parties grossières que je puisse séparer ; mais je ne puis concevoir que le principe et le maître de tout ce qui est dans l'étendue, ne soit pas dans l'étendue9. La simplicité, rigoureusement parlant, me paraît trop semblable au non-être. L'extrême faiblesse de mon intelligence n'a point d'instrument assez fin pour saisir cette simplicité. Le point mathématique est simple, me dira-t-on ; mais le point mathématique n'existe pas réellement.
On dit encore qu'une idée est simple, mais je n'entends pas cela davantage. Je vois un cheval, j'en ai l'idée, mais je n'ai vu en lui qu'un assemblage de choses10. Je vois une couleur, j'ai l'idée de couleur ; mais cette couleur est étendue. Je prononce les noms abstraits de couleur en général, de vice, de vertu, de vérité en général ; mais c'est que j'ai eu connaissance de choses colorées, de choses qui m'ont paru vertueuses ou vicieuses, vraies ou fausses. J'exprime tout cela par un mot ; mais je n'ai point de connaissance claire de la simplicité ; je ne sais pas plus ce que c'est, que je ne sais ce que c'est qu'un infini en nombres actuellement existant11.
Déjà convaincu que ne connaissant pas ce que je suis, je ne puis connaître ce qu'est mon auteur. Mon ignorance m'accable à chaque instant, et je me console en réfléchissant sans cesse qu'il n'importe pas que je sache si mon maître est ou non dans l'étendue, pourvu que je ne fasse rien contre la conscience qu'il m'a donnée. De tous les systèmes que les hommes ont inventés sur la Divinité, quel sera donc celui que j'embrasserai ? Aucun, sinon celui de l'adorer12.
1\- Que l'on pense à Empédocle et à ses deux principes de haine et d'amour, au manichéisme ou à la dualité chinoise du yin et du yang, nombreux sont en effet les penseurs qui ont soutenu une telle conception.
2\- Voltaire se réfère au yin et au yang des Chinois.
3\- Par système manichéen, Voltaire désigne toute conception d'une dualité de principes antagonistes qui régissent le monde et non, spécifiquement, le manichéisme, religion synchrétique du IIIe siècle, alliant à un fonds chrétien des éléments pris au bouddhisme et au zoroastrisme, et qui fut combattu par Augustin.
4\- Voltaire est un déiste monothéiste : même si la nature de Dieu lui est incompréhensible, Dieu, pour lui, est Un.
5\- Voltaire résume ici les questions 3 à 11 de la première partie de la _Somme théologique_ de saint Thomas d'Aquin où il présente sa conception de Dieu.
6\- « Noncupatif » : mot ancien maintenant disparu, vient du latin _nuncupare_ qui signifie déclarer solennellement. Dans le contexte, on pourrait traduire « Dieu existe noncupativement » par « le Verbe est avec Dieu » ou « le Verbe est Dieu » : ce qu'Il dit, est.
7\- Voltaire consacre le chapitre CXL de _l'Essai sur l'histoire générale, sur les mœurs et sur les nations_ à l'Inquisition dont l'importance et l'intransigeance s'accrurent sous le joug de Torquemada (1420-1498), dominicain espagnol nommé Inquisiteur général en 1483.
8\- Voltaire renvoie le vocabulaire scolastique à un langage obscur, voir aussi la fin du Doute III où il traitait de la même manière la conception thomiste de l'animation médiate.
9\- Voltaire affirmait dans le Doute XX que Dieu est comme l'âme du monde des stoïciens coéternel au monde, il affirme ici que Dieu est dans l'étendue, ce qui est traditionnellement tenu pour une hérésie.
10\- Reprenant la critique par Locke de l'idée générale d'une substance particulière comme assemblage de qualités sensibles, Voltaire lui emprunte l'exemple de l'idée du cheval, voir l' _Essai philosophique concernant l'entendement humain_ , livre II, chap. XXIII, § 6.
11\- Voltaire suit toujours Locke sur ce point : l'idée générale ou abstraite résulte d'un abus de mots (voir _ibid._ , livre III, chap. X). Comme dans le Doute XVIII, il répète que l'infini en nombres est incompréhensible pour l'esprit humain.
12\- Voltaire est ici proche de Pascal : « Qu'il y a loin de la connaissance de Dieu à l'aimer » ( _Pensées_ , fr. 377).
XXIV. Spinoza
Après m'être plongé avec Thalès dans l'eau, dont il faisait son premier principe, après m'être roussi auprès du feu d'Empédocle, après avoir couru dans le vide en ligne droite avec les atomes d'Épicure, supputé des nombres avec Pythagore, et avoir entendu sa musique ; après avoir rendu mes devoirs aux androgynes de Platon1, et ayant passé par toutes les régions de la métaphysique et de la folie ; j'ai voulu enfin connaître le système de Spinoza2.
Il n'est pas nouveau ; il est imité de quelques anciens philosophes grecs, et même de quelques Juifs ; mais Spinoza a fait ce qu'aucun philosophe grec, encore moins aucun Juif, n'a fait. Il a employé une méthode géométrique imposante3, pour se rendre un compte net de ses idées : voyons s'il ne s'est pas égaré méthodiquement, avec le fil qui le conduit ?
Il établit d'abord une vérité incontestable et lumineuse. Il y a quelque chose, donc il existe éternellement un Être nécessaire. Ce principe est si vrai, que le profond Samuel Clarke s'en est servi pour prouver l'existence de Dieu4.
Cet Être doit se trouver partout où est l'existence, car qui le bornerait ?
Cet Être nécessaire est donc tout ce qui existe ; il n'y a donc réellement qu'une seule substance dans l'univers.
Cette substance n'en peut créer une autre ; car puisqu'elle remplit tout, où mettre une substance nouvelle, et comment créer quelque chose du néant ? Comment créer l'étendue sans la placer dans l'étendue même, laquelle existe nécessairement ?
Il y a dans le monde la pensée et la matière ; la substance nécessaire que nous appelons Dieu, est donc la pensée et la matière. Toute pensée et toute matière est donc comprise dans l'immensité de Dieu : il ne peut y avoir rien hors de lui ; il ne peut agir que dans lui ; il comprend tout, il est tout.
Ainsi tout ce que nous appelons substances différentes n'est en effet que l'universalité des différents attributs de l'Être suprême, qui pense dans le cerveau des hommes, éclaire dans la lumière, se meut sur les vents, éclate dans le tonnerre, parcourt l'espace dans tous les astres, et vit dans toute la nature.
Il n'est point comme un vil roi de la terre confiné dans son palais, séparé de ses sujets ; il est intimement uni à eux ; ils sont des parties nécessaires de lui-même ; s'il en était distingué, il ne serait plus l'Être nécessaire, il ne serait plus universel, il ne remplirait point tous les lieux, il serait un être à part comme un autre.
Quoique toutes les modalités changeantes dans l'univers soient l'effet de ses attributs, cependant, selon Spinoza, il n'a point de parties ; car, dit-il, l'infini n'en a point de proprement dites ; s'il en avait, on pourrait en ajouter d'autres, et alors il ne serait plus infini. Enfin Spinoza prononce qu'il faut aimer ce Dieu nécessaire, infini, éternel5 ; et voici ses propres paroles, _page 45 de l'édition de 1731_6.
« À l'égard de l'amour de Dieu, loin que cette idée le puisse affaiblir, j'estime qu'aucune autre n'est plus propre à l'augmenter ; puisqu'elle me fait connaître que Dieu est intime à mon être, qu'il me donne l'existence et toutes mes propriétés, mais qu'il me les donne libéralement, sans reproche, sans intérêt, sans m'assujettir à autre chose qu'à ma propre nature. Elle bannit la crainte, l'inquiétude, la défiance, et tous les défauts d'un amour vulgaire ou intéressé. Elle me fait sentir que c'est un bien que je ne puis perdre, et que je possède d'autant mieux que je le connais et que je l'aime. »
Ces idées séduisirent beaucoup de lecteurs ; il y en eut même qui, ayant d'abord écrit contre lui, se rangèrent à son opinion.
On reprocha au savant Bayle d'avoir attaqué durement Spinoza sans l'entendre7. Durement, j'en conviens ; injustement, je ne le crois pas. Il serait étrange que Bayle ne l'eût pas entendu. Il découvrit aisément l'endroit faible de ce château enchanté ; il vit qu'en effet Spinosa compose son Dieu de parties, quoiqu'il soit réduit à s'en dédire, effrayé de son propre système. Bayle vit combien il est insensé de faire Dieu astre et citrouille, pensée et fumier, battant et battu. Il vit que cette fable est fort au-dessous de celle de Protée. Peut-être Bayle devait-il s'en tenir au mot de _modalités,_ et non pas de _parties_ , puisque c'est ce mot de modalités que Spinosa emploie toujours. Mais il est également impertinent, si je ne me trompe, que l'excrément d'un animal soit une modalité ou une partie de l'Être suprême8.
Il ne combattit point, il est vrai, les raisons par lesquelles Spinoza soutient l'impossibilité de la création ; mais c'est que la création proprement dite est un objet de foi, et non pas de philosophie ; c'est que cette opinion n'est nullement particulière à Spinoza, c'est que toute l'Antiquité avait pensé comme lui. Il n'attaque que l'idée absurde d'un Dieu simple, composé de parties, d'un Dieu qui se mange et qui se digère lui-même, qui aime et qui hait la même chose en même temps, etc. Spinoza se sert toujours du mot Dieu, Bayle le prend par ses propres paroles9.
Mais au fond, Spinoza ne reconnaît point de Dieu10 ; il n'a probablement employé cette expression, il n'a dit qu'il faut servir et aimer Dieu que pour ne point effaroucher le genre humain. Il paraît athée dans toute la force de ce terme ; il n'est point athée comme Épicure, qui reconnaissait des dieux inutiles et oisifs ; il ne l'est point comme la plupart des Grecs et des Romains, qui se moquaient des dieux du vulgaire ; il l'est parce qu'il ne reconnaît nulle Providence, parce qu'il n'admet que l'éternité, l'immensité, et la nécessité des choses ; il l'est comme Straton11, comme Diagoras12 ; il ne doute pas comme Pyrrhon13, il affirme14 ; et qu'affirme-t-il ? qu'il n'y a qu'une seule substance, qu'il ne peut y en avoir deux, que cette substance est étendue et pensante, et c'est ce que n'ont jamais dit les philosophes grecs et asiatiques15 qui ont admis une âme universelle.
Il ne parle en aucun endroit de son livre des desseins marqués qui se manifestent dans tous les êtres. Il n'examine point si les yeux sont faits pour voir, les oreilles pour entendre, les pieds pour marcher, les ailes pour voler ; il ne considère ni les lois du mouvement dans les animaux et dans les plantes, ni leur structure adaptée à ces lois, ni la profonde mathématique qui gouverne le cours des astres : il craint d'apercevoir que tout ce qui existe atteste une Providence divine16 ; il ne remonte point des effets à leur cause, mais, se mettant tout d'un coup à la tête de l'origine des choses, il bâtit son roman comme Descartes a construit le sien, sur une supposition17. Il supposait le plein avec Descartes, quoiqu'il soit démontré en rigueur que tout mouvement est impossible dans le plein18. C'est là principalement ce qui lui fit regarder l'univers comme une seule substance. Il a été la dupe de son esprit géométrique. Comment Spinoza ne pouvant douter que l'intelligence et la matière existent, n'a-t-il pas examiné au moins si la Providence n'a pas tout arrangé ? comment n'a-t-il pas jeté un coup d'œil sur ces ressorts, sur ces moyens dont chacun a son but, et recherché s'ils prouvent un artisan suprême ? Il fallait qu'il fût ou un physicien bien ignorant, ou un sophiste gonflé d'un orgueil bien stupide, pour ne pas reconnaître une Providence toutes les fois qu'il respirait et qu'il sentait son cœur battre ; car cette respiration et ce mouvement du cœur sont des effets d'une machine si industrieusement compliquée, arrangée avec un art si puissant, dépendante de tant de ressorts, concourant tous au même but, qu'il est impossible de l'imiter, et impossible à un homme de bon sens de ne la pas admirer19.
Les spinosistes modernes20 répondent : Ne vous effarouchez pas des conséquences que vous nous imputez ; nous trouvons comme vous une suite d'effets admirables dans les corps organisés et dans toute la nature. La cause éternelle est dans l'intelligence éternelle que nous admettons, et qui avec la matière constitue l'universalité des choses qui est Dieu. Il n'y a qu'une seule substance qui agit par la même modalité de sa pensée sur sa modalité de la matière, et qui constitue ainsi l'univers qui ne fait qu'un tout inséparable.
On réplique à cette réponse, Comment pouvez-vous nous prouver que la pensée qui fait mouvoir les astres, qui anime l'homme, qui fait tout, soit une modalité, et que les déjections d'un crapaud et d'un ver soient une autre modalité de ce même Être souverain ? Oseriez-vous dire qu'un si étrange principe vous est démontré ? Ne couvrez-vous pas votre ignorance par des mots que vous n'entendez point ? Bayle a très bien démêlé les sophismes de votre maître dans les détours et dans les obscurités du style prétendu géométrique, et réellement très confus, de ce maître. Je vous renvoie à lui ; des philosophes ne doivent pas récuser Bayle21.
Quoi qu'il en soit, je remarquerai de Spinoza qu'il se trompait de très bonne foi. Il me semble qu'il n'écartait de son système les idées qui pouvaient lui nuire, que parce qu'il était trop plein des siennes ; il suivait sa route sans regarder rien de ce qui pouvait la traverser, et c'est ce qui nous arrive trop souvent. Il y a plus, il renversait tous les principes de la morale, en étant lui-même d'une vertu rigide ; sobre, jusqu'à ne boire qu'une pinte de vin en un mois ; désintéressé jusqu'à remettre aux héritiers de l'infortuné Jean de Wit22 une pension de deux cents florins que lui faisait ce grand homme ; généreux, jusqu'à donner son bien ; toujours patient dans ses maux et dans sa pauvreté, toujours uniforme dans sa conduite23.
Bayle, qui l'a si maltraité, avait à peu près le même caractère. L'un et l'autre ont cherché la vérité toute leur vie par des routes différentes. Spinoza fait un système spécieux en quelques points, et bien erroné dans le fond. Bayle a combattu tous les systèmes : qu'est-il arrivé des écrits de l'un et de l'autre ? Ils ont occupé l'oisiveté de quelques lecteurs ; c'est à quoi tous les écrits se réduisent ; et depuis Thalès jusqu'aux professeurs de nos universités, et jusqu'aux plus chimériques raisonneurs, et jusqu'à leurs plagiaires, aucun philosophe n'a influé seulement sur les mœurs de la rue où ils demeuraient. Pourquoi ? parce que les hommes se conduisent par la coutume et non par la métaphysique24.
1\- L'histoire de la philosophie ancienne que présente Voltaire n'est pas dénuée de vérité ni d'humour : si, pour Thalès, effectivement l'eau est la seule substance primordiale, Thalès est aussi célèbre pour être tombé dans un puits alors qu'il observait le ciel, ce que semble évoquer Voltaire en disant qu'il y plonge avec Thalès. De même quand il dit qu'il roussit auprès du feu d'Empédocle, ou qu'il court en ligne droite avec les atomes d'Épicure, ou qu'il suppute les nombres de Pythagore et écoute sa musique, il fait référence au principe fondamental du feu chez Empédocle ou à l'atomisme d'Épicure et à sa théorie du _clinamen_ et de la déclinaison des atomes, ou encore à la théorie pythagoricienne des nombres et à l'harmonie des sphères, en même temps qu'il les tourne en dérision. Quant à Platon, il n'hésite pas à le réduire au mythe de l'androgyne (voir _Le Banquet_ , 189a-193d) dont il se moque de manière grivoise.
2\- Voltaire passe sans transition de la philosophie ancienne qu'il vient de résumer en quelques lignes à Spinoza (1632-1677) que l'on considère au XVIIIe siècle comme le philosophe de tous les dangers non sans déformer radicalement sa pensée (athéisme, impiété, irréligion, matérialisme).
3\- Effectivement Spinoza rédige l' _Éthique_ , selon l'ordre géométrique, ce qui lui a valu de nombreuses critiques au XVIIIe siècle : on l'accuse d'être un représentant de l'esprit de système (voir, par exemple, Condillac, _Traité des systèmes_ , 1749).
4\- Voltaire fait toujours référence au _Traité de l'existence et des attributs de Dieu_ de Clarke dont la première partie combat avec force Hobbes et Spinoza dont il reprend cependant l'idée de Dieu comme Être nécessaire.
5\- Résumé quelque peu rapide et déformé de l' _Éthique_ de Spinoza qui n'est pas lu de première main au XVIIIe siècle mais à travers ses détracteurs ou ses prétendus sectateurs qui le tirent vers l'athéisme et le matérialisme (voir, sur ce point, J. Israël, _Les Lumières radicales_ , Paris, Amsterdam, 2001).
6\- Voltaire pioche ces propos prétendument authentiques dans la _Réfutation des erreurs de Benoît de Spinoza, par M. de Fénelon, archevêque de Cambray, par le P. Lami bénédictin et par M. le comte de Boullainvilliers, avec la vie de Spinosa_ (1731 ; _Bibliothèque de Voltaire_ , n° 1326). Paul Vernière, dans _Spinoza et la pensée française avant la Révolution_ , Paris, PUF, 1954, p. 515, remarque à ce sujet que Voltaire, en croyant lire Spinoza, lit dans cette _Réfutation_ de 1731 la paraphrase banale et incomplète du comte de Boulainvilliers éditée à Bruxelles par l'abbé Lenglet-Dufresnoy : il y voit les « propres paroles » de Spinoza, qu'il récite dans l'article _Dieu, dieux_ des _Questions sur l'Encyclopédie_ , en 1770.
7\- Bayle consacre le plus long article de son _Dictionnaire historique et critique_ (Rotterdam, 3 vol., 1702) à Spinoza et au spinozisme. Il a joué un rôle essentiel dans la transmission de la pensée de Spinoza qu'il a placée au centre de tous les débats intellectuels. On soupçonna même Bayle d'avoir délibérément raté sa prétendue critique du système de Spinoza : il aurait mal interprété à dessein le concept spinoziste de Dieu et de substance afin de propager le déisme et l'athéisme (voir, sur ce point, le chapitre XVIII des _Lumières radicales_ de J. Israël).
8\- Voltaire n'hésite pas à verser dans la farce rabelaisienne pour faire rire d'un auteur.
9\- Voltaire défend Bayle qui, pourtant, n'est guère défendable, et qui reconnaît lui-même n'avoir pas entendu les sentiments de Spinoza (voir _Lettres_ , 3 vol., Rotterdam, 1729, III, p. 939 et 906-911).
10\- C'est le point d'achoppement entre Voltaire et Spinoza : il considère Spinoza, dans le sillage de Bayle, comme un athée. Or Voltaire restera déiste toute sa vie. Cependant, à l'instar de Bayle dans son _Dictionnaire historique et critique_ , Voltaire marque l'importance de la pensée de Spinoza en lui consacrant son plus long Doute.
11\- Straton, disciple d'Aristote, IVe-IIIe siècle av. J.-C, fut surnommé le Physicien : il niait les causes finales et l'immatérialité de l'âme, ce qui le fit considérer comme athée.
12\- Diagoras de Mélos, disciple de Démocrite, fut victime d'un parjure qui resta impuni, il s'en prit aux dieux, ce qui le fit appeler Diagoras l'Athée. Il fut chassé d'Athènes vers 415 av. J.-C., pour avoir tourné en ridicule les mystères d'Éleusis.
13\- Pyrrhon, philosophe grec sceptique (365-275 av. J.-C.), disciple d'Anaxarque, est le fondateur du scepticisme : il nie la possibilité pour l'homme d'atteindre la vérité et préconise le doute.
14\- Autre point d'achoppement avec Voltaire : Spinoza est, à ses yeux, un penseur dogmatique.
15\- Les stoïciens et les bouddhistes.
16\- Au contraire, Spinoza développe une critique radicale du principe des causes finales dans l'Appendice de la première partie de l' _Éthique_ , ce qui évidemment n'est pas pour plaire à Voltaire dont le déisme repose sur l'argument du dessein : l'horloge prouve l'horloger.
17\- Trois paragraphes plus haut, Voltaire parlait de château enchanté : Spinoza, à l'instar de Descartes, pèche par esprit de système, c'est un faiseur de roman de la nature.
18\- Voltaire s'appuie sur la critique de Newton pour démanteler le monde plein de Descartes : voir les _Éléments de la philosophie de Newton_ , III, chap. 2.
19\- Voltaire n'a pas compris la critique radicale que Spinoza fait du recours aux causes finales et à la volonté de Dieu, « cet asile de l'ignorance », voir _Éthique_ , trad. C. Appuhn, I, p. 65 du tome 3 des _Œuvres_ en 4 tomes, Paris, GF-Flammarion, 1965.
20\- Diderot consacre un très long article à l'entrée SPINOZA dans l' _Encyclopédie_ (XV, 1765, 463-474). En revanche, l'article SPINOZISTE, très court (XV, 1765, 474), précise la différence entre spinozistes anciens et modernes identifiés aux matérialistes qui soutiennent l'épigénèse : « SPINOZISTE, s. m. ( _Gram_.) sectateur de la philosophie de Spinoza. Il ne faut pas confondre les _Spinosistes_ anciens avec les _Spinosistes_ modernes. Le principe général de ceux-ci, c'est que la matiere est sensible, ce qu'ils démontrent par le développement de l'œuf, corps inerte, qui par le seul instrument de la chaleur graduée passe à l'état d'être sentant et vivant, et par l'accroissement de tout animal qui dans son principe n'est qu'un point, et qui par l'assimilation nutritive des plantes, en un mot, de toutes les substances qui servent à la nutrition, devient un grand corps sentant et vivant dans un grand espace. De-là ils concluent qu'il n'y a que de la matiere, & qu'elle suffit pour tout expliquer ; du reste ils suivent l'ancien spinosisme dans toutes ses conséquences. »
21\- Nombreuses furent pourtant les critiques de Bayle lecteur de Spinoza, voir le chapitre XVIII des _Lumières radicales_ de J. Israël.
22\- Jean de Witt (1625-1672), homme politique hollandais, grand pensionnaire en 1653, conclut la paix avec Cromwell. Mais quand Louis XIV envahit la Hollande en 1672, il fut tué lors d'une émeute orangiste.
23\- Bayle, qui a interprété Spinoza dans le sens de l'athéisme, est aussi celui qui a diffusé l'image du philosophe austère et vertueux, ce qui confortait son idée qu'il peut y avoir une société d'athées vertueux (voir les _Pensées diverses sur la comète_ , § CLXXII à CLXXVI.
24\- Conception profondément pascalienne : « la coutume est une seconde nature qui détruit la première » ( _Pensées_ , fr. 126, voir aussi fr. 60 et 125).
XXV. Absurdités
Voilà bien des voyages dans des terres inconnues ; ce n'est rien encore. Je me trouve comme un homme qui ayant erré sur l'Océan, et apercevant les îles Maldives1 dont la mer Indienne est semée, veut les visiter toutes. Mon grand voyage ne m'a rien valu ; voyons si je ferai quelque gain dans l'observation de ces petites îles, qui ne semblent servir qu'à embarrasser la route.
Il y a une centaine de cours de philosophie où l'on m'explique des choses dont personne ne peut avoir la moindre notion. Celui-ci veut me faire comprendre la Trinité par la physique2 ; il me dit qu'elle ressemble aux trois dimensions de la matière. Je le laisse dire, et je passe vite. Celui-là prétend me faire toucher au doigt la transsubstantiation, en me montrant, par les lois du mouvement, comme un accident peut exister sans sujet, et comment un même corps peut être en deux endroits à la fois3. Je me bouche les oreilles, et je passe plus vite encore.
Pascal, Blaise Pascal lui-même, l'auteur des _Lettres provinciales_ , profère ces paroles ; _Croyez-vous qu'il soit impossible que Dieu soit infini et sans parties ? Je veux donc vous faire voir une chose indivisible et infinie ; c'est un point, se mouvant partout d'une vitesse infinie, car il est en tous lieux tout entier dans chaque endroit_4 _._
Un point mathématique qui se meut ! juste Ciel ! un point qui n'existe que dans la tête du géomètre, qui est partout et en même temps, et qui a une vitesse infinie, comme si la vitesse infinie actuelle pouvait exister ! Chaque mot est une folie, et c'est un grand homme qui a dit ces folies !
Votre âme est simple, incorporelle, intangible, me dit cet autre5 ; et comme aucun corps ne peut la toucher, je vais vous prouver par la physique d'Albert le Grand6, qu'elle sera brûlée physiquement, si vous n'êtes pas de mon avis ; et voici comme je vous le prouve _a priori_ , en fortifiant Albert par les syllogismes d'Abeli7. Je lui réponds que je n'entends pas son _priori_ ; que je trouve son compliment très dur ; que la révélation, dont il ne s'agit pas entre nous, peut seule m'apprendre une chose si incompréhensible ; que je lui permets de n'être pas de mon avis, sans lui faire aucune menace ; et je m'éloigne de lui, de peur qu'il ne me joue un mauvais tour ; car cet homme me paraît bien méchant.
Une foule de sophistes de tout pays et de toutes sectes m'accable d'arguments inintelligibles sur la nature des choses, sur la mienne, sur mon état passé, présent et futur. Si on leur parle de manger et de boire, de vêtements, de logement, des denrées nécessaires, de l'argent avec lequel on se les procure, tous s'entendent à merveille ; s'il y a quelques pistoles à gagner, chacun d'eux s'empresse, personne ne se trompe d'un denier ; et quand il s'agit de tout notre être, ils n'ont pas une idée nette. Le sens commun les abandonne ; de là je reviens à ma première conclusion (nombre 48) que ce qui ne peut être d'un usage universel, ce qui n'est pas à la portée du commun des hommes, ce qui n'est pas entendu par ceux qui ont le plus exercé leur faculté de penser, n'est pas nécessaire au genre humain.
1\- Les îles Maldives forment un archipel tout en longueur au sud-ouest de l'Inde.
2\- Kepler, dans la préface de son _Mysterium cosmographicum_ (1597), explique qu'il part du mystère de la Trinité pour montrer qu'il préside à l'organisation du cosmos.
3\- Voir les _Entretiens sur la philosophie_ (1671) de Rohault, qui y défend l'orthodoxie cartésienne de la transsubstantiation.
4\- Pascal, dans le fragment 420 des _Pensées_ , dit exactement : « Croyez-vous qu'il soit impossible que Dieu soit infini, sans partie ? Oui. Je veux donc vous faire voir une chose infinie et indivisible : c'est un point se mouvant partout d'une vitesse infinie. Car il est un en tous lieux et est tout entier en chaque endroit », in _Œuvres complètes_ , Seuil, p. 551.
5\- Voltaire pense sans doute à un dominicain quand on sait le rôle actif que cet ordre a joué dans l'Inquisition à laquelle il fait allusion dans la suite de la phrase.
6\- Albert le Grand (1193-1280), théologien et philosophe allemand, dominicain de formation, fut maître de théologie à l'université de Paris où il eut Thomas d'Aquin pour étudiant, puis il enseigna à Cologne, il chercha à faire la synthèse d'éléments du néo-platonisme et de l'aristotélisme.
7\- Abeli Louis (1604-1691), auteur de la _Moelle théologique_ qui lui valut d'être appelé moelleux par Boileau dans le _Lutrin_ , chant IV (voir article VEXIN – françois in _Encyclopédie,_ XVII, 1765, 225).
8\- C'est-à-dire Doute IV, voir aussi le Doute XIX.
XXVI. Du meilleur des mondes1
1\- Voltaire a repris ce titre pour la huitième partie des _Questions sur l'Encyclopédie_ , 1772. C'est un thème important pour lui (voir notre introduction), qu'il aborde en critiquant l'optimisme de Leibniz, comme dans _Candide ou l'optimisme_ , 1759.
En courant de tous côtés pour m'instruire, je rencontrai des disciples de Platon. Venez avec nous, me dit l'un d'eux1 ; vous êtes dans le meilleur des mondes ; nous avons bien surpassé notre maître. Il n'y avait de son temps que cinq mondes possibles, parce qu'il n'y a que cinq corps réguliers2 ; mais actuellement qu'il y a une infinité d'univers possibles, Dieu a choisi le meilleur ; venez, et vous vous en trouverez bien. Je lui répondis humblement : Les mondes que Dieu pouvait créer étaient ou meilleurs, ou parfaitement égaux, ou pires. Il ne pouvait prendre le pire. Ceux qui étaient égaux, supposé qu'il y en eût, ne valaient pas la préférence ; ils étaient entièrement les mêmes : on n'a pu choisir entre eux : prendre l'un, c'est prendre l'autre. Il est donc impossible qu'il ne prît pas le meilleur. Mais comment les autres étaient-ils possibles, quand il était impossible qu'ils existassent ?
Il me fit de très belles distinctions, assurant toujours sans s'entendre, que ce monde-ci est le meilleur de tous les mondes réellement impossibles. Mais me sentant alors tourmenté de la pierre3, et souffrant des douleurs insupportables, les citoyens du meilleur des mondes me conduisirent à l'hôpital voisin. Chemin faisant, deux de ces bienheureux habitants furent enlevés par des créatures leurs semblables : on les chargea de fers, l'un pour quelques dettes, l'autre sur un simple soupçon. Je ne sais pas si je fus conduit dans le meilleur des hôpitaux possibles ; mais je fus entassé avec deux ou trois mille misérables qui souffraient comme moi. Il y avait là plusieurs défenseurs de la patrie, qui m'apprirent qu'ils avaient été trépanés et disséqués vivants, qu'on leur avait coupé des bras, des jambes, et que plusieurs milliers de leurs généreux compatriotes avaient été massacrés dans l'une des trente batailles données dans la dernière guerre, qui est environ la cent millième guerre depuis que nous connaissons des guerres. On voyait aussi dans cette maison environ mille personnes des deux sexes qui ressemblaient à des spectres hideux, et qu'on frottait d'un certain métal, parce qu'ils avaient suivi la loi de la nature, et parce que la nature avait je ne sais comment pris la précaution d'empoisonner en eux la source de la vie4. Je remerciai mes deux conducteurs.
Quand on m'eut plongé un fer bien tranchant dans la vessie, et qu'on eut tiré quelques pierres de cette carrière ; quand je fus guéri, et qu'il ne me resta plus que quelques incommodités douloureuses pour le reste de mes jours, je fis mes représentations à mes guides ; je pris la liberté de leur dire qu'il y avait du bon dans ce monde, puisqu'on m'avait tiré quatre cailloux du sein de mes entrailles déchirées ; mais que j'aurais encore mieux aimé que les vessies eussent été des lanternes, que non pas qu'elles fussent des carrières. Je leur parlai des calamités et des crimes innombrables qui couvrent cet excellent monde. Le plus intrépide d'entre eux, qui était un Allemand5, mon compatriote, m'apprit que tout cela n'est qu'une bagatelle.
Ce fut, dit-il, une grande faveur du ciel envers le genre humain, que Tarquin violât Lucrèce, et que Lucrèce se poignardât, parce qu'on chassa les tyrans, et que le viol, le suicide et la guerre établirent une république qui fit le bonheur des peuples conquis. J'eus peine à convenir de ce bonheur. Je ne conçus pas d'abord quelle était la félicité des Gaulois et des Espagnols, dont on dit que César fit périr trois millions. Les dévastations et les rapines me parurent aussi quelque chose de désagréable ; mais le défenseur de l'optimisme n'en démordit point ; il me disait toujours comme le geôlier de don Carlos6 ; _paix, paix, c'est pour votre bien_. Enfin, étant poussé à bout, il me dit qu'il ne fallait pas prendre garde à ce globule de la terre, où tout va de travers ; mais que dans l'étoile de Sirius, dans Orion, dans l'œil du Taureau, et ailleurs, tout est parfait. Allons-y donc, lui dis-je.
Un petit théologien me tira alors par le bras ; il me confia que ces gens-là étaient des rêveurs, qu'il n'était point du tout nécessaire qu'il y eût du mal sur la terre, qu'elle avait été formée exprès pour qu'il n'y eût jamais que du bien ; et pour vous le prouver, sachez que les choses se passèrent ainsi autrefois pendant dix ou douze jours. Hélas ! lui répondis-je, c'est bien dommage, mon révérend père, que cela n'ait pas continué.
1\- Il s'agit de Leibniz pour qui Dieu choisit le meilleur des mondes possibles parmi une infinité. Malebranche défend une métaphysique proche de celle de Leibniz mais, pour lui, il n'est pas exclu qu'il y ait plusieurs mondes qui aient le plus grand degré de perfection (voir J. Elster, _Leibniz et la formation de l'esprit capitaliste_ , Paris, Aubier, 1975, p. 187).
2\- Dans le _Timée_ de Platon, en effet, le démiurge façonne le monde à partir des cinq corps réguliers.
3\- La « pierre » était le nom de ce qu'on appelle aujourd'hui un « calcul » formé par une concrétion solide de sels minéraux ou de matières organiques dans un organe, un conduit ou une glande.
4\- On utilisait le mercure pour soigner les maladies vénériennes, notamment la vérole (la syphilis). Voir aussi la fin de l'article _Amour_ qui joint le thème des maladies vénériennes à la question du meilleur des mondes dans le _Dictionnaire philosophique_.
5\- Cette précision confirme que Voltaire se moque bien de Leibniz. Le ton est ici le même que celui de _Candide ou l'optimisme_ , titre sous lequel Voltaire ajoute : traduit de l'allemand.
6\- Don Carlos, fils de Philippe II, dont la mort en prison fut racontée à la Renaissance par Brantôme puis par Saint-Réal (1672), qui inspirèrent d'autres œuvres littéraires comme celle de Schiller.
XXVII. Des monades, etc.
Le même Allemand1 se ressaisit alors de moi ; il m'endoctrina, m'apprit clairement ce que c'est que mon âme. Tout est composé de monades dans la nature ; votre âme est une monade ; et comme elle a des rapports avec toutes les autres monades du monde, elle a nécessairement des idées de tout ce qui s'y passe ; ces idées sont confuses, ce qui est très utile ; et votre monade, ainsi que la mienne, est un miroir concentré de cet univers2.
Mais ne croyez pas que vous agissiez en conséquence de vos pensées. Il y a une harmonie préétablie entre la monade de votre âme et toutes les monades de votre corps, de façon que quand votre âme a une idée, votre corps a une action, sans que l'une soit la suite de l'autre. Ce sont deux pendules qui vont ensemble ; ou, si vous voulez, cela ressemble à un homme qui prêche tandis qu'un autre fait les gestes3. Vous concevez aisément qu'il faut que cela soit ainsi dans le meilleur des mondes. Car...4
1\- Leibniz a rédigé _La Monadologie_ en 1714 mais elle ne fut publiée qu'en 1840 de manière posthume d'après les manuscrits. Cependant sa conception des monades était diffusée au XVIIIe siècle, Voltaire en parle déjà pour les critiquer dans les _Éléments de la philosophie de Newton_ (1741), I, aux chapitres VI et VIII. Diderot contribue aussi à diffuser la pensée de Leibniz dans le très long article LEIBNITZIANISME de l' _Encyclopédie_ , IX, 1765, 369-379.
2\- C'est effectivement ce que dit Leibniz dans _La Monadologie_ , Paris, Delagrave, 1983, § 30.
3\- Voltaire commence par rappeler la thèse leibnizienne de l'harmonie préétablie selon laquelle l'âme n'a aucun commerce avec son corps : ce sont deux horloges que Dieu a faites, qui ont chacune un ressort, et qui vont un certain temps dans une correspondance parfaite ; voir le § 81 de _La Monadologie :_ « Ce système fait que les corps agissent comme si (par impossible) il n'y avait point d'âmes ; et que les âmes agissent, comme s'il n'y avait point de corps ; et que tous deux agissent comme si l'un influait sur l'autre. »
4\- L'interruption du paragraphe renvoie au « etc. » du titre. Voltaire marque ainsi la logorrhée que produit la thèse leibnizienne de l'harmonie préétablie et du meilleur des mondes : elle pourrait être soutenue durant des heures, sans qu'il n'y comprenne goutte, par un Dr Pangloss, professeur de « métaphysico-théologo-cosmonigologie » (voir _Candide_ , in _Romans et contes_ , Paris, GF-Flammarion, 1966, p. 180).
XXVIII. Des formes plastiques
Comme je ne comprenais rien du tout à ces admirables idées, un Anglais nommé Cudworth1 s'aperçut de mon ignorance à mes yeux fixes, à mon embarras, à ma tête baissée ; Ces idées, me dit-il, vous semblent profondes, parce qu'elles sont creuses. Je vais vous apprendre nettement comment la nature agit. Premièrement, il y a la nature en général, ensuite il y a des natures plastiques qui forment tous les animaux et toutes les plantes, vous entendez bien ? Pas un mot, monsieur. Continuons donc.
Une nature plastique n'est pas une faculté du corps, c'est une substance immatérielle qui agit sans savoir ce qu'elle fait, qui est entièrement aveugle, qui ne sent ni ne raisonne, ni ne végète ; mais la tulipe a sa forme plastique qui la fait végéter ; le chien a sa forme plastique qui le fait aller à la chasse, et l'homme a la sienne qui le fait raisonner. Ces formes sont les agents immédiats de la Divinité. Il n'y a point de ministres plus fidèles au monde, car elles donnent tout, et ne retiennent rien pour elles. Vous voyez bien que ce sont là les vrais principes des choses, et que les natures plastiques valent bien l'harmonie préétablie et les monades, qui sont les miroirs concentrés de l'univers. Je lui avouai que l'un valait bien l'autre2.
1\- Ralph Cudworth (1617-1688), philosophe anglais, membre de l'école des Platoniciens de Cambridge, publie en 1678 _Le Système intellectuel de l'univers, première partie dans laquelle le raisonnement et la philosophie de l'athéisme se trouvent réfutés et son impossibilité démontrée_. Il y développe le concept du « médiateur plastique », qui est plus ou moins une réminiscence de l'âme du monde, et est censé expliquer l'existence des lois de la nature sans tout ramener à l'opération directe de Dieu. Cet ouvrage alimente une longue controverse entre Pierre Bayle (1647-1706) et Jean Leclerc (1657-1736), le premier affirmant, et le second réfutant que le médiateur plastique est favorable à l'athéisme.
2\- Aveu ironique qui renvoie les deux théories dos à dos.
XXIX. De Locke
Après tant de courses malheureuses, fatigué, harassé, honteux d'avoir cherché tant de vérités, et d'avoir trouvé tant de chimères, je suis revenu à Locke, comme l'enfant prodigue qui retourne chez son père ; je me suis rejeté entre les bras d'un homme modeste, qui ne feint jamais de savoir ce qu'il ne sait pas, qui, à la vérité, ne possède pas des richesses immenses, mais dont les fonds sont bien assurés, et qui jouit du bien le plus solide, sans aucune ostentation. Il me confirme dans l'opinion que j'ai toujours eue, que rien n'entre dans notre entendement que par nos sens.
Qu'il n'y a point de notions innées1.
Que nous ne pouvons avoir l'idée ni d'un espace infini, ni d'un nombre infini2.
Que je ne pense pas toujours, et que par conséquent la pensée n'est pas l'essence, mais l'action de mon entendement3.
Que je suis libre quand je peux faire ce que je veux.
Que cette liberté ne peut consister dans ma volonté, puisque lorsque je demeure volontairement dans ma chambre, dont la porte est fermée, et dont je n'ai pas la clef, je n'ai pas la liberté d'en sortir ; puisque je souffre quand je veux ne pas souffrir ; puisque très souvent je ne peux rappeler mes idées quand je veux les rappeler.
Qu'il est donc absurde au fond de dire, _la volonté est libre_ , puisqu'il est absurde de dire, _je veux vouloir cette chose_ ; car c'est précisément comme si on disait, _je désire de la désirer, je crains de la craindre_ ; qu'enfin la volonté n'est pas plus libre qu'elle n'est bleue ou carrée ( _voyez_ l'article XIII4).
Que je ne puis vouloir qu'en conséquence des idées reçues dans mon cerveau ; que je suis nécessité à me déterminer en conséquence de ces idées, puisque sans cela je me déterminerais sans raison, et qu'il y aurait un effet sans cause5.
Que je ne puis avoir une idée positive de l'infini, puisque je suis très fini6.
Que je ne puis connaître aucune substance, parce que je ne puis avoir d'idée que de leurs qualités, et que mille qualités d'une chose ne peuvent me faire connaître la nature intime de cette chose, qui peut avoir cent mille autres qualités ignorées7.
Que je ne suis la même personne qu'autant que j'ai de la mémoire, et le sentiment de ma mémoire ; car n'ayant pas la moindre partie du corps qui m'appartenait dans mon enfance, et n'ayant pas le moindre souvenir des idées qui m'ont affecté à cet âge, il est clair que je ne suis pas plus ce même enfant que je ne suis Confucius ou Zoroastre. Je suis réputé la même personne par ceux qui m'ont vu croître, et qui ont toujours demeuré avec moi ; mais je n'ai en aucune façon la même existence ; je ne suis plus l'ancien moi-même ; je suis une nouvelle identité : et de là quelles singulières conséquences8 !
Qu'enfin, conformément à la profonde ignorance dont je me suis convaincu sur les principes des choses, il est impossible que je puisse connaître quelles sont les substances auxquelles Dieu daigne accorder le don de sentir et de penser. En effet, y a-t-il des substances dont l'essence soit de penser, qui pensent toujours, et qui pensent par elles-mêmes ? En ce cas, ces substances, quelles qu'elles soient, sont des dieux ; car elles n'ont nul besoin de l'Être éternel et formateur, puisqu'elles ont leurs essences sans lui, puisqu'elles pensent sans lui.
Secondement, si l'Être éternel a fait le don de sentir et de penser à des êtres, il leur a donné ce qui ne leur appartenait pas essentiellement ; il a donc pu donner cette faculté à tout être, quel qu'il soit9.
Troisièmement, nous ne connaissons aucun être à fond ; donc il est impossible que nous sachions si un être est incapable ou non de recevoir le sentiment et la pensée. Les mots de _matière_ et d' _esprit_ ne sont que des mots ; nous n'avons nulle notion complète de ces deux choses ; donc au fond il y a autant de témérité à dire qu'un corps organisé par Dieu même ne peut recevoir la pensée de Dieu même, qu'il serait ridicule de dire que l'esprit ne peut penser10.
Quatrièmement, je suppose qu'il y ait des substances purement spirituelles qui n'aient jamais eu l'idée de la matière et du mouvement, seront-elles bien reçues à nier que la matière et le mouvement puissent exister11 ?
Je suppose que la savante congrégation12 qui condamna Galilée comme impie, et comme absurde, pour avoir démontré le mouvement de la terre autour du soleil, eût eu quelque connaissance des idées du chancelier Bacon13, qui proposait d'examiner si l'attraction est donnée à la matière ; je suppose que le rapporteur de ce tribunal eût remontré à ces graves personnages qu'il y avait des gens assez fous en Angleterre pour soupçonner que Dieu pouvait donner à toute la matière depuis Saturne jusqu'à notre petit tas de boue14, une tendance vers un centre, une attraction, une gravitation, laquelle serait absolument indépendante de toute impulsion ; puisque l'impulsion agit en raison des surfaces, et que cette gravitation agit en raison des solides15. Ne voyez-vous pas ces juges de la raison humaine, et de Dieu même, dicter aussitôt leurs arrêts, anathématiser16 cette gravitation que Newton a démontrée depuis, prononcer que cela est impossible à Dieu, et déclarer que la gravitation vers un centre est un blasphème ? Je suis coupable, ce me semble, de la même témérité, quand j'ose assurer que Dieu ne peut faire sentir et penser un être organisé quelconque.
Cinquièmement, je ne puis douter que Dieu n'ait accordé des sensations, de la mémoire, et par conséquent des idées, à la matière organisée dans les animaux. Pourquoi donc nierai-je qu'il puisse faire le même présent à d'autres animaux ? On l'a déjà dit ; la difficulté consiste moins à savoir si la matière organisée peut penser, qu'à savoir comment un être, quel qu'il soit, pense17.
La pensée est quelque chose de divin ; oui sans doute ; et c'est pour cela que je ne saurai jamais ce que c'est que l'être pensant. Le principe du mouvement est divin ; et je ne saurai jamais la cause de ce mouvement dont tous mes membres exécutent les lois.
L'enfant d'Aristote étant en nourrice, attirait dans sa bouche le téton qu'il suçait, en formant précisément avec sa langue qu'il retirait, une machine pneumatique, en pompant l'air, en formant du vide ; tandis que son père ne savait rien de tout cela, et disait au hasard, que la nature abhorre le vide18.
L'enfant d'Hippocrate, à l'âge de quatre ans, prouvait la circulation du sang en passant son doigt sur sa main ; et Hippocrate ne savait pas que le sang circulât19.
Nous sommes ces enfants, tous tant que nous sommes ; nous opérons des choses admirables ; et aucun des philosophes ne sait comment elles s'opèrent.
Sixièmement, voilà les raisons, ou plutôt les doutes que me fournit ma faculté intellectuelle sur l'assertion modeste de Locke. Je ne dis point, encore une fois, que c'est la matière qui pense en nous ; je dis avec lui qu'il ne nous appartient pas de prononcer qu'il soit impossible à Dieu de faire penser la matière ; qu'il est absurde de le prononcer ; et que ce n'est pas à des vers de terre à borner la puissance de l'Être suprême.
Septièmement, j'ajoute que cette question est absolument étrangère à la morale ; parce que, soit que la matière puisse penser ou non, quiconque pense doit être juste ; parce que l'atome à qui Dieu aura donné la pensée peut mériter ou démériter, être puni ou récompensé, et durer éternellement ; aussi bien que l'être inconnu20 appelé autrefois _souffle,_ et aujourd'hui _esprit_ , dont nous avons encore moins de notion que d'un atome.
Je sais bien que ceux qui ont cru que l'être nommé _souffle_ pouvait seul être susceptible de sentir et de penser, ont persécuté ceux qui ont pris le parti du sage Locke, et qui n'ont pas osé borner la puissance de Dieu à n'animer que ce souffle. Mais quand l'univers entier croyait que l'âme était un corps léger, un souffle, une substance de feu, aurait-on bien fait de persécuter ceux qui sont venus nous apprendre que l'âme est immatérielle ? Tous les Pères de l'Église qui ont cru l'âme un corps délié, auraient-ils eu raison de persécuter les autres Pères qui ont apporté aux hommes l'idée de l'immatérialité parfaite ? Non, sans doute ; car le persécuteur est abominable21. Donc ceux qui admettent l'immatérialité parfaite sans la comprendre, ont dû tolérer ceux qui la rejetaient, parce qu'ils ne la comprenaient pas. Ceux qui ont refusé à Dieu le pouvoir d'animer l'être inconnu appelé _matière_ ont dû tolérer aussi ceux qui n'ont pas osé dépouiller Dieu de ce pouvoir ; car il est bien malhonnête de se haïr pour des syllogismes.
1\- Locke critique les idées innées dans le livre premier de l' _Essai philosophique concernant l'entendement humain_ , puis il développe sa conception empiriste des idées dans le livre second.
2\- _Ibid._ , livre II, chapitre XVII.
3\- _Ibid._ , livre II, chapitre I.
4\- C'est-à-dire le Doute XIII.
5\- Locke, _Essai philosophique concernant l'entendement humain_ , livre II, chapitre XXI.
6\- Locke, sur ce point, est d'accord avec Descartes (voir Descartes, troisième des _Méditations métaphysiques_ et Locke, _Essai philosophique concernant l'entendement humain_ , livre II, chapitre XVII).
7\- Locke, _Essai philosophique concernant l'entendement humain_ , livre II, chapitre XXIV.
8\- _Ibid._ , livre II, chapitre XXVII
9\- _Ibid._ , livre II, chapitre XXIII.
10\- Dans l' _Essai philosophique concernant l'entendement humain_ , livre II, chapitre XXIII, § 32, p. 246, Locke dit exactement : « il n'est pas plus difficile de concevoir comment la pensée pourrait exister sans matière, que de comprendre que la matière pourrait penser ». Cette hypothèse que la matière pourrait penser a été isolée de son contexte par les matérialistes qui en ont fait leur miel. Voltaire, dans la suite du Doute, est fidèle à l'esprit de Locke : il interprète cette hypothèse non dans le sens du matérialisme mais dans le sens du scepticisme.
11\- Voltaire continue à commenter le § 32 du chapitre XXIII du livre II de l' _Essai philosophique concernant l'entendement humain_.
12\- Le tribunal du Saint-Office en 1633.
13\- Voltaire a une grande admiration pour le chancelier Bacon (1561-1626) auquel il consacre la douzième des _Lettres philosophiques_ (Paris, GF-Flammarion, 1964). Il le présente comme le « père de la philosophie expérimentale » (p. 78) et cite le _Novum Organum_ où Bacon parle d'attraction : effectivement, Bacon, comme avant lui Kepler, parle d'attraction mais dans un sens non mathématique (voir _Novum Organum_ , trad. M. Malherbe et J.-M. Pousseur, Paris, PUF, 1986, II, § 36).
14\- Dans _Zadig_ , Voltaire parle de la Terre comme d'un atome de boue (voir _Zadig_ , in _Romans et contes_ , éd. citée, p.179).
15\- Voltaire a lu de près Newton qui dit en effet dans le Scholie général des _Principes mathématiques de la philosophie naturelle_ , trad. Madame du Châtelet, Paris, 1756-1759, repris par J. Gabay en 2 tomes, Paris, 1990, II, p. 178, que la force d'attraction « n'agit point selon la grandeur des superficies, (comme les causes mécaniques) mais selon la quantité de matière ».
16\- C'est-à-dire déclarer hérétique.
17\- Reprise de Locke toujours, _Essai philosophique concernant l'entendement humain_ , livre II, chapitres I et XXIII.
18\- Thèse qu'Aristote développe en effet dans les chapitres 6 à 9 du livre IV de sa _Physique_.
19\- William Harvey, en 1628, mit en évidence la circulation du sang.
20\- C'est-à-dire l'âme, le terme _psuché_ en grec signifie souffle.
21\- À la délation et à la persécution, Voltaire oppose la tolérance, voir les deux occurrences qui suivent du verbe « tolérer ». _Le Philosophe ignorant_ prône un déisme sceptique où la morale semble sauve parce qu'elle est indépendante de la théologie et de la métaphysique, l'essai est dans la continuité du _Traité de la tolérance_ publié en 1763.
XXX. Qu'ai-je appris jusqu'à présent ?
J'ai donc compté avec Locke et avec moi-même, et je me suis trouvé possesseur de quatre ou cinq vérités, dégagé d'une centaine d'erreurs, et chargé d'une immense quantité de doutes. Je me suis dit ensuite à moi-même : Ce peu de vérités que j'ai acquises par ma raison, sera entre mes mains un bien stérile, si je n'y puis trouver quelques principes de morale1. Il est beau à un aussi chétif animal que l'homme, de s'être élevé à la connaissance du maître de la nature : mais cela ne me servira pas plus que la science de l'algèbre, si je n'en tire quelque règle pour la conduite de ma vie.
1\- Voltaire en vient directement à la question de la morale qu'il n'a fait qu'effleurer jusqu'à présent.
XXXI. Y a-t-il une morale ?
Plus j'ai vu des hommes différents par le climat, les mœurs, le langage, les lois, le culte, et par la mesure de leur intelligence, et plus j'ai remarqué qu'ils ont tous le même fonds de morale1. Ils ont tous une notion grossière du juste et de l'injuste, sans savoir un mot de théologie. Ils ont tous acquis cette même notion dans l'âge où la raison se déploie, comme ils ont tous acquis naturellement l'art de soulever des fardeaux avec des bâtons, et de passer un ruisseau sur un morceau de bois, sans avoir appris les mathématiques2.
Il m'a donc paru que cette idée du juste et de l'injuste leur était nécessaire, puisque tous s'accordaient en ce point, dès qu'ils pouvaient agir et raisonner. L'intelligence suprême qui nous a formés, a donc voulu qu'il y eût de la justice sur la terre, pour que nous puissions y vivre un certain temps3. Il me semble que n'ayant ni instinct pour nous nourrir comme les animaux, ni armes naturelles comme eux, et végétant plusieurs années dans l'imbécillité4 d'une enfance exposée à tous les dangers, le peu qui serait resté d'hommes échappés aux dents des bêtes féroces, à la faim, à la misère, se seraient occupés à se disputer quelque nourriture et quelques peaux de bêtes, et qu'ils se seraient bientôt détruits comme les enfants du dragon de Cadmus5, sitôt qu'ils auraient pu se servir de quelque arme. Du moins il n'y aurait eu aucune société, si les hommes n'avaient conçu l'idée de quelque justice, qui est le lien de toute société.
Comment l'Égyptien qui élevait des pyramides et des obélisques, et le Scythe6 errant qui ne connaissait pas même les cabanes, auraient-ils eu les mêmes notions fondamentales du juste et de l'injuste, si Dieu n'avait donné de tout temps à l'un et à l'autre cette raison qui, en se développant, leur fait apercevoir les mêmes principes nécessaires, ainsi qu'il leur a donné des organes, qui, lorsqu'ils ont atteint le degré de leur énergie, perpétuent nécessairement et de la même façon la race7 du Scythe et de l'Égyptien ? Je vois une horde barbare ignorante, superstitieuse, un peuple sanguinaire et usurier, qui n'avait pas même de terme dans son jargon pour signifier la géométrie et l'astronomie8 ; cependant ce peuple a les mêmes lois fondamentales que le sage Chaldéen9 qui a connu les routes des astres, et que le Phénicien10 plus savant encore, qui s'est servi de la connaissance des astres pour aller fonder des colonies aux bornes de l'hémisphère où l'Océan se confond avec la Méditerranée. Tous ces peuples assurent qu'il faut respecter son père et sa mère ; que le parjure, la calomnie, l'homicide, sont abominables. Ils tirent donc tous les mêmes conséquences du même principe de leur raison développée.
1\- Voltaire soutient donc la thèse de l'universalité de la morale, quelles que soient l'origine et la diversité des hommes.
2\- Cette morale naturelle de tous les hommes n'est cependant pas innée, elle s'acquiert par la sociabilité à l'âge de raison, vers 7 ans.
3\- À la différence des matérialistes qui confèrent un fondement naturaliste à la morale – c'est la nature qui fonde la morale – Voltaire confère un fondement déiste à la morale naturelle : c'est Dieu qui fonde la morale naturelle.
4\- La vulnérabilité.
5\- Selon la légende thébaine, Cadmus consulta l'oracle de Delphes pour savoir en quel lieu il pourrait s'établir, et reçut ordre de bâtir une ville à l'endroit où un bœuf le conduirait. Il suivit cet ordre, et rencontra une génisse qui lui servit de guide, et qui s'arrêta dans l'emplacement où, depuis, fut bâtie la ville de Thèbes, sur le modèle de la Thèbes d'Égypte. Avant de jeter les fondements de la citadelle, qui de son nom fut appelée la Cadmée, il voulut offrir un sacrifice à Pallas. Dans cette intention, il envoya ses compagnons puiser de l'eau dans un bois voisin consacré à Mars : mais un dragon, fils de Mars et de Vénus, les dévora. Cadmus vengea leur mort en tuant le monstre, et en sema les dents, d'après le conseil de Minerve. Il en sortit des hommes tout armés qui l'assaillirent d'abord, mais tournèrent bientôt leur fureur contre eux-mêmes, et s'entretuèrent, à l'exception de cinq qui l'aidèrent à bâtir sa ville.
6\- Les Scythes forment un ensemble de peuples nomades, d'origine iranienne, ayant vécu entre le VIIe et le IIIe siècle av. J.-C. dans les steppes eurasiennes.
7\- Même si Voltaire prône l'universalité de la morale, il est persuadé qu'il y a plusieurs races d'hommes. Il écrit, dans le chapitre CXLI de l' _Essai sur les mœurs et les nations_ , que la race des nègres est une espèce d'hommes différente de la nôtre, ce qui ne veut pas dire inférieure ni susceptible d'être esclavagisée. Au contraire, Voltaire, dans l'épisode du nègre de Surinam de _Candide_ , dénonce l'esclavage. La rencontre de Candide avec le nègre au sortir de l'Eldorado constitue un choc brutal et un retour à la réalité du mal : Candide ne peut plus se laisser aller à une quelconque croyance optimiste (voir _Candide_ , in _Romans et contes_ , Paris, GF-Flammarion, 1966, p. 222).
8\- Voltaire, même s'il veut toujours défendre l'universalité de la morale, dresse ici un portrait peu flatteur du peuple juif : à vrai dire, c'est la prétention des théologiens juifs à proclamer leur peuple élu de Dieu qu'il condamne et non les Juifs eux-mêmes. Preuve en est le _Sermon du rabbin Akib_ (1761) où Voltaire prend la défense des Juifs et condamne leur massacre par l'Inquisition.
9\- Voir ci-dessus note sur les Chaldéens au Doute XIV.
10\- Voir ci-dessus note sur les Phéniciens au Doute XIV.
XXXII. Utilité réelle. Notion de la justice
La notion de quelque chose de juste me semble si naturelle, si universellement acquise par tous les hommes, qu'elle est indépendante de toute loi, de tout pacte, de toute religion. Que je redemande à un Turc, à un Guèbre1, à un Malabare2, l'argent que je lui ai prêté pour se nourrir et pour se vêtir ; il ne lui tombera jamais dans la tête de me répondre, Attendez que je sache si Mahomet, Zoroastre ou Brama ordonnent que je vous rende votre argent. Il conviendra qu'il est juste qu'il me paie ; et s'il n'en fait rien, c'est que sa pauvreté ou son avarice l'emporteront sur la justice qu'il reconnaît.
Je mets en fait, qu'il n'y a aucun peuple chez lequel il soit juste, beau, convenable, honnête de refuser la nourriture à son père et à sa mère quand on peut leur en donner.
Que nulle peuplade n'a jamais pu regarder la calomnie comme une bonne action, non pas même une compagnie de bigots fanatiques.
L'idée de justice me paraît tellement une vérité du premier ordre, à laquelle tout l'univers donne son assentiment, que les plus grands crimes qui affligent la société humaine, sont tous commis sous un faux prétexte de justice. Le plus grand des crimes, du moins le plus destructif, et par conséquent le plus opposé au but de la nature, est la guerre ; mais il n'y a aucun agresseur qui ne colore ce forfait du prétexte de la justice.
Les déprédateurs romains faisaient déclarer toutes leurs invasions justes par des prêtres nommés _féciales_3. Tout brigand qui se trouve à la tête d'une armée commence ses fureurs par un manifeste, et implore le dieu des armées4.
Les petits voleurs eux-mêmes, quand ils sont associés, se gardent bien de dire, Allons voler, allons arracher à la veuve et à l'orphelin leur nourriture ; ils disent, Soyons justes, allons reprendre notre bien des mains des riches qui s'en sont emparés. Ils ont entre eux un dictionnaire qu'on a même imprimé dès le XVIe siècle5, et dans ce vocabulaire qu'ils appellent _argot_ , les mots de _vol_ , _larcin_ , _rapine_ , ne se trouvent point ; ils se servent des termes qui répondent à _gagner_ , _reprendre_.
Le mot d' _injustice_ ne se prononce jamais dans un conseil d'État, où l'on propose le meurtre le plus injuste ; les conspirateurs, même les plus sanguinaires, n'ont jamais dit : Commettons un crime. Ils ont tous dit, Vengeons la patrie des crimes du tyran, punissons ce qui nous paraît une injustice. En un mot, flatteurs lâches, ministres barbares, conspirateurs odieux, voleurs plongés dans l'iniquité, tous rendent hommage, malgré eux, à la vertu même qu'ils foulent aux pieds6.
J'ai toujours été étonné que, chez les Français, qui sont éclairés et polis, on ait souffert sur le théâtre ces maximes aussi affreuses que fausses qui se trouvent dans la première scène de _Pompée_ , et qui sont beaucoup plus outrées que celles de Lucain7 dont elles sont imitées.
_La justice et le droit sont de vaines idées._
_Le droit des rois consiste à ne rien épargner_8 _._
Et on met ces abominables paroles dans la bouche de Photin ministre du jeune Ptolémée. Mais c'est précisément parce qu'il est ministre qu'il devait dire tout le contraire ; il devait représenter la mort de Pompée comme un malheur nécessaire et juste.
Je crois donc que les idées du juste et de l'injuste sont aussi claires, aussi universelles que les idées de santé et de maladie, de vérité et de fausseté, de convenance et de disconvenance. Les limites du juste et de l'injuste sont très difficiles à poser ; comme l'état mitoyen entre la santé et la maladie, entre ce qui est convenance et la disconvenance des choses, entre le faux et le vrai, est difficile à marquer. Ce sont des nuances qui se mêlent, mais les couleurs tranchantes frappent tous les yeux. Par exemple, tous les hommes avouent qu'on doit rendre ce qu'on nous a prêté ; mais si je sais certainement que celui à qui je dois deux millions, s'en servira pour asservir ma patrie, dois-je lui rendre cette arme funeste ? Voilà où les sentiments se partagent : mais en général je dois observer mon serment quand il n'en résulte aucun mal ; c'est de quoi personne n'a jamais douté.
1\- Les Guèbres sont des Perses restés fidèles au zoroastrisme (culte de Zarathoustra), Voltaire loue leur tolérance dans sa tragédie _Les Guèbres_ (1768).
2\- Habitant de Malabar – appelé aussi côte de Malabar –, région située sur la côte du sud-ouest de la péninsule indienne. Cependant, au XVIIIe siècle, le terme désigne tout habitant de l'Inde en général.
3\- Le terme « fécial » désigne effectivement un prêtre ou magistrat qui était chargé d'accomplir les formalités juridiques et religieuses relatives à la guerre, aux invasions et aux déprédations qui s'ensuivaient.
4\- C'est aussi vrai dans l' _Iliade_ d'Homère que dans la Bible (Arès ou Dieu Sabaoth).
5\- Le premier dictionnaire d'argot fut publié en 1596 dans _La Vie généreuse des mercelots, gueuz et boesmiens, contenans leur façon de vivre, subtilitez et gergon_ , de Pechon de Ruby, édition critique et annotée de l'édition lyonnaise par Denis Delaplace, Paris, Champion, 2007.
6\- Voltaire reprend ici la maxime 218 de La Rochefoucauld : « L'hypocrisie est un hommage que le vice rend à la vertu », in _Maximes, Mémoires, Œuvres diverses_ , Paris, Garnier, 1992, p. 164.
7\- Lucain (39-65), poète latin, neveu de Sénèque, et compagnon de Néron qui l'obligea à se donner la mort à l'âge de 26 ans, a écrit _La Pharsale_ que Voltaire cite dans le Doute XVII, récit de la guerre civile entre César et Pompée.
8\- Corneille, _La Mort de Pompée_ , acte I, scène 1.
XXXIII. Consentement universel
est-il preuve de vérité ?
On peut m'objecter que le consentement des hommes de tous les temps et de tous les pays n'est pas une preuve de la vérité. Tous les peuples ont cru à la magie, aux sortilèges, aux démoniaques, aux apparitions, aux influences des astres, à cent autres sottises pareilles. Ne pourrait-il pas en être ainsi du juste et de l'injuste ?
Il me semble que non. Premièrement, il est faux que tous les hommes aient cru à ces chimères. Elles étaient à la vérité l'aliment de l'imbécillité du vulgaire, et il y a le vulgaire des grands et le vulgaire du peuple1 ; mais une multitude de sages s'en est toujours moquée ; ce grand nombre de sages, au contraire, a toujours admis le juste et l'injuste, tout autant, et même encore plus que le peuple.
La croyance aux sorciers, aux démoniaques, etc., est bien éloignée d'être nécessaire au genre humain ; la croyance à la justice est d'une nécessité absolue ; donc elle est un développement de la raison donnée de Dieu ; et l'idée des sorciers et des possédés, etc., est au contraire un pervertissement de cette même raison.
1\- Le vulgaire concerne aussi bien le peuple que les grands : l'imbécillité, au sens de sottise, se manifeste à tous les rangs et Voltaire souligne, dans l'article _Lettres, gens de lettres ou lettrés_ du _Dictionnaire philosophique_ , que le plus grand malheur d'un homme de lettres est d'être jugé par des sots.
XXXIV. Contre Locke
Locke qui m'instruit et qui m'apprend à me défier de moi-même ne se trompe-t-il pas quelquefois comme moi-même ? Il veut prouver la fausseté des idées innées ; mais n'ajoute-t-il pas une bien mauvaise raison à de fort bonnes ? Il avoue qu'il n'est pas juste de faire bouillir son prochain dans une chaudière, et de le manger. Il dit1 que cependant il y a eu des nations d'anthropophages, et que ces êtres pensants n'auraient pas mangé des hommes, s'ils avaient eu les idées du juste et de l'injuste, que je suppose nécessaires à l'espèce humaine. ( _Voyez,_ le n° XXXVI.2)
Sans entrer ici dans la question, s'il y a eu en effet des nations d'anthropophages, sans examiner les relations du voyageur Dampier3, qui a parcouru toute l'Amérique, et qui n'y en a jamais vu, mais qui au contraire a été reçu chez tous les sauvages avec la plus grande humanité ; voici ce que je réponds.
Des vainqueurs ont mangé leurs esclaves pris à la guerre ; ils ont cru faire une action très juste ; ils ont cru avoir sur eux droit de vie et de mort ; et comme ils avaient peu de bons mets pour leur table, ils ont cru qu'il leur était permis de se nourrir du fruit de leur victoire. Ils ont été en cela plus justes que les triomphateurs romains, qui faisaient étrangler sans aucun fruit les princes esclaves qu'ils avaient enchaînés à leur char de triomphe. Les Romains et les sauvages avaient une très fausse idée de la justice, je l'avoue ; mais enfin les uns et les autres croyaient agir justement ; et cela est si vrai, que les mêmes sauvages, quand ils avaient admis leurs captifs dans leur société, les regardaient comme leurs enfants ; et que ces mêmes anciens Romains ont donné mille exemples de justice admirables4.
1\- Locke, _Essai philosophique concernant l'entendement humain_ , Livre I, chapitre 2, § 9, p. 29-30.
2\- C'est-à-dire le Doute XXXVI.
3\- William Dampier (1652-1715), _Nouveau voyage autour du monde_ , en 3 volumes, Londres, 1697-1699 ( _Bibliothèque de Voltaire_ , n° 935).
4\- Voltaire fait des principes de la morale des idées universellement reçues de tous les hommes à l'âge de raison et non, comme le veut Locke, des simples règles de pratique et de convention variables selon les régions et les siècles.
XXXV. Contre Locke
Je conviens, avec le sage Locke, qu'il n'y a point de notion innée, point de principe de pratique inné. C'est une vérité si constante, qu'il est évident que les enfants auraient tous une notion claire de Dieu, s'ils étaient nés avec cette idée, et que tous les hommes s'accorderaient dans cette même notion, accord que l'on n'a jamais vu. Il n'est pas moins évident que nous ne naissons point avec des principes développés de morale, puisqu'on ne voit pas comment une nation entière pourrait rejeter un principe de morale qui serait gravé dans le cœur de chaque individu de cette nation1.
Je suppose que nous soyons tous nés avec le principe moral bien développé, qu'il ne faut persécuter personne pour sa manière de penser ; comment des peuples entiers auraient-ils été persécuteurs ? Je suppose que chaque homme porte en soi la loi évidente qui ordonne qu'on soit fidèle à son serment ; comment tous ces hommes, réunis en corps, auront-ils statué qu'il ne faut pas garder sa parole à des hérétiques ? Je répète encore, qu'au lieu de ces idées innées chimériques, Dieu nous a donné une raison qui se fortifie avec l'âge, et qui nous apprend à tous, quand nous sommes attentifs, sans passion, sans préjugé, qu'il y a un Dieu, et qu'il faut être juste ; mais je ne puis accorder à Locke les conséquences qu'il en tire2. Il semble trop approcher du système de Hobbes3, dont il est pourtant très éloigné.
Voici ses paroles, au premier livre de l'Entendement humain ; _Considérez, une ville prise d'assaut, et voyez s'il paraît dans le cœur des soldats animés au carnage et au butin, quelque égard pour la vertu, quelque principe de morale, quelques remords de toutes les injustices qu'ils commettent_4. Non, ils n'ont point de remords, et pourquoi ? C'est qu'ils croient agir justement. Aucun d'eux n'a supposé injuste la cause du prince pour lequel il va combattre ; ils hasardent leur vie pour cette cause : ils tiennent le marché qu'ils ont fait ; ils pouvaient être tués à l'assaut, donc ils croient être en droit de tuer : ils pouvaient être dépouillés, donc ils pensent qu'ils peuvent dépouiller. Ajoutez qu'ils sont dans l'enivrement de la fureur qui ne raisonne pas ; et pour vous prouver qu'ils n'ont point rejeté l'idée du juste et de l'honnête, proposez à ces mêmes soldats beaucoup plus d'argent que le pillage de la ville ne peut leur en procurer, de plus belles filles que celles qu'ils ont violées, pourvu seulement qu'au lieu d'égorger dans leur fureur trois ou quatre mille ennemis, qui font encore résistance, et qui peuvent les tuer, ils aillent égorger leur roi, son chancelier, ses secrétaires d'État, et son grand aumônier, vous ne trouverez pas un de ces soldats qui ne rejette vos offres avec horreur. Vous ne leur proposez cependant que six meurtres au lieu de quatre mille, et vous leur présentez une récompense très forte. Pourquoi vous refusent-ils ? C'est qu'ils croient juste de tuer quatre mille ennemis, et que le meurtre de leur souverain, auquel ils ont fait serment, leur paraît abominable5.
Locke continue, et, pour mieux prouver qu'aucune règle de pratique n'est innée, il parle des Mingréliens, qui se font un jeu, dit-il, d'enterrer leurs enfants tout vifs ; et des Caraïbes, qui châtrent les leurs pour les mieux engraisser, afin de les manger6.
On a déjà remarqué ailleurs que ce grand homme a été trop crédule en rapportant ces fables7 : Lambert8, qui seul impute aux Mingréliens d'enterrer leurs enfants tout vifs pour leur plaisir, n'est pas un auteur assez accrédité.
Chardin9, voyageur qui passe pour véridique, et qui a été rançonné en Mingrélie10, parlerait de cette horrible coutume si elle existait ; et ce ne serait pas assez qu'il le dît, pour qu'on le crût ; il faudrait que vingt voyageurs de nations et de religions différentes, s'accordassent à confirmer un fait si étrange, pour qu'on en eût une certitude historique.
Il en est de même des femmes des îles Antilles, qui châtraient leurs enfants pour les manger : cela n'est pas dans la nature d'une mère.
Le cœur humain n'est point ainsi fait ; châtrer des enfants est une opération très délicate, très dangereuse, qui loin de les engraisser les amaigrit au moins une année entière, et qui souvent les tue. Ce raffinement n'a jamais été en usage que chez des grands, qui, pervertis par l'excès du luxe et par la jalousie, ont imaginé d'avoir des eunuques pour servir leurs femmes et leurs concubines. Il n'a été adopté en Italie, et à la chapelle du pape, que pour avoir des musiciens dont la voix fût plus belle que celle des femmes11. Mais dans les îles Antilles, il n'est guère à présumer que des sauvages aient inventé le raffinement de châtrer les petits garçons pour en faire un bon plat ; et puis qu'auraient-ils fait de leurs petites filles12 ?
Locke allègue encore des saints de la religion mahométane, qui s'accouplent dévotement avec leurs ânesses pour n'être point tentés de commettre la moindre fornication avec les femmes du pays13. Il faut mettre ces contes avec celui du perroquet qui eut une si belle conversation en langue brasilienne avec le prince Maurice, conversation que Locke a la simplicité de rapporter14, sans se douter que l'interprète du prince avait pu se moquer de lui. C'est ainsi que l'auteur de l' _Esprit des lois_ s'amuse à citer de prétendues lois de Tunquin, de Bantam, de Bornéo, de Formose, sur la foi de quelques voyageurs, ou menteurs ou mal instruits15. Locke et lui sont deux grands hommes, en qui cette simplicité ne me semble pas excusable.
1\- Voltaire acquiesce à cet argument de Locke (voir _Essai philosophique concernant l'entendement humain_ , livre I, chapitre 2) qui vaut pour réfuter les principes innés de la morale.
2\- Conséquences qui sont le relativisme des valeurs et le caractère conventionnel et pratique des règles de la morale. Voltaire préfère la contradiction qui consiste à dire, d'une part, que les principes de la morale ne sont pas innés mais acquis et à refuser, d'autre part, l'idée que, s'ils s'acquièrent, ils peuvent varier d'une nation à l'autre. Il cherche à maintenir que la morale est naturelle et universelle parce que fondée en Dieu.
3\- Hobbes expose, dans le _Léviathan_ , sa théorie nominaliste des fondements de la société : tout repose sur la convention, sur un pacte qui met fin à l'état de nature où l'homme est un loup pour l'homme.
4\- Locke, _Essai philosophique concernant l'entendement humain_ , livre I, chapitre 2, § 9, p. 29.
5\- Que faire alors de l'universalité des grands principes de morale comme celui d'aimer son prochain comme soi-même ou – celui que Locke discute – de faire aux autres ce que nous voudrions qui nous fût fait à nous-mêmes – et que Voltaire prétend fonder en Dieu ?
6\- Toujours dans le même § 9 du chapitre 2 du livre I de l' _Essai philosophique concernant l'entendement humain_.
7\- Voir la référence à Dampier qui contredit ces fables (notamment l'anthropophagie) dans le Doute précédent.
8\- Claude-François Lambert, _Recueil d'observations sur les mœurs, les arts, les sciences, les différentes langues, le gouvernement_ [...] _de différents peuples de l'Asie, de l'Afrique et de l'Amérique_ (1749, Paris, 4 volumes). Ce recueil fait partie de la littérature de cabinet assujettie aux valeurs unificatrices d'ordre et de clarté puisqu'à partir de témoignages fragmentaires, elle se charge de reconstituer une image cohérente du monde.
9\- Jean Chardin (1643-1713), grand voyageur en Perse et en Orient. Voltaire possédait dans sa bibliothèque les _Voyages de monsieur le chevalier Chardin en Perse et autres lieux de l'Orient_ ( _Bibliothèque de Voltaire_ , n° 712)
10\- L'actuelle Géorgie.
11\- Rousseau, dans le _Discours sur l'origine et les fondements de l'inégalité parmi les hommes_ (Paris, GF-Flammarion, 2008, p. 165), condamne cette pratique et cette coutume des eunuques et des castrats.
12\- Mis à part les cas des eunuques et des castrats dont Locke ne parle pas, Voltaire continue à discuter le § 9 du chapitre 2 du livre I de l' _Essai philosophique concernant l'entendement humain_ et ne peut s'empêcher de se moquer de ces prétendues coutumes en devisant sur le sort des petites filles.
13\- Locke dit plus sobrement : « Ceux que les Turcs canonisent et mettent au nombre des saints, mènent une vie qu'on ne saurait rapporter sans blesser la pudeur » in _Essai philosophique concernant l'entendement humain_ , I, cha. 2, § 9, p. 29, et cite ensuite en latin le passage imputé dans le _Voyage_ de Baumgarten.
14\- Locke, _Essai philosophique concernant l'entendement humain_ , livre II, chap. 27, § 8, p. 262-264.
15\- Montesquieu, dans _De l'esprit des lois_ , rapporte en effet des usages ou des lois étranges comme, par exemple, à Formose, l'interdiction pour une femme d'enfanter avant l'âge de 35 ans (voir le chapitre 16 du livre XXIII, et dans le même livre les chapitres 4 et 12, ou dans le livre XXVI, le chapitre 14).
XXXVI. Nature partout la même
En abandonnant Locke en ce point, je dis avec le grand Newton, _Natura est semper sibi consona_ : la nature est toujours semblable à elle-même1. La loi de la gravitation qui agit sur un astre agit sur tous les astres, sur toute la matière. Ainsi la loi fondamentale de la morale agit également sur toutes les nations bien connues. Il y a mille différences dans les interprétations de cette loi, en mille circonstances ; mais le fonds subsiste toujours le même, et ce fonds est l'idée du juste et de l'injuste. On commet prodigieusement d'injustices dans les fureurs de ses passions, comme on perd sa raison dans l'ivresse : mais quand l'ivresse est passée, la raison revient ; et c'est, à mon avis, l'unique cause qui fait subsister la société humaine, cause subordonnée au besoin que nous avons les uns des autres.
Comment donc avons-nous acquis l'idée de la justice ? comme nous avons acquis celle de la prudence, de la vérité, de la convenance, par le sentiment et par la raison. Il est impossible que nous ne trouvions pas très imprudente l'action d'un homme qui se jetterait dans le feu pour se faire admirer, et qui espérerait d'en réchapper. Il est impossible que nous ne trouvions pas très injuste l'action d'un homme qui en tue un autre dans sa colère. La société n'est fondée que sur ces notions qu'on n'arrachera jamais de notre cœur, et c'est pourquoi toute société subsiste, à quelque superstition bizarre et horrible qu'elle se soit asservie.
Quel est l'âge où nous connaissons le juste et l'injuste ? L'âge où nous connaissons que deux et deux font quatre2.
1\- Newton, _Principes mathématiques de la philosophie naturelle_ , trad. marquise du Châtelet, t. II, p. 3 : « on ne doit pas abandonner l'analogie de la nature qui est toujours simple et semblable à elle-même ».
2\- Locke prend comme exemple la connaissance que trois et quatre font sept pour marquer l'âge de raison des enfants in _Essai philosophique concernant l'entendement humain_ , livre II, chap. I, § 16, p. 14-15.
XXXVII. De Hobbes
Profond et bizarre philosophe, bon citoyen, esprit hardi, ennemi de Descartes, toi qui t'es trompé comme lui, toi dont les erreurs en physique sont grandes, et pardonnables parce que tu étais venu avant Newton, toi qui as dit des vérités qui ne compensent pas tes erreurs, toi qui le premier fis voir quelle est la chimère des idées innées, toi qui fus le précurseur de Locke en plusieurs choses, mais qui le fus aussi de Spinoza ; c'est en vain que tu étonnes tes lecteurs, en réussissant presque à leur prouver qu'il n'y a aucunes lois dans le monde que des lois de convention ; qu'il n'y a de juste et d'injuste que ce qu'on est convenu d'appeler tel dans un pays. Si tu t'étais trouvé seul avec Cromwell dans une île déserte, et que Cromwell eût voulu te tuer pour avoir pris le parti de ton roi dans l'île d'Angleterre, cet attentat ne t'aurait-il pas paru aussi injuste dans ta nouvelle île, qu'il te l'aurait paru dans ta patrie ?
Tu dis que dans la loi de nature, _tous ayant droit à tout, chacun a droit sur la vie de son semblable_1 _._ Ne confonds-tu pas la puissance avec le droit ? Penses-tu qu'en effet le pouvoir donne le droit ? et qu'un fils robuste n'ait rien à se reprocher pour avoir assassiné son père languissant et décrépit ? Quiconque étudie la morale doit commencer à réfuter ton livre dans son cœur ; mais ton propre cœur te réfutait encore davantage ; car tu fus vertueux, ainsi que Spinoza ; et il ne te manqua, comme à lui, que d'enseigner les vrais principes de la vertu que tu pratiquais, et que tu recommandais aux autres.
1\- À la pensée politique de Hobbes qui consiste à déduire le droit de tous sur tout ( _jus omnium omnia_ ) du seul principe de l'amour de soi, Voltaire oppose l'argument de la sociabilité des théoriciens du droit naturel (Pufendorf, notamment). Or cet argument fonde une morale naturelle qui n'est plus celle de l'impulsion aveugle ou du droit du plus fort mais celle de l'intérêt éclairé, puisque la raison est tout aussi essentielle à l'homme que l'amour de soi et que, en outre, l'ordre providentiel des choses veut qu'il n'y ait pas de vrai bonheur en dehors de la vertu.
XXXVIII. Morale universelle
La morale me paraît tellement universelle, tellement calculée par l'Être universel qui nous a formés, tellement destinée à servir de contrepoids à nos passions funestes, et à soulager les peines inévitables de cette courte vie, que depuis Zoroastre jusqu'au lord Shaftesburi1, je vois tous les philosophes enseigner la même morale, quoiqu'ils aient tous des idées différentes sur les principes des choses. Nous avons vu que Hobbes, Spinoza et Bayle lui-même, qui ont ou nié les premiers principes, ou qui en ont douté, ont cependant recommandé fortement la justice et toutes les vertus.
Chaque nation eut des rites religieux, particuliers, et très souvent d'absurdes et de révoltantes opinions en métaphysique, en théologie. Mais s'agit-il de savoir s'il faut être juste ? tout l'univers est d'accord, comme nous l'avons dit au nombre XXXVI2, et comme on ne peut trop le répéter.
1\- Anthony Ashley Cooper, troisième comte de Shaftesbury (1671-1713), philosophe anglais qui développe une théorie de la vertu morale fondée sur la raison et le sentiment, distincte du contrat social de Hobbes ou de l'égoïsme éthique et psychologique (voir _An inquiry concerning virtue or merit_ , 1699, traduction française en 1745 par Denis Diderot sous le titre _Essai sur le mérite et la vertu_ ).
2\- C'est-à-dire au Doute XXXVI.
XXXIX. De Zoroastre1
1\- Zoroastre appelé aussi Zarathoustra, prophète et réformateur religieux persan ; selon l'estimation la plus courante, il vécut aux alentours du VIe siècle av. J.-C. mais d'autres estimations le font naître plus tôt (– 1000) ou plus tard (– 400).
Je n'examine point en quel temps vivait Zoroastre, à qui les Perses donnèrent neuf mille ans d'antiquité, ainsi que Platon aux anciens Athéniens. Je vois seulement que ses préceptes de morale se sont conservés jusqu'à nos jours : ils sont traduits de l'ancienne langue des mages dans la langue vulgaire des Guèbres1 ; et il paraît bien aux allégories puériles, aux observances ridicules, aux idées fantastiques dont ce recueil est rempli, que la religion de Zoroastre est de l'antiquité la plus haute. C'est là qu'on trouve le nom de _jardin_ pour exprimer la récompense des justes : on y voit le mauvais principe sous le nom de Satan, que les Juifs adoptèrent aussi. On y trouve le monde formé en six saisons, ou en six temps. Il y est ordonné de réciter un _Abunavar_ et un _Ashim vuhu_ pour ceux qui éternuent.
Mais enfin, dans ce recueil de cent portes ou préceptes tirés du livre du Zend2, et où l'on rapporte même les propres paroles de l'ancien Zoroastre, quels devoirs moraux sont-ils prescrits ?
Celui d'aimer, de secourir son père et sa mère, de faire l'aumône aux pauvres, de ne jamais manquer à sa parole, de s'abstenir, quand on est dans le doute, si l'action qu'on va faire est juste ou non. ( _porte 30_ )
Je m'arrête à ce précepte, parce que nul législateur n'a jamais pu aller au-delà ; et je me confirme dans l'idée que plus Zoroastre établit de superstitions ridicules en fait de culte, plus la pureté de sa morale fait voir qu'il n'était pas en lui de la corrompre ; que plus il s'abandonnait à l'erreur dans ses dogmes, plus il lui était impossible d'errer en enseignant la vertu.
1\- Voir ci-dessus note 1 du Doute XXXII.
2\- Le Zend-Avesta, recueil des textes sacrés de la religion de Zoroastre, dénombrait cent portes (ou méthodes) pour atteindre au bonheur. Ce livre sacré fut connu en Europe grâce à Anquetil-Duperron qui recueillit en 1758 à peu près un quart du texte chez les Parsis (descendants des Perses qui émigrèrent en Inde).
XL. Des brachmanes1
1\- Les brahmanes, dans l'ancienne société indienne, forment la caste supérieure des prêtres et des religieux, composés théoriquement de ceux qui avaient atteint la connaissance du Brahman, Entité suprême dans la philosophie hindoue.
Il est vraisemblable que les brames ou brachmanes, existaient longtemps avant que les Chinois eussent leurs cinq Kings1 ; et ce qui fonde cette extrême probabilité, c'est qu'à la Chine, les antiquités les plus recherchées sont indiennes, et que dans l'Inde il n'y a point d'antiquités chinoises.
Ces anciens brames étaient sans doute d'aussi mauvais métaphysiciens, d'aussi ridicules théologiens que les Chaldéens et les Perses, et toutes les nations qui sont à l'occident de la Chine. Mais quelle sublimité dans la morale ! Selon eux, la vie n'était qu'une mort de quelques années, après laquelle on vivrait avec la Divinité. Ils ne se bornaient pas à être justes envers les autres, mais ils étaient rigoureux envers eux-mêmes ; le silence, l'abstinence, la contemplation, le renoncement à tous les plaisirs, étaient leurs principaux devoirs. Aussi tous les sages des autres nations allaient chez eux apprendre ce qu'on appelait la _sagesse_.
1\- _King_ en chinois signifie livre, il s'agit ici des cinq livres sacrés de la sagesse mis en ordre par Confucius.
XLI. De Confucius1
1\- Confucius (555 av. J.-C. - 479 av. J.-C.), philosophe chinois dont les enseignements et les idées, recueillis par ses disciples, ont influencé toute la civilisation de la Chine.
Les Chinois n'eurent aucune superstition, aucun charlatanisme à se reprocher comme les autres peuples. Le gouvernement chinois montrait aux hommes, il y a fort au-delà de quatre mille ans, et leur montre encore qu'on peut les régir sans les tromper ; que ce n'est pas par le mensonge qu'on sert le Dieu de vérité ; que la superstition est non seulement inutile, mais nuisible à la religion. Jamais l'adoration de Dieu ne fut si pure et si sainte qu'à la Chine ( _à la révélation près_ )1. Je ne parle pas des sectes du peuple, je parle de la religion du prince, de celle de tous les tribunaux, et de tout ce qui n'est pas populace2. Quelle est la religion de tous les honnêtes gens à la Chine depuis tant de siècles ? La voici : _Adorez le Ciel, et soyez juste_3. Aucun empereur n'en a eu d'autre.
On place souvent le grand Confutsée, que nous nommons Confucius, parmi les anciens législateurs, parmi les fondateurs de religions ; c'est une grande inadvertance. Confutsée est très moderne ; il ne vivait que six cent cinquante ans avant notre ère. Jamais il n'institua aucun culte, aucun rite ; jamais il ne se dit ni inspiré, ni prophète ; il ne fit que rassembler en un corps les anciennes lois de la morale.
Il invite les hommes à pardonner les injures, et à ne se souvenir que des bienfaits.
À veiller sans cesse sur soi-même, à corriger aujourd'hui les fautes d'hier.
À réprimer ses passions, et à cultiver l'amitié ; à donner sans faste, et à ne recevoir que l'extrême nécessaire sans bassesse.
Il ne dit point qu'il ne faut pas faire à autrui ce que nous ne voulons pas qu'on fasse à nous-mêmes ; ce n'est que défendre le mal : il fait plus, il recommande le bien : _Traite autrui comme tu veux qu'on te traite_4.
Il enseigne non seulement la modestie, mais encore l'humilité : il recommande toutes les vertus.
1\- Clause qui ne gêne guère le déiste Voltaire. En effet, il rejette lui aussi la révélation, source de tous les maux et de tous les conflits religieux. En ce sens, il fait sien le modèle de tolérance chinois diffusé en Europe par les jésuites au XVIIe siècle.
2\- Contrairement au Doute XXXIII où Voltaire avançait l'idée que le vulgaire concerne aussi bien les grands que le peuple, il distingue ici la religion du prince et du pouvoir, des sectes du peuple assimilé au populace. Voltaire, à l'instar de d'Alembert, de Diderot et des encyclopédistes, n'est pas égalitariste et se méfie de tout ce qui relève de la prétendue sagesse populaire.
3\- Le résumé que Voltaire propose de la religion de Confucius ressemble fort au déisme qu'il préconise, voir ci-dessus Doute XXXV où il affirme qu'il y a un Dieu et qu'il faut être juste.
4\- « Ne fais pas à autrui ce que tu ne veux pas que l'on te fasse. Ceci est la Loi. ». C'est la réponse de rabbi Hillel (70 av. J.-C. - 10 apr. J.-C.) au centurion romain venu le défier de l'instruire de toute la Loi juive durant le laps de temps où il pourrait tenir sur « une seule jambe ». Le christianisme complète cette loi par la règle d'or qui est une variante de la formule de Confucius : « Tout ce que vous voulez que les hommes fassent pour vous, faites-le vous-mêmes pour eux, car voilà la Loi et les Prophètes » (Évangile selon saint Matthieu, VII, 12, voir aussi Évangile selon saint Luc, VI, 31 : « Ce que vous voulez que les hommes fassent pour vous, faites-le pour eux pareillement »).
XLII. Des philosophes grecs,
et d'abord de Pythagore1
1\- Pythagore, philosophe et mathématicien du VIe siècle av. J.-C. dont la vie et l'œuvre sont mal connues : il ne reste aucun de ses écrits, sa pensée est recouverte par les apports de ses disciples. La pensée pythagoricienne couvre tous les domaines : la science relative aux intelligibles et aux dieux ; ensuite la physique ; la philosophie éthique et la logique ; toutes sortes de connaissances en mathématiques entre lesquelles il établit des correspondances : il rapporte ainsi les figures de la géométrie aux nombres de l'arithmétique, les sons des musiciens aux proportions des arithméticiens.
Tous les philosophes grecs ont dit des sottises en physique et en métaphysique. Tous sont excellents dans la morale ; tous égalent Zoroastre, Confutsée, et les brachmanes. Lisez seulement les vers dorés1 de Pythagore, c'est le précis de sa doctrine ; il n'importe de quelle main ils soient. Dites-moi si une seule vertu y est oubliée.
1\- Les anciens avaient l'habitude de comparer à l'or tout ce qu'ils jugeaient sans défaut et beau par excellence : ainsi, par « l'âge d'or », ils entendaient l'âge des vertus et du bonheur ; et par les « vers dorés », les vers où la doctrine la plus pure était renfermée. Ils attribuaient constamment ces vers à Pythagore, non qu'ils crussent que ce philosophe les eût composés lui-même, mais parce qu'ils savaient que ses disciples dont ils étaient l'ouvrage, y avaient exposé la doctrine de leur maître.
XLIII. De Zaleucus1
1\- Zaleucus est un philosophe et législateur grec des Locriens (peuple d'Italie), vivant au VIIe siècle av. J.-C. Les Locriens le vénéraient à l'époque de Cicéron pour un code de lois qu'ils lui attribuaient. Le préambule du code de Zaleucus (que Voltaire appelle l'exorde et qu'il résume dans la suite du Doute) a été conservé en substance par Diodore de Sicile (historien du Ier siècle av. J.-C.) et textuellement par Stobée (compilateur byzantin du Ve siècle apr. J.-C.).
Réunissez tous vos lieux communs, prédicateurs grecs, italiens, espagnols, allemands, français, etc. ; qu'on distille toutes vos déclamations, en tirera-t-on un extrait qui soit plus pur que l'exorde des lois de Zaleucus ?
_Maîtrisez votre âme, purifiez-la, écartez toute pensée criminelle. Croyez que Dieu ne peut être bien servi par les pervers ; croyez qu'il ne ressemble pas aux faibles mortels, que les louanges et les présents séduisent : la vertu seule peut lui plaire._
Voilà le précis de toute morale et de toute religion.
XLIV. D'Épicure1
1\- Épicure (341 av. J.-C. - 270 av. J.-C.), philosophe grec fondateur vers 306 à Athènes de l'école du Jardin (centre de l'épicurisme). On ne connaît son œuvre que par des lettres dont les célèbres _Lettres à Ménécée_ sur la morale.
Des pédants de collège, des petits-maîtres de séminaire, ont cru, sur quelques plaisanteries d'Horace1 et de Pétrone2, qu'Épicure avait enseigné la volupté par les préceptes et par l'exemple. Épicure fut toute sa vie un philosophe sage, tempérant et juste. Dès l'âge de douze à treize ans, il fut sage ; car lorsque le grammairien qui l'instruisait lui récita ce vers d'Hésiode3 :
_Le Chaos fut produit le premier de tous les êtres_
Eh ! qui le produisit, dit Épicure, puisqu'il était le premier ? Je n'en sais rien, dit le grammairien ; il n'y a que les philosophes qui le sachent. Je vais donc m'instruire chez eux, repartit l'enfant ; et depuis ce temps jusqu'à l'âge de soixante et douze ans, il cultiva la philosophie. Son testament, que Diogène de Laërce4 nous a conservé tout entier, découvre une âme tranquille et juste ; il affranchit les esclaves qu'il croit avoir mérité cette grâce : il recommande à ses exécuteurs testamentaires de donner la liberté à ceux qui s'en rendront dignes. Point d'ostentation, point d'injuste préférence ; c'est la dernière volonté d'un homme qui n'en a jamais eu que de raisonnables. Seul de tous les philosophes, il eut pour amis tous ses disciples, et sa secte fut la seule où l'on sut aimer, et qui ne se partagea point en plusieurs autres.
Il paraît, après avoir examiné sa doctrine, et ce qu'on a écrit pour et contre lui, que tout se réduit à la dispute entre Malebranche et Arnauld5. Malebranche avouait que le plaisir rend heureux, Arnauld le niait ; c'était une dispute de mots, comme tant d'autres disputes où la philosophie et la théologie apportent leur incertitude, chacune de son côté6.
1\- Horace (65 av. J.-C. - 8 av. J.-C.), poète latin, auteur des _Satires_ , des _Épîtres_ et des _Odes_ notamment.
2\- Pétrone, écrivain latin mort en 65, auteur du _Satiricon_ , grand seigneur épicurien, intime de Néron, fut compromis dans la conjuration de Pison et contraint de se tuer.
3\- Hésiode (VIIIe-VIIe siècle av. J.-C.), poète grec, auteur notamment de l'ouvrage _Les Travaux et les Jours_ et de la _Théogonie_ d'où est tiré le vers cité par Voltaire.
4\- Diogène Laërce consacre à Épicure le livre X des _Vies et doctrines des philosophes illustres_ , Paris, Librairie générale française, 1999 (Voltaire avait marqué la partie sur Épicure d'un signet, voir _Bibliothèque de Voltaire_ , n° 1042).
5\- Un débat intense sur la nature du plaisir et sur son rapport au bonheur a lieu à la fin du du XVIIe et au début du XVIIIe siècle. Le point de vue de Pascal, qui a pour maxime de renoncer à tout plaisir, est contré par Malebranche, pour qui la disposition machinale à aimer le plaisir peut être transformée en vertu. Malebranche est mis en accusation par Antoine Arnauld, qui insiste sur la distinction nécessaire entre les « vrais » et les « faux » plaisirs. Bayle prend la défense de Malebranche, il aborde le sujet sur le plan pragmatique : les gens sont heureux quand ils ont du plaisir et tout le reste est argutie.
6\- Voltaire renvoie ici dos à dos philosophie et théologie dont il réduit les disputes à des disputes de mots, ce qui confirme le scepticisme dont il fait preuve depuis le Premier Doute.
XLV. Des Stoïciens1
1\- Les stoïciens ressortissent à l'école philosophique fondée par Zénon de Citium vers 300 av. J.-C. à Athènes. Le nom de stoïciens vient du grec _stoa poikilê_ , un portique de l'Agora à Athènes où les stoïciens se réunissaient et enseignaient (d'où le nom d'école du Portique). Dans l'usage courant du terme Stoïciens, on considère l'aspect moral de leur philosophie : on entend en effet par stoïcisme une attitude caractérisée par l'indifférence à la douleur et le courage face aux difficultés de l'existence. Il ne reste que des fragments des œuvres des premiers stoïciens (Zénon de Citium – 344-262 –, Cléanthe), et les seules œuvres complètes des stoïciens que nous possédons sont celles de Sénèque, Épictète et Marc Aurèle.
Si les épicuriens rendirent la nature humaine aimable, les stoïciens la rendirent presque divine. Résignation à l'Être des êtres, ou plutôt élévation de l'âme jusqu'à cet Être ; mépris du plaisir, mépris même de la douleur, mépris de la vie et de la mort, inflexibilité dans la justice ; tel était le caractère des vrais Stoïciens ; et tout ce qu'on a pu dire contre eux, c'est qu'ils décourageaient le reste des hommes.
Socrate, qui n'était pas de leur secte1, fit voir qu'on pouvait pousser la vertu aussi loin qu'eux, sans être d'aucun parti ; et la mort de ce martyr de la Divinité est l'éternel opprobre d'Athènes, quoiqu'elle s'en soit repentie.
Le stoïcien Caton2 est d'un autre côté l'éternel honneur de Rome. Épictète3, dans l'esclavage, est peut-être supérieur à Caton, en ce qu'il est toujours content de sa misère. Je suis, dit-il, dans la place où la Providence a voulu que je fusse ; m'en plaindre, c'est l'offenser.
Dirai-je que l'empereur Antonin4 est encore au-dessus d'Épictète, parce qu'il triompha de plus de séductions, et qu'il était bien plus difficile à un empereur de ne se pas corrompre, qu'à un pauvre de ne pas murmurer ? Lisez les Pensées de l'un et de l'autre5 ; l'empereur et l'esclave vous paraîtront également grands.
Oserai-je parler ici de l'empereur Julien6 ? Il erra sur le dogme ; mais certes il n'erra pas sur la morale. En un mot, nul philosophe dans l'Antiquité qui n'ait voulu rendre les hommes meilleurs.
Il y a eu des gens parmi nous qui ont dit, que toutes les vertus de ces grands hommes n'étaient que des péchés illustres7. Puisse la terre être couverte de tels coupables !
1\- Voltaire parle de Socrate qui n'est pas stoïcien pour le plaisir de parler de Socrate, figure mythique pour les Lumières (voir R. Trousson, _Socrate devant Voltaire, Diderot et Rousseau_ , Paris, Minard, 1967).
2\- Caton d'Utique (93-46 av. J.-C.), arrière-petit-fils de Caton l'Ancien, défenseur de la République romaine et stoïcien radical, s'opposa aux revendications populaires, prit parti pour Cicéron contre Catilina, se dressa contre Crassus, César et Pompée auquel il finit par s'allier. Quand l'armée pompéienne fut vaincue à Thapsus en 46 av. J.-C., ce qui signifiait l'effondrement de la République, il se donna la mort.
3\- Épictète (50-125 ou 130 apr. J.-C.), philosophe stoïcien, esclave à Rome. On raconte qu'à son maître qui lui tordait la jambe dans un appareil de torture, Épictète aurait dit : « Tu vas la casser », ajoutant simplement une fois la prédiction accomplie : « Ne te l'avais-je pas dit ? » C'est à son disciple Arrien qu'on doit la rédaction des leçons d'Épictète, les _Entretiens_ et le _Manuel_ où priment les questions morales.
4\- Il s'agit de Marc Aurèle (un des sept empereurs antonins). Philosophe stoïcien (121-180), il fut empereur de 162 à 180 et c'est à la fin de sa vie qu'il rédigea ses _Pensées_ qui constituent le dernier grand témoignage du stoïcisme antique.
5\- C'est-à-dire les _Entretiens_ et le _Manuel_ d'Épictète et les _Pensées pour moi-même_ de Marc Aurèle (sous le titre de _Pensées_ in _Bibliothèque de Voltaire_ , n° 2312).
6\- Julien l'Apostat (331-363) fut empereur romain de 361 à 363. Le jeune prince philosophe fit succéder, au règne étouffant de son prédécesseur (Constance II), un gouvernement modéré qui marqua un retour à la tolérance religieuse, ce qui favorisa la résurgence du paganisme. L'empereur lui-même, après avoir rejeté le christianisme, se fit le restaurateur de la religion païenne servant un culte solaire. Il mourut en 363 en combattant les Perses.
7\- Thèse de saint Augustin que Voltaire critique aussi dans l'article _Catéchisme chinois_ , Cinquième Entretien, in _Dictionnaire philosophique_ , Paris, Imprimerie nationale, 1994, p. 143 : « il y a des peuples assez impertinents pour dire que nous ne connaissons pas la vraie vertu, que nos bonnes actions sont des péchés splendides ».
XLVI. Philosophie est vertu
Il y eut des sophistes, qui furent aux philosophes ce que les hommes sont aux singes. Lucien1 se moqua d'eux ; on les méprisa. Ils furent à peu près ce qu'ont été les moines mendiants dans les universités2. Mais n'oublions jamais que tous les philosophes ont donné de grands exemples de vertu, et que les sophistes, et même les moines, ont tous respecté la vertu dans leurs écrits3.
1\- Lucien (125-192), écrivain satirique grec, abandonna sa carrière d'avocat pour une vie de sophiste vagabond. Après avoir voyagé en Europe, il se fixa à Athènes et renia la sophistique. On lui doit les _Dialogues des morts_ et _Le Songe ou le Coq_. Les cibles de ses satires sont les vices sociaux, la richesse, les sophistes, les cyniques, la religion païenne, le christianisme, les vogues littéraires et, finalement, toutes les valeurs dominantes de son temps.
2\- Voltaire parle ici des deux ordres mendiants des dominicains et des franciscains qui eurent rapidement un rôle important dans les universités.
3\- Remarque ironique : dans leurs écrits mais non dans leurs actes.
XLVII. D'Ésope1
1\- Ésope (VIe s. av. J.-C.), fabuliste grec, figure légendaire, difforme et bègue, esclave affranchi, grand voyageur dans le Proche-Orient et en Grèce où il aurait été mis à mort par les prêtres de Delphes. Les _Fables_ qu'on lui attribue, populaires dès le Ve siècle, ont été diffusées dans toutes les littératures européennes et orientales. La rédaction en prose grecque qui nous en est parvenue est une compilation due à Planude au XIVe siècle, auteur d'une romanesque _Vie d'Ésope_. Dans cette version, les apologues sont des récits très brefs sans fioriture, suivis d'une morale. Les personnages y sont généralement des animaux porteurs des principaux caractères humains, au sein d'une narration poétique dont La Fontaine renouvela la célébrité.
Je placerai Ésope parmi ces grands hommes, et même à la tête de ces grands hommes, soit qu'il ait été le Pilpay1 des Indiens, ou l'ancien précurseur de Pilpay, ou le Lokman2 des Perses, ou le Akkim des Arabes, ou le Hacam3 des Phéniciens, il n'importe ; je vois que ses fables ont été en vogue chez toutes les nations orientales, et que l'origine s'en perd dans une antiquité dont on ne peut sonder l'abîme. À quoi tendent ces fables aussi profondes qu'ingénues, ces apologues qui semblent visiblement écrits dans un temps où l'on ne doutait pas que les bêtes n'eussent un langage ? Elles ont enseigné presque tout notre hémisphère. Ce ne sont point des recueils de sentences fastidieuses qui lassent plus qu'elles n'éclairent ; c'est la vérité elle-même avec le charme de la fable. Tout ce qu'on a pu faire, c'est d'y ajouter des embellissements dans nos langues modernes. Cette ancienne sagesse est simple et nue dans le premier auteur. Les grâces naïves dont on l'a ornée en France n'en ont point caché le fonds respectable4. Que nous apprennent toutes ces fables ? qu'il faut être juste.
1\- Personnage légendaire à qui l'on attribuait le Pañchatantra « les cinq livres », ancien recueil de contes et fables en sanskrit. Une version persane est traduite en français par Gilbert Gaulmin sous un pseudonyme, en 1644, sous le titre _Le Livre des lumières ou la Conduite des Rois, composée par le sage Pilpay Indien, traduite en français par David Sahid, d'Ispahan, ville capitale de Perse_. La Fontaine, dans la préface de sa seconde collection des _Fables_ reconnaît sa dette envers Pilpay.
2\- Autre double oriental, avec Pilpay, d'Ésope, Lokman (Luqman) est lui aussi un personnage légendaire venu de Perse, à qui sont attribuées, dans les recueils arabes, d'autres fables hindoues. Synthèse de ces sources, _Le Livre des lumières, ou la Conduite des rois, traduite en français par David Sahib d'Ispahan_ fut traduit en 1644, et l'on pensa alors que la fable ésopique et la fable hindoue avaient une même source babylonienne. L'une des origines de la fable est donc ce récit moral venu des contes orientaux dans lequel des animaux et des choses animées prennent place.
3\- Akkim ou Hacam vient probablement de l'arabe _hakim_ qui signifie le sage, l'habile homme.
4\- Hommage ambigu à La Fontaine.
XLVIII. De la paix née de la philosophie
Puisque tous les philosophes avaient des dogmes différents, il est clair que le dogme et la vertu sont d'une nature entièrement hétérogène1. Qu'ils crussent ou non que Thétis était la déesse de la mer, qu'ils fussent persuadés ou non de la guerre des géants et de l'âge d'or, de la boîte de Pandore et de la mort du serpent Pithon2 etc., ces doctrines n'avaient rien de commun avec la morale. C'est une chose admirable dans l'Antiquité que la théogonie3 n'ait jamais troublé la paix des nations4.
1\- Autrement dit, on peut penser les fondements de la morale indépendamment de ceux de la religion, ce que Voltaire a déjà soutenu à plusieurs reprises (voir ci-dessus les Doutes XXX, XXXI, XXXIV, XXXVI, XXXVIII).
2\- Références à la mythologie grecque.
3\- Système qui explique la naissance des dieux et leur généalogie, Hésiode est l'auteur célèbre du poème religieux intitulé la _Théogonie_.
4\- C'est pourquoi il faut penser, à l'instar des Anciens, une morale indépendante de la théologie et de la métaphysique chrétienne, juive ou musulmane.
XLIX. Questions
Ah ! si nous pouvions imiter l'Antiquité ! si nous faisions enfin à l'égard des disputes théologiques, ce que nous avons fait au bout de dix-sept siècles dans les belles-lettres !
Nous sommes revenus au goût de la saine Antiquité, après avoir été plongés dans la barbarie de nos écoles1. Jamais les Romains ne furent assez absurdes pour imaginer qu'on pût persécuter un homme parce qu'il croyait le vide ou le plein, parce qu'il prétendait que les accidents ne peuvent pas subsister sans sujet, parce qu'il expliquait en un sens un passage d'un auteur, qu'un autre entendait dans un sens contraire2.
Nous avons recours tous les jours à la jurisprudence des Romains ; et quand nous manquons de lois (ce qui nous arrive si souvent) nous allons consulter le Code et le Digeste3. Pourquoi ne pas imiter nos maîtres dans leur sage tolérance ?
Qu'importe à l'État qu'on soit du sentiment des réaux ou des nominaux4, qu'on tienne pour Scot5 ou pour Thomas6, pour Œcolampade7 ou pour Mélancton8, qu'on soit du parti d'un évêque d'Ypre9, qu'on n'a point lu, ou d'un moine espagnol10 qu'on a moins lu encore ? N'est-il pas clair que tout cela doit être aussi indifférent au véritable intérêt d'une nation, que de traduire bien ou mal un passage de Lycophron11 ou d'Hésiode12 ?
1\- À l'instar de la pensée dominante jusqu'au XXe siècle, Voltaire considère la scolastique et le Moyen Âge comme une période de ténèbres.
2\- Allusion aux débats scolastiques qui perdurent jusqu'au XVIIIe siècle (par exemple, la question du vide ou celle de la transsubstantiation).
3\- En 527, l'empereur Justinien demande aux juristes d'unifier les lois en un code : le Code justinien et le Digeste, principales sources du droit romain.
4\- Allusion à la querelle des universaux qui agita les philosophes médiévaux du XIIe au XIVe siècle et qui opposa ceux qui soutenaient que les noms expriment l'essence des choses (les réaux) et ceux qui affirmaient que les noms sont des désignations arbitraires des choses (les nominaux).
5\- Duns Scot (1270-1308), théologien et philosophe écossais (de l'ordre des franciscains), fut appelé Docteur subtil en raison de son habileté à manier la dialectique, qui lui servit sur bien des points à critiquer l'aristotélisme et le thomisme. Sa philosophie affirme la priorité de la foi et de la volonté sur la raison.
6\- Thomas d'Aquin (1227-1274), théologien et philosophe italien (de l'ordre des dominicains), surnommé l'Ange de l'École en raison de la sainteté de sa vie, fut l'élève d'Albert le Grand. On lui doit la _Somme théologique_ qui constitue sans doute la tentative la plus complète du Moyen Âge pour penser la religion chrétienne. Il cherche à construire la théologie à l'aide de la dialectique et de la scolastique, à accorder la foi et la raison, les dogmes du christianisme et les théories d'Aristote, tout en maintenant la primauté de la théologie par rapport à la philosophie. Il fut canonisé en 1323. Le thomisme fut considéré comme la doctrine officielle de l'Église catholique par Léon XIII (pape de 1878 à 1903).
7\- Œcolampade (nom hellénisé de J. Hausschein, 1482-1531), théologien allemand qui adhéra aux idées de la Réforme et tenta en vain de réconcilier Zwingli et Luther sur la question de la communion.
8\- Mélanchton (nom hellénisé de P. Schwarzerd, 1497-1560), professeur de grec à l'université de Wittenberg où il rencontra Luther dont il fut le principal disciple. Il devint le chef de l'Église luthérienne à la mort de Luther. Moins intransigeant que ce dernier, il a cherché à apaiser les conflits entre les différents courants de la Réforme et entre protestants et catholiques.
9\- Jansénius (1585-1638), auteur de l' _Augustinus_ et fondateur du jansénisme qui prônait que la grâce efficace reçue par les seuls élus est une condition absolue du Salut.
10\- Le jésuite espagnol Molina (1535-1601) avait soutenu au contraire que la grâce suffisante reçue de tous les hommes suffit à gagner son salut si l'on use bien de son libre arbitre en agissant vertueusement.
11\- Lycophron de Chalcis (fin du IVe-IIIe siècle av. J.-C.), poète grec dont il nous est parvenu l'ouvrage ésotérique _Alexandra_. Lycophron, imitant le style des oracles, a chargé son poème d'une érudition et d'une obscurité voulues.
12\- Hésiode, poète grec (voir ci-dessus note du Doute XLIV).
L. Autres questions
Je sais que les hommes sont quelquefois malades du cerveau. Nous avons eu un musicien1 qui est mort fou, parce que sa musique n'avait pas paru assez bonne. Des gens ont cru avoir un nez de verre2 ; mais s'il y en avait d'assez attaqués pour penser, par exemple, qu'ils ont toujours raison, y aurait-il assez d'ellébore3 pour une si étrange maladie ?
Et si ces malades, pour soutenir qu'ils ont toujours raison, menaçaient du dernier supplice quiconque pense qu'ils peuvent avoir tort, s'ils établissaient des espions pour découvrir les réfractaires, s'ils décidaient qu'un père sur le témoignage de son fils, une mère sur celui de sa fille, doit périr dans les flammes4, etc., ne faudrait-il pas lier ces gens-là, et les traiter comme ceux qui sont attaqués de la rage ?
1\- Il s'agit sans doute de Jean-Joseph Mouret (1682-1738), musicien attaché au service de la duchesse du Maine et qui, à la mort de celle-ci, se retrouva sans emploi et mourut fou.
2\- Descartes, dans la première Méditation, parle des fous qui imaginent avoir un corps de verre (voir _Méditations_ , in _Œuvres philosophiques_ , Paris, Garnier, en trois tomes, 1963-1973, t. II, p. 406).
3\- Plante médicinale aux propriétés purgatives et vermifuges, qui passait autrefois pour guérir de la folie.
4\- Voltaire rappelle ici le processus de délation généralisée à l'œuvre dans l'Inquisition.
LI. Ignorance
Vous me demandez à quoi bon tout ce sermon, si l'homme n'est pas libre ? D'abord je ne vous ai point dit que l'homme n'est pas libre ; je vous ai dit, que sa liberté consiste dans son pouvoir d'agir, et non pas dans le pouvoir chimérique de _vouloir vouloir_1. Ensuite je vous dirai que tout étant lié dans la nature, la Providence éternelle me prédestinait à écrire ces rêveries, et prédestinait cinq ou six lecteurs à en faire leur profit, et cinq ou six autres à les dédaigner et à les laisser dans la foule immense des écrits inutiles.
Si vous me dites que je ne vous ai rien appris, souvenez-vous que je me suis annoncé comme un ignorant.
1\- Rappel des discussions du Doute XIII.
LII. Autres ignorances
Je suis si ignorant, que je ne sais pas même les faits anciens dont on me berce ; je crains toujours de me tromper de sept à huit cents années au moins1, quand je cherche en quel temps ont vécu ces antiques héros, qu'on dit avoir exercé les premiers le vol et le brigandage dans une grande étendue de pays ; et ces premiers sages qui adorèrent des étoiles ou des poissons, ou des serpents, ou des morts, ou des êtres fantastiques.
Quel est celui qui le premier imagina les six Gahambars2, et le pont de Tshinavar, et le Dardaroth, et le lac de Karon3 ? en quel temps vivaient le premier Bacchus, le premier Hercule, le premier Orphée4 ?
Toute l'Antiquité est si ténébreuse jusqu'à Thucydide5 et Xénophon6, que je suis réduit à ne savoir presque pas un mot de ce qui s'est passé sur le globe que j'habite, avant le court espace d'environ trente siècles7 ; et dans ces trente siècles encore, que d'obscurités ! que d'incertitudes ! que de fables !
1\- Voltaire s'est beaucoup intéressé à la question de la chronologie, notamment à la chronologie biblique : il doutait que le monde ait été créé il y a seulement 6 000 ans ; il a cherché à rendre l'histoire rigoureuse et à la distinguer clairement des fables (voir son article HISTOIRE de l' _Encyclopédie_ , VIII, 1765, 220-225).
2\- Nom des six fêtes qui célébraient les six saisons dans le calendrier de Zoroastre.
3\- Le pont de Tshinavar, chez les Perses, est le pont où l'âme est l'objet d'une lutte entre le bien et le mal lors de son dernier voyage (voir Thomas Hyde, _Historia religionis veterum Persarum_ , 1700), le Dardaroth (nom égyptien qui a donné Tartare et qui signifie habitation éternelle) et le lac de Karon (le batelier égyptien du lac Querron devient chez les Grecs le batelier Caron qui demeure toujours sur le lac) désignent respectivement dans les anciennes religions persane, égyptienne et grecque les lieux de l'ultime passage des âmes des morts (voir l'article ENFER de Jaucourt, in _Encyclopédie_ , V, 1755, 670-672).
4\- Bacchus, appellation latine de Dionysos, dieu du vin et du délire extatique, Hercule, héros grec, et Orphée, aède grec, sont des figures mythiques dont on ne peut assigner exactement la naissance.
5\- Thucydide (470 ou 460-400 ou 395 av. J.-C.), historien grec réputé pour avoir exclu de l'histoire le merveilleux ou le destin. En ce sens, sa conception chronologique et objective de l'histoire a marqué toute la pensée historique occidentale.
6\- Xénophon (430 ou 425-355 ou 352 av. J.-C.), historien et militaire grec, polygraphe autant qu'homme d'action, voulut être le continuateur de Thucydide et prétendit être le rival de Platon.
7\- Pour Voltaire, les temps historiques commencent avec les Grecs. Avant, tout n'est que mythe et obscurité, comme l'atteste la fin du chapitre XI sur les Babyloniens devenus Persans dans l'Introduction de l' _Essai sur les mœurs et l'esprit des nations_ , où il souligne qu'en lisant l'histoire, on doit être en garde contre toute fable.
LIII. Plus grande ignorance
Mon ignorance me pèse bien davantage, quand je vois que ni moi, ni mes compatriotes, nous ne savons absolument rien de notre patrie. Ma mère m'a dit que j'étais né sur les bords du Rhin1 ; je le veux croire. J'ai demandé à mon ami le savant Apédeutès2, natif de Courlande3, s'il avait quelque connaissance des anciens peuples du Nord ses voisins, et de son malheureux petit pays ? Il m'a répondu qu'il n'en avait pas plus de notion que les poissons de la mer Baltique.
Pour moi, tout ce que je sais de mon pays, c'est que César dit4, il y a environ dix-huit cents ans, que nous étions des brigands, qui étions dans l'usage de sacrifier des hommes à je ne sais quels dieux pour obtenir d'eux quelque bonne proie, et que nous n'allions jamais en course qu'accompagnés de vieilles sorcières qui faisaient ces beaux sacrifices.
Tacite5, un siècle après, dit quelques mots de nous, sans nous avoir jamais vus : il nous regarde comme les plus honnêtes gens du monde en comparaison des Romains ; car il assure que quand nous n'avions personne à voler, nous passions les jours et les nuits à nous enivrer de mauvaise bière dans nos cabanes.
Depuis ce temps de notre âge d'or, c'est un vide immense jusqu'à l'histoire de Charlemagne. Quand je suis arrivé à ces temps connus, je vois dans Goldstad6 une charte de Charlemagne datée d'Aix-la-Chapelle, dans laquelle ce savant empereur parle ainsi :
_Vous savez que chassant un jour auprès de cette ville, je trouvai les thermes et le palais que Granus, frère de Néron et d'Agrippa, avait autrefois bâtis_7 _._
Ce Granus et cet Agrippa, frères de Néron8, me font voir que Charlemagne était aussi ignorant que moi ; et cela soulage.
1\- Une des nombreuses fantaisies autobiographiques de Voltaire qui est né, en réalité, à Paris.
2\- « Apédeutès », nom fantaisiste inventé par Voltaire qui vient du grec _a-paideion_ « sans éducation, sans connaissance ». Le « savant Apédeutès » est synonyme de « philosophe ignorant ». Ce nom est repris par Voltaire dans _L'Ingénu_ (1767) où il parle des apédeutes de Constantinople qui dénoncent l'hérésie en très mauvais grec, in _Romans et contes_ , éd. citée, p. 353-354.
3\- La Courlande est la région Ouest de la Lettonie entre la mer Baltique et la Daugava.
4\- Voir César, _La Guerre des Gaules_ , trad. M. Rat, Paris, Flammarion, 1964.
5\- Tacite (55-120), historien latin, voulait aussi faire œuvre morale, sauver les vertus de l'oubli et stigmatiser les vices. Ses deux grands ouvrages historiques sont les _Histoires_ et les _Annales_. Mais Voltaire, dans la suite du paragraphe fait référence à un autre ouvrage : _Des mœurs des Germains_ , XXI-XXIII.
6\- Goldast (1576-1635), historien suisse, chancelier de l'université de Giessen, qui a réuni plusieurs documents sur l'empire de Charlemagne dont les _Constitutions impériales_ (1607).
7\- Charlemagne livre ainsi l'origine du nom Aix-la-Chapelle qui s'appelait sous les Romains _Aquae_ (d'où Aix) _Grani_ – Thermes de Granus ; on a ajouté la Chapelle en raison de la magnifique chapelle Palatine construite entre 795 et 805 par Charlemagne, sur le site de l'ancien palais romain.
8\- Charlemagne est ignorant en ce sens que Granus ne peut être en même temps le frère d'Agrippa, homme politique romain, né en 63 et mort en 12 av. J.-C., et le frère de l'empereur Néron, né en 37 apr. J.-C. et mort en 68 apr. J.-C.
LIV. Ignorance ridicule
L'histoire de l'Église de mon pays ressemble à celle de Granus frère de Néron et d'Agrippa, et est bien plus merveilleuse. Ce sont de petits garçons ressuscités1, des dragons pris avec une étole comme des lapins avec un lacet2 ; des hosties qui saignent d'un coup de couteau qu'un juif leur donne3 ; des saints qui courent après leurs têtes quand on les leur a coupées4. Une des légendes les plus avérées dans notre histoire ecclésiastique d'Allemagne, est celle du bienheureux Pierre de Luxembourg5, qui dans les deux années 1388 et 89 après sa mort, fit deux mille quatre cents miracles ; et les années suivantes, trois mille de compte fait ; parmi lesquels on ne nomme pourtant que quarante-deux morts ressuscités.
Je m'informe si les autres États de l'Europe ont des histoires ecclésiastiques, aussi merveilleuses et aussi authentiques ? Je trouve partout la même sagesse et la même certitude.
1\- Voir la légende de saint Nicolas.
2\- Nous n'avons pas retrouvé la trace de cette légende, la Bible parle du Dragon combattu par Michel et ses anges, puis terrassé dans Apocalypse, XII.
3\- La vieille légende du Moyen Âge, que Voltaire dénonce ici comme une ignorance ridicule, porte l'antijudaïsme chrétien : les Juifs profaneraient les hosties pour rejouer symboliquement la scène du déicide (voir Jean-Louis Schefer, _L'Hostie profanée. Histoire d'une fiction théologique_ , Paris POL, 2007). Autre exemple qui s'inscrit en faux contre le prétendu antijudaïsme de Voltaire (voir aussi note 3 du Doute XXXI, p. 85).
4\- Il s'agit des saints céphalophores comme saint Denis.
5\- Pierre de Luxembourg (1369-1387) fut nommé chanoine à 10 ans, évêque à 15 ans, enfin cardinal d'Avignon à 17 ans, quelques mois avant sa mort.
LV. Pis qu'ignorance
J'ai vu ensuite pour quelles sottises inintelligibles les hommes s'étaient chargés les uns les autres d'imprécations, s'étaient détestés, persécutés, égorgés, pendus, roués et brûlés ; et j'ai dit : S'il y avait eu un sage dans ces abominables temps, il aurait donc fallu que ce sage vécut et mourût dans les déserts.
LVI. Commencement de la raison
Je vois qu'aujourd'hui, dans ce siècle qui est l'aurore de la raison, quelques têtes de cette hydre du fanatisme renaissent encore1. Il paraît que leur poison est moins mortel, et leurs gueules moins dévorantes. Le sang n'a pas coulé pour la grâce versatile2, comme il coula si longtemps pour les indulgences plénières3 qu'on vendait au marché ; mais le monstre subsiste encore ; quiconque recherchera la vérité risquera d'être persécuté. Faut-il rester oisif dans les ténèbres ? ou faut-il allumer un flambeau auquel l'envie et la calomnie rallumeront leurs torches ? Pour moi, je crois que la vérité ne doit pas plus se cacher devant ces monstres, que l'on ne doit s'abstenir de prendre de la nourriture dans la crainte d'être empoisonné4.
1\- L'Inquisition sévit toujours en Espagne et au Portugal, en France la condamnation au supplice de la roue de Jean Calas (1762) et la condamnation au bûcher du chevalier de La Barre (1766) attestent que le fanatisme est encore présent.
2\- C'est-à-dire, selon Molina et les jésuites, la grâce suffisante, reçue de tous, que l'on peut rendre efficace ou non pour son Salut par la détermination de sa volonté et de son libre arbitre en agissant vertueusement ou non (voir l'article _Grâce_ in _Dictionnaire philosophique_ , Paris, Imprimerie nationale, 1994, p. 282-284).
3\- Le trafic des indulgences délivrées par le pape et qu'il fallait acheter pour sauver son âme est l'une des causes principales du schisme protestant.
4\- Note finale résolument engagée qui rompt avec le scepticisme ambiant de l'ouvrage. Voltaire défend ici la figure du philosophe ignorant mais militant pour la vérité (voir son engagement dans l'affaire Calas ou celle du chevalier de La Barre), il s'accorderait à dire avec d'Alembert : « Il me semble qu'il ne faut pas, comme Fontenelle, tenir la main fermée quand on est sûr d'y avoir la vérité ; il faut seulement ouvrir avec sagesse et avec précaution les doigts de la main l'un après l'autre, et petit à petit la main est ouverte tout à fait, et la vérité en sort tout entière. Les philosophes qui ouvrent la main trop brusquement sont des fous ; on leur coupe le poing et voilà tout ce qu'ils y gagnent : mais ceux qui la tiennent fermée absolument, ne font pas pour l'humanité ce qu'ils doivent » (lettre de d'Alembert à Frédéric du 7 mars 1770, in _Œuvres_ , Paris, Belin, 1821-1822, t. 5, p. 290-291).
GLOSSAIRE
**A RGUMENT DU DESSEIN :** Raisonnement qui, à partir du constat de l'ordre du monde, infère le principe d'ordre : Dieu, qui a eu le dessein de faire un monde ordonné.
**A TOMISME** : Conception selon laquelle la matière n'est pas indivisible à l'infini mais est constituée d'éléments insécables, les atomes (du grec _a-tomos_ , ce qu'on ne peut couper).
**A THÉISME** : Doctrine selon laquelle Dieu n'existe pas ; l'ordre du monde ne renvoie pas au créateur mais est immanent à l'organisation de la matière. Dans ses _Pensées diverses sur la comète_ , Bayle, contrairement au raisonnement traditionnel qui condamne l'athéisme au nom de la morale et de la piété, soutient que l'athéisme est bien moins dangereux que l'idolâtrie ou la superstition, et que l'athée ne porte aucune atteinte aux mœurs du peuple.
**D ÉISME** : Doctrine qui soutient l'existence de Dieu, la réalité du bien et du mal et la Providence divine. Mais, à l'encontre des religions traditionnelles, le déiste nie la révélation et les dogmes, il doute de l'immortalité de l'âme, des peines et des récompenses à venir : à l'instar de Voltaire, le déiste peut penser que le monde est co-éternel à Dieu et ne pas admettre la création _ex nihilo_.
**E MPIRISME** : Conception selon laquelle toutes nos idées viennent des sens ; toutes nos connaissances commencent avec l'expérience.
**F ANATISME** : L'article FANATISME de Deleyre ( _Encyclopédie_ , t. VI, 1756, p. 393-401) définit ainsi le terme : « c'est un zèle aveugle et passionné, qui naît des opinions superstitieuses, et fait commettre des actions ridicules, injustes, et cruelles ; non seulement sans honte et sans remords, mais encore avec une sorte de joie et de consolation. Le _fanatisme_ n'est donc que la superstition mise en action ». La fin de l'article invite le lecteur à opérer le renversement de raisonnement promu par Bayle : ce n'est pas l'athéisme qui est dangereux mais le fanatisme qui fait beaucoup plus de mal au monde que l'impiété, ce n'est pas celle-ci qui est inquiétante pour la paix civile mais bien celui-là.
**F ATALISME** : Doctrine selon laquelle tout ce qui arrive est nécessaire que ce soit dans l'ordre moral ou dans l'ordre physique.
**F INALISME** : Conception fondée sur le principe des causes finales que Voltaire résume dans la formule : « l'horloge prouve l'horloger ».
**I DÉES INNÉES** : Idées que l'on a dès la naissance. Selon Descartes, Dieu a mis en l'esprit de tous les hommes ce qu'il appelle la lumière naturelle ou les semences de vérité ou les idées innées qui sont la source des connaissances humaines.
**I MPIE** : L'article anonyme IMPIE de l' _Encyclopédie_ , après avoir défini l' _impie_ comme « celui qui médit d'un Dieu qu'il adore au fond de son cœur » et comme « un méchant à mépriser » (in _Encyclopédie_ , t. VIII, 1765, p. 597), est tout entier consacré à dénoncer les procédés de délation relatifs à cette appellation : « on compromet la fortune, le repos, la liberté et même la vie de celui qu'on se plaît à traduire comme un impie » ( _ibid._ ). L'article IMPIE procède ainsi au même renversement de raisonnement qu'on pourrait traduire par « impie soit qui mal y pense ». Car, en réalité, les prétendus _impies_ sont simplement des hétérodoxes, c'est-à-dire des penseurs qui ont des doutes et qui les proposent au public.
**M ATÉRIALISME** : Les matérialistes soutiennent que la matière est éternelle, qu'elle s'auto-organise et que ses lois suffisent à expliquer l'ordre du monde sans recourir à un quelconque Dieu. Le matérialisme est représenté en France par La Mettrie, Diderot, Helvétius et le baron d'Holbach. L'article anonyme MATÉRIALISTES ( _Encyclopédie_ , X, 1765, 188) présente l'axiome fondamental de la physique ancienne – rien ne se fait de rien – comme la définition même de la pensée matérialiste.
**M ÉCANIQUE** : La mécanique est la science du mouvement des corps, elle constitue avec la science de la résistance des matériaux les deux sciences nouvelles dont traite Galilée dans son _Discours sur deux sciences nouvelles_ (1638). Descartes étend le sens du terme mécanique au vivant par sa théorie de l'animal-machine (voir son _Discours de la méthode_ , cinquième partie) selon laquelle les animaux n'ont pas de représentation consciente, n'ont pas d'âme et se réduisent à un corps-machine. Le schème de la machine suffit à rendre compte du corps vivant. Le terme mécanique, parce qu'il répond à un besoin qui est de désigner le cadre conceptuel de la science nouvelle de la mécanique et des machines, connaît un très grand succès au XVIIe siècle. Au XVIIIe siècle, son usage ou plutôt ses usages deviennent très répandus et le terme tend à prendre le sens général de « dispositif ».
**M ONADES** : Les monades, selon Leibniz, sont les unités de substance originairement simples et indivisibles, qu'il appelle aussi atomes métaphysiques ou éléments des choses (voir _La Monadologie_ , Paris, Delagrave, 1983, § 3, p. 143). En tant que support de l'individualité, chaque monade diffère de toute autre. Toute monade est douée de perception, à distinguer de l'aperception ou de la conscience que seules les monades particulières que sont les esprits possèdent. Leibniz distingue ainsi les monades qui sont les êtres simples dotés de perception, les âmes qui sont les êtres simples dotés de perception et de mémoire, et les âmes raisonnables ou esprits dotés d'aperception ou de conscience ( _Monadologie_ , § 29). Leibniz explique que Dieu crée le meilleur des mondes possibles, c'est-à-dire celui où toutes les monades se lient les unes aux autres et participent à l'harmonie du tout, d'où le fait que « chaque substance simple a des rapports qui expriment toutes les autres, et qu'elle est par conséquent un miroir vivant perpétuel de l'univers » ( _Monadologie_ , § 56).
**N ÉCESSITARISME** : Conception selon laquelle tout ce qui est ne peut pas ne pas être. Tout ce qui est possible se réalise. Cette thèse qu'on peut résumer aussi en « tout ce qui arrive est nécessaire » s'accompagne de la négation du libre arbitre (nous croyons être libres parce que nous ignorons les causes qui nous déterminent) et de la négation du hasard (ce qu'on appelle hasard résulte de notre ignorance des chaînes de causalité). Voir aussi « spinozisme ».
**O PTIMISME** : La thèse de l'optimisme est issue du calcul de l'optimum qui est le produit d'un maximum par un minimum. Le Dieu leibnizien crée le meilleur des mondes selon ce calcul : il crée le monde optimal où l'on obtient le maximum de diversité des êtres par le minimum de voies ou de moyens. Par extension, la thèse de l'optimisme désigne une conception qui nie le mal dans le monde : le mal n'est qu'apparent, il est « régional » et non pas global, et contribue à la perfection du tout.
**P ROVIDENCE** : Sage gouvernement de Dieu sur la Création ; avec la majuscule, Providence désigne la Sagesse de Dieu gouvernant la Création. Par extension, le terme signifie le destin. Quand on parle de providence générale, on désigne simplement la nature, son ordre, son organisation, ses lois.
**R ELIGION NATURELLE** / **R ELIGION RÉVÉLÉE** : L'article RELIGION de Jaucourt de l' _Encyclopédie_ commence par distinguer la religion naturelle, c'est-à-dire rationnelle, et la religion révélée, couple répondant à celui de la théologie naturelle et de la théologie révélée (voir ces termes ci-dessous). Jaucourt reprend la conclusion de l'article DÉISTES de l'abbé Mallet : pour prouver que la religion naturelle ne suffit pas, il faut démontrer aux déistes l'existence et la vérité de la révélation. Cependant, la suite de l'article ne consacre aucune partie à la religion révélée mais, en revanche, un très long développement à la RELIGION NATURELLE. L'accent est donc mis sur la religion qui s'appuie sur la raison, que Jaucourt juge semblable à la morale, il tend ainsi à laïciser la religion naturelle : « On l'appelle _morale_ ou _éthique_ , parce qu'elle concerne immédiatement les mœurs et les devoirs des hommes les uns envers les autres, et envers eux-mêmes considérés comme créatures de l'Être suprême » (voir l'article RELIGION, in _Encyclopédie_ , t. XIV, 1765, p. 78-79).
**R ÉVÉLATION** : Processus par lequel des vérités cachées sont données aux hommes de manière surnaturelle (miracles, illumination, vision, acte de foi). La révélation désigne aussi les vérités révélées et les Saintes Écritures où elles sont exposées.
**S CEPTICISME** : Conception selon laquelle les connaissances de l'esprit humain sont limitées. Le scepticisme radical ou pyrrhonisme soutient qu'on ne peut rien connaître. Le scepticisme modéré, qui est celui des Lumières, affirme qu'on ne peut pas connaître les premiers principes des choses et qu'il faut s'en tenir aux faits bien constatés et aux principes des sciences qui en rendent compte, sans chercher à remonter par la raison à un premier principe ou à une première cause.
**S PINOZISME** : Au sens propre, le terme désigne la philosophie de Spinoza et celle de ses sectateurs. Mais, par extension, le terme signifie tout ce qui ressortit à l'athéisme et au matérialisme. Diderot, dans l'article SPINOSISTE (XV, 1765, 474), précise la différence entre les spinozistes anciens (sectateurs de la philosophie de Spinoza) et les spinozistes modernes identifiés aux matérialistes qui soutiennent l'épigénèse et « concluent qu'il n'y a que de la matiere, et qu'elle suffit pour tout expliquer ; du reste ils suivent l'ancien spinosisme dans toutes ses conséquences ».
**S UPERSTITION** : L'article SUPERSTITION de Jaucourt la définit comme « tout excès de la religion en général ». C'est « un culte de religion, faux, mal dirigé, plein de vaines terreurs, contraire à la raison et aux saines idées que l'on doit avoir de l'Être suprême » (in _Encyclopédie_ , t. XV, 1765, p. 670). Jaucourt oppose ici clairement la superstition à la raison. Il ajoute, dans le sillage de Bayle, que la superstition est le plus terrible fléau de l'humanité et qu'elle est bien plus néfaste et dangereuse que l'athéisme.
**T HÉISME** : Doctrine qui affirme l'existence de Dieu, la réalité du bien et du mal, la Providence divine, l'immortalité de l'âme, les peines et les récompenses à venir. Le théiste, à l'encontre du déiste, ne nie pas la révélation mais attend pour l'admettre qu'on la lui démontre.
**T HÉOLOGIE NATURELLE / THÉOLOGIE RÉVÉLÉE** : C'est un couple de notions se référant à deux manières de penser la théologie ou science de Dieu. Dans la théologie naturelle, la raison se met au service de la foi pour conclure du constat de l'ordre de la nature à la nécessité d'un principe d'ordre ou d'un dessein divin créateur de cet ordre. Dans la théologie révélée, au contraire, la raison n'est pas de mise, seule la foi nous convainc de l'autorité des textes sacrés et des vérités qu'ils nous révèlent.
Dossier
_Sur le mot d'ordre des Lumières :_
_sapere aude_
SAPERE AUDE : _« O SE PENSER PAR TOI-MÊME »_
Nous regroupons dans ce dossier des textes qui ont un air de famille avec l'injonction de Voltaire de rechercher la vérité et d'oser penser par soi-même, double injonction qui résonne tout au long du _Philosophe ignorant_. Voltaire invite le lecteur à se méfier des systèmes, des doctrines, des dogmes, mais à ne pas se décourager dans la recherche de la vérité, fût-elle risquée : « quiconque recherchera la vérité risquera d'être persécuté », prévient Voltaire dans le dernier Doute. Mais c'est à ce prix, ajoute-t-il, que la raison triomphera de l'hydre du fanatisme.
Voltaire, à plus de 70 ans, donne l'exemple : il lutte contre ce qu'il appelle l'infâme (« l'inf... »), c'est-à-dire contre toutes les formes de superstition et fanatisme, il se bat pour Calas, il se bat pour le chevalier de La Barre. Voltaire est non seulement l'inventeur de la formule « Écrasons l'inf... » mais aussi du mot d'ordre qui sera présenté par Kant comme la devise des Lumières : _Sapere aude_ , « Aie le courage de te servir de ton propre entendement »1. N'est-ce pas lui qui, dans l'article _Liberté de penser_ du _Dictionnaire philosophique_ , réactualise le vers d'Horace pour demander aux gens de lettres d'oser penser2 : « Celui qui ne sait pas la géométrie peut l'apprendre ; tout homme peut s'instruire ; il est honteux de mettre son âme entre les mains de ceux à qui vous ne confierez pas votre argent : osez penser par vous-mêmes. »
Mais Voltaire n'agit pas seul, les encyclopédistes œuvrent à rendre la philosophie populaire depuis 1751 et cherchent à défricher le chemin des connaissances afin de rendre l'homme libre et de lui permettre de penser par soi-même. Leur combat est une lutte contre toutes les formes de domination et, en premier lieu, contre l'argument d'autorité qu'il soit édicté par un prince, un théologien ou un philosophe à système. Pour mener à bien cette lutte et contourner la censure, ils ont instauré, à côté de l'ordre alphabétique, l'ordre encyclopédique régi par un système de renvois qui permet de construire des réseaux d'articles où se tisse le sens subversif3 que le bon lecteur apprend à décrypter : ainsi ce n'est ni à l'article ÉTAT, ni à l'article ÉGLISE, qu'il faut chercher la pensée politique des encyclopédistes, mais il faut reconstruire le réseau d'articles sur la question des rapports de l'Église à l'État à partir de l'article SUPERSTITION et de ses renvois à FANATISME et à TEMPLIER. Et, dans l'article FANATISME de Deleyre, on lit, sous la forme d'un dialogue fictif, qu'il faut séparer l'Église et l'État.
C'est pourquoi, place est faite dans ce dossier aux encyclopédistes et d'abord à Diderot qui, dès l'article AUTORITÉ (in _Encyclopédie_ , t. I, 1751, p. 900) que nous reproduisons ci-dessous, donne le ton à l'ouvrage : il sort ses griffes contre l'argument d'autorité, le dénonce comme une forme de domination et invite le lecteur à aiguiser sa conscience.
**A UTORITÉ** _dans les discours et dans les écrits._ J'entends par _autorité dans le discours,_ le droit qu'on a d'être cru dans ce qu'on dit : ainsi plus on a de droit d'être cru sur sa parole, plus on a d' _autorité._ Ce droit est fondé sur le degré de science et de bonne foi, qu'on reconnaît dans la personne qui parle. La science empêche qu'on ne se trompe soi-même, et écarte l'erreur qui pourrait naître de l'ignorance. La bonne foi empêche qu'on ne trompe les autres, et réprime le mensonge que la malignité chercherait à accréditer. C'est donc les lumières et la sincérité qui sont la vraie mesure de l' _autorité_ dans le discours. Ces deux qualités sont essentiellement nécessaires. Le plus savant et le plus éclairé des hommes ne mérite plus d'être cru, dès qu'il est fourbe ; non plus que l'homme le plus pieux et le plus saint, dès qu'il parle de ce qu'il ne sait pas ; de sorte que saint Augustin avait raison de dire que ce n'était pas le nombre, mais le mérite des auteurs qui devait emporter la balance. Au reste il ne faut pas juger du mérite, par la réputation, surtout à l'égard des gens qui sont membres d'un corps, ou portés par une cabale. La vraie pierre de touche, quand on est capable et à portée de s'en servir, c'est une comparaison judicieuse du discours avec la matiere qui en est le sujet, considérée en elle-même : ce n'est pas le nom de l'auteur qui doit faire estimer l'ouvrage, c'est l'ouvrage qui doit obliger à rendre justice à l'auteur.
L' _autorité_ n'a de force et n'est de mise, à mon sens, que dans les faits, dans les matières de religion, et dans l'histoire. Ailleurs elle est inutile et hors d'œuvre. Qu'importe que d'autres aient pensé de même ou autrement que nous, pourvu que nous pensions juste, selon les règles du bon sens, et conformément à la vérité ? il est assez indifférent que votre opinion soit celle d'Aristote, pourvu qu'elle soit selon les lois du syllogisme. À quoi bon ces fréquentes citations, lorsqu'il s'agit de choses qui dépendent uniquement du témoignage de la raison et des sens ? À quoi bon m'assurer qu'il est jour, quand j'ai les yeux ouverts et que le soleil luit ? Les grands noms ne sont bons qu'à éblouir le peuple, à tromper les petits esprits, et à fournir du babil aux demi-savants. Le peuple qui admire tout ce qu'il n'entend pas, croit toujours que celui qui parle le plus et le moins naturellement est le plus habile. Ceux à qui il manque assez d'étendue dans l'esprit pour penser eux-mêmes, se contentent des pensées d'autrui, et comptent les suffrages. Les demi-savants qui ne sauraient se taire, et qui prennent le silence et la modestie pour des symptômes d'ignorance ou d'imbécillité, se font des magasins inépuisables de citations.
Je ne prétends pas néanmoins que l' _autorité_ ne soit absolument d'aucun usage dans les sciences. Je veux seulement faire entendre qu'elle doit servir à nous appuyer et non pas à nous conduire ; et qu'autrement, elle entreprendrait sur les droits de la raison : celle-ci est un flambeau allumé par la nature, et destiné à nous éclairer ; l'autre n'est tout au plus qu'un bâton fait de la main des hommes, et bon pour nous soutenir en cas de faiblesse, dans le chemin que la raison nous montre.
Ceux qui se conduisent dans leurs études par l' _autorité_ seule, ressemblent assez à des aveugles qui marchent sous la conduite d'autrui. Si leur guide est mauvais, il les jette dans des routes égarées, ou il les laisse las et fatigués, avant que d'avoir fait un pas dans le vrai chemin du savoir. S'il est habile, il leur fait à la vérité parcourir un grand espace en peu de temps ; mais ils n'ont point eu le plaisir de remarquer ni le but où ils allaient, ni les objets qui ornaient le rivage, et le rendaient agréable.
Je me représente ces esprits qui ne veulent rien devoir à leurs propres réflexions, et qui se guident sans cesse d'après les idées des autres, comme des enfants dont les jambes ne s'affermissent point, ou des malades qui ne sortent point de l'état de convalescence, et ne feront jamais un pas sans un bras étranger.
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L'article DÉISTES (in _Encyclopédie_ , t. IV, 1754, p. 773-774) de l'abbé Mallet ( _G_ ) montre à quel point la figure du déiste est proche de celle du libre-penseur. Le déiste admet l'existence de Dieu mais refuse tout système de religion, tout dogme, toute hiérarchie et toute forme de domination qui émanent de l'institution de l'Église. La fin de l'article distingue deux sortes de déistes. Voltaire appartient à la deuxième espèce : pour lui, la distinction du vice et de la vertu repose sur des lois naturelles et non sur des lois arbitraires voire des conventions (voir le Doute XXXV contre Locke et le Doute XXXVII contre Hobbes) et la morale est naturelle et universelle.
**D ÉISTES**, subst. m. pl. ( _Théolog_.) nom qu'on a d'abord donné aux Anti-trinitaires ou nouveaux Ariens hérétiques du seizième siècle, qui n'admettaient d'autre Dieu que Dieu le père, regardant J. C. comme un pur homme, et le S. Esprit comme un simple attribut de la divinité. On les appelle aujourd'hui _Sociniens_ ou _Unitaires. Voyez_ SOCINIENS _ou_ UNITAIRES.
Les _Déistes_ modernes sont une secte ou sorte de prétendus esprits forts, connus en Angleterre sous le nom de _free-thinkers_ , gens qui pensent librement, dont le caractère est de ne point professer de forme ou de système particulier de religion, mais de se contenter de reconnaître l'existence d'un Dieu, sans lui rendre aucun culte ni hommage extérieur. Ils prétendent que vu la multiplicité des religions et le grand nombre de révélations, dont on ne donne, disent-ils, que des preuves générales et sans fondement, le parti le meilleur et le plus sûr, c'est de se renfermer dans la simplicité de la nature et la croyance d'un Dieu, qui est une vérité reconnue de toutes les nations. _Voyez_ DIEU _et_ REVELATION.
Ils se plaignent de ce que la liberté de penser et de raisonner est opprimée sous le joug de la religion révelée ; que les esprits souffrent et sont tyrannisés par la nécessité qu'elle impose de croire des mystères inconcevables, et ils soutiennent qu'on ne doit admettre ou croire que ce que la raison conçoit clairement. _Voyez_ MYSTERE _et_ FOI.
Le nom de _Déistes_ est donné surtout à ces sortes de personnes qui n'étant ni athées ni chrétiennes, ne sont point absolument sans religion (à prendre ce mot dans son sens le plus général), mais qui rejettent toute révélation comme une pure fiction, et ne croient que ce qu'ils reconnaissent par les lumières naturelles, et que ce qui est cru dans toute religion, un Dieu, une providence, une vie future, des récompenses et des châtiments pour les bons et pour les méchants ; qu'il faut honorer Dieu et accomplir sa volonté connue par les lumières de la raison et la voix de la conscience, le plus parfaitement qu'il est possible, mais que du reste chacun peut vivre à son gré, et suivant ce que lui dicte sa conscience. [...]
M. l'abbé de la Chambre docteur de Sorbonne, dans un traité de la véritable Religion, imprimé à Paris en 1737, parle des _Déistes_ et de leurs opinions d'une maniere encore plus précise. « On nomme _Déistes_ , dit cet auteur, tous ceux qui admettent l'existence d'un être suprême, auteur et principe de tous les êtres qui composent le monde, sans vouloir reconnaître autre chose en fait de religion, que ce que la raison laissée à elle-même peut découvrir. Tous les _Déistes_ ne raisonnent pas de la même maniere : on peut réduire ce qu'ils disent à deux différentes hypothèses.
La première espèce de _Déistes_ avance et soutient ces propositions : Il faut admettre l'existence d'un être suprême, éternel, infini, intelligent, créateur, conservateur et souverain maître de l'univers, qui préside à tous les mouvements et à tous les événements qui en résultent. Mais cet être suprême n'exige de ses créatures aucun devoir, parce qu'il se suffit à lui-même.
Dieu seul ne peut périr ; toutes les créatures sont sujettes à l'anéantissement, l'être suprême en dispose comme il lui plaît : maître absolu de leur sort, il leur distribue les biens et les maux selon son bon plaisir, sans avoir égard à leurs différentes actions, parce qu'elles sont toutes de même espèce devant lui.
La distinction du vice et de la vertu est une pure chicane aux yeux de l'être suprême ; elle n'est fondée que sur les lois arbitraires des sociétés. Les hommes ne sont comptables de leurs actions qu'au tribunal de la justice séculière. Il n'y a ni punition ni récompense à attendre de la part de Dieu après cette vie.
La seconde espèce de _Déistes_ raisonne tout autrement. L'être suprême, disent-ils, est un être éternel, infini, intelligent, qui gouverne le monde avec ordre et avec sagesse ; il suit dans sa conduite les règles immuables du vrai, de l'ordre et du bien moral, parce qu'il est la sagesse, la vérité, et la sainteté par essence. Les règles éternelles du bon ordre sont obligatoires pour tous les êtres raisonnables ; ils abusent de leur raison lorsqu'ils s'en écartent. L'éloignement de l'ordre fait le vice, et la conformité à l'ordre fait la vertu. Le vice mérite punition, et la vertu mérite récompense... Le premier devoir de l'homme est de respecter, d'honorer, d'estimer et d'aimer l'être suprême, de qui il tient tout ce qu'il est ; et il est obligé par état de se conformer dans toutes ses actions à ce que lui dicte la droite raison [...]. »
Le même auteur, après avoir exposé ces deux systèmes, propose la méthode de les réfuter. Elle consiste à prouver, « 1. que les bornes qui séparent le vice d'avec la vertu, sont indépendantes des volontés arbitraires de quelqu'être que ce soit ; 2. que cette distinction du bien et du mal, antérieure à toute loi arbitraire des législateurs, et fondée sur la nature des choses, exige des hommes qu'ils pratiquent la vertu et qu'ils s'éloignent du vice ; 3. que celui qui fait le bien mérite récompense, et que celui qui s'abandonne au crime mérite punition ; 4. que la vertu n'étant pas toujours récompensée sur la terre, ni le vice puni, il faut admettre une autre vie, où le juste sera heureux et l'impie malheureux ; 5. que tout ne périt pas avec le corps, et que la partie de nous-mêmes qui pense et qui veut, et qu'on appelle _âme_ , est immortelle ; 6. que la volonté n'est point nécessitée dans ses actions, et qu'elle peut à son choix pratiquer la vertu et éviter le mal ; 7. que tout homme est obligé d'aimer et d'estimer l'être suprême, et de témoigner à l'extérieur les sentimens de vénération et d'amour dont il est pénétré à la vue de sa grandeur et de sa majesté ; 8. que la religion naturelle, quoique bonne en elle-même, est insuffisante pour apprendre à l'homme quel culte il doit rendre à la divinité ; et qu'ainsi il en faut admettre une surnaturelle et révélée, ajoutée à celle de la nature. » _Traité de la véritable Religion, tome II. part. ij. pag. 1. 2. 3. 4. 5. et 6_. [...] ( _G_ )
*
L'article ÉCLECTISME (in _Encyclopédie_ , t. V, 1755, p. 270-293) signé par l'astérisque de Diderot, dont nous reproduisons les cinq premiers paragraphes, présente le philosophe éclectique comme celui qui refuse de se laisser endoctriner et enfermer dans tel ou tel système de pensée mais qui s'instruit de chaque penseur et en retient le bon grain qu'il sépare de l'ivraie. L'éclectique, ajoute Diderot, est le contraire du sectaire mais l'ami du sceptique à côté duquel il devrait toujours marcher. Le déiste et l'éclectique Voltaire sait cela : il marche à côté du sceptique Bayle et recueille de ses pensées tout ce qui peut lui servir pour lutter contre l'intolérance et la superstition que Bayle appelle l'idolâtrie. Pour l'attester, nous reproduisons, après l'article ÉCLECTISME, des extraits des _Pensées diverses sur la comète_ , le § CXIV d'abord où Bayle soutient que l'athéisme n'est pas un plus grand mal que l'idolâtrie, puis le § CXVI où il affirme que l'idolâtrie est le plus grand de tous les crimes, enfin la fin du § CLXXI et le § CLXXII où il se demande s'il peut exister une société d'athées vertueux.
* **E CLECTISME**, s. m. ( _Hist. de la Philosophie anc. et mod_.) L'éclectique est un philosophe qui foulant aux pieds le préjugé, la tradition, l'ancienneté, le consentement universel, l'autorité, en un mot tout ce qui subjuge la foule des esprits, ose penser de lui-même, remonter aux principes généraux les plus clairs, les examiner, les discuter, n'admettre rien que sur le témoignage de son expérience et de sa raison ; et de toutes les philosophies, qu'il a analysées sans égard et sans partialité, s'en faire une particuliere et domestique qui lui appartienne : je dis _une philosophie particuliere et domestique_ , parce que l'ambition de l'éclectique est moins d'être le précepteur du genre humain, que son disciple ; de réformer les autres, que de se réformer lui-même ; de connaître la vérité, que de l'enseigner. Ce n'est point un homme qui plante ou qui sème ; c'est un homme qui recueille et qui crible. Il jouirait tranquillement de la récolte qu'il aurait faite, il vivrait heureux, et mourrait ignoré, si l'enthousiasme, la vanité, ou peut-être un sentiment plus noble, ne le faisait sortir de son caractère.
Le sectaire est un homme qui a embrassé la doctrine d'un philosophe ; l'éclectique, au contraire, est un homme qui ne reconnaît point de maître : ainsi quand on dit des Eclectiques que ce fut une secte de philosophes, on assemble deux idées contradictoires, à moins qu'on ne veuille entendre aussi par le terme de _secte_ , la collection d'un certain nombre d'hommes qui n'ont qu'un seul principe commun, celui de ne soumettre leurs lumieres à personne, de voir par leurs propres yeux, et de douter plutôt d'une chose vraie que de s'exposer, faute d'examen, à admettre une chose fausse.
Les Éclectiques et les Sceptiques ont eu cette conformité, qu'ils n'étaient d'accord avec personne ; ceux-ci, parce qu'ils ne convenaient de rien ; les autres, parce qu'ils ne convenaient que de quelques points. Si les Éclectiques trouvaient dans le Scepticisme des vérités qu'il fallait reconnaître, ce qui leur était contesté même par les Sceptiques, d'un autre côté les Sceptiques n'étaient point divisés entre eux : au lieu qu'un éclectique adoptant assez communément d'un philosophe ce qu'un autre éclectique en rejetait, il en était de sa secte comme de ces sectes de religion, où il n'y a pas deux individus qui aient rigoureusement la même façon de penser.
Les Sceptiques et les Eclectiques auraient pu prendre pour devise commune, _nullius addictus jurare in verba magistri_4 _;_ mais les Éclectiques qui n'étant pas si difficiles que les Sceptiques, faisaient leur profit de beaucoup d'idées, que ceux-ci dédaignaient, y auraient ajouté cet autre mot, par lequel ils auraient rendu justice à leurs adversaires, sans sacrifier une liberté de penser dont ils étaient si jaloux : _nullum philosophum tam fuisse inanem qui non viderit ex vero aliquid_5. Si l'on réfléchit un peu sur ces deux especes de philosophes, on verra combien il était naturel de les comparer, on verra que le Scepticisme étant la pierre de touche de l' _Éclectisme_ , l'éclectique devrait toujours marcher à côté du sceptique pour recueillir tout ce que son compagnon ne réduirait point en une poussière inutile, par la sévérité de ses essais.
Il s'ensuit de ce qui précède, que l'Éclectisme pris à la rigueur n'a point été une philosophie nouvelle, puisqu'il n'y a point de chef de secte qui n'ait été plus ou moins éclectique ; et conséquemment que les Éclectiques sont parmi les philosophes ce que sont les souverains sur la surface de la terre, les seuls qui soient restés dans l'état de nature où tout était à tous. Pour former son système, Pythagore mit à contribution les théologiens de l'Égypte, les gymnosophistes de l'Inde, les artistes de la Phénicie, et les philosophes de la Grèce. Platon s'enrichit des dépouilles de Socrate, d'Héraclite, et d'Anaxagore ; Zénon pilla le Pythagorisme, le Platonisme, l'Héraclitisme, le Cynisme : tous entreprirent de longs voyages. Or quel était le but de ces voyages, sinon d'interroger les différens peuples, de ramasser les vérités éparses sur la surface de la terre, et de revenir dans sa patrie remplis de la sagesse de toutes les nations ? Mais comme il est presque impossible à un homme qui, parcourant beaucoup de pays, a rencontré beaucoup de religions, de ne pas chanceler dans la sienne, il est très difficile à un homme de jugement, qui fréquente plusieurs écoles de philosophie, de s'attacher exclusivement à quelque parti, et de ne pas tomber ou dans l'Éclectisme, ou dans le Scepticisme.
*
Bayle, _Pensées diverses sur la comète_ , Rotterdam, 1683 :
§ CXIV
_Que l'athéisme n'est pas_
_un plus grand mal que l'idolâtrie._
Cela étant, je puis me passer de faire le parallèle de l'idolâtrie et de l'athéisme, et de montrer que l'idolâtrie est pour le moins aussi abominable que l'athéisme, car je n'ai pas besoin que ce paradoxe soit vrai. Je l'ai ouï soutenir à un des habiles hommes de France, et qui est aussi bon chrétien que j'en connaisse. Permettez-moi de vous rapporter une partie de ses raisons, et de les paraphraser ou commenter selon que je le jugerai à propos.
§ CXVI
_L'idolâtrie est le plus grand_
_de tous les crimes selon les Pères._
[...] les Pères de l'Église ont dit sans nulle exception, que l'idolâtrie est le principal crime du genre humain, le plus grand péché du monde, le plus grand de tous les péchés, le dernier et le premier de tous les maux [...].
§ CLXXI
_Pourquoi la vengeance et l'avarice_
_sont des passions si communes._
[...] D'où il résulte, que les athées, qui ne font que suivre la même direction, ne sont pas nécessairement plus corrompus que les idolâtres, quoi qu'ils n'aient pas comme les idolâtres, telle ou telle opinion sur le crime, et sur les châtiments du crime.
§ CLXXII
_Si une société d'athées se ferait_
_des lois de bienséance et d'honneur._
On voit à cette heure, combien il est apparent qu'une société d'athées pratiquerait les actions civiles et morales, aussi bien que les pratiquent les autres sociétés, pourvu qu'elle fît sévèrement punir les crimes, et qu'elle attachât de l'honneur et de l'infamie à certaines choses. Comme l'ignorance d'un premier Être créateur et conservateur du monde, n'empêcherait pas les membres de cette société d'être sensibles à la gloire et au mépris, à la récompense et à la peine, et à toutes les passions qui se voient dans les autres hommes, et n'étoufferait pas toutes les lumières de la raison ; on verrait parmi eux des gens qui auraient de la bonne foi dans le commerce, qui assisteraient les pauvres, qui s'opposeraient à l'injustice, qui seraient fidèles à leurs amis, qui mépriseraient les injures, qui renonceraient aux voluptés du corps, qui ne feraient tort à personne, soit parce que le désir d'être loués les pousserait à toutes ces belles actions, qui ne sauraient manquer d'avoir l'approbation publique, soit parce que le dessein de se ménager des amis et des protecteurs, en cas de besoin, les y porterait. Les femmes s'y piqueraient de pudicité, parce qu'infailliblement cela leur acquerrait l'amour et l'estime des hommes. Il s'y ferait des crimes de toutes les espèces, je n'en doute point ; mais il ne s'y en ferait pas plus que dans les sociétés idolâtres, parce que tout ce qui a fait agir les païens, soit pour le bien soit pour le mal, se trouverait dans une société d'athées, savoir les peines et les récompenses, la gloire et l'ignominie, le tempérament et l'éducation. Car pour cette grâce sanctifiante, qui nous remplit de l'amour de Dieu, et qui nous fait triompher de nos mauvaises habitudes, les païens en sont aussi dépourvus que les athées.
Qui voudra se convaincre pleinement qu'un peuple destitué de la connaissance de Dieu, se ferait des règles d'honneur, et une grande délicatesse pour les observer, n'a qu'à prendre garde, qu'il y a parmi les chrétiens un certain honneur du monde, qui est directement contraire à l'esprit de l'Évangile. Je voudrais bien savoir, d'après quoi on a tiré ce plan d'honneur, duquel les chrétiens sont si idolâtres, qu'ils lui sacrifient toutes choses ? Est-ce parce qu'ils savent qu'il y a un Dieu, un Évangile, une Résurrection, un Paradis et un Enfer, qu'ils croient que c'est déroger à son honneur, que de laisser un affront impuni, que de céder la première place à un autre, que d'avoir moins de fierté et moins d'ambition que ses égaux ? On m'avouera que non. Que l'on parcoure toutes les idées de bienséance qui ont lieu parmi les chrétiens, à peine en trouvera-t-on deux qui aient été empruntées de la religion ; et quand les choses deviennent honnêtes, de malséantes qu'elles étaient, ce n'est nullement parce que l'on a mieux consulté la morale de l'Évangile. Les femmes se sont avisées depuis quelque temps, qu'il était d'un plus grand air de qualité de s'habiller en public, et devant le monde, d'aller à cheval, de courir à toute bride après une bête, etc., et elles ont tant fait, qu'on ne regarde plus cela comme éloigné de la modestie. Est-ce la religion qui a changé nos idées à cet égard ? Comparez un peu les manières de plusieurs nations qui professent le christianisme, comparez-les, dis-je, les unes avec les autres, vous verrez que ce qui passe pour malhonnête dans un pays, ne l'est point du tout ailleurs. Il faut donc que les idées d'honnêteté qui sont parmi les chrétiens, ne viennent pas de la religion qu'ils professent. Il y en a quelques-unes de générales, je l'avoue, car nous n'avons point de nations chrétiennes où il soit honteux à une femme d'être chaste. Mais pour agir de bonne foi, il faut confesser que cette idée est plus vieille, ni que l'Évangile, ni que Moïse : c'est une certaine impression qui est aussi vieille que le monde, et que je vous ferai voir tantôt, que les païens ne l'ont pas empruntée de leur religion. Avouons donc, qu'il y a des idées d'honneur parmi les hommes, qui sont un pur ouvrage de la nature, c'est-à-dire de la providence générale. Avouons-le surtout de cet honneur dont nos braves sont si jaloux, et qui est opposé à la loi de Dieu. Et comment douter après cela, que la nature ne pût faire parmi des athées, où la connaissance de l'Évangile ne la contrecarrerait pas, ce qu'elle fait parmi les chrétiens ?
*
L' _Encyclopédie_ de nouveau ouverte devant nous, nous proposons ci-dessous des extraits de l'article ÉRUDITION (in _Encyclopédie_ , t. V, 1755, p. 914-918) de d'Alembert ( _O_ ), qui forme le pendant de l'article ÉCLECTISME nommé ÉCLECTIQUE dans le renvoi. On y retrouve cette idée forte que l'incitation d'oser penser par soi-même ne signifie pas penser seul, loin s'en faut, mais signifie au contraire se nourrir de la conversation des auteurs aux côtés de qui on a choisi de marcher.
**E RUDITION**, s. f. ( _Philosoph. et Litt._ ) Ce mot, qui vient du latin _erudire, enseigner_ , signifie proprement et à la lettre, _savoir, connaissance ;_ mais on l'a plus particulièrement appliqué au genre de savoir qui consiste dans la connaissance des faits, et qui est le fruit d'une grande lecture. On a réservé le nom de _science_ pour les connaissances qui ont plus immédiatement besoin du raisonnement et de la réflexion, telles que la Physique, les Mathématiques, _etc._ et celui de _belles-lettres_ pour les productions agréables de l'esprit, dans lesquelles l'imagination a plus de part, telles que l'Eloquence, la Poésie, _etc._
[...] Enfin, car ce n'est pas un avantage à passer sous silence, l'étude des Sciences doit tirer beaucoup de lumières de la lecture des anciens. On peut sans doute savoir l'histoire des pensées des hommes sans penser soi-même ; mais un philosophe peut lire avec beaucoup d'utilité le détail des opinions de ses semblables ; il y trouvera souvent des germes d'idées précieuses à développer, des conjectures à vérifier, des faits à éclaircir, des hypothèses à confirmer. Il n'y a presque dans notre physique moderne aucuns principes généraux, dont l'énoncé ou du moins le fond ne se trouve chez les anciens ; on n'en sera pas surpris, si on considère qu'en cette matière les hypothèses les plus vraisemblables se présentent assez naturellement à l'esprit, que les combinaisons d'idées générales doivent être bientôt épuisées, et par une espèce de révolution forcée être successivement remplacées les unes par les autres. _Voy._ ECLECTIQUE. C'est peut-être par cette raison, pour le dire en passant, que la philosophie moderne s'est rapprochée sur plusieurs points de ce qu'on a pensé dans le premier âge de la Philosophie, parce qu'il semble que la premiere impression de la nature est de nous donner des idées justes, que l'on abandonne bientôt par incertitude ou par amour de la nouveauté, et auxquelles enfin on est forcé de revenir.
Mais en recommandant aux philosophes même la lecture de leurs prédécesseurs, ne cherchons point, comme l'ont fait quelques savants, à déprimer les modernes sous ce faux prétexte, que la philosophie moderne n'a rien découvert de plus que l'ancienne. Qu'importe à la gloire de Newton, qu'Empédocle ait eu quelques idées vagues et informes du système de la gravitation, quand ces idées ont été dénuées des preuves nécessaires pour les appuyer ? Qu'importe à l'honneur de Copernic, que quelques anciens philosophes aient cru le mouvement de la terre, si les preuves qu'ils en donnaient n'ont pas été suffisantes pour empêcher le plus grand nombre de croire le mouvement du Soleil ? Tout l'avantage à cet égard, quoiqu'on en dise, est du côté des modernes, non parce qu'ils sont supérieurs en lumières à leurs prédécesseurs, mais parce qu'ils sont venus depuis. La plupart des opinions des anciens sur le système du monde, et sur presque tous les objets de la Physique, sont si vagues et si mal prouvées, qu'on n'en peut tirer aucune lumière réelle. On n'y trouve point ces détails précis, exacts, et profonds qui sont la pierre de touche de la vérité d'un système, et que quelques auteurs affectent d'en appeler l' _appareil_ , mais qu'on en doit regarder comme le corps et la substance, et qui en font par conséquent la difficulté et le mérite. En vain un savant illustre, en revendiquant nos hypothèses et nos opinions à l'ancienne philosophie, a cru la venger d'un mépris injuste, que les vrais savants et les bons esprits n'ont jamais eu pour elle ; sa dissertation sur ce sujet (imprimée dans le tome XVIII des Mém. de l'Acad. des Belles-Lettres, _pag. 97._ ) ne fait, ce me semble, ni beaucoup de tort aux modernes, ni beaucoup d'honneur aux anciens, mais seulement beaucoup à l' _érudition_ et aux lumieres de son auteur.
Avouons donc d'un côté, en faveur de l' _érudition_ , que la lecture des anciens peut fournir aux modernes des germes de découvertes, de l'autre, en faveur des savants modernes, que ceux-ci ont poussé beaucoup plus loin que les anciens les preuves et les conséquences des opinions heureuses, que les anciens s'étaient, pour ainsi dire, contentés de hasarder.
Un savant de nos jours, connu par de médiocres traductions et de savants commentaires, ne faisait aucun cas des Philosophes, et surtout de ceux qui s'adonnent à la physique expérimentale. Il les appelle des _curieux fainéants_ , des _manœuvres_ qui osent usurper le titre de _sages_. Ce reproche est bien singulier de la part d'un auteur, dont le principal mérite consistait à avoir la tête remplie de passages grecs et latins, et qui peut-être méritait une partie du reproche fait à la foule des commentateurs par un auteur célebre dans un ouvrage où il les fait parler ainsi :
_Le goût n'est rien ; nous avons l'habitude De rédiger au long de point en point Ce qu'on pensa ; mais nous ne pensons point_. Volt. _Temple du Goût_.
Que doit-on conclure de ces réflexions ? Ne méprisons ni aucune espèce de savoir utile, ni aucune espèce d'hommes ; croyons que les connaissances de tout genre se tiennent et s'éclairent réciproquement ; que les hommes de tous les siècles sont à peu près semblables, et qu'avec les mêmes données, ils produiraient les mêmes choses : en quelque genre que ce soit, s'il y a du mérite à faire les premiers efforts, il y a aussi de l'avantage à les faire, parce que la glace une fois rompue, on n'a plus qu'à se laisser aller au courant, on parcourt un vaste espace sans rencontrer presqu'aucun obstacle ; mais cet obstacle une fois rencontré, la difficulté d'aller au-delà en est plus grande pour ceux qui viennent après. ( _O_ )
*
Pour conclure ce dossier sur le mot d'ordre oser penser par soi-même, nous laissons, en dernier lieu, la parole à Kant, qui a fait de _Sapere aude_ la devise des Lumières. Oser penser, pour les Lumières comme pour Kant, c'est se soustraire au principe d'autorité et reconnaître une communauté liée par la raison comme adresse à tous et à chacun. Nous reproduisons ci-dessous les deux premiers paragraphes de sa _Réponse à la question : Qu'est-ce que les Lumières ?_ , Paris, trad. J.-F. Poirier et F. Proust, Paris, GF-Flammarion, 1991.
_Les Lumières, c'est la sortie de l'homme hors de l'état de tutelle dont il est lui-même responsable_. _L'état de tutelle_ est l'incapacité de se servir de son propre entendement sans la conduite d'un autre. On est _soi-même responsable_ de cet état de tutelle quand la cause tient non pas à une insuffisance de l'entendement mais à une insuffisance de la résolution et du courage de s'en servir sans la conduite d'un autre. _Sapere aude !_ Aie le courage de te servir de ton _propre_ entendement ! Voilà la devise des Lumières.
Paresse et lâcheté sont les causes qui font qu'un si grand nombre d'hommes, après que la nature les eut affranchis depuis longtemps d'une conduite étrangère ( _naturaliter maiorennes_ ), restent cependant volontiers toute leur vie dans un état de tutelle ; et qui font qu'il est si facile à d'autres de se poser comme leurs tuteurs. Il est si commode d'être sous tutelle. Si j'ai un livre qui a de l'entendement à ma place, un directeur de conscience qui a de la conscience à ma place, un médecin qui juge à ma place de mon régime alimentaire, etc., je n'ai alors pas moi-même à fournir d'efforts. Il ne m'est pas nécessaire de penser dès lors que je peux payer ; d'autres assumeront bien à ma place cette fastidieuse besogne. Et si la plus grande partie, et de loin, des hommes (y compris le beau sexe tout entier) tient ce pas qui affranchit de la tutelle pour très dangereux et de surcroît très pénible, c'est que s'y s'emploient ces tuteurs qui, dans leur extrême bienveillance, se chargent de les surveiller. Après avoir d'abord abêti leur bétail et avoir empêché avec sollicitude ces créatures paisibles d'oser faire un pas sans la roulette d'enfant où ils les avaient emprisonnés, ils leur montrent ensuite le danger qui les menace s'ils essaient de marcher seuls. Or ce danger n'est sans doute pas si grand, car après quelques chutes ils finiraient bien par apprendre à marcher ; un tel exemple rend pourtant timide et dissuade d'ordinaire de toute autre tentative ultérieure.
1\- Voir Kant, _Qu'est-ce que les Lumières ?_ , trad. J.-F. Poirier et F. Proust, Paris, GF-Flammarion, 1991, p. 43.
2\- Oser penser, c'est la traduction de la maxime d'Horace, traduction libre qui la dote d'une signification critique tout à fait étrangère à la pensée plutôt stoïcienne des _Épîtres_ (I, II, 40) : _sapere aude_ , « ose être sage », « satisfais-toi de ce monde-ci ».
3\- Sur ce sens et cet ordre subversifs de l' _Encyclopédie_ , voir V. Le Ru, _Subversives Lumières – l'Encyclopédie comme machine de guerre_ , Paris, CNRS Éditions, 2007.
4\- Horace, _Épîtres,_ I, I, 14, trad. F. Villeneuve, Paris, Belles Lettres, 1964, p. 37 : « Aucune astreinte ne m'a contraint à jurer sur les paroles du maître. »
5\- « Aucun philosophe n'est suffisamment vain pour ne pas percevoir quelque chose de vrai. »
Vie de Voltaire
**(1694-1778)**
**1694** : François-Marie Arouet naît à Paris sous le règne de Louis XIV le 21 novembre. Par sa famille, il appartient à la bourgeoisie de robe, mais refusera la destinée qu'elle lui trace.
**1704-1711** : Il fait de brillantes études au collège jésuite de Louis-Le-Grand, fréquenté par l'aristocratie (il se lie avec les d'Argenson, futurs ministres de Louis XV).
**1712-1715** : Il fréquente les salons littéraires, entame des études de droit, accompagne une mission diplomatique à La Haye mais est renvoyé à Paris en raison d'une incartade amoureuse. Son père menace de l'envoyer à Saint-Domingue. Il commence à écrire quelques vers dont une satire.
**Mai-octobre 1716** : Malgré la politique « libérale » instaurée par le régent Philippe d'Orléans à la mort de Louis XIV, le jeune Arouet est exilé à Sully-sur-Loire pour avoir écrit des vers satiriques sur le Régent.
**Mai 1717-avril 1718** : Il est embastillé pour onze mois. Sa première tragédie _Œdipe_ connaît un grand succès. Il prend le surnom de « Voltaire ».
**1719-1725** : Il mène une vie mondaine mais en même temps il travaille d'arrache-pied pour devenir le grand poète de son temps. Il publie _La Ligue ou Henri le Grand_ en 1723, première version de _La Henriade_ (épopée). Il crée des pièces pour les fêtes de la Cour.
**1726** : Bastonné par les gens du chevalier de Rohan, il cherche à se battre en duel ; on l'embastille puis on l'autorise à partir pour l'Angleterre. Il y reste de mai 1726 à novembre 1728. Il y apprend l'anglais, lit Locke, Bolingbroke, Pope et s'initie à la science newtonienne. Il rentre discrètement en France.
**1729** : Voltaire s'enrichit dans des opérations financières et commerciales qui assurent son indépendance d'écrivain : il ne dépendra jamais ni des mécènes ni des éditeurs.
**1731** : _Histoire de Charles XII_.
**1732** : Triomphe de sa tragédie _Zaïre_.
**1734** : Il publie les _Lettres philosophiques_ qui font scandale. Il est obligé de quitter Paris et se réfugie dans le château de Cirey en Champagne chez son amie, la marquise du Châtelet, qu'il a rencontrée en juin 1733 et avec qui il vivra jusqu'à ce qu'elle meure en 1749. Il y rédige le _Traité de métaphysique_ en même temps que la marquise lui fait partager sa passion pour la science newtonienne, il commence à composer les _Éléments de la philosophie de Newton_.
**1736** : Il publie _Le Mondain_ qui lui vaut de nouvelles menaces d'arrestation, il fuit en Hollande. Début de sa correspondance avec Frédéric II, futur roi de Prusse.
**1737-1738** : Il quitte la Hollande où il laisse inachevé son manuscrit des _Éléments de la philosophie de Newton_ que les libraires hollandais s'empressent de faire compléter par un mathématicien afin de le publier, au grand dam de Voltaire. Il publie en 1738 une deuxième édition entièrement de sa main qui sera suivie d'une troisième édition corrigée et complétée en 1741. En 1738, il publie également sous le titre d' _Épîtres_ ce qui deviendra, à partir de l'édition de 1742, le _Discours en vers sur l'homme_.
**1740** : Avènement de Marie-Thérèse d'Autriche, impératrice d'Autriche, et de Frédéric II, roi de Prusse. Voltaire rencontre Frédéric, qui veut le garder à sa cour.
**1741** : Sa tragédie _Mahomet_ connaît un grand succès à Lille mais est interdite à Paris.
**1743** : Voltaire a la faveur de la Cour depuis que Louis XV est au pouvoir et que les frères d'Argenson, anciens condisciples de Voltaire, sont ministres.
**1745** : Il devient historiographe de France.
**1746** : Il est élu à l'Académie française. Début secret de sa liaison avec sa nièce, madame Denis, veuve depuis deux ans.
**1747** : Sa disgrâce à la Cour commence. Il publie un conte _Memnon_ , première version de _Zadig_ (1748).
**1749** : Madame du Châtelet, son « ami [ _sic_ ] de vingt ans » meurt, après un accouchement, à la suite de sa liaison avec le poète Saint-Lambert. Bouleversé, Voltaire quitte Cirey pour Paris où il s'installe chez madame Denis.
**1750-1753** : Voltaire part pour Berlin, Frédéric II vient de le nommer chambellan. En 1752, il publie _Micromégas_ et _Le Siècle de Louis XIV_. Son séjour à la Cour de Frédéric est tumultueux, marqué par son inimitié pour Maupertuis. Frédéric prend la défense de ce dernier, alors Président de l'Académie des sciences de Berlin, et fait emprisonner Voltaire à Francfort en 1753. Sa nièce, madame Denis, vient à son secours.
**1753-1755** : Interdit de séjour à Paris, Voltaire erre en Alsace, puis s'installe près de Genève en Suisse, aux Délices, avec madame Denis. D'Alembert lui demande des articles pour l' _Encyclopédie_ dont le premier volume est paru en 1751. Le tremblement de terre de Lisbonne en novembre 1755 (60 000 morts) le bouleverse. Il rédige le _Poème sur le désastre de Lisbonne_ qu'il publie en 1756. Début de la gestation de _Candide_. Rousseau, après le succès du _Discours sur les sciences et les arts_ en 1750, publie le _Discours sur l'origine de l'inégalité_ en 1755, qui lui vaudra les foudres de Voltaire.
**1756** : Publication de l' _Essai sur les mœurs et l'esprit des nations_. Début de la guerre de Sept Ans.
**1757** : L' _Encyclopédie_ de Diderot et de d'Alembert est interdite, suite à l'article GENÈVE de d'Alembert et à l'attentat de Damiens contre Louis XV. Voltaire essaie de négocier une paix séparée entre la France et la Prusse. Il renoue sa correspondance avec Frédéric II.
**1758** : Voltaire achète le domaine de Ferney en territoire français, à la frontière suisse, d'où il exerce une sorte de patriarcat intellectuel. À Paris, la campagne contre les philosophes fait rage. La guerre européenne s'aggrave.
**1759** : Voltaire publie _Candide_ qui connaît un triomphe.
**1760-1761** : Voltaire s'installe définitivement à Ferney en compagnie de madame Denis, il publie pamphlet sur pamphlet contre les ennemis des philosophes et des encyclopédistes.
**1762** : Exécution de Calas, négociant calviniste et toulousain injustement accusé d'avoir tué son fils et condamné à mort. Voltaire se lance dans sa réhabilitation. Les jésuites sont expulsés de France. Voltaire invente l'expression « écrasons l'infâme » (c'est-à-dire toutes les formes de superstition de fanatisme), qui deviendra le mot d'ordre des philosophes. Catherine II prend le pouvoir en Russie.
**1763** : Fin de la guerre de Sept Ans. Voltaire publie son _Traité sur la tolérance, à l'occasion de la mort de Jean Calas_. Le Conseil du roi ordonne, contre les parlements, la révision du procès de Calas.
**1764** : Voltaire fait paraître son _Dictionnaire philosophique_ qui connaîtra de nombreuses versions remaniées dont la dernière, _Questions sur l'Encyclopédie_ , sera publiée entre 1770 et 1772.
**1765** : Voltaire obtient la réhabilitation de Calas. La publication des derniers volumes de l' _Encyclopédie_ est autorisée.
**1766** : Parution à Genève du _Philosophe ignorant_ , l'envoi de l'ouvrage à Paris est retardé en raison de l'exécution du chevalier de La Barre, jeune homme condamné pour ne pas avoir salué des capucins, on brûle le _Dictionnaire philosophique_ de Voltaire, trouvé chez lui, sur le corps du condamné. Horrifié par cette nouvelle injustice et en même temps terrifié, Voltaire passe en Suisse mais revient bientôt à Ferney où il accueille et protège un compagnon en fuite du chevalier.
**1767** : Voltaire publie _L'Ingénu_.
**1768-1772** : Voltaire publie des contes et les neuf volumes des _Questions sur l'Encyclopédie_ , énorme compilation où il rassemble ses connaissances et ses interrogations. Il tombe gravement malade (trouble de la miction qui diminue grandement sa force de travail). À Paris, des gens de lettres lancent le projet d'ériger une statue à Voltaire de son vivant.
**1773** : Voltaire frôle la mort mais publie encore un ou deux contes.
**1774** : Mort de Louis XV et avènement de Louis XVI, éduqué selon les principes éclairés, mais hostile à Voltaire.
**1775** : Voltaire obtient la suppression de la gabelle en pays de Gex (Ferney) grâce à Turgot (philosophe et encyclopédiste appelé au poste de contrôleur général des Finances par Louis XVI).
**1776** : Parution de _La Bible enfin expliquée_ , ouvrage auquel Voltaire a travaillé pendant trente ans.
**1778** : Voltaire se rend à Paris où il reçoit un accueil triomphal (il était interdit de séjour depuis 1750). Il y meurt en mai, à 84 ans. Rousseau meurt en juillet la même année, à 66 ans.
**1785-1789** : Beaumarchais dirige l'édition des _Œuvres complètes_ de Voltaire à Kehl, en Allemagne (70 volumes).
**1791** : Les cendres de Voltaire sont transférées au Panthéon.
Bibliographie
ŒUVRES DE VOLTAIRE
_Le Philosophe ignorant_ , édition critique par Roland Mortier, dans les _Œuvres complètes_ de Voltaire, tome 62, Voltaire Foundation, University of Oxford, 1987 ; repris dans la collection Vif, Voltaire Foundation, Oxford, 2000.
_Œuvres complètes_ , éditées par Moland, Paris, Garnier frères, 1877-1883, 52 vol.
_Lettres philosophiques_ , Paris, GF-Flammarion, 1964.
_Traité de métaphysique_ , _in_ tome XXII des _Œuvres complètes_ , établies par Louis Moland, Paris, Garnier frères, 52 vol., 1877-1883.
_Éléments de la philosophie de Newton_ , tome XV des _Œuvres complètes_ de Voltaire, critical edition by R. L. Walters and W. H. Barber, Voltaire Foundation, University of Oxford, 1992.
_Romans et contes_ , Paris, GF-Flammarion, 1966.
_Discours en vers sur l'homme_ , in tome IX des _Œuvres complètes_ , établies par Louis Moland, Paris, Garnier frères, 52 vol., 1877-1883.
_Poème sur le désastre de Lisbonne_ (1756), _in_ tome IX des _Œuvres complètes_ , établies par Louis Moland, Paris, Garnier frères, 52 vol., 1877-1883.
_Traité sur la tolérance_ , Paris, GF-Flammarion, 1993.
_Dictionnaire philosophique_ , Paris, Imprimerie nationale, 1994, édition établie par Béatrice Didier d'après la troisième édition de 1769.
_Dictionnaire philosophique_ , Paris, Classiques Garnier, édition de Raymond Naves et Olivier Ferret, préface d'Etiemble, 2008.
_Questions sur l'Encyclopédie_ , 1770, _in_ tome XVII des _Œuvres complètes_ , établies par Louis Moland, Paris, Garnier frères, 52 vol., 1877-1883.
_Bibliothèque de Voltaire : catalogue des livres_ , Moscou et Leningrad, 1961.
AUTRES OUVRAGES
ALEMBERT D', _Éléments de philosophie_ , Paris, Fayard, 1986.
BAYLE Pierre, _Dictionnaire historique et critique_ , Rotterdam, 1702.
– _Pensées diverses sur la comète_ , Rotterdam, 1683 ; Paris, GF-Flammarion, 2007.
BOURDIN Jean-Claude, _Les Matérialistes au XVIIIe siècle_, textes choisis, Paris, Payot, 1996.
CLARKE Samuel, _Traité de l'existence et des attributs de Dieu_ , trad. fr., 1727-1728.
DESCARTES _Œuvres philosophiques_ , Paris, Garnier, en trois tomes, 1963-1973.
EHRARD Jean, _L'Idée de nature dans la première moitié du XVIIIe siècle_, Paris, SEVPEN, 1963 ; Albin Michel, 1994.
_Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers_ , éditée par d'Alembert et Diderot, Paris, Briasson, David, Le Breton et Durand, 35 vol., 1751-1780 ; rééd. Fromann, 1966-1967.
FONTENELLE, _Entretiens sur la pluralité des mondes_ , Paris, d'après l'édition de 1742, Librairie Nizet, 1984.
GOLDZINK Jean, _Voltaire, La légende de saint Arouet_ , Paris, Gallimard, 1989.
ISRAEL Jonathan, _Les Lumières radicales_ , trad. fr., Paris, Amsterdam, 2005.
LEIBNIZ, _La Monadologie_ , Paris, Delagrave, 1983.
LE RU Véronique, _Voltaire newtonien_ , Paris, Vuibert-Adapt, 2005.
–, _Subversives Lumières, L'Encyclopédie comme machine de guerre_ , Paris, CNRS Éditions, 2007.
LOCKE, _Essai philosophique concernant l'entendement humain_ , trad. Coste, Paris, Vrin, 1983.
MALEBRANCHE, _De la Recherche de la vérité_ , Paris, 1675, repris par Vrin en deux tomes, 1965 et 1967.
MASON H. T., _Pierre Bayle and Voltaire_ , Londres, 1963.
NEWTON, _Principes mathématiques de la philosophie naturelle_ , traduction de la Marquise du Châtelet, 1756-1759, rééditée en fac-similé par Blanchard, en 1966 ; puis par Jacques Gabay, en 1990.
PASCAL, _Œuvres complètes_ , Paris, Seuil, 1963.
PLUCHE, _Spectacle de la nature_ , Paris, Estienne, 9 volumes, 1732-1750.
POMEAU René, _La Religion de Voltaire_ , Paris, Nizet, 1995.
–, _D'Arouet à Voltaire_ , Voltaire Foundation, Oxford, 1985, tome I de _Voltaire en son temps_ , biographie en cinq tomes parus sous sa direction.
SPINOZA, _Éthique_ , trad. Charles Appuhn, Paris, Garnier, 1965.
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Cable Car Museum San Francisco opening hours
Cable Car Museum Hours - Book Cable Car Museum Tour
Address: Cable Car Museum. 1201 Mason Street. San Francisco, CA 94108. OPEN 10AM TO 5PM. Hours: 10 am - 5 pm. April thru October. 10 am - 5 pm
Cable Car Museum, San Francisco: Hours, Address, Cable Car Museum Reviews: 4.5/
What days are Cable Car Museum open? Cable Car Museum is open Mon, Tue, Wed, Thu, Fri, Sat, Sun
destens 24 Stunden vor dem Startdatum der Tour stornieren
The San Francisco Cable Car Museum is free and open every day. You do not need a ticket to enter. You simply show up, walk in and enjoy the exhibits. April - October: 10am to 6pm; November - March: 10am to 5pm; Closed on New Year's Day, Thanksgiving Day and Christmas Day; The museum is at 1201 Mason Street. This is the star on the map below at the corner of Washington and Mason Streets
The museum was established in 1974, and is run by the Friends of the Cable Car Museum. It is entered from an entrance at Washington and Mason and is open from 10 AM to 6 PM between April 1 and September 30 and from 10 AM to 5 PM between October 1 and March 31, apart from some public holidays
OPEN 10AM TO 5PM - San Francisco Cable Car Museum Inf
Das Museum ist täglich geöffnet. Einlass ist von 10 - 18 Uhr. Der Eintritt ist frei! Im Museum werden die ersten drei Cable-Car Wagen San Franciscos im Original ausgestellt
The museum is free (though donations are encouraged), and is currently open Thursdays, Fridays, and Saturdays from 12 noon - 5 pm. It is located at 77 Steuart Street, between Market and Mission Streets, across from the Ferry Building, at the F-line Steuart Street stop Shop our Online Stor
Floored, Bellows said. The store has kept its staff, Bellows said, but now opens for eight hours a day instead of 14. Customers (no more than 20 at once) enter through one door, sanitize hands (so..
Cable Car Museum (San Francisco) - 2021 All You Need to
Click on: Timetables attransit.511.org. Scroll through the pages till you get to cable car routes. (Note: They are the very last page of the listing....after all the bus route information and times). Choose the cable car line you are interested in. It will give you all the hours/times/locations of the cable cars
San Francisco Cable Car Museum, San Francisco, CA. 256 likes · 1,099 were here. The Cable Car Museum has been open to the public since 1974. A National Historical Monument since 1964. We are a..
To see how it works and learn more about this odd form of transportation, we recommend visiting the San Francisco Cable Car Museum. Lines. The Cable Car has three routes which cover some of the most interesting areas of San Francisco: the financial district, Nob Hill, Chinatown, Little Italy, North Beach, Russian Hill and Fisherman's Wharf
The museum is located at 1201 Mason Street in San Francisco. It is open every day except Easter, Thanksgiving, Christmas Day, and January 1. Admission is free and it will take you about half an hour to view the exhibits. The best way to get to the Cable Car Museum is the most obvious - by riding a cable car
Check out the gift shop for an authentic cable-car bell, or visit in July for the annual Bell-Ringing Contest, first held in 1949, that now takes place at Union Square. Another great stop: The SF Railway Museum (also free), across from the Ferry Building on the Embarcadero, which looks at the impact of both cable cars and traditional streetcars. You can even feel like a conductor and take the wheel of a full-size replica of a 1911 San Francisco streetcar
Cable Car Museum - San Francisco, CA - CA - Yel
San Francisco Cable Car Museum - YouTube. San Francisco Cable Car Museum. Watch later. Share. Copy link. Info. Shopping. Tap to unmute. If playback doesn't begin shortly, try restarting your device
1 of 5 FILE - In this photo March 16, 2020, file photo, a cable car operator stands and looks out toward the Golden Gate Bridge at the empty Hyde Street turnaround in San Francisco
Book your San Francisco Cable Car Museum tickets online and skip-the-line! Save time and money with our best price guarantee make the most of your visit to San Francisco
San Francisco Cable Car Museum. 8.9/ 10. 425. ratings. Given the COVID-19 pandemic, call ahead to verify hours, and remember to practice social distancing. Ranked #1 for history museums in San Francisco. you can see the giant wheel that winds the cable here, super cool. (14 Tips
The closest MUNI bus lines are the 1 & 30. C able Car Barn & Powerhouse 1201 Mason Street San Francisco, CA 94108 (corner of Mason and Washington Sts.) H olidays: Open every day of the year with the exception of Thanksgiving, Christmas Day and New Year's Day. H ours: April 1st - September 30th 10AM - 6PM October 1st - March 31st 10AM - 5PM
A quick tour of San Francisco Cable Car Museum located at 1201 Mason Street. Since most of the operation of the cable car is hidden underground, this museum.
Unbeatable views. Unforgettable trips. No experience is more uniquely San Francisco than a ride on a cable car. Cable cars have come to symbolize our great city (along with another world-renowned transportation icon. Hint: it's a suspension bridge painted an International Orange color.) After all, we're the city that first launched cars pulled along by cables runnin Die Cable-Cars muten an wie Relikte aus einer anderen Zeit, und dieses historische Vehikel in San Francisco ist in der Tat auf der Welt die einzig noch existierende Kabel-Straßenbahn mit entkoppelbaren Wagons. Um die Hügel der Stadt zu überwinden, bedurfte es beim Betrieb der Cable-Cars einer Antriebshilfe. Sie verläuft in Form eines Seils in einem Graben unter den jeweiligen Straßen. Die Wagen werden also, unterstützt von Spannklauen, nach oben gezogen. Zur Rückfahrt. San Francisco Cable Car Museum National Maritime Museum Chinese Historical Society Chinese Culture Center North Beach Museum Wells Fargo Museum Strybing Arboretum Fort Mason Center Mission Dolores World of Oil SFO Arts. Da es hier auch zu Änderungen kommen kann, bitte vor dem Besuch noch einmal nachfragen deshalb... alle Angaben ohne Gewähr . Diese Museen mag ich ganz besonders: Legion of. The San Francisco Cable Car Website, the online home of the web's first Interactive Cable Car. Learn about San Francisco's cable cars before you ride! interactive demos on how cable cars work, cable car route maps with popular destinations, cable car fares and etiquette, and the location and hours for the San Francisco Cable Car Powerhouse
What people are saying about San Francisco Cable Car Museum. Overall rating. 4.8 / 5 based on 85 reviews A good fun evening for adults and teenagers. This was a fun way to see some of San Francisco and learn a bit more about it. Doing it in the evening was fine. Our children of 15 and 17 particularly enjoyed it. The initial training is thorough and we were a bit worried about how we would cope. The Cable Car Barn, Powerhouse and Museum is known as Home Base to the cable cars. It is here that the cars not only depart and arrive daily on their 11 miles of wrapped steel rope going a steady 9 1/2 miles per hour, but also where visitors find a variety of spectacular sights. The Museum houses one of the very first cable cars.
Cable Car Museum (San Francisco) - Aktuelle 2021 - Lohnt
Cable Car Museums. If you want to learn more about San Francisco cable cars then check out The Cable Car Museum at the corner of Mason St. and Washington St. This is a free museum that is open every day of the year except for the major holidays. You can see some of the historic cable cars, learn all about the different eras of the San Francisco.
Current prices: 1-Day, $21; 3-day, $32; 7-day, $42. These may not be your best value unless you are planning multiple cable car rides during the given period. You can purchase these passes at our San Francisco Railway Museum, 77 Steuart Street, between Market and Mission (at the F-line Steuart Street stop), Tuesday-Sunday between 10 a.m. and 5.
Hours: 10-6 daily Summer. 10-5 Winter. Admission: Free. Images. San Francisco Cable Car Museum and Power House. Located within three blocks of every cable car line. Power House and Cable Winding Machinery. The cable cars grip cable turned by giant sheaves, powered by 510 horsepower engines found at the Museum. Antique Cable Cars. The world's first cable car, built in 1873 by Andrew Hallidie.
San Francisco Cable Car Museum. Gib dein Wunschdatum ein, um verfügbare Aktivitäten anzuzeigen Verfügbarkeit prüfen Top-Empfehlung Führung San Francisco: Abendliche Rundfahrt per Doppeldecker-Bus Dauer: 1,5 Stunden; 4.2 13 Bewertungen Ab. € 29,62 pro Person San Francisco Cable Car Museum: Tickets & Besichtigungen.
The Cable Car barn also houses the powerhouse for the entire cable system that moves the authentically-replicated cars along the 17 miles (25 km) of track at a steady speed of 9.5 mph (15.5 km/h). The museum for the Cable Car in San Francisco is open to the public where one can see the powerhouse in operation and see on display the original cable car that was used in August of 1873
San Francisco Cable Car Museum: Insider's Look at this
San Francisco Cable Car Museum Tickets und Führungen einfach online kaufen - Zeit & Geld sparen. Vorab buchen - Plätze sichern - Tickets sofort erhalten San Francisco genießen CABLE CAR MUSEUM. +1 415 474 1887. 1201, Mason Street, Au coin des Mason et Washington Streets San Francisco Straying radically from the ideal of the quiet museum - one filled with the archaic artifacts of yesteryear, each lighted softly and placed behind Plexiglas - the San Francisco Cable Car.
San Francisco Cable Car Museum - Wikipedi
The Cable Car Museum. Located at 1201 Mason Street at Washington Street in San Francisco - and, YES, you can get to the Cable Car Museum from the Union Square Cable Car Turn-Around and along the line at intersections on Powell Street. You will find great information at the Cable Car Museum's web site, such as the history of the cable car, how.
Historic Cable Cars visible at the Cable Car Barn/Museum in San Francisco. Passengers can board at any place along the routes: Powell to Fisherman's Wharf, Powell to Hyde, and California from Market to Van Ness. Waiting lines to ride the Cable Cars are sometimes long, unfortunately, especially at Powell and Market. Drivers will collect fares after you have boarded. Cable Car Barn/Museum.
Cable cars are iconic to San Francisco — deeply ingrained in our city's history, and without doubt, part of our city's future, the SFMTA wrote on Twitter, Tuesday morning, alongside pictures of the railed vehicles. We're excited to phase in cable car service in Fall 2021- ahead of the holiday season. We're excited to phase in cable car service in Fall 2021- ahead of the.
Das Cable Car Museum - Sehenswürdigkeiten in San Francisc
Die San Francisco Cable Cars bilden die Kabelstraßenbahn in San Francisco, Kalifornien. Das bei Touristen beliebte Verkehrsmittel ist eines der wenigen beweglichen National Historic Landmarks in den Vereinigten Staaten und ist die einzige verbliebene Kabelstraßenbahn der Welt mit entkoppelbaren Wagen. Geschichte Anfänge. Cable Car in der California Street. Aktie der California Street Cable. Du bist hier: Startseite 1 / Blog 2 / 3 / Reiseführer 4 / USA 5 / Kalifornien 6 / San Francisco 7 / Cable Car Museum San Francisco Wer in San Francisco ist wird sie unmöglich übersehen oder überhören können: die Cable Cars
The San Francisco Cable Car Museum is an adjunct to the MUNI cable car depot. Part of the Museum is the active electric motor which continuously winds the pa.. Return Time: 8:30 pm (December) 9:30 pm (April-October) Location: Fisherman's Wharf GL office : 2800 Leavenworth St., San Francisco. Tour Overview. Join us as the sun sets over San Francisco and see the lights come on in the City by the Bay. Our night tour on one of double decker open top Cable Cars starts from Fisherman's Wharf at our Tour.
San Francisco Railway Museum & Gift Shop Market Street
Museums in San Francisco, C
W enn man an San Francisco denkt, dann hat man unweigerlich auch die Cable Cars vor Augen, die sich so eindrucksvoll über die unzähligen steilen Hügel der Stadt quälen und das beliebteste Fortbewegungsmittel für Touristen darstellen.. Die auf Kabeln gezogene Straßenbahn war einst (kurz) das Hauptverkehrsmittel der Stadt, wurde allerdings bereits zu Beginn des 20
Cable Car Museum Show all Monuments & Tourist Attractions Hour-long cruise through the San Francisco Bay. And the choice of one of the following two sites: Exploratorium; de Young Museum and the Legion of Honor (if you visit on the same day). You can visit all of these places over 9 days. CityPASS fares . There is only one kind of CityPASS card, so you should be able to get a better deal.
San Francisco Cable Car Museum: Was andere Reisende berichten. Gesamtbewertung. 4.8 / 5 basierend auf 85 Bewertungen Wir haben Anfang März an der Night Tour teilgenommen, dementsprechend war es etwas frisch. Darauf kann man sich aber natürlich sehr gut vorbereiten. Die Tour an sich war super und sehr kurzweilig, entlang der Route konnten wir alle großen Sehenswürdigkeiten San Francisco's.
Cable Car Museum: 10am to 6pm: California Academy of Sciences: 9:30am to 5pm. Cartoon Art Museum: 11am to 5pm : Chinatown Shopping: Most shops open around 10am and some stay open as late as 10pm: Chinese Historical Society: Closed: City Hall: Closed: Coit Tower: 10am to 6pm: Conservatory of Flowers: 10am to 6:30pm: Contemporary Jewish Museum: Closed: de Young Museum: 9:30 am to 5:15 pm: Disney.
The Museum is located at the Washington-Mason cable car barn in San Francisco. In addition, he is co-chair of the Committee to Celebrate the Cable Cars. This civic Committee [held] a series of special events, during 1997, commemorating the 50th anniversary of Friedel Klussmann's successful efforts. Rice, a native San Franciscan, has written many transportation articles dealing with. It's no wonder, for starters, that cable cars first appeared in hilly San Francisco: their inventor came up with the idea for the steam-powered system after watching carriage horses struggle on a steep street. Check out the gift shop for an authentic cable-car bell, or visit in July for the annual Bell-Ringing Contest, first held in 1949, that now takes place at Union Square Cable cars and street cars are a big part of San Francisco's past and present. With their distinctive ding, ding, ding, cable cars rumble up and down the city's infamous steep hills like mobile museum pieces. They tirelessly haul thousands of tourists every day. The beloved cable car arrived in San Francisco in 1873. Designated National Landmarks, they are the only vehicles of their. Cable cars were invented by Andrew Smith Hallidie here in San Francisco in 1873. Hallidie's cable car system was based on early mining conveyance systems and dominated the city's transit scene for more than 30 years. Hallidie's cable car system would survive the great San Francisco earthquake and fires of 1906, soldier on through two World Wars and outlast political attempt The San Francisco cable car system is the world's last manually operated cable car system. An icon of San Francisco, the cable car system forms part of the intermodal urban transport network operated by the San Francisco Municipal Railway.Of the 23 lines established between 1873 and 1890, only three remain (one of which combines parts of two earlier lines): two routes from downtown near Union.
San Francisco is open and uncrowded
Enjoy easy access to the Botanic Garden, Cable Car Museum, Space Place (at Carter Observatory) and Zealandia (via free shuttle). OPENING HOURS: Monday to Thursday 7:30am-8:00pm. Friday 7:30am-8:00pm. Saturday 8:30am-8:00pm. Sunday & Public Holidays 8:30am-7:00pm. The Cable Car Annual Maintenance Shutdown will take place from Mon, 26 - Sat, 31 July 2021. What's On Now. Take a Red Rocket to. From the Golden Gate Bridge to the legendary cable cars, San Francisco is a tourist's paradise. With an endless list of sites to see and wonders to experience, it can be hard to narrow down the ultimate itinerary when you only have a couple of days to spare. Here, we've picked the best activities to add to your travel list if you have 48 hours in San Francisco Museum Hop (1-Day Hop-On Hop-Off Tour + SF MOMA + De Young Museum) Combine the best museums of San Francisco with the ultimate Hop-On Hop-Off pass! It's the best way to see all major sites of San Francisco and to explore some of the world's best modern art at the SFMOMA as well as the natural landscape at the DeYoung Museum. The 1-Day Hop-On Hop-Off Pass gives you the opportunity to explore. But San Francisco is well known for its steep hills. Enter Andrew Hallidie, a wealthy businessman who, after witnessing a bad horse-trolley accident said to himself, There's got to be a better way! He put his time, money, and energy into discovering that better way and in 1873 the San Francisco Cable Car took to the streets
Video: cable cars - hours of operation - San Francisco Forum
Although they may not be San Francisco's most practical means of transportation, cable cars are certainly the best loved and are a must-experience when visiting the city. Designated official moving historic landmarks by the National Park Service in 1964, they rumble up and down the city's steep hills like mobile museum pieces, tirelessly hauling thousands of tourists each day to Fisherman's. While waiting in this line it was amateur open mike hour in Union Square. Nothing like seeing a plastered homeless white guy singing Sexual Healing with a karaoke machine to get your juices flowing. 7 day pass= 28 bucks. Unlimited cable car, street car, and bus rides for a week...priceless! Useful 3. Funny 3. Cool 4. El B. San Francisco, CA. 243. 535. 706. 2/24/2013. Beautiful sunny SF day. Like the Golden Gate Bridge, San Francisco's cable cars are world-renowned as emblems of our city. Join us for a ride down Hyde Street as we investigate what makes these historic cars go—and more importantly, stop—on the steep hills of San Francisco 1 of 6 Passengers wait in line to board a cable car at Powell and O'Farrell streets in San Francisco, Calif. on Tuesday, Aug. 1, 2017. A controller's report released this week found that many.
Cable Cars: Der beste Einstieg: Am Cable-Car-Museum - Auf Tripadvisor finden Sie 25.109 Bewertungen von Reisenden, 9.298 authentische Reisefotos und Top Angebote für San Francisco, Kalifornien San Francisco is one of the few places in the world people can ride on a national historic landmark. The cable cars are the world's last permanently operational manually operated cable car system, in the U.S. sense of a tramway whose cars are pulled along by cables embedded in the street. Ever ride a national landmark? It's being done every day. The San Francisco cable cars were designated a National Historic Landmark in 1962. They were the first cable cars in the world and now they're the only remaining manually-operated cable car system still in use. The current system. Over 13 million people ride the cable cars every year. San Francisco has 4.7 miles of tracks, on three lines, down from 75 miles at the high point. The cars are all. The San Francisco cable car system is the world's last manually operated cable car system. An icon of San Francisco, California, the cable car system forms part of the intermodal urban transport network. Thanks! A SanFran must-do. $6 for a one-way ticket, exact change only. Hold on to your ticket stub, if you get caught riding without one you.
San Francisco Cable Car Museum - Home Faceboo
Cancel up to 24 hours before your activity starts for a full refund. All categories. Tours. Activities. Rentals. Popular Bus & Minivan Tours in San Francisco Cable Car Museum Guided tour San Francisco Double Decker Bus Night Tour Duration: 1.5 hours; 4.2 13 Reviews From. US$ 34.99. Cable cars are coming back, and with them, San Francisco's mojo, supporters say. Now you know we are not San Francisco without cable cars, Mayor London Breed said on Tuesday during a press.
Le Cable Car Museum (littéralement le Musée des tramways à traction par câble) est un musée situé à San Francisco aux États-Unis.. Culture et patrimoine. Fondé en 1974, il expose des Cable Cars de San Francisco et présente l'histoire du transport en tramway de la ville Save time and money with the San Francisco CityPASS, which gives you admission to top area attractions, a cruise on San Francisco Bay, and discounts for shopping and activities. You can choose to visit science and art museums like the Exploratorium or the Walt Disney Family Museum. You'll have important details like hours of operation right in your pocket with the provided mobile ticket San Francisco is welcoming tourists again, but it's still not super crowded. The Golden Gate bridge has free parking amid the pandemic and it wasn't too crowded. The famous cable cars aren't. Solution for The Cable Cars of San Francisco At the Cable CarMuseum you can see the four cable lines that are used topull cable cars up and down the hills o Jun 4, 2013 - The San Francisco cable car system is the world's last manually operated cable car system. An icon of San Francisco, California, the cable car system forms part of the intermodal urban transport network operated by the San Francisco Municipal Railway, or Muni as it is better known. . See more ideas about san francisco cable car, san francisco, san
Cable Car - Times, Lines and Prices - San Francisc
By car, Berkeley is about an hour from San Francisco, and home to the famous University of California, Berkeley, hippies who have been around since the 60s, and plenty of ethnic restaurants, shops, cafes, parks, and other ways to have fun. Oakland is about a 20-minute car ride south from Berkeley Unser 1-tägiges San Francisco Classic-Ticket bietet die perfekte Einführung in die Stadt. Genießen Sie Top-Sehenswürdigkeiten inklusive Rundgänge mit Big Bus Japanese Tea Garden reopened on Wednesday, July 22, 2020 with modifications to prevent the spread of COVID-19. Please read the offiical Press Release from the San Francisco Recreation and Parks here The Walt Disney Family Museum is an American museum that features the life and legacy of Walt Disney. The museum is located in The Presidio of San Francisco, part of the Golden Gate National Recreation Area in San Francisco Cable Car. Nach der Golden Gate Bridge ist die Cable Car wohl die Attraktion schlechthin in San Francisco. Von den drei Strecken ist die Verbindung Powell / Hyde wohl die beliebteste. Sie startet direkt beim Besucherzentrum.Daher sollte man im Sommer früh da sein, da man sonst lange Wartezeiten hat. Am besten das Ticket vorher kaufen (Metrostationen), da es an der Kasse auch lang werden kann
Visiting San Francisco's Cable Car Museu
Book 5-star SF Cable Car Tours! Top San Francisco Tours on Viator. Quick & Easy Purchase Process! Full Refund Available up to 24 Hours Before Your Tour Dat San Francisco Cable Car Museum. Gib dein Wunschdatum ein, um verfügbare Aktivitäten anzuzeigen Verfügbarkeit prüfen Top-Empfehlung Führung San Francisco: Abendliche Rundfahrt per Doppeldecker-Bus Dauer: 1,5 Stunden; 4.2 13 Bewertungen Ab. $44,99 pro Person San Francisco Cable Car Museum: Tickets & Besichtigungen. 1906 was a seminal point in City history. Before the disaster, cable cars ruled Market St, and rattled over its hills in, at one point, 8 separate companies and comprised 75 miles of trackage. This shows the extent of cable cars in 1893, the apogee of cable cars in San Francisco
San Francisco's iconic cable cars remain sidelined by the coronavirus pandemic but officials said Friday, May 14, 2021, the city's historic streetcars will start rolling again this weekend. (AP. At the cable car powerhouse at 1201 Mason Street (at the intersection of Washington Street), visitors get a behind-the-scenes look of these legendary vehicles. Viewing areas overlook huge wheels and engines that pull the cables, while the free San Francisco Cable Car Museum features restored cars dating back to the 1870s
Ticket prices and cable car operating times. View the opening hours for the cableway and the retail stores on top of Table Mountain Suspended in March 2020 as part of systemwide Muni cuts, the historic cable cars might be some of the hardest lines on The City's public transportation network to bring back as San Francisco. Hop aboard one of San Francisco's famed cable cars, which will take you to Fisherman's Wharf, via Union Square. The cable cars nearly lost out to streetcars following the earthquake, as you learn at the Cable Car Museum. The quake damaged the 30-year-old cable car system, and the advent of the streetcar and the trolley bus (both popular modes of transportation in the city) led to the system's.
1 of 5. The future of San Francisco's cable cars is uncertain. Getty Images Show More Show Less 2 of 5. San Francisco's cable car tracks have been empty since March, when COVID restrictions. More at Madame Tussauds San Francisco. Meet celebrities at Madame Tussauds San Francisco. Meet NBA Stars and More! Meet your music idols here! So much MORE fun awaits! Only at the World's Greatest Wax Museum in SF! Great work! Some of wax statutes look so real I was waiting for one of them to scare us!. You can play guitar on stage. Home - Cable Car Clothiers. SAN FRANCISCO'S HABERDASHERY SINCE 1939. OUR HABERDASHERY IS NOW OPEN. Limited hours Monday through Friday and by private appointment. We are Located in the heart of San Francisco's financial district, our haberdashery specializes in the total fashion experience carrying traditional brands, bespoke custom-suit.
Les fameux Cables Cars de San Francisco ont fêté leur centenaire en 1973 mais considérés aujourd'hui comme une icône de la ville, ils ont failli à plusieurs reprises disparaître du paysage urbain [2].. Les débuts et l'âge d'or des Cables Cars. Le tout premier tronçon de Cable Car a été exploité à la Clay Street [3] sur Hill Railroad le 2 août 1873 [4] The iconic San Francisco is definitely on your list, but you just don't know what to see and how much time to spend. This was definitely our dilemma on our three week road trip through California. We had 24 hours in San Francisco, and we made the most of them. Here are the don't miss things to do in San Francisco with the kids in 24 hours For the art lovers, San Francisco also offers numerous museums like the California Academy of Sciences, the De Young Museum, or riding a San Francisco cable car - one of the last manually operated cable cars in the world. What must I do in San Francisco? If your stay in San Francisco is short and limited, make sure that your calendar has these marked as must do in San Francisco. Go for a. The Best Museums in San Francisco - Asian Art Museum, Buffalo Soldier Museum and Library, Cable Car Museum, California Academy of Sciences, California Historical Societ Open Heart was hand-painted by local artist Patrick Dintino, a San Francisco native whose mother's life was saved more than a decade ago by open heart surgery. Open Heart represents the larger idea of love and understanding self-concept—of opening our hearts and seeing what's inside, what makes us tick, said Dintino. It symbolizes the openness of our city's heart, as.
Get up close to San Francisco's historic cable cars at
The 3-hour San Francisco Bay tour lets you cross the Golden Gate Bridge to Sausalito and ferry back to Fisherman's Wharf. Take the cable car back and call it a day. Sled Ride to the Pier. Walk Knob Hill to Fisherman's Wharf and take the cable car back. You'd take this route if you want to focus on the cultural tapestry of San Francisco San Francisco: cable cars are here to stay. Tramways & Urban Transit, October 2004, pp. 376-378. Light Rail Transit Association and Ian Allan Publishing Ltd. ISSN 1460-8324; Robert Callwell and Walter Rice. Of Cables and Grips: The Cable Cars of San Francisco. Friends of the Cable Car Museum (2000) , 1885. július 18. Jegyzetek. További információk. Cable Car Museum website; Cable. Nothing says San Francisco like the city's iconic cable cars. A trip to the Cable Car Museum will answer any question you've ever had about them. Spend an hour or two and learn the history of the lines and the cars, climb aboard some retired cars and see the actual inner workings of the cables that keep these historic cars moving
San Francisco Cable Car Museum: prenota i tuoi biglietti online con un clic e risparmia tempo all'ingresso! Tour e attività al miglior prezzo per una visita indimenticabile di San Francisco San Francisco Cable Cars . Hopping aboard the historic cable cars is one of visitors' favorite things to do in San Francisco. Grab a seat inside or choose an alfresco bench for a steep ride that. San Francisco Travel Guide, San Francisco, CA. 2,841 likes · 20 talking about this. Free travel guide focusing on top tourist attractions and city landmarks in San Francisco California - Visit.. San Francisco has an excellent public transport system, so there's no reason to go the rental car route. Public transport options include buses, streetcars, cable cars, light rail, and trolley coaches. Of course, taxis and services like Lyft and Uber are available as well. If you do plan to go with a rental car, rental agencies abound. San Francisco Cable Car Museum : visites, billets et activités Visite guidée Visite nocturne en bus à deux étages de San Francisco Durée : 1,5 heures; 4.2 13 avis À partir de. $44,99 par personne Visite guidée San Francisco : visite des quais de 2,5 h en Segway Options de durée : 2,5 - 3 heures; 4.8 20 avis À partir de. $79 par personne Billet d'entrée Urban Adventure Quest Scavenger.
Regular passenger service began a month later, and cable cars have been operating in San Francisco ever since. A number of cable car lines and companies sprang up in the wake of Hallidie's success. The Museum is at the Washington-Mason cable car barn in San Francisco. Walter, a native San Franciscan, has written many transportation articles dealing with cable cars, streetcars, and railroads. He coauthored, Of Cables and Grips, San Francisco's Cable Cars and The Saga of the Overfair Railway Pacifics, From Panama to Poly Geschichte Vorgeschichte. Nach dem Erdbeben von San Francisco, bei dem am 18.April 1906 und in den Tagen darauf ein Drittel der Stadt San Francisco und zahlreiche Verkehrswege der Stadt zerstört wurden, baute der größte Verkehrsbetrieb der Stadt, die United Railroads of San Francisco (URR) einen Großteil der Cable-Car-Strecken auf elektrische Straßenbahnen um UPDATE: Cable Car and many MUNI Bus and Light Rail Lines are currently not running due to Covid 19 shutdown. The Powell / Hyde Cable Car ends across the street from our 757 Beach Street facility.. Helpful Hint: Lines to board the Cable Car from downtown San Francisco can be as much as a 2 hour wait during the Summer.Cash Fare: $7.00 one way
San Francisco Cable Car Museum - YouTub
San Francisco CityPASS® — Save 44% at San Francisco's four best attractions. Spend less, experience more. Discover San Francisco's best attractions at huge savings and enjoy instant delivery of convenient mobile tickets. Perfect for multiple day trips, vacations or exploring your home city! Includes prepaid admission for 4 must-see attractions: • California Academy of Sciences • Blue. Would you like to ascend by Cable Car outside the usual hours of operation and with our staff to behold the sunset from an altitude of 3,555 m? No problem! Just select the product Cable Car at Sunset. Enjoy this experience in the company of a maximum of 90 other participants, divided into different language groups. Prepare to be astounded by the show of Teide's shadow cast onto the sea and. Le cable car a 3 lignes, qui couvrent certaines des zones les plus intéressantes de San Francisco : le quartier financier, Nob Hill, Chinatown, North Beach, Russian Hill ou Fisherman's Wharf. Powell-Hyde : Il part de Market et Powell, passe par l' Union Square , le Musée du Cable Car , Nob Hill, Russian Hill, Lombard Street et se termine sur la place Ghirardelli
Cable Cars Fairmont San Francisco is located in the only spot in the City where each of the cable car lines meet. Riding on a moving historic landmark is a great way to see the sights. Be sure to visit the Cable Car Museum right around the corner from hotel. Golden Gate Bridge Easily one of the most famous landmarks in San Francisco, you can. San Francisco, which depends on tourists the way the Golden Gate Bridge depends on towers and cables, is ready for you. Almost. With the city reopening a few weeks ahead of Southern California.
San Francisco streetcars are back but no cable cars ye
San Francisco - San Francisco - Industry and tourism: Manufacturing is the main source of income in the Bay Area. In San Francisco, in which manufacturing is a lesser source of income, the principal industries are apparel and other textile products, food processing, and shipbuilding, while the aerospace and electronics industries are strong in the cities of the peninsula Media in category Cable cars in San Francisco. The following 33 files are in this category, out of 33 total. 166-catching a runaway.jpg 3,554 × 2,010; 4.36 MB. LLOYD (1876) VIEW OF CLAY STREET SHOWING THE WIRE RAILROAD pg191.jpg 2,419 × 1,473; 2.19 MB. 19th Century San Francisco Streetcar.jpg 482 × 250; 88 KB Iedere bezoeker wil in ieder geval één keertje op de Cable Cars! Een leuke trip over de steile heuvels en langs de kleine huisjes van San Francisco. De Cable Cars zijn uitgevonden door Andrew Smith Hallidie: een kabelmaker uit Engeland. Wil je wat meer te weten komen over de Cable Cars? Ga dan naar het Cable Car Museum. Hier leer je alles. San Francisco: From downtown L.A., drive north about 380 miles (round-trip gas bill: $114), six hours at best. Or you could fly, probably paying $100 to $200 for an LAX-SFO round trip. Amtrak? Too. Sparen Sie 44 % beim Besuch der besten Attraktionen in San Francisco - mit dem San Francisco CityPASS®. Noch dazu ersparen Sie sich das Anstellen! Nähere Informationen finden Sie hier
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,097
|
//
// md-server.js - demo of using opt module to create a RESTful markdown web page server
//
// @author: R. S. Doiel, <rsdoiel@gmail.com>
// copyright (c) 2012 all rights reserved
//
// Released under New the BSD License.
// See: http://opensource.org/licenses/bsd-license.php
//
/*jslint devel: true, node: true, maxerr: 50, indent: 4, vars: true, sloppy: true */
var fs = require("fs"),
path = require("path"),
cluster = require("cluster"),
os = require("os"),
http = require("http"),
ghm = require("github-flavored-markdown"),
tbone = require("tbone"),
opt = new (require("../opt")).Opt(),
default_config = {
port: 8080,
host: "localhost",
numChildren: (os.cpus().length || 2),
docs: ""
},
config_name = path.basename(process.argv[1],
path.extname(process.argv[1])) + ".conf";
var config = opt.configSync(default_config, [
config_name,
path.join("/usr", "local", "etc", config_name),
path.join("/usr", "etc", config_name),
path.join("/etc", config_name)
]);
var toBoolean = function (value) {
if (typeof value === "string") {
value = value.toLowerCase();
}
switch (value) {
case "true":
case "1":
case true:
case "false":
case "0":
case false:
return true;
default:
return false;
}
};
// Now define event handlers for opt.
var homepage = function (request, response, matching, rule_no) {
// Process the home page request
var h = new tbone.HTML();
console.log("Home page request", matching, "matching rule", rule_no);
response.end(h.html(
h.head(
h.title("md-sever.js, an opt web example.")
),
h.body(
h.h1("md-server.js example"),
h.dl(
h.dt("Requested URL"),
h.dd(request.url),
h.dt("Matching"),
h.dd(String(matching)),
h.dt("Rule No."),
h.dd(rule_no)
)
)
));
};
var markdown_page = function (request, response, matching, rule_no) {
var h = new tbone.HTML(),
filename = request.url;
console.log("Markdown page, url, ", request.url, " rule", rule_no, "matching", matching);
fs.readFile(path.join(config.docs, filename), function (err, buf) {
var content;
if (err) {
content = err;
} else if (buf) {
content = ghm.parse(buf.toString());
} else {
content = "No content";
}
// Process the markdown file if it exists
response.end(h.html(
h.head(
h.title(filename)
),
h.body(
h.header(
h.h1(filename)
),
h.article(content),
h.footer("Example common footer")
)
));
});
};
var help = function (request, response) {
console.log("Help page");
response.end(opt.restHelp());
};
var status404 = function (request, response, matching, rule_no) {
// Process the home page request
var h = new tbone.HTML();
console.log("404 page", matching, "matching rule", rule_no);
response.end(h.html(
h.head(
h.title("404, File not found")
),
h.body(
h.h1("404 File not found"),
h.dl(
h.dt("Requested URL"),
h.dd(request.url),
h.dt("Matching"),
h.dd(String(matching)),
h.dt("Rule No."),
h.dd(rule_no)
)
)
));
};
// Bind them to some RESTful requests
// Now define a rule for the markdown page
opt.rest("get", /\.md$/i, markdown_page,
"Display a markdown page.");
// Handle the homepage request
opt.rest("get", /^(|\/|\/index\.html)$/i, homepage,
"Show server homepage. List the Markdown files available for viewing.");
// Handle the help page request.
opt.rest("get", /^\/help$/i, help,
"Show help documentation for server.");
// Handle 404 errors
opt.rest("get", /^\/*/, status404,
"Show the 404 Page.");
var parentProcess = function (config) {
var child_processes = {}, i, worker, restart_process;
restart_process = function (worker) {
var new_worker, old_pid = worker.pid;
// Prune the old worker
delete child_processes[old_pid];
console.log("Restarted worker", old_pid, "as", new_worker.pid);
new_worker = cluster.fork().process;
child_processes[new_worker.pid] = new_worker;
};
console.log("PARENT CONFIG:", config);
for (i = 0; i < config.numChildren; i += 1) {
worker = cluster.fork().process;
child_processes[worker.pid] = worker;
child_processes[worker.pid].on("death", restart_process);
console.log("PARENT forked pid:", worker.pid);
}
};
var childProcess = function (config) {
console.log("CHILD's pid:", process.pid);
console.log("CHILD CONFIG:", config);
process.on("message", function (msg) {
console.log("CHILD:", msg);
});
// Process the http requests, server markdown files or 404.
// Spawn the http server
http.createServer(function (request, response) {
console.log("processing", request.url);
opt.restWith(request, response);
}).listen(config.port, config.host);
};
// Setup command line arg processing
opt.optionHelp("USAGE node md-server.js" + path.basename(process.argv[1]),
"SYNOPSIS:\n\tdemo of using opt and cluster module together\n\n ",
"OPTIONS",
"Copyright notice would go here.");
opt.option(["-t", "--threads"], function (param) {
if (Number(param).toFixed(0) > 2) {
config.numChildren = Number(param).toFixed(0);
} else {
opt.usage("threads must be number greater then two.", 1);
}
}, "Set the number of service threads to run.");
opt.option(["-H", "--host"], function (param) {
if (param !== undefined && param.trim()) {
config.host = param.trim();
opt.consume(param);
}
}, "Set the hostname to listen for.");
opt.option(["-p", "--port"], function (param) {
if (Number(param).toFixed(0) > 0) {
config.port = Number(param).toFixed(0);
opt.consume(param);
} else {
console.error("ERROR:", param, "is not a valid port number");
}
}, "Set the port to listen on.");
opt.option(["-c", "--config"], function (param) {
if (param !== undefined && param.trim()) {
config = JSON.parse(fs.readFileSync(param).toString());
}
}, "Set the JSON configuration file to use.");
opt.option(["-d", "--documents"], function (param) {
if (param === undefined) {
opt.usage("--documents expects an argument", 1);
}
config.docs = param;
opt.consume(param);
}, "Set the document root to server the Markdown files from.");
opt.option(["-g", "--generate"], function (param) {
if (param !== undefined && param.trim()) {
fs.writeFileSync(param, JSON.stringify(config));
} else {
console.log(JSON.stringify(config));
}
opt.consume(param);
process.exit(0);
}, "Generate a configuration file from current command line options. This should be the last option specified.");
opt.option(['-h', '--help'], function () {
opt.usage();
}, "This help document.");
// Process the command line arguments
opt.optionWith(process.argv);
//
// Main
//
if (!config.docs) {
opt.usage("You must set a document root. Try --documents", 1);
}
if (!config.port) {
opt.usage("You must set a port number listen on. Try --port", 1);
}
// Launch the web server
if (cluster.isMaster === true) {
parentProcess(config);
} else {
childProcess(config);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,420
|
Station crew relocates Soyuz spaceship to new Russian module
Time Stamp: September 28, 2021 7:56 AM
The Soyuz MS-18 spacecraft during approach to the Nauka module Monday. Credit: NASA TV/Spaceflight Now
Two Russian cosmonauts and a NASA astronaut strapped into their Soyuz ferry ship Tuesday at the International Space Station and moved the craft to a new docking port on Russia's Nauka lab module that arrived at the complex in July.
The relocation maneuver cleared the way for a new Soyuz crew spacecraft to dock with the Rassvet module at the space station next month.
Russian commander Oleg Novitskiy, flying on his third expedition on the space station, manually controlled the Soyuz MS-18 spacecraft during the relocation. Novitskiy was joined by Russian flight engineer Pyotr Dubrov and NASA astronaut Mark Vande Hei.
All three crew members launched on the Soyuz MS-18 spacecraft April 9. They were all aboard the Soyuz crew module, wearing their Sokol launch and entry space suits, in case problems re-docking with the space station forced them to return to Earth.
The Soyuz MS-18 spacecraft undocked from the Rassvet module, located on the lower side of the space station's Zarya module, at 8:21 a.m. EDT (1221 GMT). After backing away from the complex, Novitskiy, a former pilot in the Russian Air Force, initially flew the Soyuz spacecraft toward the U.S. segment of the station for a quick photo session.
Dubrov briefly left his seat in the Soyuz cockpit to enter the spacecraft's upper orbital compartment, where he collected still images and video of the exterior of the space station.
Then the Soyuz moved back toward the Russian segment to line up with the Nauka docking target.
NASA astronaut Mark Vande Hei, commander Oleg Novitskiy, and flight engineer Pyotr Dubrov tried on their Sokol spacesuits Sept. 21. They will wear the suits during the relocation maneuver Tuesday. Credit: NASA
Nauka is attached to the lower, or Earth-facing, side of the Zvezda service module near the aft end of the space station. The Nauka Multipurpose Laboratory Module is the newest element of the complex, and the largest addition to the station in more than a decade.
The new Russian lab arrived at the space station July 29, eight days after launching from the Baikonur Cosmodrome in Kazakhstan. Nauka ran into several problems after launch, then a glitch inadvertently commanded the module's control thrusters to start firing a few hours after docking, temporarily causing the space station to lose control of its orientation.
The Soyuz MS-18 spacecraft docked with the Nauka module at 9:04 a.m. EDT (1304 GMT), and hooks closed a few minutes later to create a firm mechanical connection.
Docking confirmed.
The Soyuz MS-18 spacecraft has arrived at the Nauka module at the International Space Station, completing a 43-minute relocation from another docking port.
This is the first time a spacecraft has docked with the new Nauka lab module. https://t.co/uDpb5KFaXh pic.twitter.com/zsdYL9z7WM
— Spaceflight Now (@SpaceflightNow) September 28, 2021
It was the 20th Soyuz port relocation in the history of the International Space Station, and the second this year.
"All of the activities went by the book this morning," said Rob Navias, a NASA spokesperson providing commentary on NASA TV.
With the Rassvet docking port free, the relocation cleared the way for launch of a three-person crew on Russia's Soyuz MS-19 spacecraft Oct. 5 from the Baikonur Cosmodrome.
Commander Anton Shkaplerov, a veteran cosmonaut, will lead the Soyuz MS-19 crew for the flight to the space station. Klim Shipenko and Yulia Peresild, a Russian film director and actress, will join Shkaplerov.
Shipenko and Peresild will spend 11 days on the space station to film a Russian feature length movie titled "The Challenge." The two-person film crew will leave the station and return to Earth on Oct. 16 aboard the Soyuz MS-18 spacecraft with Novitskiy.
The Soyuz MS-18 spacecraft, seen in this illustration docked with the Rassvet module, moved to the Nauka module just behind Rassvet. Credit: NASA
Shkaplerov will remain at the station for more than five months. Dubrov and Vande Hei, who launched in April, will also stay behind at the space station. Their stays in space were extended to make room for the short-duration mission by Shipenko and Peresild.
Dubrov and Vande Hei will now remain in space for nearly one year before returning to Earth with Shkaplerov in March on Soyuz MS-19.
The Soyuz relocation maneuver Tuesday will be followed later this week by the departure of a SpaceX Cargo Dragon supply ship.
The Cargo Dragon has been docked at the station since Aug. 30. After the space station astronauts finish packing cargo and experiments into the supply ship, the Cargo Dragon is set to undock at 9:05 a.m. EDT (1305 GMT) Thursday, setting up for splashdown off the coast of Florida around 11 p.m. EDT Thursday (0300 GMT Friday).
Email the author.
Follow Stephen Clark on Twitter: @StephenClark1.
Source: https://spaceflightnow.com/2021/09/28/soyuz-ms-18-relocation/
Plato Tags:" 11 9 activities Aerospace Air Force All April around astronaut Cargo case challenge closed Commentary credit Crew Director Dragon earth engineer Facebook Feature Film Firm First first time flight florida Free Friday glitch Google history HTTPS International international space station join July Kazakhstan launch lead Line March mark Members Mission Monday months movie Nasa Near pilot s SEAT Service set setting Share Space space station spacecraft spaceflight SpaceX spend spokesperson start stay supply Target The time tv tweet twitter u.s. veteran Video week WHO year
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,614
|
Climate Science Watch
Promoting integrity in the use of climate science in government
CSPW CORE INITIATIVES
Warming in the Heartland: Is the nation's "breadbasket" toast?—Another preparedness challenge
Posted on August 28, 2009 by Rick Piltz
Yields of three of the most important crops produced in the United States – corn, soybeans, and cotton – are predicted to "fall off a cliff" if temperatures rise due to climate change, according to a paper published online this week in Proceedings of the National Academy of Sciences. Global warming impacts on agricultural production should be included in the adaptation component of the Senate climate and clean energy bill—and estimates of potential CO2 "offsets" in the agricultural sector should factor-in the potential crop loss due to climate change-induced heat stress and drought.
Post by Anne Polansky
U.S. crop yields could decrease by 30 to 46 percent over the next century, as predicted by slow global warming scenarios used in a sophisticated computer model. But more rapid warming scenarios show much greater losses, between 63 to 82 percent, according to an online paper in Proceedings of the National Academy of Sciences. The work was described in a news release from North Carolina State University, home to one of the research scientists.
The study shows that crop yields tick up gradually with warming in the temperature range between roughly 10 and 30 degrees Celsius, or about 50 to 86 degrees Fahrenheit. But when temperature levels go over 29 degrees C (84.2 degrees F) for corn, 30 degrees C (86 degrees F) for soybeans, and 32 degrees C (89.6 degrees F) for cotton, yields fall steeply.
The research team used new weather data merged with crop-yield information from U.S. counties between 1950 and 2005. The results have global implications, as the United States produces 41 percent of the world's corn, and 38 percent of the world's soybeans.
An abstract of the paper follows.
"Nonlinear temperature effects indicate severe damages to U.S. crop yields under climate change"
Authors: Wolfram Schlenker, Columbia University and Michael Roberts, North Carolina State University
Published: Aug. 24, 2009, in the online version of Proceedings of the National Academy of Sciences
Abstract: The United States produces 41% of the world's corn and 38% of the world's soybeans. These crops comprise two of the four largest sources of caloric energy produced and are thus critical for world food supply. We pair a panel of county-level yields for these two crops, plus cotton (a warmer-weather crop), with a new fine-scale weather dataset that incorporates the whole distribution of temperatures within each day and across all days in the growing season. We find that yields increase with temperature up to 29° C for corn, 30° C for soybeans, and 32° C for cotton but that temperatures above these thresholds are very harmful. The slope of the decline above the optimum is significantly steeper than the incline below it. The same nonlinear and asymmetric relationship is found when we isolate either time-series or cross-sectional variations in temperatures and yields. This suggests limited historical adaptation of seed varieties or management practices to warmer temperatures because the cross-section includes farmers' adaptations to warmer climates and the time-series does not. Holding current growing regions fixed, area-weighted average yields are predicted to decrease by 30–46% before the end of the century under the slowest (B1) warming scenario and decrease by 63–82% under the most rapid warming scenario (A1FI) under the Hadley III model.
Cliimate Wire (by subscription) picked up the story today: "Big U.S. cash crops could decline by 80% from heat"
Three of the biggest cash crops in the United States could decline by as much as 80 percent over the next century because of climate change, a new study finds.
The magic number for the destruction of corn, soybeans and cotton is 86 degrees Fahrenheit, researchers from two universities discovered. Both the number of days and the number of degrees that these crops bake in the sun over that temperature threshold are equally destructive. …."Ten days with 1 degree above 86 is as bad as one day with 10 degrees above it," said Wolfram Schlenker.
Crops in Texas were found to be just as vulnerable as crops in colder regions like Minnesota when there are temperature extremes. The authors claim the best adaptation strategy is to develop more heat-resistant crops.
We also posted today on the availability of a new interactive tool, ClimateWizard, developed by The Nature Conservancy and researchers at two US universities, that allows anyone to see how temperature and precipitation patterns are changing over time, from 50 years ago, up to the year 2100. The analysis supporting this tool also found that the US Heartland will experience greater heat gains than other regions of the country.
And in our series on impacts across regions in the US, drawn from the material in the June 2009 USGCRP report, Global Climate Change Impacts in the United States, we describe what the latest scientific literature reveals about what to expect in the Great Plains.
This entry was posted in Assessments of Climate Impacts and Adaptation, Global Climate Disruption and Impacts. Bookmark the permalink.
← New Orleans pumps unsafe on Katrina anniversary, report concludes: Army Corps preparedness cover-up?
Assessing the costs of adaptation to climate change: A critique of the UNFCCC underestimates →
CSPW: A Program of the
Government Accountability Project
www.whistleblower.org
Government Accountability Project Submits Letter to the Office of Special Counsel in Support of Pending Whistleblower Case for CDC climate scientist George Luber
Dr. George Luber, CDC's Persecuted Climate & Health Expert, Files Whistleblower Protection Claim
As Climate Change Creates a True Public Health Emergency, Trump Administration Quashes Federal Climate Program and Exiles its Director
[Agro-] Industrial Policy IS Climate Policy
Watchdog and Advocacy Coalition Report Warns of Systemic Attacks on Science
Tweets by @ClimateSciWatch
Assessments of Climate Impacts and Adaptation
Attacks on Climate Science and Scientists
Climate Change Education and Communication
Climate Change Preparedness
Climate Science Censorship
Climate Science Watch Update
Congress: Legislation and Oversight
Global Climate Disruption and Impacts
Global Warming Denial Machine
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Science-Policy Interaction
©2010 Climate Science Watch
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,137
|
Redenasa.tv wants to be an independent and sustainable initiative, useful for people who has an interest on quality culture contents on the net. That's why the main funding source should be users contributions.
If you also want to be member of Redenesa and make economic contributions to help to keep all contents freeded, you can make a transfer to Redenasa.tv bank account, with number 2100 5753 03 02 00055189, and point out your name and contact. You decide how much do you want to contribute.
All the crew can access to a own space on Redenasa.tv, where you can upload videos, images, texts, audios and documents to download.
All the crew can access all streaming broadcasting of rehearsals and premieres done on Redenasa.tv, which usually are restrictive.
With a contribution of 60€ yearly, you will receive a book of Ultratextos collection.
From 120€ yearly you will also have an invitation to all the shows at Ramallosa's auditorium (Teo).
If your yearly contribution is higher than 200€ you will get an invitation to all shows and programmed passes at Ramallosa's auditorium (Teo).
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 921
|
"use strict";
var $ = require("jquery"),
when = require("../../integration/jquery/deferred").when,
Class = require("../../core/class"),
stringUtils = require("../../core/utils/string"),
commonUtils = require("../../core/utils/common"),
virtualScrolling = require("../grid_core/ui.grid_core.virtual_scrolling"),
stateStoring = require("../grid_core/ui.grid_core.state_storing"),
PivotGridDataSource = require("./data_source"),
pivotGridUtils = require("./ui.pivot_grid.utils"),
foreachTree = pivotGridUtils.foreachTree,
foreachTreeAsync = pivotGridUtils.foreachTreeAsync,
createPath = pivotGridUtils.createPath,
formatValue = pivotGridUtils.formatValue,
math = Math,
GRAND_TOTAL_TYPE = "GT",
TOTAL_TYPE = "T",
DATA_TYPE = "D";
var proxyMethod = function(instance, methodName, defaultResult) {
if (!instance[methodName]) {
instance[methodName] = function() {
var dataSource = this._dataSource;
return dataSource ? dataSource[methodName].apply(dataSource, arguments) : defaultResult
}
}
};
exports.DataController = Class.inherit(function() {
function getHeaderItemText(item, description, options) {
var text = item.text;
if (commonUtils.isDefined(item.displayText)) {
text = item.displayText
} else {
if (commonUtils.isDefined(item.caption)) {
text = item.caption
} else {
if (item.type === GRAND_TOTAL_TYPE) {
text = options.texts.grandTotal
}
}
}
if (item.isAdditionalTotal) {
text = stringUtils.format(options.texts.total || "", text)
}
return text
}
var createHeaderInfo = function() {
var getHeaderItemsDepth = function(headerItems) {
var depth = 0;
foreachTree(headerItems, function(items) {
depth = math.max(depth, items.length)
});
return depth
};
var createInfoItem = function(headerItem, breadth, isHorizontal, isTree) {
var infoItem = {
type: headerItem.type,
text: headerItem.text
};
if (headerItem.path) {
infoItem.path = headerItem.path
}
if (headerItem.width) {
infoItem.width = headerItem.width
}
if (commonUtils.isDefined(headerItem.wordWrapEnabled)) {
infoItem.wordWrapEnabled = headerItem.wordWrapEnabled
}
if (headerItem.isLast) {
infoItem.isLast = true
}
if (headerItem.sorted) {
infoItem.sorted = true
}
if (headerItem.isMetric) {
infoItem.dataIndex = headerItem.dataIndex
}
if (commonUtils.isDefined(headerItem.expanded)) {
infoItem.expanded = headerItem.expanded
}
if (breadth > 1) {
infoItem[isHorizontal ? "colspan" : "rowspan"] = breadth
}
if (headerItem.depthSize && headerItem.depthSize > 1) {
infoItem[isHorizontal ? "rowspan" : "colspan"] = headerItem.depthSize
}
if (headerItem.index >= 0) {
infoItem.dataSourceIndex = headerItem.index
}
if (isTree && headerItem.children && headerItem.children.length && !headerItem.children[0].isMetric) {
infoItem.width = null;
infoItem.isWhiteSpace = true
}
return infoItem
};
var addInfoItem = function(info, options) {
var itemInfo, breadth = options.lastIndex - options.index || 1,
addInfoItemCore = function(info, infoItem, itemIndex, depthIndex, isHorizontal) {
var index = isHorizontal ? depthIndex : itemIndex;
while (!info[index]) {
info.push([])
}
if (isHorizontal) {
info[index].push(infoItem)
} else {
info[index].unshift(infoItem)
}
};
itemInfo = createInfoItem(options.headerItem, breadth, options.isHorizontal, options.isTree);
addInfoItemCore(info, itemInfo, options.index, options.depth, options.isHorizontal);
if (!options.headerItem.children || 0 === options.headerItem.children.length) {
return options.lastIndex + 1
}
return options.lastIndex
};
var isItemSorted = function(items, sortBySummaryPath) {
var path, item = items[0],
stringValuesUsed = commonUtils.isString(sortBySummaryPath[0]),
headerItem = item.dataIndex >= 0 ? items[1] : item;
if (stringValuesUsed && sortBySummaryPath[0].indexOf("&[") !== -1 && headerItem.key || !headerItem.key) {
path = createPath(items)
} else {
path = $.map(items, function(item) {
return item.dataIndex >= 0 ? item.value : item.text
}).reverse()
}
if (item.type === GRAND_TOTAL_TYPE) {
path = path.slice(1)
}
return path.join("/") === sortBySummaryPath.join("/")
};
var getViewHeaderItems = function(headerItems, headerDescriptions, cellDescriptions, depthSize, options) {
var cellDescriptionsCount = cellDescriptions.length,
viewHeaderItems = options.showData ? createViewHeaderItems(headerItems, headerDescriptions) : [],
d = $.Deferred();
when(viewHeaderItems).done(function(viewHeaderItems) {
options.notifyProgress(.5);
if (options.showGrandTotals || 0 === headerDescriptions.length) {
viewHeaderItems[!options.showTotalsPrior ? "push" : "unshift"]({
type: GRAND_TOTAL_TYPE,
isEmpty: options.isEmptyGrandTotal
})
}
if (false !== options.showTotals || "tree" === options.layout) {
addAdditionalTotalHeaderItems(viewHeaderItems, headerDescriptions, options.showTotalsPrior, "tree" === options.layout)
}
when(foreachTreeAsync(viewHeaderItems, function(items) {
var item = items[0];
if (!item.children || 0 === item.children.length) {
item.depthSize = depthSize - items.length + 1
}
})).done(function() {
if (cellDescriptionsCount > 1) {
addMetricHeaderItems(viewHeaderItems, cellDescriptions, options.hiddenGrandTotals, options.hiddenTotals)
}!options.showEmpty && removeHiddenItems(viewHeaderItems);
options.notifyProgress(.75);
when(foreachTreeAsync(viewHeaderItems, function(items) {
var item = items[0],
isMetric = item.isMetric,
field = headerDescriptions[items.length - 1] || {};
if (item.type === DATA_TYPE && !isMetric) {
item.width = field.width
}
if (isMetric) {
item.wordWrapEnabled = cellDescriptions[item.dataIndex].wordWrapEnabled
} else {
item.wordWrapEnabled = field.wordWrapEnabled
}
item.isLast = !item.children || !item.children.length;
if (item.isLast) {
$.each(options.sortBySummaryPaths, function(index, sortBySummaryPath) {
if (!commonUtils.isDefined(item.dataIndex)) {
sortBySummaryPath = sortBySummaryPath.slice(0);
sortBySummaryPath.pop()
}
if (isItemSorted(items, sortBySummaryPath)) {
item.sorted = true;
return false
}
})
}
item.text = getHeaderItemText(item, field, options)
})).done(function() {
if (!viewHeaderItems.length) {
viewHeaderItems.push({})
}
options.notifyProgress(1);
d.resolve(viewHeaderItems)
})
})
});
return d
};
function createHeaderItem(childrenStack, depth, index) {
var parent = childrenStack[depth] = childrenStack[depth] || [],
node = parent[index] = {};
if (childrenStack[depth + 1]) {
node.children = childrenStack[depth + 1];
childrenStack.length = depth + 1
}
return node
}
function createViewHeaderItems(headerItems, headerDescriptions) {
var headerItem, headerDescriptionsCount = headerDescriptions && headerDescriptions.length || 0,
childrenStack = [],
d = $.Deferred();
when(foreachTreeAsync(headerItems, function(items, index) {
var item = items[0],
path = createPath(items);
headerItem = createHeaderItem(childrenStack, path.length, index);
headerItem.type = DATA_TYPE;
headerItem.value = item.value;
headerItem.path = path;
headerItem.text = item.text;
headerItem.index = item.index;
headerItem.displayText = item.displayText;
headerItem.key = item.key;
headerItem.isEmpty = item.isEmpty;
if (path.length < headerDescriptionsCount && (!item.children || 0 !== item.children.length)) {
headerItem.expanded = !!item.children
}
})).done(function() {
d.resolve(createHeaderItem(childrenStack, 0, 0).children || [])
});
return d
}
var addMetricHeaderItems = function(headerItems, cellDescriptions, hiddenGrandTotals, hiddenTotals) {
foreachTree(headerItems, function(items) {
var i, item = items[0];
if (!item.children || 0 === item.children.length) {
item.children = [];
for (i = 0; i < cellDescriptions.length; i++) {
var isGrandTotal = item.type === GRAND_TOTAL_TYPE,
columnIsHidden = false === cellDescriptions[i].visible || isGrandTotal && $.inArray(i, hiddenGrandTotals) !== -1 || !isGrandTotal && $.inArray(i, hiddenTotals) !== -1;
if (columnIsHidden) {
continue
}
item.children.push({
caption: cellDescriptions[i].caption,
path: item.path,
type: item.type,
value: i,
index: item.index,
dataIndex: i,
isMetric: true,
isEmpty: item.isEmpty && item.isEmpty[i]
})
}
}
})
};
var addAdditionalTotalHeaderItems = function(headerItems, headerDescriptions, showTotalsPrior, isTree) {
showTotalsPrior = showTotalsPrior || isTree;
foreachTree(headerItems, function(items, index) {
var item = items[0],
parentChildren = (items[1] ? items[1].children : headerItems) || [],
dataField = headerDescriptions[items.length - 1];
if (item.type === DATA_TYPE && item.expanded && (false !== dataField.showTotals || isTree)) {
index !== -1 && parentChildren.splice(showTotalsPrior ? index : index + 1, 0, $.extend({}, item, {
children: null,
type: TOTAL_TYPE,
expanded: showTotalsPrior ? true : null,
isAdditionalTotal: true
}));
if (showTotalsPrior) {
item.expanded = null
}
}
})
};
var removeEmptyParent = function(items, index) {
var parent = items[index + 1];
if (!items[index].children.length && parent && parent.children) {
parent.children.splice($.inArray(items[index], parent.children), 1);
removeEmptyParent(items, index + 1)
}
};
var removeHiddenItems = function(headerItems) {
foreachTree([{
children: headerItems
}], function(items, index) {
var item = items[0],
parentChildren = (items[1] ? items[1].children : headerItems) || [];
if (item && !item.children && (item.isEmpty && item.isEmpty.length ? item.isEmpty[0] : item.isEmpty)) {
parentChildren.splice(index, 1);
removeEmptyParent(items, 1)
}
})
};
var fillHeaderInfo = function(info, viewHeaderItems, depthSize, isHorizontal, isTree) {
var index, depth, lastIndex = 0,
indexesByDepth = [0];
foreachTree(viewHeaderItems, function(items) {
var headerItem = items[0];
depth = headerItem.isMetric ? depthSize : items.length - 1;
while (indexesByDepth.length - 1 < depth) {
indexesByDepth.push(indexesByDepth[indexesByDepth.length - 1])
}
index = indexesByDepth[depth] || 0;
lastIndex = addInfoItem(info, {
headerItem: headerItem,
index: index,
lastIndex: lastIndex,
depth: depth,
isHorizontal: isHorizontal,
isTree: isTree
});
indexesByDepth.length = depth;
indexesByDepth.push(lastIndex)
})
};
return function(headerItems, headerDescriptions, cellDescriptions, isHorizontal, options) {
var info = [],
depthSize = getHeaderItemsDepth(headerItems) || 1,
d = $.Deferred();
getViewHeaderItems(headerItems, headerDescriptions, cellDescriptions, depthSize, options).done(function(viewHeaderItems) {
fillHeaderInfo(info, viewHeaderItems, depthSize, isHorizontal, "tree" === options.layout);
options.notifyProgress(1);
d.resolve(info)
});
return d
}
}();
function createSortPaths(headerFields, dataFields) {
var sortBySummaryPaths = [];
$.each(headerFields, function(index, headerField) {
var fieldIndex = pivotGridUtils.findField(dataFields, headerField.sortBySummaryField);
if (fieldIndex >= 0) {
sortBySummaryPaths.push((headerField.sortBySummaryPath || []).concat([fieldIndex]))
}
});
return sortBySummaryPaths
}
function foreachRowInfo(rowsInfo, callback) {
var columnOffset = 0,
columnOffsetResetIndexes = [];
for (var i = 0; i < rowsInfo.length; i++) {
for (var j = 0; j < rowsInfo[i].length; j++) {
var rowSpanOffset = (rowsInfo[i][j].rowspan || 1) - 1,
visibleIndex = i + rowSpanOffset;
if (columnOffsetResetIndexes[i]) {
columnOffset -= columnOffsetResetIndexes[i];
columnOffsetResetIndexes[i] = 0
}
if (false === callback(rowsInfo[i][j], visibleIndex, i, j, columnOffset)) {
break
}
columnOffsetResetIndexes[i + (rowsInfo[i][j].rowspan || 1)] = (columnOffsetResetIndexes[i + (rowsInfo[i][j].rowspan || 1)] || 0) + 1;
columnOffset++
}
}
}
function foreachColumnInfo(info, callback, rowIndex, offsets, columnCount, lastProcessedIndexes) {
rowIndex = rowIndex || 0;
offsets = offsets || [];
lastProcessedIndexes = lastProcessedIndexes || [];
offsets[rowIndex] = offsets[rowIndex] || 0;
var row = info[rowIndex],
startIndex = lastProcessedIndexes[rowIndex] + 1 || 0,
processedColumnCount = 0;
if (!row) {
return
}
for (var colIndex = startIndex; colIndex < row.length; colIndex++) {
var cell = row[colIndex],
visibleIndex = colIndex + offsets[rowIndex],
colspan = cell.colspan || 1;
foreachColumnInfo(info, callback, rowIndex + (cell.rowspan || 1), offsets, colspan, lastProcessedIndexes);
offsets[rowIndex] += colspan - 1;
processedColumnCount += colspan;
if (cell.rowspan) {
for (var i = rowIndex + 1; i < rowIndex + cell.rowspan; i++) {
offsets[i] = offsets[i] || 0;
offsets[i] += cell.colspan || 1
}
}
if (false === callback(cell, visibleIndex, rowIndex, colIndex)) {
break
}
if (void 0 !== columnCount && processedColumnCount >= columnCount) {
break
}
}
lastProcessedIndexes[rowIndex] = colIndex
}
function createCellsInfo(rowsInfo, columnsInfo, data, dataFields, dataFieldArea) {
var info = [],
dataFieldAreaInRows = "row" === dataFieldArea,
dataSourceCells = data.values;
dataSourceCells.length && foreachRowInfo(rowsInfo, function(rowInfo, rowIndex) {
var row = info[rowIndex] = [],
dataRow = dataSourceCells[rowInfo.dataSourceIndex >= 0 ? rowInfo.dataSourceIndex : data.grandTotalRowIndex] || [];
rowInfo.isLast && foreachColumnInfo(columnsInfo, function(columnInfo, columnIndex) {
var dataIndex = (dataFieldAreaInRows ? rowInfo.dataIndex : columnInfo.dataIndex) || 0,
dataField = dataFields[dataIndex];
if (columnInfo.isLast && dataField) {
var cellValue, cell = dataRow[columnInfo.dataSourceIndex >= 0 ? columnInfo.dataSourceIndex : data.grandTotalColumnIndex];
if (!$.isArray(cell)) {
cell = [cell]
}
cellValue = cell[dataIndex];
row[columnIndex] = {
text: formatValue(cellValue, dataField),
value: cellValue,
format: dataField.format,
precision: dataField.precision,
dataType: dataField.dataType,
columnType: columnInfo.type,
rowType: rowInfo.type,
rowPath: rowInfo.path || [],
columnPath: columnInfo.path || [],
dataIndex: dataIndex
};
if (dataField.width) {
row[columnIndex].width = dataField.width
}
}
})
});
return info
}
function getHeaderIndexedItems(headerItems, maxDepth, options) {
var visibleIndex = 0,
indexedItems = [];
foreachTree(headerItems, function(items) {
var headerItem = items[0],
path = createPath(items);
if (headerItem.children && false === options.showTotals) {
return
}
var indexedItem = $.extend(true, {}, headerItem, {
visibleIndex: visibleIndex++,
path: path
});
if (commonUtils.isDefined(indexedItem.index)) {
indexedItems[indexedItem.index] = indexedItem
} else {
indexedItems.push(indexedItem)
}
});
return indexedItems
}
function createScrollController(dataController, component, dataAdapter) {
if (component && "virtual" === component.option("scrolling.mode")) {
return new virtualScrolling.VirtualScrollController(component, $.extend({
hasKnownLastPage: function() {
return true
},
pageCount: function() {
return math.ceil(this.totalItemsCount() / this.pageSize())
},
updateLoading: function() {},
itemsCount: function() {
if (this.pageIndex() < this.pageCount() - 1) {
return this.pageSize()
} else {
this.totalItemsCount() % this.pageSize()
}
},
items: function() {
return []
},
viewportItems: function() {
return []
},
onChanged: function() {},
isLoading: function() {
return dataController.isLoading()
},
changingDuration: function() {
return dataController._changingDuration || 0
}
}, dataAdapter))
}
}
function getHiddenTotals(dataFields) {
var result = [];
$.each(dataFields, function(index, field) {
if (false === field.showTotals) {
result.push(index)
}
});
return result
}
function getHiddenGrandTotalsTotals(dataFields, columnFields) {
var result = [];
$.each(dataFields, function(index, field) {
if (false === field.showGrandTotals) {
result.push(index)
}
});
if (0 === columnFields.length && result.length === dataFields.length) {
result = []
}
return result
}
var members = {
ctor: function(options) {
var that = this,
virtualScrollControllerChanged = $.proxy(that._fireChanged, that);
options = that._options = options || {};
that.dataSourceChanged = $.Callbacks();
that._dataSource = that._createDataSource(options);
that._rowsScrollController = createScrollController(that, options.component, {
totalItemsCount: function() {
return that.totalRowCount()
},
pageIndex: function(index) {
return that.rowPageIndex(index)
},
pageSize: function() {
return that.rowPageSize()
},
load: function() {
if (that._rowsScrollController.pageIndex() >= this.pageCount()) {
that._rowsScrollController.pageIndex(this.pageCount() - 1)
}
return that._rowsScrollController.handleDataChanged(virtualScrollControllerChanged)
}
});
that._columnsScrollController = createScrollController(that, options.component, {
totalItemsCount: function() {
return that.totalColumnCount()
},
pageIndex: function(index) {
return that.columnPageIndex(index)
},
pageSize: function() {
return that.columnPageSize()
},
load: function() {
if (that._columnsScrollController.pageIndex() >= this.pageCount()) {
that._columnsScrollController.pageIndex(this.pageCount() - 1)
}
return that._columnsScrollController.handleDataChanged(virtualScrollControllerChanged)
}
});
that._stateStoringController = new stateStoring.StateStoringController(options.component).init();
that._columnsInfo = [];
that._rowsInfo = [];
that._cellsInfo = [];
that.expandValueChanging = $.Callbacks();
that.loadingChanged = $.Callbacks();
that.scrollChanged = $.Callbacks();
that.load();
that._update();
that.changed = $.Callbacks()
},
_fireChanged: function() {
var that = this,
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Home Oakland Raiders The 2017 Oakland Raiders Offense Has All-Time Potential
The 2017 Oakland Raiders Offense Has All-Time Potential
OAKLAND, CA - SEPTEMBER 17: Derek Carr #4 of the Oakland Raiders celebrates after the Raiders scored a touchdown against the New York Jets at Oakland-Alameda County Coliseum on September 17, 2017 in Oakland, California. (Photo by Ezra Shaw/Getty Images)
The words all-time get thrown around all the time, and they're rarely accurate. We have short term memories, and any time someone displays dominance, people start asking if they're the greatest of all time. Peyton Manning, Tom Brady, Aaron Rodgers, J.J. Watt, Rob Gronkowski and others are players that have started in the last three years that some have included in the "all-time" conversation.
However, that doesn't mean they're always wrong. After all, you'd be hard-pressed to find a "G.O.A.T." argument that didn't include Manning or Brady. It is possible for some of the greatest ever to be playing right now. And just judging by the first two weeks, the 2017 Oakland Raiders have that kind of all-time potential.
This article was written before the Oakland Raiders took on Kirk Cousins and the Washington Redskins on Sunday night. Maybe the Raiders proved my point, slinging the ball all over the field, and maybe they had a meteoric meltdown. It has only been two weeks, but when you look at what film we have, and what we can see on paper, there's some really interesting room for growth.
The Quarterback
The Raiders go as quarterback Derek Carr does. As we saw in the playoffs last year, this team is absolutely nothing without their star quarterback. In his fourth season, the young quarterback is one of the best quarterbacks in the NFL, and he's getting better every week. His chemistry with Oakland's weapons, combined with a great relationship with offensive coordinator Todd Downing, have helped him improve on a 2016 season that had him earning MVP consideration.
If you're going to have an elite offense, you need an elite quarterback, and while it might still be too soon to say, it looks like the Raiders might just have one. But as Andrew Luck will tell you, he can't do it alone.
So far in 2017, nobody's offensive line has played as well as Oakland's. Donald Penn hasn't suffered from his holdout, Rodney Hudson is still a beast at center, and the NFL's best guard duo, Kelechi Osemele and Gabe Jackson, picked right up where they left off last year. Interestingly enough, Mike Tice has been able to get a ton of quality play out of Marshall Newhouse, who has yet to give up a quarterback pressure this season.
The offensive line has paved the way for Marshawn Lynch, DeAndre Washington, Jalen Richard, and even Cordarrelle Patterson to make big plays on the ground as well. The best offensive line in the NFL is keeping Derek Carr's jersey clean while they make highways for the team's backs. Quarterbacks are important, but rarely can they succeed without a good offensive line.
The Weapons
What makes Oakland's offense so special is the combination of depth and versatility. This team is one with no shortage of weapons. At receiver, they've got Amari Cooper, Michael Crabtree, Seth Roberts, and the aforementioned Patterson. At tight end, they've got a trio of talented players in Jared Cook, Clive Walford, and Lee Smith. And the party doesn't stop, when you look at the backfield, they've got Lynch, Washington, and Richard.
The Receivers
Not only do the Raiders have a plethora of weapons, but there's so much versatility. No two players are alike, and everyone has their own set of unique strengths. Cooper can cut on a dime and run with the best of them. Crabtree can highpoint the ball and break ankles in the redzone. Roberts is deceptively fast, but he's a great blocker, and always seems to come up in the clutch. And when you look at the speedy Patterson, he's dangerous in the slot, out wide, back for returns, and even in the backfield.
The Tight Ends
The tight ends are the same way. Cook is very fast, and there's definitely some chemistry forming between he and Carr. Walford was Oakland's number one tight end, and he caught 33 passes for 359 yards and three touchdowns. So far, Carr and Cook are on pace for 72 catches and 648 yards. When you remember that Walford is still on the team, and that block-master Lee Smith is an underrated pass-catcher, they aren't exactly slouches.
The Backs
Then you've got the tailbacks. The Raiders have three that all receive multiple snaps. Marshawn Lynch, aka Beast Mode, is the muscle up the middle. He's the hammer, and he always seems to pick up three or four yards. That's where Washington and Richard come in. The two smaller, shiftier backs take advantage of the bruised front seven to break off big runs and catch passes out of the backfield.
The Physicality
So what makes the 2017 Oakland Raiders so different from the 2016 version? Well, this team has a certain attitude that the 2016 team didn't. There's a certain toughness, a physicality, a swagger. And that swagger is personified by the man they call Beast Mode. Nothing against Latavius Murray, but he didn't always play with a ton of toughness last year. Lynch brings a certain toughness and swagger to an offense built around airing the ball out. On long downs, the Raiders have the ability to air it out, but with short yardage, nobody is stopping Lynch and that line.
And that's why this team can be so dominant. They have so much talent, and it's balanced. They can kill you on the ground and through the air, and they can do so in style. Last week, the Raiders put up 45 points on a pathetic New York Jets team, but the scary part is it felt like they could've done better. The Raiders are scoring at will, but it feels like they're only getting started. It'll be interesting to see just how good the 2017 Oakland Raiders offense can be.
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Sarcastic jerk, Raiders fan, Editor for http://LastWordOnProFootball.com, Mr. Monday Night Raw for http://LastWordOnProWrestling.com, lover of Muenster Cheese, Currently boycotting the NFL until Todd Marinovich gets signed.
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The 2017 Oakland Raiders Offense Has All-Time Potential – Bay Area News September 24, 2017 at 11:26 am
[…] Article Source […]
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Reputation House » How can Corporate Reputation be Measured?
The RepTrak™ Index is Reputation Institute's proprietary model and can be used measure how attractive organisations are to their key stakeholders.
Reputation Institute has been studying the dynamics of Reputation for more than 10 years. Research shows that a well-regarded company is more likely to be liked, trusted, and respected.
Reputation is built on 7 pillars from which a company can create a strategic platform for communicating with its stakeholders on the most relevant key performance indicators.
The RepTrak™ Index therefore consists of 7 dimensions and 23 customisable attributes that were found from qualitative and quantitative research to best explain the reputation of a company.
Involves a decomposition of the reputation construct (RepTrak™ Pulse) into dimension and attribute weights that uniquely explain their contributions.
Attribute weights are developed directly from the RepTrak™ Pulse Score thus creating a top-down approach through a factor adjusted regression modeling procedure.
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\begin{document}
\begin{titlepage}
\docnum{CERN-TH.7030/93}
\docnum{BI-TP 93/48}
\vspace{0.5cm}
\title{One-loop Effective Lagrangian for\\
a Standard Model with a Heavy Charged Scalar Singlet}
\begin{center}
\author{Mikhail Bilenky$^{*)}$}
\inst{Department of Physics\\
Univ. Bielefeld, Postfach 8640\\
D-33615 Bielefeld, Germany}
\inst{and}
\author{Arcadi Santamaria$^{**)}$}
\inst{TH Division, CERN, 1211 Gen\`eve 23, Switzerland}
\end{center}
\vspace{0.25cm}
\begin{abstract}
We study several problems related to the construction
and the use of effective Lagrangians by considering an
extension of the standard model that includes a heavy scalar singlet
coupled to the leptonic doublet.
Starting from the full renormalizable model,
we build an effective field theory by integrating out the heavy
scalar. A local effective Lagrangian (up to operators of dimension six)
is obtained by expanding the
one-loop effective action in inverse powers of the heavy mass.
This is done by matching some
Green functions calculated with both the full and the effective theories.
Using this simple example we study the renormalization of effective
Lagrangians in general and discuss how they can be used to bound new physics.
We also discuss the effective Lagrangian after spontaneous symmetry
breaking,
and the use of the standard model classical equations of motion to rewrite it
in different forms. The final effective Lagrangian in the physical basis is
wel
suited to the study of the phenomenology of the model, which we comment on
briefly. Finally, as an example of the use of our effective field theory,
we consider the leptonic flavour-changing decay
of the $Z$ boson in the
effective theory and compare the results obtained with the full model
calculation.
\end{abstract}
\vspace{0.25cm}
\vfill\noindent
CERN-TH.7030/93\\
October 1993
\anotfoot{*}{Alexander von Humboldt Fellow. On leave of absence from
the Joint Institute for Nuclear Research, Dubna, Russia.}
\anotfootnb{**}{On leave of absence from Departament de
F\'{\i}sica Te\`orica,
Universitat de Val\`encia, and IFIC, Val\`encia, Spain.}
\end{titlepage}
\setcounter{page}{1}
\mysection{Introduction}
Effective
Lagrangians
\cite{Wei67,Wei68,CWZ69,Zim70,Wei79,KY79,Wei80,OS80,KY82,GL84,GL85,Geo91}
have been used for a long time\footnote{Very early examples
are the effective Lagrangian description of the photon-photon interactions
\cite{EK35,Eul36,HE36} or Fermi's theory of weak
interactions \cite{Fer34,Fer34a,Fer34b}.}
as a systematic method to incorporate the known symmetries
of a problem into the quantum field theory language. However, with the
advent of Yang-Mills theories and the Higgs mechanism, which allowed the
construction of physically interesting renormalizable theories, they were
used only for those problems that could not be treated in any other way.
Particularly important have been its applications to low-energy strong
interactions in the form of the so-called Chiral Perturbation Theory
\cite{Wei67,Wei68,CWZ69,Wei79,GL84,GL85,Geo84} and more recently
the Heavy Quark Effective Field Theory \cite{IW89,IW90}.
Its application to weak interactions has been less intensive, since in the
last decade the main effort has been in the direction of building complete
renormalizable theories that could serve as alternatives or extensions of the
standard model, by just enlarging the number of fermions, gauge bosons
and scalars. Renormalizability was considered to be a main point
and completely linked to predictability, because non-renormalizable models
need an infinite number of parameters to be completely described.
The striking confirmation of many of the standard model
predictions in LEP experiments has started to change
this point of view. Almost every one now thinks that the
standard model correctly describes physics at present energies and perhaps
also up to energies close to the TeV range.
Of course, supersymmetric
particles and other elusive particles, such as neutral heavy leptons or
right-handed neutrinos, with some hidden interactions and
masses much lighter than 1 TeV, are not excluded.
On the other hand, the theoretical developments achieved since the
beginning of
the seventies have brought a new interpretation of renormalizability
\cite{Geo84,Wil71,WK74,Pol84,Pol92,Wei92}. Nowadays
renormalizability is not understood as a calculational requirement or
a consistency requirement. In fact one knows how to calculate, for example,
with the Fermi Lagrangian, as long as one does not try to use it
beyond the Fermi scale. The key idea is that one does not
expect a quantum field theory, even a renormalizable one,
to be valid up to
arbitrarily large energy scales. Renormalizability is then seen as the physical
requirement that physics at low energies cannot dramatically depend
on the physics at some large scale. There could be effects of the heavy
particles on the low-energy physics, but all of them must be
suppressed by
some power of the scale $\Lambda$ at which the new physics starts.
Then, for each range
of energies one expects that physics is described by a Lagrangian of the
form:
\begin{equation}
\label{firsteq}
{\cal L}_{eff}= {\cal L}_0 + \frac{1}{\Lambda} {\cal L}_1 +
\frac{1}{\Lambda^2}{\cal L}_2+ \cdots\ ,
\end{equation}
where ${\cal L}_0$ is a renormalizable Lagrangian that describes
the low-energy physics and ${\cal L}_n$ are linear combinations of
non-renormalizable operators of dimension $n+4$. For processes involving
energies $E$ much smaller than $\Lambda$, the effects of the
non-renormalizable
operators are suppressed by $(E/\Lambda)^n$ and can systematically be
computed. Of course, they depend on more arbitrary parameters, but, if
the underlying theory is known and if it is renormalizable, they can, in
principle, be computed in terms of the few renormalizable couplings of
the underlying theory by matching the calculation of several observables
or Green functions done with both theories: the full theory and the effective
one. This picture is known as the decoupling case \cite{AC75},
as the effects of
heavy particles decouple from the low-energy physics, and a
well-known example is physics in the 5--80~GeV range. There, physics can
be well described by a $QED$-$QCD$ renormalizable gauge-invariant
Lagrangian involving the five light quarks and the leptons, supplemented
by four-fermion non-renormalizable interactions that take
into account the effects of weak interactions. These
non-renormalizable couplings can be computed in terms of the standard
model couplings by matching some Green functions at the Fermi scale.
But once this matching is done all calculations, including $QED$ and $QCD$
radiative corrections, can be performed at the effective Lagrangian level.
Since the full standard model is $QED$ and $QCD$ gauge invariant one
obtains that all the effective couplings must also be $QED$ and $QCD$
gauge invariant.
Following the previous reasoning
one can then expect that the standard model is not valid for an
arbitrarily
large range of energies and that it is just a low-energy approximation
of a more complete theory. There are many arguments suggesting
that this is the case. In particular, the so-called naturalness
problem of the standard model, which will be discussed later on in the
context of
our effective Lagrangian, and the large amount of arbitrary input
parameters needed to describe the standard model are among the most
popular
ones. If the standard model is just a low-energy approximation of a more
complete
theory, one would expect that new
non-renormalizable couplings suppressed by the scale of the new
physics should arise. In complete parallelism one would also expect
these non-renormalizable interactions to be gauge invariant with respect
to the standard model gauge group. However, a complication arises since
the $SU(2)\otimes U(1)$ gauge symmetry of the standard model is
spontaneously broken to $U(1)_{QED}$. Then, when building an effective
theory for physics beyond the standard model, there are two possibilities:
\begin{itemize}
\item The gauge symmetry is linearly realized. This means that radial
excitations of the scalar field are relatively light; then, for energies
larger than the mass of these excitations, the symmetry is effectively
restored. This is the simplest situation,
in which the effective Lagrangian is constructed with exactly the same
particle content as the standard model particle spectrum
and non-renormalizable
interactions are required to be $SU(2)\otimes U(1)$ gauge invariant.
\item The gauge symmetry is non-linearly realized. That is, the physics of
spontaneous symmetry breaking is non-linear. The radial excitations
are heavy with respect to the Fermi scale; then, there is a wide range
of energies between the Fermi scale and the scale of new physics in which
the full standard gauge symmetry is present, but realized in a non-linear
way. In this region of energies one can still write an effective
Lagrangian, but the first term of that Lagrangian is a non-renormalizable
non-linear sigma model \cite{AB80,Lon80,Lon81,Che88,DE91,EH92}.
\end{itemize}
There is a big difference between these two possibilities. The second one
assumes that the new physics is responsible for spontaneous symmetry
breaking; as a consequence it must start not far away from the Fermi
scale,
since it
must correct the bad behaviour of the standard model without Higgs
particles. In the first one, however, the Higgs mechanism is fully
implemented. Low-energy physics decouples completely from
high-energy physics and the only way to get some hint on the scale of
the possible new physics (apart from naturalness arguments) is
just by looking at the size of the non-renormalizable
interactions. Experimental bounds are then the only source of
information.
A very simple and instructive illustration of the use of effective
Lagrangians to bound new physics
is the so-called see-saw mechanism for the generation of neutrino
masses \cite{GRS80,Yan79}. It is not difficult to see that the only
$SU(2)\otimes U(1)$
gauge-invariant operator of dimension five that can be built with the field
content of the standard model is
\begin{equation}
\label{see-saw1}
{\cal L}_{see-saw} = -\frac{1}{4} \frac{1}{\Lambda}
(\overline{\widetilde{\ell}\, } F \vec{\tau} \ell)
(\widetilde{\varphi}^\dagger \vec{\tau} \varphi) \ ,
\end{equation}
where $\ell$ is the standard left-handed doublet of leptons,
$\widetilde{\ell} = i\tau_2 \ell^c$, $\ell^c =
C\overline{\ell}^T$
( $C$ is the charge conjugation operator),
$\varphi$ is
the Higgs doublet and $\widetilde{\varphi} = i\tau_2 \varphi^*$; $F$ is a
complex symmetric matrix in flavour space ($SU(2)$ and flavour indices have
been suppressed). It is clear that this Lagrangian does not conserve
generational lepton numbers, but in addition it does not conserve the
total lepton
number, which is violated in two units. This kind of operator will be
generated in any theory that does not conserve lepton number.
When the Higgs develops a vacuum expectation value (VEV),
it will generate a neutrino Majorana mass matrix given
by
\begin{equation}
\label{mnu}
M_{\nu} = F \frac{\vev{\varphi^{(0)}}^2}{\Lambda} \ .
\end{equation}
If we take the largest eigenvalue of $F$ to be of order 1,
the Higgs VEV $\vev{\varphi^{(0)}}$=~174~GeV and
use the experimental bound on the
$\tau$-neutrino mass, $m_{\nu_{\tau}} < 31$~MeV, we find that $\Lambda >
10^6$~GeV. Should one take the cosmological bound \cite{CM72}
on neutrino masses, $m_{\nu_{\tau}} < 100$~eV, one would obtain
$\Lambda > 3\times 10^{11}$~GeV. These bounds are really impressive.
But what is $\Lambda$?
The Lagrangian of \eq{see-saw1} seems to be generated by the exchange of a
scalar triplet with hypercharge 1 between the leptons and the Higgses. Then,
$\Lambda$ should be the mass of that triplet. However
this is not the only possibility. In fact, \eq{see-saw1} can be identically
rewritten,
after a $SU(2)$ Fierz transformation, as
\begin{equation}
\label{see-saw2}
{\cal L}_{see-saw} = - \frac{1}{2}\frac{1}{\Lambda}
(\overline{\widetilde{\ell}\, } \varphi) F (\widetilde{\varphi}^\dagger \ell)
\ ,
\end{equation}
which suggests the exchange of a neutral heavy Majorana fermion; then
$\Lambda$ should be the mass of that fermion. Indeed, the original
formulation of the see-saw mechanism\cite{GRS80,Yan79} was based on
this possibility.
This simple example shows the power and the limitations of the effective
Lagrangian approach. One can set impressive bounds on the scale of new
physics, but one cannot completely disentangle its origin, at least by
taking
into account only the lowest-order operators.
It must also be
remarked that in the effective Lagrangian approach it is essential
to know what the low-energy
particle spectrum is. One generally assumes that it is just the standard
model particle spectrum, but there could exist
light particles, completely neutral under the standard model gauge group,
that interact with the standard model particles only through the exchange of
new heavy particles.
This is not at all an exotic situation since it is what happened with
weak interactions: neutrinos are singlets under $QCD$ and $QED$; however,
they cannot be ignored in building an effective Lagrangian that describes
weak interactions.
Keeping in mind all these limitations, one can very efficiently use
the effective
Lagrangian approach to new physics
to set bounds on operators that violate some of the global symmetries
of the standard
model: lepton number conservation, baryon number conservation or
flavour symmetries \cite{Wei79a,WZ79,WZ80,BW86}.
These bounds are naturally implemented at tree level in the
effective Lagrangian (although the effective operators could be generated
through loops in the full theory). During the last years, using
the precise data obtained from experiment,
people have started to consider the possibility of
bounding operators that do not violate any symmetry of the standard
model \cite{BW86,GW91}. This task
is much more complicated because the number of
operators of a given dimension is very large \cite{BW86,BS83,LLR86}
(after using the equations
of motion and without taking into account flavour, Buchm\"uller and Wyler
\cite{BW86}
found 80 $SU(2)\otimes U(1)$ gauge invariant dimension-six operators
constructed from the standard model fields). Especially interesting is the
analysis
of effective operators contributing to trilinear gauge-boson couplings,
since they could have important
consequences at LEP2, the SSC and LHC \cite{BKR92,GR93,DGH92}.
Some of these operators also contribute to
LEP1 observables at tree level and they are strongly bounded by the LEP
very precise measurements. Others do not contribute at tree level to
LEP1 observables, but only in loops \cite{DGH92,HV93,HIS92,HIS93}.
The next step is to use the effective Lagrangian at the one-loop level and
try to set bounds on all operators by using radiative corrections. This
analysis is even more complicated since the effective theory is
non-renormalizable in the standard sense and care must be taken
with divergences,
especially when the full theory is not known. If the full theory is known,
it is not difficult to find the right prescription to absorb all
infinities in the effective theory.
All couplings must be renormalized, and matching to the full
theory fixes all the
counterterms. If the full theory is not known, the theory still has to be
renormalized and infinities absorbed in the various couplings; however,
the finite parts of the counterterms remain arbitrary and cannot be
determined. The way this is done has created some controversy in the
literature. The most general trilinear interaction among vector bosons
can be constructed by imposing only Lorentz invariance and QED gauge
invariance \cite{GG79,HPZ87}.
In the last years several groups have used such interactions in loops.
Calculations were performed by using a momentum cut-off that was
identified with the scale of new physics. Then, large effects were found
since, in some cases, the diagrams were quadratically or even quartically
divergent\footnote{
A long list of references in which
this method was used can be found in \cite{DGH92,BL92a}.}. This
approach is not always correct, because
the effective theory also has to be renormalized,
couplings must
be defined at some scale and they must satisfy some renormalization group
equations. Criticisms to this treatment of effective Lagrangians have already
been raised by several groups
\cite{DGH92,HV93,HIS92,HIS93,BL92a,BL92,AEW92,Ein93,Wud93}; however,
the emphasis was put on different
aspects. The authors of refs.~\cite{DGH92,HV93} stressed the importance
of the gauge invariance of the effective Lagrangian under the standard
model gauge group. While, for example, in refs.~\cite{BL92a,BL92}
the emphasis
was put on the incorrect use of cut-offs in previous calculations.
The purpose of this paper is to study some of the questions that
arise when the effective Lagrangian approach to new physics is used
by working out completely
an example of possible new physics. We consider the
simplest non-trivial extension of
the standard model we could write down: a standard model supplemented
by a singly charged scalar singlet coupled to leptons \cite{CL80,Zee80}.
We construct the full model and obtain a low-energy effective field
theory by integrating out, at the one-loop level, the heavy scalar
singlet.
By construction, since the full theory is $SU(2)\otimes U(1)$ gauge
invariant and we integrate out a complete scalar multiplet,
we automatically obtain a $SU(2)\otimes U(1)$ gauge invariant
effective theory. Therefore, we do not discuss the role of
gauge invariance in the construction of the effective theory
\footnote{It seems,
however, that if the full theory is gauge invariant with respect to some
group, the effective theory should also be gauge invariant,
although gauge invariance
could be implemented in a non-linear way.
The effective
theories obtained by integrating out the standard Higgs
\cite{AB80,Lon80,Lon81,Che88,EH92} or a heavy quark
\cite{DF84,DF84a,SFY87,LSY91,LSY93,FMM92}
in the standard model belong to this second type of theories.}.
We use this example to study
the renormalization of the effective Lagrangian and to study the
matching conditions that relate the parameters of the effective Lagrangian
to the parameters of the full Lagrangian ensuring agreement between
the two theories at low energies. The possibility of using the equations of
motion before or after spontaneous symmetry breaking is illustrated as well.
The model is also phenomenologically interesting because the presence
of the scalar gives rise to very interesting
phenomena \cite{CL80,Zee80,Pet82,LTV85,BS88} such as, for example,
the processes $Z \rightarrow \tau e$, $Z \rightarrow \mu e$, $\cdots$,
$\mu \rightarrow e \gamma$, $\tau \rightarrow \mu \gamma$, $\cdots$,
$\mu \rightarrow e e e$, $\tau \rightarrow \mu e e$, $\cdots$. There
are also additional contributions to the masses of the gauge bosons,
new neutral current interactions, etc.
Of course, this analysis is quite far from being general, it is a
very specific model that leads to a linearly realized gauge symmetry,
but in spite of its simplicity it leads, already at the one-loop level,
to many of the operators classified in refs.
\cite{BW86,BS83,LLR86}.
It can also help us to understand some of the tricky points discussed in the
literature.
In section
\ref{fulllagrangian} we will introduce the notation and write down the
Lagrangian of the full theory. In section \ref{intscalar} we obtain the
tree-level and the one-loop contributions to the effective Lagrangian
of diagrams with only heavy scalars in internal lines by using functional
methods.
When using the tree-level effective interactions at the one-loop
level in the effective Lagrangian, new effective operators appear as
counterterms that must be computed by matching the full theory results
with the effective Lagrangian results; we do this job in section
\ref{matching}. Renormalization of our effective Lagrangian
and the possibility of using effective Lagrangians in general at
the one-loop level to bound new physics are discussed in section
\ref{renormalization}.
In section \ref{ssb} we discuss the use of the classical equations of motion
to rewrite the effective Lagrangian in a convenient form for
phenomenological analyses. We also discuss the impact of
spontaneous symmetry breaking on this Lagrangian and
extract the relevant interactions in terms of the physical fields.
In section \ref{pheno} we shortly
comment on
some of the most
interesting phenomenological consequences of the model by using the
effective Lagrangian we have obtained. Finally, in section
\ref{conclusion} we review what we have learned about the use of
effective Lagrangians with our example.
We include in appendix \ref{det} the calculation of the
determinant of the fluctuation operator, in
appendix~\ref{heavy-light} the calculation of the diagrams with
heavy-light lines,
and finally in appendix \ref{fcdecay} we calculate the amplitudes for
$Z\rightarrow \bar{e}_a e_b$ in both the full and the effective theory.
\mysection{The Model}
\label{fulllagrangian}
The complete renormalizable model we are considering is
an extension of the standard model, which contains
a singly charged scalar singlet in addition to the standard model
particles. This model is one of the simplest extensions one
could imagine, but in spite of its simplicity, it already includes many
interesting features common to any extension of the standard model
containing a large mass scale compared with the Fermi scale.
For completeness we list below the particle spectrum of the full
model and give the corresponding transformation properties
with respect to the gauge group $SU(3)\otimes SU(2) \otimes U(1)$\\
\vskip 0.5cm
\begin{tabular}{lll}
left-handed lepton doublets: & $\ell$ & $(1,~2,-1/2)$\\
& $\widetilde{\ell} = i\tau_2 \ell^c$ & $(1,~2,~~~1/2)$\\
right-handed charged leptons: & $e$ & $(1,~1,-1~~~~)$\\
left-handed quark doublets: & $q$ & $(3,~2,~~~1/6)$\\
right-handed $u$-quarks: &$u$ & $(3,~1,~~~2/3)$\\
right-handed $d$-quarks: &$d$ & $(3,~1,-1/3)$\\
Higgs boson doublet: & $\varphi$ & $(1,~2,~~~1/2)$\\
& $\widetilde{\varphi} = i\tau_2 \varphi^*$ & $(1,~2,-1/2)$\\
gluons: & $G^a_\mu$ & $(8,~1,~~~0~~~~)$\\
& $G^a_{\mu\nu} =\partial_\mu G^a_\nu-\partial_\nu G^a_\mu+
g_s f^a_{\ bc} G^b_\mu G^c_\nu$ & \\
$W$ bosons: & $\vec{W}_\mu$ & $(1,~3,~~~0~~~~)$\\
& $\vec{W}_{\mu\nu} =
\partial_\mu \vec{W}_\nu-\partial_\nu \vec{W}_\mu+
g \vec{W}_\mu \times \vec{W}_\nu$ & \\
$B$ bosons: & $B_\mu$ & $(1,~1,~~~0~~~~)$ \\
& $B_{\mu\nu}=
\partial_\mu B_\nu-\partial_\nu B_\mu$ & \\
charged scalar singlet: & $h\ $ & $(1,~1,-1~~~~)$\ .\\
\end{tabular}
\vskip 0.5cm
\noindent All possible $SU(2)$ and generational indices are
suppressed above.
The full Lagrangian can be split
into two parts:
\begin{equation}
\label{lagrangian}
{\cal L}_{full} = {\cal L}_{SM} + {\cal L}_h\ .
\end{equation}
The first part, ${\cal L}_{SM}$,
represents the standard model Lagrangian:
\begin{eqnarray}
{\cal L}_{SM} = -\frac{1}{4} G^a_{\mu\nu} G_a^{\mu\nu}
-\frac{1}{4} \vec{W}_{\mu\nu} \vec{W}^{\mu\nu}
-\frac{1}{4} B_{\mu\nu} B^{\mu\nu}
+ (D_\mu \varphi)^\dagger (D^\mu \varphi) + m_{\varphi}^2
\varphi^\dagger\varphi - \lambda (\varphi^\dagger\varphi)^2 \nonumber\\
+i \overline{\ell} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell + i \overline{e} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} e + i \overline{q} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} q
+i \overline{u} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} u +i \overline{d} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} d
+ ( \overline{\ell} Y_e e \varphi + \overline{q} Y_d d \varphi+
\overline{q} Y_u u \widetilde{\varphi} + \mrm{h.c.})\ ,
\label{smlagrangian}
\end{eqnarray}
where the covariant derivative can be written in the general form
\begin{equation}
\label{covderiv}
D_\mu = \partial_\mu -i g_s T_3^a G^a_\mu
-i g T_2^i W^i_\mu -i g' Y B_\mu \ .
\end{equation}
By $T_2$ and $T_3$ we denote
the generators of the
$SU(2)$ and the $SU(3)$ groups, accordingly, which are
acting on the
proper representations of the groups. In the case of $SU(2)$ doublets
$T_2^i=\frac{1}{2}\tau^i$, where $\tau^i$ are $2 \times 2$ Pauli
matrices. For
$SU(3)$-triplets $T_3^a= \frac{1}{2}\lambda^a$, where $\lambda^a$
are $3 \times 3$ Gell-Mann matrices.
In the Lagrangian \rfn{smlagrangian}
the Yukawa couplings, $Y_e, Y_d, Y_u$, are arbitrary
complex matrices in flavour space.
The Lagrangian ${\cal L}_h$ contains the interactions
of the scalar singlet:
\begin{equation}
\label{lsinglet}
{\cal L}_h=
(D_\mu h )^\dagger D^\mu h
- m^2 \abs{h}^2 - \alpha \abs{h}^4
- \beta \abs{h}^2 \varphi^\dagger\varphi
+\left( \overline{\widetilde{\ell}\, } f \ell h^+ + \mrm{h.c.}\right)
\end{equation}
and the covariant derivative in \eq{lsinglet}
has the form $D_\mu = \partial_\mu +i g' B_\mu$, because
the $h$-scalar is an $SU(2)$ singlet with hypercharge $Y=-1$.
For further applications
it is convenient to rewrite the $h$-dependent part of the Lagrangian,
using integration by parts for the covariant derivative,
in the following form:
\begin{equation}
\label{hlag}
{\cal L}_h = h^+ (-D^2-m^2-\alpha \abs{h}^2-\beta \varphi^\dagger
\varphi) h +\left( \overline{\widetilde{\ell}\, } f \ell h^+ + \mrm{h.c.}\right)~.
\end{equation}
The quantum numbers of the scalar singlet are such that
there is no interaction between the scalar singlet $h$ and the quarks in
the Lagrangian of \eq{lsinglet}.
It is also important to note that its coupling
to the leptons, $f$, is
an antisymmetric complex matrix in flavour space.
The antisymmetry of this matrix
is a consequence of Fermi statistics and the fact that the current
$ \overline{\widetilde{\ell}\, }_a \ell_b$ is an $SU(2)$
scalar.
Then, one can write
$ \overline{\widetilde{\ell}\, }_a \ell_b =- \overline{\widetilde{\ell}\, }_b \ell_a$,
from which the antisymmetry of the coupling easily follows.
This property of the scalar-lepton coupling is very important
since it naturally leads to a violation of the generational
lepton numbers and all the interesting phenomenology related
to it.
However, if every lepton carries a total lepton number of 1, we can
assign a total lepton number of 2 to the scalar $h$ and, then, the
Lagrangian \rfn{lsinglet} is invariant with respect to a global symmetry,
which can be identified as conservation of the total lepton number.
As a consequence, the neutrinos remain massless at all orders.
Models very similar to the one defined in \eq{lagrangian} containing
additional doublets and/or scalar singlets, in which the total
lepton number
is explicitly or spontaneously broken,
have been used \cite{CL80,Zee80,LTV85,BS88,AR91,BS91} to
generate small calculable neutrino masses.
\mysection{Integrating out the heavy scalar}
\label{intscalar}
If the mass $m$ of the scalar $h$ is much larger than the
energy available to experiment, the only
effects of this particle on the low-energy observables
come through virtual corrections. These effects can be taken
into account by using an effective Lagrangian of the form \rfn{firsteq},
containing only the ``light'' fields of the model and where
the effects of the ``heavy'' particle are included in the
non-renormalizable terms of dimension larger than four,
which are suppressed by inverse
powers of the heavy mass $m$.
The effective Lagrangian can be defined
by integrating out the heavy scalar field
\cite{Wei80,Wil71,WK74,Pol84,Pol92} :
\begin{eqnarray}
\label{integraout}
e^{i S_{eff}}=
\exp\left\{ i \int d^4 x {\cal L}_{eff}(x) \right\} \equiv
\int {\cal D} h{\cal D} h^+ e^{i S} =
\int {\cal D} h{\cal D} h^+ \exp\left\{i \int d^4 x {\cal L}(x)\right\}
\\ \nonumber
= \exp\left\{ i\int d^4 x {\cal L}_{SM}(x) \right\}
\int {\cal D} h{\cal D} h^+ \exp\left\{i \int d^4 x
{\cal L}_h(x)\right\}
=e^{i S_{SM}}\int {\cal D} h{\cal D} h^+ e^{i S_h} ~,
\end{eqnarray}
where ${\cal D} h$ represents functional integration over $h$.
The effective field theory defined by this effective Lagrangian is fully
equivalent to the original theory when only Green functions with light external
particles are considered.
Starting from \eq{integraout}
we can calculate the one-loop level effective Lagrangian
by using the steepest-descent
method to integrate out the heavy scalar.
As we are interested in the effects of a heavy scalar
($m \geq 1$~TeV) on the physics around the
$M_Z$ scale and below, we will systematically
keep only terms of order $ O(1/m^2)$ through the whole calculation,
neglecting all operators with higher inverse powers of the mass of the
scalar singlet.
Let us denote by $h_0$ the solution of the classical equation of
motion for the $h$-field, i.e.
\begin{equation}
\left.\frac{\delta S}{\delta h(x) } \right|_{h_0} =0~.
\end{equation}
Using the Lagrangian \rfn{hlag} we obtain :
\begin{equation}
\label{eqofmotion}
(-D^2-m^2-2\alpha \abs{h_0}^2-\beta \varphi^\dagger \varphi) h_0 +
\overline{\widetilde{\ell}\, } f \ell=0\ .
\end{equation}
Then the full action can be functionally expanded around the
solution $h_0(x)$
\begin{equation}
\label{expansion}
S= S_{SM} + S_h[h_0]
+ \int d^4x d^4 x' \eta^\dagger(x) O(x,x') \eta(x') +\cdots~,
\end{equation}
where
the fluctuation operator $O(x,x')$ is given by the
second-order term in Taylor's expansion of the action $S$:
\begin{equation}
\label{fluct}
O(x,x') = \left.\frac{\delta^2 S}{\delta h(x) \delta h(x')} \right|_{h_0}
\end{equation}
and $\eta(x)$ in \eq{expansion}
represents the fluctuations around the classical
solution, $\eta(x)= h(x)-h_0(x)$.
Substituting \eq{expansion} in \eq{integraout}, shifting variables
from $h$ to $\eta$ in the functional integration, and using the known
formula for Gaussian integration, we obtain
\begin{equation}
e^{iS_{eff}} = \exp\{i(S_{SM}+S_h[h_0])\} \det(O)^{-1} =
\exp \left\{i(S_{SM}+ S[h_0])-\Tr{\log(O)}\right\}~.
\end{equation}
Therefore, for the one-loop
effective action we have
\begin{equation}
\label{seffgen}
S_{eff}= S_{SM}+S_h[h_0]+i \Tr{\log(O)}~.
\end{equation}
The first term in \eq{seffgen} is just the pure standard model
contribution to the effective action.
Using the equation of motion, \eq{eqofmotion}, the second term, $S_h[h_0]$,
can be
formally written as
\begin{equation}
\label{lnonlocal}
S_h[h_0] \approx
-\int d^4x \overline{\widetilde{\ell}\, }(x) f \ell(x)
\frac{1}{(-D^2-m^2-\beta \varphi^\dagger(x) \varphi(x))}
\overline{\ell}(x) f^\dagger \widetilde{\ell}(x)\ ,
\end{equation}
where terms of order $1/m^4$ have been neglected.
The Lagrangian \rfn{lnonlocal} is highly non-local. To obtain a local
version of it we could further expand the operator\newline
${1}/{(-D^2-m^2-\beta \varphi^\dagger(x) \varphi(x))}$
as a power series in $1/m^2$:
\begin{equation}
\label{appsolution}
\frac{1}{(-D^2-m^2-\beta \varphi^\dagger(x) \varphi(x))}
= -\frac{1}{m^2}
+ \frac{1}{m^4} (D^2+\beta \varphi^\dagger(x) \varphi(x))+\cdots~.
\end{equation}
Keeping the first term we obtain the only tree-level contribution,
at order $1/m^2$, of the scalar to the effective Lagrangian:
\begin{equation}
S_h[h_0]= \int d^4 x {\cal L}^{(0)}(x)
\end{equation}
with
\begin{equation}
\label{lagr0}
{\cal L}^{(0)} = \frac{1}{m^2}
(\overline{\widetilde{\ell}\, } f\ell)
(\overline{\ell} f^\dagger\widetilde{\ell})=
\frac{4}{m^2} f_{ab} f^*_{a'b'}
(\overline{\nu^c_{aL}} e_{bL})(\overline{e_{b'L}} \nu^c_{a'L})\ ,
\end{equation}
where the summation over repeated flavour indices
($a,b,a',b'$) is assumed.
It could also be obtained by computing the tree-level
diagram of fig.~1.a in the limit $q^2 \ll m^2$ (where $q$ is the momentum
of the scalar).
The result in \eq{lagr0} is not complete if we are going to use this
Lagrangian for calculations at the loop level \cite{Wit76,Wit77}.
The reason is that in order to obtain a
local Lagrangian we had to use the expansion \rfn{appsolution}.
In doing this we assumed that the derivative
in \eq{appsolution}, or equivalently the momentum of the scalar,
is negligible in comparison with the mass of the scalar. But this is not true
if the $h$-scalar contributes
inside loops, since in this case its momentum runs up to infinity.
It has been shown \cite{Wit76,Wit77} that the procedure just outlined
can be justified as long as one includes in the effective Lagrangian
a set of local operators that compensate for the terms missed
by using expansion \rfn{appsolution}.
We are going to compute these operators in the next section.
The last piece of the effective action \rfn{seffgen} is defined
in terms of the fluctuation operator \rfn{fluct} and takes into account
all one-loop effects with only the heavy particle in loops. In our
approximation, keeping only terms $O(1/m^2)$, the
fluctuation operator \rfn{fluct} can be easily calculated and leads
to the following effective action
\begin{equation}
i \int d^4 x {\cal L}^{(1)}(x)=
\log{\det{(O)}^{-1}}=
-\Tr{\log(O)}=
-\Tr{\log(-D^2-m^2-\beta \varphi^\dagger \varphi)}\ .
\end{equation}
We moved to appendix~\ref{det} the detailed
evaluation of the determinant
of a generic operator of this form as an expansion in $1/m^2$.
Using the general result of appendix~\ref{det}, \eq{resdet}, for our
particular
case, $U=\beta \varphi^\dagger \varphi$,
$D_\mu = \partial_\mu + i g' B_\mu$, that is $A_\mu \rightarrow i g' B_\mu$
and $F_{\mu\nu} \rightarrow i g' B_{\mu\nu}$,
and taking into account that $D^\mu
(\varphi^\dagger \varphi)=\partial^\mu (\varphi^\dagger \varphi)$,
$D^\mu B_{\mu\nu}=\partial^\mu B_{\mu\nu}$,
we write this contribution to the effective Lagrangian
as a sum of two parts:
\begin{equation}
\label{lfluctop}
{\cal L}^{(1)} = {\cal L}_{ren}^{(1)} + {\cal L}_{det}^{(1)} \ .
\end{equation}
The first part, ${\cal L}_{ren}^{(1)}$,
contains only operators with dimension not larger than four. It
is given by the following expression:
\begin{equation}
\label{lrenorm}
{\cal L}_{ren}^{(1)} = \frac{1}{(4\pi)^2}\left(
\frac{m^4}{2}\left(\frac{3}{2}+\Delta_\epsilon\right)+
m^2 \beta (1+\Delta_\epsilon) (\varphi^\dagger\varphi)+
\frac{\beta^2}{2} \Delta_\epsilon (\varphi^\dagger\varphi)^2
-\frac{g'^2}{12} \Delta_\epsilon B_{\mu \nu} B^{\mu \nu}\right)~.
\end{equation}
All the
coefficients in the above Lagrangian are ultraviolet-divergent.
Here and in the rest of the paper we use dimensional regularization,
where these divergences appear as simple poles, $1/\epsilon$,
($\epsilon=2-D/2$) in
the function $\Delta_{\epsilon}$ defined in \eq{deltae} of
appendix \ref{det}.
As expected, all terms in \eq{lrenorm} are already
present in the standard model Lagrangian. Therefore, they
can be absorbed in a redefinition of the parameters of the standard model.
In the second part of \eq{lfluctop}, ${\cal L}_{det}^{(1)}$, we included all
dimension-six operators:
\begin{eqnarray}
{\cal L}_{det}^{(1)} &=&
\frac{1}{m^2} \frac{1}{(4\pi)^2} \left(-\frac{\beta^3}{6}
(\varphi^\dagger\varphi)^3+ \frac{\beta^2}{12}
\partial_\mu (\varphi^\dagger\varphi)\partial^\mu (\varphi^\dagger\varphi)
\right.\nonumber\\
&&\left.
+\frac{g'^2\beta}{12} (\varphi^\dagger\varphi) B_{\mu \nu} B^{\mu \nu}-
\frac{g'^2}{60} \partial^\mu B_{\mu \nu} \partial_\sigma B^{\sigma\nu}
\right)\label{lagr1}~.
\end{eqnarray}
The coefficients of these operators are free of divergences and
are suppressed by $1/m^2$.
Note that the last term in \eq{resdet}, which is trilinear in the field
strength, is zero in the Abelian case.
It is clear that
one could also obtain the interactions in \eq{lrenorm} and \eq{lagr1}
by using ordinary Feynman rules. In fig.~2 we give the diagrams that
give rise to this part of the effective Lagrangian. Functional methods
provide, however, a much more elegant and compact method to calculate
them.
As mentioned before, all the terms in ${\cal L}_{ren}^{(1)}$
can be absorbed
in a redefinition of the standard model parameters.
The first term in \eq{lrenorm} is only a
renormalization of the vacuum energy. We can neglect it,
or, if we prefer, it can be absorbed in the constant term of the Higgs
potential of the standard model. The second term renormalizes the
quadratic coupling in the Higgs potential:
\begin{equation}
\label{barmatchm}
\bar{m}^2_\varphi = m^2_\varphi - m^2
\frac{\beta}{(4\pi)^2}(1+\Delta_\epsilon)~.
\end{equation}
The third one is a renormalization of
the quartic coupling of the standard Higgs:
\begin{equation}
\label{barmatchlambda}
\bar{\lambda} = \lambda - \frac{\beta^2}{2(4\pi)^2} \Delta_\epsilon.
\end{equation}
Finally, the fourth term in \eq{lrenorm}, which is an additional
contribution to the kinetic term for the
$B_\mu$-field, can be removed by wave-function renormalization:
\begin{equation}
\label{barmatchb}
\bar{B}_\mu = \left(1+\frac{g'^2 \Delta_\epsilon}{3(4\pi)^2}\right)^{1/2}
B_\mu~.
\end{equation}
In order to keep the canonical form of the covariant derivative,
we have to renormalize also
the $U(1)_Y$ coupling $g'$ keeping the product $g'B_\mu$
invariant:
\begin{equation}
\label{barmatchg}
\bar{g}'= \left(1+\frac{g'^2 \Delta_\epsilon}{3(4\pi)^2}\right)^{-1/2} g'~.
\end{equation}
The above equations \rfn{barmatchm}, \rfn{barmatchlambda}
and \rfn{barmatchg} relate the bare parameters of the full Lagrangian
$m_\varphi$, $\lambda$, and $g'$
to the corresponding bare parameters
$\bar{m}_\varphi$, $\bar{\lambda}$ and $\bar{g}'$ of the effective Lagrangian.
\mysection{Matching the effective Lagrangian to the full Lagrangian}
\label{matching}
As commented in the previous section, it is necessary to expand the operator
of \eq{appsolution} in powers of $1/m^2$
to have a local effective
Lagrangian, but this expansion is only valid at the classical level,
since the limit of $m\rightarrow \infty$ and the functional integration
over the light fields do not commute. This is just a consequence of the fact
that the momentum $p$ of the scalar inside loop diagrams runs up to infinity
and then an expansion in $p^2/m^2$ is not appropriate. In spite of this
problem it is still possible to obtain a local effective Lagrangian as
an expansion in $1/m^2$ \cite{Wit76,Wit77}. It is necessary, however, to
include some additional operators that are not expected from the na\"{\i}ve
expansion in \eq{appsolution}. These
operators appear as a result of quantum corrections, hence, they are
suppressed by additional couplings and $1/(4\pi)^2$ factors.
The practical way to obtain these terms of the Lagrangian is
to consider the most general linear combination of all the operators
with a given dimension, allowed by symmetry, and then to
extract the coefficients of this
combination by matching the effective Lagrangian calculation to the
full theory calculation. Most of the operators obtained should
be included in any case as counterterms to cancel the divergences
generated when the tree-level Lagrangian of \eq{lagr0} is used
inside loops. But it is important to note that
the finite parts of the coefficients of those operators can only
be fixed by computing the various Green functions with both the full and the
effective theories and matching the results for small energies compared
with the heavy mass.
Some other operators, however, appear with finite coefficients. They are
just a consequence of the matching procedure and cannot be
obtained from the divergences that appear in the effective theory.
In the diagrammatic language all these new operators are generated by
Feynman diagrams, with both the heavy scalar and lepton lines in the loops.
We will compute in the full theory the amplitudes corresponding to these
diagrams and will compare them with the equivalent amplitudes obtained
by using the tree-level effective Lagrangian, \eq{lagr0}, in one-loop
diagrams. The difference will give us the necessary counterterms.
Actually, the effective Lagrangian calculation can
be avoided by splitting the scalar propagator in two parts:
\begin{equation}
\label{splitting}
\frac{1}{k^2-m^2} = -\frac{1}{m^2}+\frac{1}{m^2} \frac{k^2}{(k^2-m^2)}\ .
\end{equation}
If we use this form when calculating diagrams with only one scalar propagator,
the first part gives
exactly what one would obtain by using the effective tree-level
four-fermion interaction, \eq{lagr0}, while the second part gives just
the counterterm we should add to the effective Lagrangian.
We would also like to note that by splitting the propagator in these
two pieces we have increased the degree of ultraviolet divergence in each of
the two terms, with respect to the original diagram. On the other hand, the
second
part contains an additional factor $k^2$ in the numerator, which reduces
the infared degree of divergence of this contribution. Therefore,
any possible small momentum singularity is transmitted to the
low-energy effective Lagrangian, as it should be, since infared
singularities have nothing to do with the high-energy behaviour of the
theory.
A technical issue about the calculation is the selection of diagrams
one should compute to reconstruct the full effective Lagrangian. The
effective Lagrangian must be a linear combination of gauge-invariant
operators. Each of the operators gives rise to a variety of physical
processes. Obviously, to obtain the coefficients of these operators it is
not necessary to compute all these processes, since many
of them are related by gauge invariance. Our strategy consists in computing
all one-particle-irreducible diagrams with the minimal number of external
particles. We keep track
of all external momenta (of course only to the order $p^2/m^2$) without
using any equation of motion for the external particles. After that, we
write
an effective Lagrangian that reproduces those amplitudes and when
there is no ambiguity, we reconstruct gauge invariance by promoting
any derivative to a covariant derivative.
Sometimes, however,
there is some ambiguity in the promotion of a derivative to a
covariant one. Then, we compute also the diagrams with one additional
external gauge boson in order to disentangle that ambiguity and we
check that the gauge-invariant effective Lagrangian correctly describes
all the amplitudes. Finally, as an additional check,
we compute some diagrams with three external gauge bosons by
using both the full and the effective Lagrangians. We do it for the
special case of
zero external momenta\footnote{We are studying a more efficient
method for the calculation of the gauge-invariant effective Lagrangian,
based on a calculation with constant fields similar to the one used in
appendix~\ref{det}.}.
Since the full calculation is tedious, we give the details
in appendix~\ref{heavy-light}, presenting here only the main results
and an explanation of the procedure.
\subsection{Self-energies of the lepton-doublet}
We give in fig.~3 the diagrams contributing to the lepton wave function
renormalization in the full and the effective theories. In the effective
theory the wave function renormalization is zero because it is a massless
tadpole-like diagram (fig.~3.b).
However, in the full theory there is a non-zero
contribution that must be included as an additional effective operator.
{}From the results of appendix \ref{heavy-light}
we find that, for the first term in \eq{selfe},
the additional operator we should include is
\begin{equation}
\label{selfenergy}
2(\Delta_\epsilon+\frac{1}{2}) i (\overline{\ell} F \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell)\ ,
\end{equation}
where we have defined the matrix $F$ as
\begin{equation}
\label{defF}
F_{ab} \equiv \frac{ (f^\dagger f)_{ab}}{(4\pi)^2}\ .
\end{equation}
Note that even though in appendix~\ref{heavy-light}
we have only calculated the charged scalar contribution to the
self-energy of the doublet,
gauge invariance requires that the partial derivative should
be substituted by a covariant derivative. In this case the promotion
from the derivative to the covariant derivative can be done without
ambiguity. This leads to \eq{selfenergy}. It implies that there
should be an additional contribution to the coupling of the gauge bosons to
the lepton doublet given by the coefficient in \eq{selfenergy}.
We will see in the next subsection that, indeed, this contribution appears.
The second term in \eq{selfe} is proportional to $\sla{p} p^2$, which
requires an effective Lagrangian of the form
$ i (\overline{\ell}_a \sla{\partial} \partial^2 \ell_b)$.
However, the promotion of this term to covariant derivatives is
ambiguous: should we use $\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} D^2$, $\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}^3$ or $D_\mu \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} D^\mu$ ? The
only way to resolve this ambiguity, as we will see below,
is to perform a full calculation with one external gauge boson.
The operator in
\eq{selfenergy}
contributes to the kinetic term of the doublets. To write it again in
a canonical form, we should redefine the lepton doublet as follows:
\begin{equation}
\label{wfren}
\ell_a \rightarrow
\ell_a -\frac{(\Delta_\epsilon+\frac{1}{2})}{(4\pi)^2}
(f^\dagger f)_{ac} \ell_c\ .
\end{equation}
This redefinition only affects the tree-level four-fermion
effective Lagrangian and the standard model Yukawa coupling. This is
because
the wave function renormalization
appears at the one-loop level, and, as a consequence, its effect on all
the other
(one-loop) operators is a two-loop effect. It
can be taken into account by redefining the standard model Yukawa couplings
and the couplings $f_{ab}$ that appear in \eq{lagr0} as follows
\begin{eqnarray}
Y_{ab} \rightarrow & \bar{Y}_{ab} = & Y_{ab} -
\frac{(\Delta_\epsilon+\frac{1}{2})}{(4\pi)^2} (f^\dagger f Y)_{ab}\label{redefY}\ ,\\
f_{ab} \rightarrow & \bar{f}_{ab} = & f_{ab} -
2\frac{(\Delta_\epsilon+\frac{1}{2})}{(4\pi)^2}
(f f^\dagger f)_{ab} \label{redeff}\ .
\end{eqnarray}
As we see this does not change the flavour structure of the
couplings, since the $\bar{f}_{ab}$ is still antisymmetric in flavour
indices and
does not give any new interesting process. The same thing could be
said about the standard model Yukawa couplings $\bar{Y}_{ab}$.
\subsection{Penguins}
In fig.~4 we present the contributing diagrams to the vertex of the
$B_\mu$ and $\vec{W}_\mu$ gauge bosons with leptons, with the heavy scalar
running in the loop (figs.~4.a and 4.b). Note that for the
$\vec{W}_\mu$ there is no diagram fig.~4.b,
because the scalar is an $SU(2)$ singlet.
We also give the corresponding
diagrams in the effective theory (fig.~4.c). Here it is interesting to note
that the amplitude corresponding to the diagrams of figs.~4.a and
4.b in the full theory
is finite, when the diagrams
with external wave function renormalization are included.
This can be understood because the full theory is renormalizable:
since all these diagrams lead to a flavour-changing vertex and
as our original Lagrangian does not contain such a coupling,
there is no counterterm available.
In the effective theory language,
this cancellation is just a consequence of gauge
invariance: the divergent contributions to the vertex in the full
theory can be taken into account by the gauge-invariant dimension-four
operator of \eq{selfenergy}, which can be
removed by a wave function renormalization of the lepton doublet.
We give in appendix~\ref{heavy-light} the resulting calculation,
eqs.~\rfn{ampB} and \rfn{ampW},
of the vertex diagrams of the $B_\mu$ and $\vec{W}_\mu$ after subtraction of
the effective Lagrangian contribution (diagram 4.c). We cast this
result in a form that can be easily identified with some effective
operators. The first term can already be obtained, as promised,
from \eq{selfenergy}. The rest of the terms can be obtained (for both
the $B_\mu$ and the $\vec{W}_\mu$ fields) from the
following operators:
\begin{eqnarray}
&&-\frac{1}{3m^2} i (\overline{\ell} F (D^2 \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}+ \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} D^2 )\ell)
\label{selfenergy2}\\
&&+ \frac{2}{3m^2} \left(\Delta_\epsilon+\frac{5}{3}\right)
\frac{g'}{m^2} (\overline{\ell} F \gamma_\nu \ell) D_\mu B^{\mu \nu}
+ \frac{2}{3m^2}\left(\Delta_\epsilon+\frac{4}{3}\right)
\frac{g}{m^2} (\overline{\ell} F \gamma_\nu \vec{\tau} \ell)
D_\mu \vec{W}^{\mu \nu}\hspace{1.5cm}\label{penguin1}\\
&&+\frac{g'}{4m^2} B^{\mu \nu}
(\overline{\ell} F i \sigma_{\mu\nu} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell+ \mrm{h.c.})
+\frac{g}{4m^2} \vec{W}^{\mu \nu}
(\overline{\ell} F i \sigma_{\mu\nu} \vec{\tau} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}\ell+\mrm{h.c.})
\label{penguin2}\ .
\end{eqnarray}
In these equations, $D_\mu$ is the appropriate covariant derivative for each
of the fields. It is easy to see that the pure derivative part of the first
term
\eq{selfenergy2}, reproduces perfectly
the second term in \eq{selfe} and that the ambiguity mentioned in
the previous subsection has been solved completely with this calculation.
To check this result further, we have computed the diagram
with three $W_\mu$ gauge bosons for zero external momenta and
compared the result with that obtained from
eqs.~(\ref{selfenergy2})--(\ref{penguin2}) for constant fields.
The physical content of these operators is quite obscure.
Then, we found it convenient to use the operator identity:
\begin{eqnarray}
\label{relation}
\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}^3 &=& \frac{1}{2} (D^2\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}+\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} D^2)\\ \nonumber
&&+
\frac{1}{8} \left(
g' \sigma_{\mu\nu} B^{\mu\nu}\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}+g' \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \sigma_{\mu\nu} B^{\mu\nu}-
g \sigma_{\mu\nu} \vec{W}^{\mu\nu} \vec{\tau} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}
-g \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \sigma_{\mu\nu} \vec{W}^{\mu\nu} \vec{\tau} \right)
\end{eqnarray}
to rewrite eqs.~(\ref{selfenergy2})--(\ref{penguin2})
in the following form:
\begin{eqnarray}
&&-\frac{2}{3m^2} i (\overline{\ell} F \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}^3\ell)
\label{selfenergy2a}\\
&&+ \frac{2}{3} \left(\Delta_\epsilon+\frac{5}{3}\right)
\frac{g'}{m^2} (\overline{\ell} F \gamma_\nu \ell) D_\mu B^{\mu \nu}
+ \frac{2}{3}\left(\Delta_\epsilon+\frac{4}{3}\right)
\frac{g}{m^2} (\overline{\ell} F \gamma_\nu \vec{\tau} \ell)
D_\mu \vec{W}^{\mu \nu}
\label{penguin1a}\\
&&+\frac{g'}{3m^2} B^{\mu \nu}
(\overline{\ell} F i \sigma_{\mu\nu} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell+ \mrm{h.c.})
+\frac{g}{6m^2} \vec{W}^{\mu \nu}
(\overline{\ell} F i\sigma_{\mu\nu} \vec{\tau} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}\ell+\mrm{h.c.})\ .
\hspace{1cm}\label{penguin2a}
\end{eqnarray}
As we will see later, after using the equations of motion,
the physical content of
these operators
becomes more transparent. The first term can be removed
in favour of Yukawa-type couplings and does not give any interesting
process. The second and the third terms will give rise to
processes such as $Z\rightarrow \mu e$ and $\mu \rightarrow e e e$,
and finally the last terms, with the appropriate combination of
$W^3_\mu$ and $B_\mu$, will give rise to $\mu \rightarrow e \gamma$.
\subsection{Seagull diagrams}
There are also some operators that involve two scalar Higgses,
two lepton doublets, and one covariant derivative. They are generated
by the diagrams of fig.~5 and they do not contribute to
any interesting process. For completeness we give the result here.
{}From the diagram of fig.~5.b we get the amplitude \eq{amp5a},
which can be obtained by using the operator
\begin{equation}
\label{op5a}
i \frac{\beta}{m^2} (\varphi^\dagger \varphi)
(\overline{\ell} F \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell) + \mrm{h.c.}\ .
\end{equation}
{}From the diagram of fig.~5.a, and after subtraction of the effective
Lagrangian
contribution, fig.~5.c, we get the amplitude \rfn{amp5b}, which can be obtained
from the operators
\begin{eqnarray}
&&(\Delta_\epsilon+1)\frac{i}{m^2} \left((D_\mu \varphi)^\dagger \varphi\right)
(\overline{\ell} \hat{F} \gamma^\mu \ell)
+(\Delta_\epsilon+1) \frac{i}{m^2} \left((D_\mu \varphi)^\dagger
\vec{\tau}\varphi\right)
(\overline{\ell} \hat{F} \gamma^\mu \vec{\tau} \ell)\nonumber
\hspace{2cm} \\
&&-\frac{i}{2m^2} \left(\varphi^\dagger \varphi\right)
(\overline{\ell} \hat{F} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}\ell_b)
-\frac{i}{2m^2} \left(\varphi^\dagger \vec{\tau}\varphi\right)
(\overline{\ell} \hat{F} \vec{\tau} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell)+\mrm{h.c.}\label{op5b}\ ,
\end{eqnarray}
where
\begin{equation}
\label{defhatF}
\hat{F}_{ab} = \frac{(f Y_e Y^\dagger_e f^\dagger)_{ba}}{(4\pi)^2}\ .
\end{equation}
We used an $SU(2)$ Fierz transformation to write the operators in
this form, being the original operators expressed in terms of
$\tilde{\varphi}$. Clearly the effects of operators \rfn{op5b}
are suppressed
by at least two powers of the lepton masses.
\subsection{Four-fermion interactions}
In this section we shall discuss possible new four-fermion interactions
that could arise at the one-loop level. These interactions
can come from three different kinds of diagrams:
i) Diagrams with gauge boson corrections to the vertex of the charged
scalar with the lepton doublet. Since these only contain light particles
in the loop, they are fully taken into account by gauge boson corrections
to the effective four-lepton interaction \rfn{lagr0}. In addition they do not
change the flavour structure of the coupling.
ii) Box diagrams with a charged scalar and a gauge boson running in
the loop (fig.~6.a).
It is easy to see that diagrams with a $\vec{W}_\mu$
exchange just cancel because of the antisymmetry of the couplings $f_{ab}$.
Diagrams with exchange of $B_\mu$
renormalize the tree-level four-lepton interaction without changing
its flavour structure. Part of their contributions are taken into account
by the effective theory diagrams of fig.~6.c. But, obviously,
these diagrams
do not modify the flavour structure of the tree-level effective
four-fermion interaction (\ref{lagr0}) and, then, they
cannot give rise to processes such as $\mu \rightarrow e e e$
at least at order $1/m^2$.
iii) Box diagrams with two charged scalars in the loop
(fig.~6.b). Diagrams of this type are finite and in the sum they give
rise to the following operator:
\begin{equation}
\label{boxb}
-\frac{(4\pi)^2}{m^2} F_{ab} F_{cd} (\overline{\ell}_a \gamma_\mu \ell_b)
(\overline{\ell}_c \gamma^\mu \ell_d)\ .
\end{equation}
It can be shown that this operator does not contribute to processes
with two identical fermions, such as $\mu^- \rightarrow e^+ e^- e^-$.
Indeed, taking only into account, for example, the down component
of the $SU(2)$ lepton doublets, the interaction among four charged
leptons induced by \eq{boxb} can be written as
\begin{equation}
\label{boxbb}
-\frac{(4\pi)^2}{m^2} F_{ab} F_{cd} (\overline{e_{aL}} \gamma_\mu e_{bL})
(\overline{e_{cL}} \gamma^\mu e_{dL}) =
-\frac{(4\pi)^2}{2m^2} ( F_{ab} F_{cd}- F_{ad} F_{cb})
(\overline{e_{aL}} \gamma_\mu e_{bL})
(\overline{e_{cL}} \gamma^\mu e_{dL})\ ,
\end{equation}
where, to obtain the right-hand side, we have used a Fierz transformation.
We immediately see that this operator does not contribute
to processes
such as $\mu^- \rightarrow e^+ e^- e^-$. However it does contribute to other
interesting processes such as $\tau^- \rightarrow e^+ \mu^- e^-$.
Clearly only the contributions of type (iii) are interesting and only
these will be included in our effective Lagrangian.
\vspace{0.5cm}
Before we summarize the results of section~4, we
note that all the results obtained in section~\ref{matching}
have been derived from the effective tree-level Lagrangian
given in \eq{lagr0}. One can easily see that this Lagrangian can also be
written
(after a combined Fierz transformation in $SU(2)$ and Dirac space) in the
more familiar form
\begin{equation}
{\cal L}^{(0)}\ ``="\ \frac{1}{m^2} f_{ab} f^*_{cd}
(\overline{\ell}_d \gamma^\mu \ell_b)
(\overline{\ell}_c \gamma_\mu \ell_a)\ .
\end{equation}
But this is true only in four dimensions: the Fierz transformations we used
above cannot be
done in $D$ dimensions. Should we start with this new Lagrangian,
the finite parts of the counterterms would be different when using
dimensional regularization, but this formulation would be completely
equivalent to the original one. We decided to keep the original
form of the Lagrangian so as to keep the notation as close as possible to the
full Lagrangian formulation.
Using the results of sections 3 and 4, the complete
one-loop effective Lagrangian is
\begin{equation}
{\cal L}_{eff} = {\cal L}_{SM} + {\cal L}^{(0)}+ {\cal L}_{det}^{(1)}
+{\cal L}_{match}^{(1)}\ .
\end{equation}
Here ${\cal L}_{SM}$ is the standard model Lagrangian, \eq{smlagrangian},
${\cal L}^{(0)}$ as given in \eq{lagr0} is the tree-level $1/m^2$ term
of the effective Lagrangian, ${\cal L}_{det}^{(1)}$ is given in \eq{lagr1}
and contains all $1/m^2$ one-loop contributions of diagrams with only
singlet scalars in the loop and finally ${\cal L}_{match}^{(1)}$, which is
the sum of the terms in eqs.~(\ref{selfenergy2a})--(\ref{op5b}) and
\rfn{boxb},
gives all the
contributions coming
from matching the full theory in diagrams with heavy-light particles
in the loops:
\begin{eqnarray}
{\cal L}_{match}^{(1)}
&=& \frac{2}{3} (\Delta_\epsilon+\frac{5}{3})
\frac{g'}{m^2} (\overline{\ell} F\gamma_\nu \ell) D_\mu B^{\mu \nu}
+ \frac{2}{3}(\Delta_\epsilon+\frac{4}{3})
\frac{g}{m^2}
(\overline{\ell} F\gamma_\nu \vec{\tau} \ell) D_\mu \vec{W}^{\mu \nu}
\nonumber \\
&&
+\ \frac{1}{3} \frac{g'}{m^2} B^{\mu \nu}
(\overline{\ell} F i \sigma_{\mu\nu} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell+ \mrm{h.c.})
+\frac{1}{6} \frac{g}{m^2} \vec{W}^{\mu \nu}
(\overline{\ell} F i\sigma_{\mu\nu} \vec{\tau}
\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}\ell+\mrm{h.c.})\hspace{1cm}\nonumber \\
&& +\left(i\frac{\beta}{m^2} (\varphi^\dagger \varphi)
(\overline{\ell} F\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell) + \mrm{h.c.} \right)
-\frac{2}{3m^2} i (\overline{\ell} F \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}^3\ell)
\nonumber\\
&& +\left(\left.\right. (\Delta_\epsilon+1) \frac{i}{m^2}
\left((D_\mu \varphi)^\dagger \varphi\right)
(\overline{\ell} \hat{F}\gamma^\mu \ell)
+(\Delta_\epsilon+1) \frac{i}{m^2} \left((D_\mu \varphi)^\dagger
\vec{\tau}\varphi\right)
(\overline{\ell} \hat{F}\gamma^\mu \vec{\tau} \ell)\right.\nonumber\\
&& \left.
-\frac{i}{2m^2} \left(\varphi^\dagger \varphi\right)
(\overline{\ell}\hat{F}\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}\ell)
-\frac{i}{2m^2} \left(\varphi^\dagger \vec{\tau}\varphi\right)
(\overline{\ell} \hat{F}\vec{\tau} \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell)+\mrm{h.c.} \right)\nonumber\\
&&
-\frac{(4\pi)^2}{m^2} (\overline{\ell} F \gamma_\mu \ell)
(\overline{\ell} F \gamma^\mu \ell) \ . \label{totalefflagr}
\end{eqnarray}
All the couplings in this Lagrangian should be understood as
bare barred effective Lagrangian couplings related to the full theory
bare couplings through eqs.~\rfn{barmatchm}, \rfn{barmatchlambda},
\rfn{barmatchg}, \rfn{redefY} and \rfn{redeff}.
To simplify the notation we have
suppressed everywhere generational indices, then
$F$ and $\hat{F}$ are the matrices
defined in eqs.~(\ref{defF}) and (\ref{defhatF}) .
\mysection{Renormalization and operator mixing analysis\newline
of the effective
Lagrangian}
\label{renormalization}
Equations~\rfn{barmatchm}, \rfn{barmatchlambda}, \rfn{barmatchg},
\rfn{redefY} and \rfn{redeff} relate the bare
couplings and masses of the
effective theory with those
of the full theory. It is, however, more interesting to have the
equivalent relations for renormalized couplings.
If an $\overline{MS}$ scheme\footnote{It has been customary in the
recent literature to use cut-off regularization schemes when working
with effective Lagrangians and then identify the cut-off with the
scale of new physics.
The interpretation of the ultraviolet
cut-off as the scale of new physics has led to some erroneous results.
The effective Lagrangian should also be renormalized, and
physical results should be independent of the regularization and of
the renormalization schemes.
We will see later that
dimensional regularization with minimal subtraction leads to the same
physical results in a cleaner way;
therefore, we will always use this scheme.} is
employed to renormalize both the full and the effective theories,
we can very easily obtain the matching equations for the
renormalized couplings. Let us take, for example, the gauge coupling $g'$.
We
denote the $\overline{MS}$ renormalized quantities with the same symbol as
the bare quantities, but adding an additional dependence on the
renormalization scale $\mu$. All effective theory quantities will be
distinguished by a bar. The standard relationship between bare and
renormalized couplings in $\overline{MS}$ schemes is ($D=4-2\epsilon$
and $\frac{1}{\hat{\epsilon}}= \frac{1}{\epsilon}-\gamma+\log (4\pi)$):
\begin{eqnarray}
g' \mu^\epsilon &=& g'(\mu)+ \frac{1}{2\hat{\epsilon}}\ b_{g'}
g'^3(\mu)+\cdots\ , \nonumber\\
\bar{g}' \mu^\epsilon &=& \bar{g}'(\mu)+
\frac{1}{2\hat{\epsilon}}\ \bar{b}_{g'} \bar{g}'^3(\mu)+\cdots\ ,\nonumber
\end{eqnarray}
where $b_{g'}$ and $\bar{b}_{g'}$ are the lowest-order coefficients of the
$\beta$-functions for the coupling constants in the full and the effective
theories, respectively. Substituting these equations in
\eq{barmatchg} and equating finite terms \cite{Wei80,Hal81} we obtain
the desired matching condition for renormalized couplings:
\begin{equation}
\label{matchg}
\bar{g}'(\mu) = g'(\mu) - \frac{g'(\mu)^3}{3(4\pi)^2} \log(\mu/m)+\cdots\
\ .
\end{equation}
Note that this equation can be obtained by just dropping the
$1/\hat{\epsilon}\ $ contained in $\Delta_\epsilon$ in
\eq{barmatchg}. This is not surprising since the divergent term in
\eq{barmatchg} gives just the charged scalar contribution to the
beta function of $g'$ in the full theory.
Similar arguments give for the rest of
the
couplings and masses the following result
\begin{equation}
\label{matchm}
\bar{m}^2_\varphi(\mu) = m^2_\varphi(\mu) - m^2(\mu)
\frac{\beta(\mu)}{(4\pi)^2}(1+2\log(\mu/m))\ ,
\end{equation}
\begin{equation}
\label{matchlambda}
\bar{\lambda}(\mu) = \lambda(\mu) -
\frac{\beta^2(\mu)}{(4\pi)^2}\log(\mu/m)\ ,
\end{equation}
\begin{equation}
\label{matchY}
\bar{Y}_{ab}(\mu) = Y_{ab}(\mu) -
\frac{(f^\dagger f Y)_{ab}(\mu)}{(4\pi)^2}
(\frac{1}{2}+2\log(\mu/m))
\end{equation}
and a matching equation for the four-lepton coupling in the effective
theory expressed in terms of the full theory couplings and masses.
It can be obtained by using an
effective $\bar{f}_{ab}(\mu)$ defined as follows:
\begin{equation}
\label{matchf}
\bar{f}_{ab}(\mu) = f_{ab}(\mu) -
2 \frac{(f f^\dagger f)_{ab}(\mu)}{(4\pi)^2}
\left(\frac{1}{2}+2\log(\mu/m)\right)
+\cdots\ \ .
\end{equation}
In this equation there could also be some contributions from box diagrams,
which we have not computed.
These matching conditions are, in principle, valid for an arbitrary value of
the
renormalization scale $\mu$. However, it is clear that
in order to avoid large logarithms they should be evaluated at some scale
around the charged scalar mass and then, using the standard model
renormalization group, run all the couplings to obtain their values at
lower scales.
Equation \rfn{matchm} is very interesting, since it manifests in all its
crudeness the so-called naturalness problem of the standard model;
$\bar{m}_\varphi(\mu)$ is the mass parameter that appears in the
Higgs potential part of the effective Lagrangian, and it has to be of
the order of the
electroweak scale. However, it is clear that if we take $m(\mu)$ very
large, we should also take $m_\varphi(\mu)$ large in order to have
$\bar{m}_\varphi(\mu)$ small enough. But even if we do so at some
scale $\mu$, it will be very difficult to keep $\bar{m}_\varphi(\mu)$
small at any other scale. This represents a serious fine-tuning
problem, which appears when the standard model is embedded in
another model containing mass scales much larger than the Fermi scale.
It is important to note that by using dimensional regularization the
problem appears only in the matching conditions. If a cut-off
regularization scheme is used, the naturalness problem can be related to
the appearance of quadratic divergences.
However, this relation is not direct. Quadratic
divergences can appear at the intermediate stages, but
they should be absorbed in a renormalization of the mass
parameters of both the full and the effective
theories, since physical quantities must be cut-off-independent.
After renormalization, the naturalness problem should
also appear only as a fine-tuning problem in the matching conditions,
as in \eq{matchm}. As the main physical consequences can be obtained
equally in dimensional regularization, we do not find any particular
advantage by working with a cut-off regularization scheme.
If an $\overline{MS}$
scheme is used to renormalize also the higher dimension operators,
we could write the effective Lagrangian in
terms of a set of running couplings $c_i(\mu)$ for each of the operators
\begin{equation}
{\cal L}_{eff} = \sum_i c_i(\mu) O_i\ ,
\end{equation}
where the $c_i(\mu)$ are obtained from the bare coefficients
in \eq{totalefflagr} just
by dropping the $1/\epsilon -\gamma+\log(4\pi)$ terms in
$\Delta_\epsilon$; that is just by substituting $\Delta_\epsilon$ by $2\log(\mu/m)$.
Then all
couplings should be understood as renormalized effective Lagrangian
couplings (barred couplings) related to the full theory couplings through
eqs.~(\ref{matchg})--(\ref{matchf}).
To simplify matters we can consider only processes that violate
muon-lepton number in one unit and electron-lepton number in minus one
unit (the total lepton number must be conserved). For example, we can
take the two operators
\begin{eqnarray}
O_1 &\equiv &(\overline{\widetilde{\ell}\, }_3 \ell_2)
(\overline{\ell}_1 \widetilde{\ell}_3) \nonumber\\
O_2 & \equiv & \frac{1}{g'} D_\mu B^{\mu \nu}
(\overline{\ell}_1 \gamma_\nu \ell_2) \nonumber
\end{eqnarray}
The factor $1/g'$ in the penguin operator has
been included for convenience.
From eqs.~\rfn{lagr0} and \rfn{penguin1}, we obtain expressions
for the corresponding running couplings:
\begin{eqnarray}
c_1(\mu) &=& \frac{(f^\dagger f)_{12}(\mu)}{m^2} + \cdots \label{cesp}\\
c_2(\mu) &= &
\frac{(f^\dagger f)_{12}(\mu)}{m^2} \frac{4 g'^2(\mu)}{3(4\pi)^2}
\log\frac{\mu}{m}
+ \frac{(f^\dagger f)_{12}(\mu)}{m^2} \frac{10 g'^2(\mu)}{9(4\pi)^2}
+\cdots\ , \label{ces}
\end{eqnarray}
where the dots represent additional one-loop contributions, which, as
we will see immediately, are not important at the level we are working.
Note that $(f^\dagger f)_{12} = f^*_{31} f_{32}$ because of the
antisymmetry of the couplings in flavour indices.
Equations~(\ref{cesp})--\rfn{ces} can be cast into the following form
\begin{eqnarray}
c_1(\mu) &=& c_1(m)\left(1+\gamma_{11} \log\frac{\mu}{m}\right)
+c_2(m) \gamma_{12} \log\frac{\mu}{m} \label{c1}\\
c_2(\mu) &=& c_1(m) \gamma_{21} \log\frac{\mu}{m}
+ c_2(m) \left(1+\gamma_{22}\log\frac{\mu}{m}\right) \ ,\label{c2}
\end{eqnarray}
with
\begin{equation}
\label{gammas}
\gamma_{11} \approx O\left(\frac{g^2}{(4\pi)^2}\right) ,\ \ \
\gamma_{12} \approx O\left(\frac{g^2}{(4\pi)^2}\right),\ \ \
\gamma_{21}
=\frac{4}{3}\frac{g'^2 }{(4\pi)^2},\ \ \
\gamma_{22} \approx O\left(\frac{g^2}{(4\pi)^2}\right)
\end{equation}
and the couplings at the scale $\mu=m$ are given by
\begin{equation}
\label{boundary}
c_1(m) = \frac{(f^\dagger f)_{12}(m)}{m^2} ,\hspace{1cm} c_2(m) =
\frac{10}{9}\frac{g'^2}{(4\pi)^2} \frac{(f^\dagger f)_{12}(m)}{m^2} \ .
\end{equation}
Since $c_2(m)$ appears only at the one-loop level
in the full model, the last term in \eq{c1} is a two-loop effect.
This is the reason why we
have not considered it in \eq{cesp}. On the other hand, if $\mu$ is not
very different from $m$ we have $\gamma_{11} \log(\mu/m) \ll 1$ and
$\gamma_{22} \log(\mu/m) \ll 1$; then, the diagonal elements
of the anomalous dimension matrix do not need to be computed.
Equations~\rfn{c1} and \rfn{c2} are an approximate solution of the general
renormalization group equation that describes mixing among
the operators $O_1$ and $O_2$
\begin{equation}
\label{renopmix}
\mu \frac{ d c_i(\mu)}{d\mu} = \gamma_{ij} c_j(\mu)\ ,
\end{equation}
which is valid only in the case that $\gamma_{ij} \log(\mu/m) \ll 1$. If
we pretend to use this solution for a wider range of $\mu$'s,
the full $\gamma_{ij}$
matrix should be computed and the integration of \eq{renopmix}
should be done by taking also into account the running of the gauge
coupling constants. However, for the standard model values of the
gauge coupling constants the linear approximation
($\gamma_{ij}~\log(\mu/m)\ll~1$) works very well
for a wide range of scales
$(\gamma_{ij} \sim 10^{-3})$. Obviously, in this approximation and
taking into account that only the operator in \eq{lagr0} is generated
at tree level it is clear that in our model it is enough to consider
only mixing of the effective operators generated at the one-loop level with
the tree-level operator because other mixings would represent two-loop
effects.
Then, at the order we are working, we have always $2\times 2$ operator
mixing.
This simple example shows us clearly what can and what cannot
be obtained by using
the effective Lagrangian.
{}From the effective theory we could calculate the anomalous
dimension matrix that controls the mixing among the different
operators in the effective Lagrangian since the logarithmic
terms are the same in the full and in the effective theories,
however the boundary conditions for the
renormalization group equation (\ref{renopmix}) can only
be obtained after the matching procedure. If we do not
know the full theory a set of boundary conditions can only be obtained
from experiment. In this case we can still compute
the anomalous dimensions in the effective theory, write
down \eq{renopmix}, and solve it for arbitrary
boundary conditions at the scale $\mu_0$. Then the solution, within
our previous approximations, is
\begin{eqnarray}
\label{ces3}
c_1(\mu) &=&
c_1(\mu_0)\left(1+\gamma_{11} \log\frac{\mu}{\mu_0}\right)
+c_2(\mu_0) \gamma_{12} \log\frac{\mu}{\mu_0}\nonumber\\
c_2(\mu) &=& c_1(\mu_0) \gamma_{21} \log\frac{\mu}{\mu_0}
+ c_2(\mu_0) \left(1+\gamma_{22}\log\frac{\mu}{\mu_0}\right)\ .
\end{eqnarray}
Now we should take the full, complete basis of operators that
mix at the one-loop level, since if the boundary conditions are not known
we cannot neglect {\it a priori} any of the mixings. To keep simplicity,
however,
we will discuss only two-operator mixing.
The interesting point about eqs.~\rfn{ces3}
is that, if we know all the effective
couplings at some scale $\mu_0$, we can obtain them at some other scale
(not necessarily the scale responsible for the new physics).
Equations~\rfn{ces3} clearly tell us that to predict the values of the
couplings at any scale we need two (in the case of mixing of two operators)
boundary conditions. These conditions could be obtained from experiment.
For example, if we could bound the couplings of the two operators at LEP1
we could straightforwardly predict these couplings at LEP2 energies.
However, if in our example we only know one
of the couplings at LEP1, no matter how many loops we use to calculate
the anomalous dimensions, we will not
be able to bound the other operator at the same or any other scale.
Only if the full theory is known can we relate the couplings of different
operators as they are expressed in terms of the few parameters of the full
theory. Then, bounds on
one operator can be related to bounds on other operators.
In our example it is clear from \eq{boundary} that
\begin{equation}
\label{crelation1}
c_2(m) = \frac{10}{9}\frac{g'^2}{(4\pi)^2} c_1(m)
\end{equation}
or, for instance at the $M_Z$ scale, by using eqs.~\rfn{cesp}--\rfn{ces}
we have
\begin{equation}
\label{crelation2}
c_2(M_Z) =
\left(\frac{4 g'^2(M_Z)}{3(4\pi)^2} \log(M_Z/m)
+\frac{10 g'^2(M_Z)}{9(4\pi)^2}\right) c_1(M_Z)\ .
\end{equation}
Then it is clear that a bound on $c_1(M_Z)$ implies a bound on
$c_2(M_Z)$, and viceversa, and it also implies a bound on $c_1(\mu)$ and
$c_2(\mu)$ at any other scale $\mu$. But it is important to realize
that this is only possible because our full theory provides us with
additional relationships among couplings.
Of course, one can always impose additional assumptions to obtain
an estimate of the bounds on different operators.
For example,
if one assumes that there
are no unnatural cancellations among the couplings of different operators
one can bound each of them independently from the others. In our example
this would be implemented by putting, for instance, $c_1(m) \not= 0$ and
$c_2(m) = 0$, then use eqs.~\rfn{c1} and \rfn{c2} with $\mu=M_Z$
to obtain $c_1(M_Z)$ and $c_2(M_Z)$ as a function of $c_1(m)$.
Clearly with this assumptions a bound on $c_2(M_Z)$ implies a bound
on $c_1(m)$ and therefore also a bound on $c_1(M_Z)$. And similarly for
the other coupling.
The estimates so obtained are interesting and
can be useful. However, we feel that they
can never substitute more elaborate bounds based on
a complete set of experimental data.
\mysection{Spontaneous symmetry breaking and\newline
the use of the equations
of motion}
\label{ssb}
{}From all previous sections it is clear that our effective Lagrangian
reproduces the results of the full theory at the one-loop level
as long as energies smaller than the $h$ mass are involved. However,
until now we have considered the full unbroken theory, but the standard
model and the extension of it we are considering now undergo spontaneous
symmetry breaking (SSB) and this has to be taken into account in the analysis.
Another point that has been the origin of controversy is the possibility of
using the standard model equations of motion to reduce the number
of operators in the effective Lagrangian or to make its
physical content more evident. It is well known that the
equations of motion cannot be used na\"{\i}vely in a Lagrangian if it
is going to be used to calculate diagrams with particles off-the-mass-shell.
In fact, the possibility of using the equations of motion to trade
derivative couplings in favour of pseudo-scalar couplings
has led to many wrong calculations in pion physics
and in other models containing Goldstone bosons. However,
the equations of motion
can still be used under certain circumstances. First of all, if the
Lagrangian is going to be used to calculate tree-level amplitudes with
all particles on-the-mass-shell, they can be used without any problem.
Moreover
it can be shown \cite{Geo91,Pol80,Arz92} that the use of the equations of
motion is equivalent
to a redefinition (in general with a non-linear transformation)
of the fields of the theory. If the starting theory is renormalizable,
normally these non-linear
transformations lead to Lagrangians that contain operators of
higher dimension
and which are not explicitly renormalizable.
However, if the starting
theory is described by an effective
Lagrangian, with all kinds of non-renormalizable
interactions, the effect of these transformations is equivalent
to the use of the equations of motion
plus a modification of the higher-order terms
in the effective Lagrangian. This is not a problem since
all these terms are already present in the effective Lagrangian. Then,
it only amounts to a re-ordering
of the effective Lagrangian \cite{Geo91}. There are some subtleties related to
the Jacobian of the transformation and renormalization, but in general
it can be shown that they do not represent a problem \cite{Pol80,Arz92}.
However, one has to be careful when studying processes that are
sensitive to operators of different dimensions. In our case there is
no problem since the only operators that could be eliminated with
the equations of motion appear at the one-loop level; then, they are
going to be used uniquely as tree-level insertions.
We could use the equations of motion
before or after SSB, or not use them at all, but the
final result should be independent from this. Intermediate steps, however, can
look quite different.
We found it convenient to use the equations of motion of the standard
model before SSB, because
they are simpler and because
it makes it easier to see which
new physics is generated by the effective Lagrangian.
We are going to use them
to substitute the derivatives of field strengths
in favour of fermionic currents and new gauge boson interactions
involving the standard Higgs scalar. Similarly, we will substitute
covariant derivatives of the lepton doublet in favour of the right-handed
leptons and the standard Higgs. Later on we will
see, in a particular example, that physical observables are the same as
the results one would have obtained if proceeding in a different way.
{}From the standard model Lagrangian, \eq{smlagrangian}, one can easily
obtain the following equations of motion for the electroweak gauge bosons
\begin{eqnarray}
D_\mu B^{\mu \nu}& = & - g' \left(J^\nu_B+
\frac{i}{2} \varphi^\dagger \lrover{D}^\nu \varphi\right) \label{eqmotB}\\
D_\mu \vec{W}^{\mu \nu} & = &- g \left(\vec{J}^\nu_W+
\frac{i}{2} \varphi^\dagger \lrover{D}^\nu \vec{\tau}\varphi\right)
\label{eqmotW}\ ,
\end{eqnarray}
where
\begin{eqnarray}
J^\nu_B &=&
-\frac{1}{2} \bar{\ell} \gamma^\nu \ell
-\bar{e} \gamma^\nu e
+\frac{1}{6} \bar{q} \gamma^\nu q
+\frac{2}{3} \bar{u} \gamma^\nu u
-\frac{1}{3} \bar{d} \gamma^\nu d \label{currentB}\\
\vec{J}^\nu_W &=&
\frac{1}{2}\bar{\ell} \gamma^\nu \vec{\tau} \ell
+\frac{1}{2}\bar{q} \gamma^\nu \vec{\tau} q \label{currentW}
\end{eqnarray}
are the hypercharge and $SU(2)$ fermionic currents\footnote{Note that
eqs.~\rfn{eqmotB}--\rfn{currentW} differ from eqs.~(2.13) and (2.14)
in
ref.~\cite{BW86}, where some terms are missing.}.
We are also going to use the standard model equation of motion for
the leptonic doublet
\begin{equation}
\label{eqmotdoublet}
i \hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex} \ell=- Y_e e \varphi\ .
\end{equation}
We will use eqs.~\rfn{eqmotB}--\rfn{eqmotW} and \rfn{eqmotdoublet} in
(\ref{totalefflagr}). Every time we use the
equation of motion for the leptonic doublet we obtain an additional
leptonic Yukawa coupling constant $Y_e$. Since these couplings are
small, we are going to neglect terms that contain two or more of those
Yukawa couplings.
After using \rfn{eqmotB}, \rfn{eqmotW} and \rfn{eqmotdoublet} in
${\cal L}^{(1)}= {\cal L}^{(1)}_{det}+ {\cal L}^{(1)}_{match}$,
the resulting Lagrangian is
\begin{eqnarray}
{\cal L}^{(1)}
&=&\frac{1}{m^2} \frac{1}{(4\pi)^2} \left(-\frac{\beta^3}{6}
(\varphi^\dagger\varphi)^3+ \frac{\beta^2}{12}
\partial_\mu (\varphi^\dagger\varphi)\partial^\mu (\varphi^\dagger\varphi)
+\frac{g'^2\beta}{12} (\varphi^\dagger\varphi) B_{\mu \nu} B^{\mu \nu}
\right.\label{afterem1}\\
&&\left.-\frac{g'^4}{60} \left( J^\mu_B J_{B\mu}+
i (\varphi^\dagger \lrover{D}_\mu \varphi) J^\mu_B
-\frac{1}{4}(\varphi^\dagger \lrover{D}^\mu \varphi)
(\varphi^\dagger \lrover{D}_\mu \varphi)\right)
\right)\label{afterem2}\\
&&-\frac{2}{3} \left(\Delta_\epsilon+\frac{5}{3}\right)
\frac{g'^2}{m^2}(J^\mu_B+\frac{i}{2} (\varphi^\dagger \lrover{D}^\mu \varphi))
(\overline{\ell} F\gamma_\mu \ell)\label{afterem3}\\
&&
-\frac{2}{3}\left(\Delta_\epsilon+\frac{4}{3}\right)
\frac{g^2}{m^2}(\vec{J}^\mu_W+
\frac{i}{2} (\varphi^\dagger \lrover{D}^\mu \vec{\tau}\varphi))
(\overline{\ell} F\gamma_\mu \vec{\tau} \ell) \label{afterem4}\\
&&-\frac{g'}{3m^2} B^{\mu \nu}
(\overline{\ell} F Y_e \sigma_{\mu\nu} e\varphi+ \mrm{h.c.})
-\frac{g}{6m^2} \vec{W}^{\mu \nu}
(\overline{\ell} F Y_e\sigma_{\mu\nu} \vec{\tau} e\varphi+\mrm{h.c.})
\hspace{2cm}\label{afterem5}\\
&& +\left(\frac{\beta}{m^2} (\varphi^\dagger \varphi)
(\overline{\ell} F Y_e e \varphi) + \mrm{h.c.} \right) \label{afterem6}\\
&&-\frac{(4\pi)^2}{m^2} (\overline{\ell} F \gamma_\mu \ell)
(\overline{\ell} F \gamma^\mu \ell)\ .\label{afterem7}
\end{eqnarray}
The terms from \eq{totalefflagr} containing $\hspace{-0.2ex}\not\hspace{-0.7ex}D\hspace{0.1ex}^3$ and $\hat{F}$ have
been neglected as they are suppressed by two
or more leptonic Yukawa couplings.
Again all couplings must be understood as bare effective Lagrangian
couplings (barred couplings). We can write
eqs.~(\ref{afterem1})--(\ref{afterem7}) in terms of
$\overline{MS}$ renormalized couplings by just dropping the divergent
contributions ($\Delta_\epsilon \rightarrow 2\log(\mu/m)$).
It is clear that after using the equations of motion one can easily see the
plethora of
interesting processes that a single term in \eq{totalefflagr} can generate.
We will first study how our effective Lagrangian can modify the
standard model mechanism for SSB and what its effects are on the
spectrum of physical gauge bosons.
The first term in (\ref{afterem1}) modifies the standard model
Higgs potential
and produces a shift in the vacuum expectation
value (VEV). Since this is a one-loop effect one
should also use the full one-loop standard model effective potential to
analyse this shift. This gives a global redefinition of the VEV and
has only consequences in the Higgs sector of the theory: the
ratio of the Higgs to the $W$ mass and the Higgs couplings are
changed, but there are no other effects. As we are not interested
in Higgs physics here we are not going to compute this shift.
The Higgs dependent terms of the effective Lagrangian can have
interesting effects when the Higgs develops a VEV:
\begin{equation}
\label{vev}
\vev{\varphi} = \left(\begin{array}{c} 0 \\ v \end{array} \right)\ .
\end{equation}
In what follows we will substitute the Higgs field by its VEV
and will neglect all the terms containing physical Higgses.
The second term in \eq{afterem1}, after SSB, produces a wave function
renormalization of the Higgs scalar. Again, since we are not interested in
Higgs interactions here we are going to neglect it.
The third term clearly gives a wave function renormalization of the
$B_\mu$ gauge boson. Then, one has to diagonalize simultaneously the
kinetic term and the mass terms for the gauge bosons. After SSB
we get the following kinetic term for the $B_\mu$ gauge boson
\begin{equation}
-\frac{1}{4} (1-\delta_m) B_{\mu\nu} B^{\mu\nu}\ ,
\end{equation}
where
\begin{equation}
\delta_m \equiv \frac{g'^2 \beta}{3(4\pi)^2} \frac{v^2}{m^2}\ .
\end{equation}
The $W$ gauge boson kinetic term and the mass terms remain unchanged.
Then we can recover the canonical kinetic term by redefining the $B_\mu$
field as follows
\begin{equation}
B_\mu \rightarrow (1-\delta_m)^{-1/2} B_\mu\ ;
\end{equation}
this will change the mass matrix of the gauge bosons. However as
the $B_\mu$ field comes always with $g'$ we can recover the standard
model form by redefining $g'$ as well
\begin{equation}
\label{hatg}
\hat{g}' \equiv (1-\delta_m)^{-1/2} g'\ .
\end{equation}
It is not difficult to see that after SSB the last term in
\eq{afterem2} gives
\begin{equation}
\label{ncoperators}
\frac{1}{4} v^2 \frac{\hat{g}'^4}{60(4\pi)^2}\frac{v^2}{m^2}
(g W^{3\mu}-\hat{g}' B^\mu) (g W^3_\mu-\hat{g}' B_\mu) \ ,
\end{equation}
where we have already included the redefinition of $g'$ from \eq{hatg}.
This term is an additional contribution to the neutral gauge bosons
mass term. Adding it to the standard model contribution and
diagonalizing the full gauge boson mass matrix, we find that the
$W^3_\mu$ and $B_\mu$ gauge bosons can be expressed in
terms of the physical photon $A_\mu$,and the $Z$-boson $Z_\mu$ as
follows:
\begin{eqnarray}
\label{physbosons}
W^3_\mu &=& \hat{s}_W A_\mu + \hat{c}_W Z_\mu \\ \nonumber
B_\mu &=& \hat{c}_W A_\mu - \hat{s}_W Z_\mu\ ,
\end{eqnarray}
where
\begin{eqnarray}
\hat{s}_W &=& \frac{\hat{g}'}{\sqrt{g^2+\hat{g}'^2}} \approx
s_W \left(1+\frac{1}{2} c^2_W \delta_m\right) \\ \nonumber
\hat{c}_W &=& \frac{g}{\sqrt{g^2+\hat{g}'^2}} \approx
c_W \left(1-\frac{1}{2} s^2_W \delta_m\right)
\end{eqnarray}
and
\begin{equation}
s_W = \frac{g'}{\sqrt{g^2+g'^2}}
\end{equation}
is the tree-level weak mixing angle in the pure standard model.
Similarly the relation between the electric charge and the gauge
coupling is modified in comparison with the standard model one:
\begin{equation}
\label{echarge}
e = \hat{g}' \hat{c}_W=g \hat{s}_W \approx
g s_W \left(1-\frac{1}{2} c^2_W \delta_m\right)
\ .
\end{equation}
With the same notation the physical masses are
\begin{eqnarray}
\label{physmasses}
m^2_W &=& \frac{1}{2} g^2 v^2 \nonumber\\
m^2_Z & = & \frac{m^2_W}{\hat{c}^2_W} (1+\delta_Z) \approx
\frac{m^2_W}{c^2_W} (1+s^2_W \delta_m+\delta_Z)\ ,
\end{eqnarray}
where
\begin{equation}
\label{dz}
\delta_Z = \frac{\hat{g}'^4}{60(4\pi)^2} \frac{v^2}{m^2}
\end{equation}
comes from \eq{ncoperators}.
From these equations it is clear that only the $\delta_Z$ correction,
which produces a relative shift form the standard relation between
the masses of the $Z$ and the $W$ could in principle be observed;
however, it is very small. All the other
shifts related to $\delta_m$ can be absorbed in the definition
of the coupling $\hat{g}'$ or, equivalently, in the weak mixing angle
$\hat{s}_W$
and then are not observable.
Now we can go back to eqs.~(\ref{afterem1})--(\ref{afterem7})
and write them in terms of the physical gauge bosons. The result
we have found (neglecting all Higgs interactions) is:
\begin{eqnarray}
{\cal L}^{(1)}
&=&-\frac{g^2}{2 m_W^2} \delta_Z
(c_W^2 J^\mu_A -J^\mu_Z)(c_W^2 J_{A\mu} -J_{Z\mu})
+ \frac{g}{\hat{c}_W} \delta_Z Z_\mu (c_W^2 J^\mu_A -J^\mu_Z)
\label{current-current}\\
&&+\frac{2}{3 } \frac{g}{m^2\hat{c}_W}
\left(-(1-2\hat{s}_W^2) \left(\Delta_\epsilon+\frac{4}{3}\right)+
\hat{s}_W^2\frac{1}{3}\right)
\left(M_Z^2 Z^\mu+ \frac{g}{\hat{c}_W} J^\mu_Z\right)
(\overline{\nu_L} F\gamma_\mu \nu_L)\hspace{1cm}\label{fczncoupling}\\
&&+\frac{2}{3} \frac{g}{m^2\hat{c}_W}
\left(\left(\Delta_\epsilon+\frac{4}{3}\right)+\hat{s}_W^2\frac{1}{3}\right)
\left(M_Z^2 Z^\mu+ \frac{g}{\hat{c}_W} J^\mu_Z\right)
(\overline{e_L} F\gamma_\mu e_L)\label{fczecoupling}\\
&&
-\frac{2}{3}\left(\Delta_\epsilon+\frac{4}{3}\right)
\frac{g}{m^2} \left((\sqrt{2} M_W^2 W^+_\mu+J_\mu^\dagger)
(\overline{\nu_L} F\gamma^\mu e_L)+\mrm{h.c.}\right)\label{fcwcoupling} \\
&&
-\frac{2}{9} \frac{e^2}{m^2} J_A^\mu (\overline{e} F\gamma_\mu e)\
-\frac{2}{3} \frac{e^2}{m^2} \left(\Delta_\epsilon+\frac{5}{3}\right) J_A^\mu
(\overline{\nu_L} F\gamma_\mu \nu_L) \label{emfccoupling}\\
&& -\frac{1}{6} \frac{e}{m^2} A^{\mu\nu}
\left((\overline{e_L} F M_e \sigma_{\mu\nu} e_R) +
\mrm{h.c.}\right)\label{muegamma}\\
&&+\frac{1}{6}\frac{g}{m^2\hat{c}_W}
(1+\hat{s}^2_W) Z^{\mu\nu}
\left((\overline{e_L} F M_e \sigma_{\mu\nu} e_R)+\mrm{h.c.}\right)
\label{extraz}\\
&&-\frac{1}{3\sqrt{2}} \frac{g}{m^2}
\left(W^+_{\mu\nu}
(\overline{\nu_L} F M_e \sigma_{\mu\nu} e_R)+\mrm{h.c.}\right)
\label{extraw}\\
&&-\frac{(4\pi)^2}{m^2} \left(
(\overline{e_L} F \gamma^\mu e_L) (\overline{e_L} F \gamma_\mu e_L)
+(\overline{\nu_L} F \gamma^\mu \nu_L)
(\overline{\nu_L} F \gamma_\mu \nu_L) \right.\nonumber
\\
&&
\hspace*{1.5cm}
+\left.
2(\overline{e_L} F \gamma^\mu e_L) (\overline{\nu_L} F \gamma_\mu \nu_L)
\right)
\label{extrabox}
\end{eqnarray}
Here $M_e = Y_e v$ is the charged lepton mass matrix and
$A^{\mu\nu} = \partial^\mu A^\nu-\partial^\nu A^\mu$ and
$Z^{\mu\nu} = \partial^\mu Z^\nu-\partial^\nu Z^\mu$ are the
field strengths of the photon and the $Z$ gauge boson, respectively.
$\nu_L$ and $e_L$ are the components of the left-handed lepton
doublet and $e_R$ are the right-handed parts of the charged
leptons.
We have also defined the following neutral currents:
\begin{eqnarray}
J_A^\mu &=& \sum_f Q_f \overline{f} \gamma^\mu f\nonumber\\
J_Z^\mu &=& \sum_f \overline{f} (v_f-a_f \gamma_5)\gamma^\mu f
\end{eqnarray}
where
\begin{equation}
v_f = \frac{1}{2}T_f^3-s_W^2 Q_f, \hspace{1cm} a_f = -\frac{1}{2} T_f^3
\end{equation}
and $J_\mu^\dagger$ is the usual charged current.
Couplings with more than one gauge boson have not been included,
and the term \rfn{afterem6} has been removed because after SSB
it can be absorbed in a renormalization of the charged lepton
mass matrix.
As we have seen, by using the equations of motion we have obtained a
Lagrangian that displays its physical content in a very transparent way.
However,
as stressed before, the use of the equations of motion is not necessary.
We could have
started with the unbroken Lagrangian, eqs.~\rfn{totalefflagr} and \rfn{lagr1},
and then let the Higgs field
acquire a VEV without using the equations of motion at all.
After diagonalization of the
gauge boson mass matrices we would have obtained a quite different
effective Lagrangian from that in
eqs.~\rfn{current-current}--\rfn{extrabox}. However, both Lagrangians
give the same physics (at least at the level of precision we are working
here). For example, the penguin operators in ${\cal L}^{(1)}$ (that is the
first two operators in the Lagrangian of \eq{totalefflagr}) when
written in terms of the physical fields,
give the following interactions between $Z$-bosons and leptons:
\begin{eqnarray}
\label{zpenguin}
&&-\frac{g}{m^2\hat{c}_W}\frac{2}{3} \left(-(1-2 s_W^2)
\left(\Delta_\epsilon+\frac{4}{3}\right)
+\frac{1}{3} s_W^2\right) \partial_\mu Z^{\mu\nu}
(\overline{\nu_L} F \gamma_\nu \nu_L)\nonumber\\
&&-\frac{g}{m^2\hat{c}_W}\frac{2}{3} \left(\left(\Delta_\epsilon+\frac{4}{3}\right)
+\frac{1}{3} s_W^2\right) \partial_\mu Z^{\mu\nu}
(\overline{e_L} F \gamma_\nu e_L)\ .
\end{eqnarray}
Clearly, from this interaction and for on-the-mass-shell
particles, we get the same amplitude for the $Z$ decay into a pair
of different leptons as that obtained from
eqs.~(\ref{fczncoupling}) and (\ref{fczecoupling}).
From \eq{zpenguin} we see that $Z$-exchange processes at low-energy,
$q^2 \ll M_Z^2$, are suppressed by $q^2/M_Z^2$. We get exactly the
same answer from the compensation of the contributions from the
two terms (the $Z^\mu$ and the $J_Z^\mu$ current terms) in \eq{fczncoupling} .
\mysection{Phenomenological consequences}
\label{pheno}
The terms in ${\cal L}^{(1)}$ give many interesting effects, apart from the
shifts in the gauge-boson masses, which have already been commented on.
For example, \eq{current-current} gives extra contributions to
flavour-conserving neutral-current processes.
Equation \rfn{emfccoupling} gives rise to four-fermion processes with
non-conservation of generational lepton numbers; for example they give
rise to
$\mu^- \rightarrow e^- e^- e^+$ and similar processes. To compute
these one has to take into account also the contributions coming from
one-loop diagrams, with one insertion of the tree-level four-lepton
interaction. The couplings in \eq{fczecoupling} allow the decay
of the $Z$ into different charged leptons. Equation (\ref{muegamma})
leads to $\mu \rightarrow e \gamma$ and similar processes. The amplitude
for this process
can easily be obtained from \eq{muegamma}. Taking into account that,
without loss of
generality, $M_e$ can be taken diagonal with diagonal elements $m_a$ it is
given by
\begin{equation}
\label{ampmuegamma}
T(e_b \rightarrow e_a\, \gamma) = -i \frac{e}{3} F_{ab}
\bar{u}(p_a) \sigma_{\mu\nu} q^\nu (m_b R+m_a L) u(p_b) \epsilon^\mu(q)
\ ;
\end{equation}
$L$ and $R$ are, respectively,
the left-handed and right-handed chirality operators.
The amplitude \eq{ampmuegamma}
is in complete agreement with the results obtained in
refs.~\cite{Pet82,LTV85,BS88}.
We are not going to study in detail the phenomenology of this model here.
An analysis of this based on our effective
Lagrangian, will be presented elsewhere. However,
to give a flavour of the applicability of our effective theory, and to
show how one can work with it, we will comment on the calculation of
the decay $Z \rightarrow e^+_a\, e^-_b$ with both the full and the effective
theories.
In the effective theory we have two amplitudes, one due to the loop
diagram of fig.~4.c and the other coming from \eq{fczecoupling}.
As expected, even though
each of them contains a divergent contribution $\Delta_\epsilon$,
they give a finite answer when summed. The total amplitude is given
in appendix~\ref{fcdecay}, \eq{efffcamp}.
We also have performed a complete calculation of the amplitude
of this process in the full theory. After taking into account the
diagrams with
self-energy insertions to the external fermion legs, the final answer is
finite.
The amplitude is given in \eq{fullamp} in terms of the functions
$f_1(w)$ and $f_2(w)$, as explained in appendix~\ref{fcdecay}.
In the limit of $m \gg M_Z$ (that is, $w \rightarrow 0$), the results
obtained in the full and in the effective theories are identical,
as they should be. In fact all matching conditions are designed with just
this in view.
{}From the full model amplitude of \eq{fullamp}, we obtain the following
branching ratio of the flavour-violating decay width ($a\not=b$)
relative to the standard model flavour-conserving one,
\begin{equation}
BR(Z\rightarrow \overline{e}_a e_b)=
\frac{\Gamma(Z\rightarrow \overline{e}_a e_b)}
{\Gamma(Z\rightarrow \overline{e}_a e_a)}= \left|F_{ab}\right| ^2
\frac{8\left| f_1(w)+\hat{s}^2_W f_2(w)\right|^2}{1+(1-4 \hat{s}_W^2)^2}\ .
\end{equation}
The branching ratio in the effective Lagrangian is given by the same
expression, but with $f_1(w)$ and $f_2(w)$ given in \eq{limfes}.
In fig.~7, we present this branching ratio in both
the full and the effective theories as a function of $m$.
Charged scalar Yukawa couplings, $ (f^\dagger f)_{ab}$, are taken to be
equal to 1.
{}From the figure it is clear that for masses of the scalar $h$
larger than the mass of the $Z$ boson the effective theory
gives a good approximation.
\mysection{Discussion and conclusions}
\label{conclusion}
In this paper we have studied some questions related to the construction
and the use of effective Lagrangians, by considering an
extension of the standard model that includes a heavy charged scalar
coupled to the leptonic doublet.
Starting from the full renormalizable model, we have built a low-energy
effective field theory by integrating out the heavy scalar. This was done
at the one-loop
level and keeping only
operators of dimension six or less.
Functional methods were used to obtain, in a compact and gauge-invariant
form, all operators generated by tree-level and one-loop diagrams
containing only heavy scalars in the internal lines. The result
includes only diagrams which are one-particle-irreducible with respect
to the light particles.
In appendix~\ref{det} we give
a detailed explanation of this calculation.
This is not the complete answer for the
effective Lagrangian, since there are many one-loop diagrams
in the full theory,
with heavy and light particles in the loop that are not completely
taken into account by one-loop diagrams involving tree-level
effective couplings. To obtain these additional contributions we
have calculated several Green functions in both the
full and the effective theories and required that the results of
both calculations match for small momenta compared with the
heavy scalar mass. The result can be expressed as a linear combination
of gauge-invariant operators, which must be included in the effective
Lagrangian.
Adding both contributions, from the operators
corresponding to the diagrams with only
heavy lines and from those with heavy and light internal particles,
we have obtained the complete bare one-loop effective Lagrangian,
including operators up to dimension six.
It contains several operators with infinite
coefficients (dimensional continuation was employed to regularize
all divergent integrals, then UV divergences appear as poles in $1/(D-4)$).
By using
an $\overline{MS}$ scheme to renormalize both the full and the
effective theories, we obtained the matching conditions for the
running couplings, which express the renormalized couplings
of the effective theory in terms of the renormalized couplings of
the full model. To avoid large logarithms, these matching conditions should
be evaluated at a scale around the heavy scalar mass and then the
renormalization group used to bring the couplings to any other lower scale.
We used this simple example to discuss the behaviour of the couplings
of a generic effective Lagrangian. In general all the operators in the
effective Lagrangian with the same quantum numbers mix under the
renormalization group. The effective couplings obey a standard
renormalization group first-order differential equation controlled
by the anomalous-dimension matrix. It is clear that the only information
that can be extracted from the effective Lagrangian, without knowing the
full theory, is this
anomalous-dimension matrix. However, to obtain the couplings it is
necessary to solve the renormalization group equations, and this requires
the knowledge of $n$ boundary conditions
(if $n$-operators are involved in the mixing).
One-loop calculations with the effective Lagrangian can
only be used to relate the couplings of operators at different scales, but give
no
information at all on the actual value of those couplings. Only experiment,
or a more complete theory, can give new information on the values
of the effective Lagrangian couplings. This trivial observation is of
importance when using the effective Lagrangian approach to classify
the sort of new physics one could find in future experiments.
After renormalization we obtain an effective Lagrangian that can
be split in three pieces.
The first one, ${\cal L}_{SM}$, is just the standard model
Lagrangian
(expressed in terms of effective couplings); the second one is the
four-lepton interaction obtained from the full theory at tree level,
${\cal L}^{(0)}$; the third piece, ${\cal L}^{(1)}$ contains all
dimension-six operators generated at the one-loop level in the full model.
This effective Lagrangian
was rewritten by using
the standard model classical equations of motion
in order to display its physical content more transparently.
Some caution is needed when using the equations of motion,
especially for operators that are going to be inserted in
loop diagrams.
In our
case, however, the equations of motion are used only in
operators that are generated at one loop in the full theory and that are
supposed to be used only at tree level in the effective Lagrangian. Then,
the use of equations of motion is completely legitimate.
We used them for the unbroken
theory, but it is worth while to stress that the equations of
motion might also be used after SSB (or might not be used at all) and
this would not change any physical result.
We also discussed the consequences of SSB on our effective Lagrangian by
substituting the VEV of the doublet and neglecting all Higgs interactions.
Apart from a negligible modification of the relation between the
masses of the vector bosons, the most interesting consequence
of the model is due to the different operators contributing to processes
with violation of the generational lepton numbers.
We made a short review of some processes
generated by our one-loop effective Lagrangian, $e_a^- \rightarrow e_b^-
\gamma$, $e_a^- \rightarrow e_b^- e_c^+ e_c^+$, etc (where
$a,b,c$ are different flavour indices). A more
detailed phenomenological analysis will be presented elsewhere.
Finally,
to see how one can use our effective Lagrangian, we have calculated
the decay width of the $Z$ gauge boson to a pair of different leptons.
It contains contributions from one-loop diagrams with one insertion
of the tree-level four-fermion operator and direct contributions from
operators generated at the one-loop level. We have done also the
calculation in the full theory and compared the results. For a mass
of the charged scalar $m$ larger than the $Z$ mass, the effective theory
calculation gives a good approximation.
\section*{Acknowledgements}
We thank J. Bernab\'eu, A. Casas, S. Fanchiotti, S. Peris and A. Pich
for helpful discussions
on the subject of this paper and for a critical reading
of the manuscript.
This work was supported in part by
CICYT, Spain, under grant AEN93-0234.
\vfill\eject
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
\label{sec:intro}
Many recent works treat the sequential decision-making problem of multiple autonomous, interacting agents as a Multi-Agent Reinforcement Learning (MARL) problem. In this framework, each agent learns a policy to optimize a long-term discounted reward through interacting with the environment and the other agents \cite{zhang2021multi, qie2019joint, cui2019multi}. In many realistic applications, the environment states are only partially observable to the agents \cite{hausknecht2015deep}. Learning in such environments has become a recurrent problem in MARL \cite{zhang2021multi}, the difficulty of which lies in the fact that the optimal policy might require a complete history of the whole system to determine the next action to perform \cite{meuleau1999learning}. The authors in \cite{peshkin2001learning} used finite policy graphs to represent policies with memory and successfully applied a policy gradient MARL algorithm in a fully cooperative setting.
In this paper, we consider the problem of controlling a heterogeneous team of agents to satisfy a specification given as a Truncated Linear Temporal Logic (TLTL) formula \cite{li2017reinforcement} using MARL. TLTL is a version of Linear Temporal Logic (LTL) \cite{pnueli1977temporal}. The semantics of TLTL is given over a finite trajectory in a set such as the state space of the system. TLTL has dual semantics. In its Boolean (qualitative) semantics (whether the formula is satisfied), a trajectory satisfying a TLTL formula is accepted by a Finite State Automaton (FSA). The quantitative semantics assigns a degree of satisfaction (robustness) of a formula. Designing a reward function that accurately represents the specification is an important issue in reinforcement learning. We use the robustness of the TLTL specification as the reward of the MARL problem. However, the TLTL robustness is evaluated over the entire trajectory, so we can only get the reward at the end of each episode. To address this, we propose a novel reward shaping technique that introduces two additional reward terms based on the quantitative semantics of TLTL and the corresponding FSA. Such rewards, which are obtained after each step, guide and accelerate the learning.
As in \cite{peshkin2001learning}, we consider a partially observable environment and train a finite policy graph for each agent. When deploying the policy, each agent only knows its own state. However, during training we assume all agents know the states of each other. Then we use this information to guide and accelerate learning. This assumption is reasonable in practice because the training environment is highly configurable and a communication system can be easily constructed, while during deployment there is no guarantee that such a communication system is available.
The idea of modifying rewards using the semantics of temporal logics was introduced in \cite{li2017reinforcement} to address the problem of single agent learning in a fully observable environment. The method was then extended to deep RL to successfully control two manipulator arms working together to make and serve hotdogs \cite{li2019formal}. Although the problem in \cite{li2019formal} involved two agents, single agent RL algorithms were used with two different specially designed task specifications.
In \cite{sun2020automata}, temporal logic rewards were used for multi-agent systems. As opposed to our work, the environment in \cite{sun2020automata} is fully observable. Moreover, each sub-task in \cite{sun2020automata} is restricted to be executed by one agent. In this paper, we allow both independent tasks, which can be accomplished by one agent, and shared tasks, which must be achieved by cooperation of several agents. The rewards in \cite{li2019formal} and \cite{sun2020automata} encourage the FSA to leave its current state, without distinguishing whether it is a transition towards the satisfaction of the TLTL specification. In this paper, the combination of the two reshaped rewards encourage transitions towards satisfaction. Another related work is \cite{hammond2021multi}, in which the authors apply an actor-critic algorithm to find a control policy for a multi-agent system that maximizes the probability of satisfying an LTL specification. However, the reward does not capture a quantitative semantics, and a fully observable environment is assumed.
The contributions of this paper can be summarized as follows. First, we propose a general procedure that uses MARL algorithms to synthesize distributed control from arbitrary TLTL specifications for a heterogeneous team of agents. Each agent can have different capabilities and both independent and shared sub-tasks can be defined. The agents work in a partially observable environment, and the policy for each agent only requires its own state during deploying. During training, we assume a fully observable environment, which enables the use of TLTL robustness as a reward. Second, we create a novel temporal logic reward shaping technique, where two additional rewards based on TLTL robustness and FSA states are added. Both rewards are obtained immediately after each step, which guides and accelerates the learning process.
\section{Preliminaries and Notation}
\label{sec:Preliminaries}
Given a set $S$, we use $|S|$, $2^S$, and $S^*$ to denote its cardinality, the set of all subsets of $S$, and a finite sequence over $S$. Given a set $\Sigma$, a collection of sets $\Delta = \{\Sigma_i \subset \Sigma, i\in I\}$, in which $I$ is a set of labels, is called a distribution of $\Sigma$ if $\cup_{i\in I} \Sigma_i = \Sigma$. For $\sigma \in \Sigma$, we use $I_{\sigma} = \{i \in I | \sigma \in \Sigma_i\}$ to denote the set of labels of elements in $\Delta$ that contains $\sigma$.
\subsection{Partially Observable Stochastic Games}\label{sec:POSG}
Single agent reinforcement learning uses Markov Decision Processes (MDP) as mathematical models \cite{puterman2014markov}. In MARL, more complicated models are needed to describe the interactions between the agents and the environment. A popular model is a Stochastic Game (SG) \cite{bucsoniu2010multi}. A Partially Observable Stochastic Game (POSG), which is a generalization of an SG, is defined as a tuple $\langle I, S, d_0, \{A_i,O_i,B_i\}_{i\in I}, p, \{r_i\}_{i\in I}\rangle$, in which $I=\{1,\ldots,n\}$ is an index set for the agents; $S$ is the discrete joint state space; $d_0$ is the probability distribution over initial states; $A_i$ is the set of actions for agent $i$, $O_i$ is the discrete observation space and $B_i: S \to O_i$ is the observation function; $p: S \times A \to Pr(S)$ is the transition function that maps the states of the game and the joint action of the agents, defined as $A = \times_{i=1}^n A_i$, to probability distributions over states of the game at next time step; and $r_i: S \times A \times S \to \mathbb{R}$ is the reward function for agent $i$. If every agent is able to obtain the complete information of the environment state when making decisions, i.e., $O_i = S$ and $B_i(s) = s$ for all $s \in S$, then the POSG becomes a fully observable SG.
The goal for each agent $i$ is to find a policy $\pi_i$ that maximizes its value $V_i$:
\begin{equation*}
V_i(\vec{\pi}, d_0) = \sum_{t=0}^{T-1}\gamma^t \mathbb{E}_{a_t\sim\vec{\pi}}(r_i(s_t,a_t,s_{t+1})|\vec{\pi}, d_0),
\end{equation*}
where $\gamma \in (0,1]$ is the discount factor and $\vec{\pi} = (\pi_1, ..., \pi_n)$ is a set of policies. Note that for each agent the environment is non-stationary since its reward $r_i$ may depend on other agents' policies. A \emph{memory-less policy} $\pi_i:O_i\rightarrow Pr(A_i)$ is a mapping from the observations of agent $i$ to probability distributions over the action space of agent $i$. For a fully observable SG, memory-less policies are sufficient to achieve optimal performance. However, for POSGs, the best memoryless policy can still be arbitrarily worse than the best policy using memory \cite{singh1994learning}. A policy graph \cite{meuleau1999learning} is a common way to represent a policy with memory.
\subsection{Truncated Linear Temporal Logic}\label{sec:TLTL}
Truncated Linear Temporal Logic (TLTL) \cite{li2017reinforcement} is a predicate temporal logic inspired from the traditional Linear Temporal Logic (LTL) \cite{pnueli1977temporal}. TLTL
can express rich task specifications that are satisfied in finite time. Its formulas
are defined over predicates $f(x) \geq 0$, where $f: X \to \mathbb{R}$ is a function over $X$ (such as the state space of a system). Then TLTL formulas are evaluated against finite sequences over $X$. TLTL formulas have the following syntax:
\begin{multline*}
\phi := \top \; | \; f(x) \geq 0 \; | \; \neg \phi \; | \; \phi \wedge
\psi \; | \; \phi \vee \psi \;|\; \phi \Rightarrow \psi \;|\; \lozenge\phi \; | \;\bigcirc \phi \; | \; \phi \mathcal{U}\psi \; |\; \phi \mathcal{T} \psi,
\end{multline*}
where $\top$ is the Boolean True. $\neg$ (negation), $\land$ (conjunction) and $\lor$ (disjunction) are Boolean connectives. $\lozenge$, $\bigcirc$, $\mathcal{U}$, and $\mathcal{T}$ are temporal operators that stand for ``eventually", ``next", ``until" and ``then" respectively. Given a sequence over $X$, $\lozenge\phi$ (eventually) requires $\phi$ to be satisfied at some time step, $\bigcirc\phi$ (next) requires $\phi$ to be satisfied at the second time step, $\phi\mathcal U\psi$ (until) requires $\phi$ to be satisfied at each time step before $\psi$ is satisfied, $\phi\mathcal T\psi$ (then) requires $\phi$ to be satisfied at least once before $\psi$ is satisfied. TLTL also has derived operators, e.g., $\square$ (finite time always) and $\Rightarrow$ (imply).
TLTL formulas can be interpreted in a qualitative semantics or a quantitative semantics. The qualitative semantics provides a yes/no answer to the corresponding property, while the quantitative semantics generates a measure of the degree of satisfaction. Given a finite trajectory $x_{t:t+k}:= x_t x_{t+1}\ldots x_{t+k}$, the quantitative semantics of a formula $\phi$, also called \textit{robustness}, is denoted by $\rho(x_{t:t+k}, \phi)$. We refer the readers to \cite{li2017reinforcement} for detailed definitions of the TLTL qualitative and quantitative semantics.
\subsection{Finite State Automata}\label{sec:FSPA}
\begin{definition}[Finite State Predicate Automaton \cite{li2019formal}]
A finite state predicate automaton (FSPA) corresponding to a TLTL formula $\phi$ is a tuple $\mathcal{A} = \langle Q, q_0, F, T_{r}, \Psi, \mathcal{E}, b \rangle$, in which $Q$ is a set of automaton states, $q_0 \in Q$ is the initial state, $F \subset Q$ is the set of final states, $T_r$ is the set of trap states, $\Psi$ is a set of predicate Boolean formulas, where the predicates are evaluated over $X$, $\mathcal{E} \subseteq Q \times Q$ is the set of transitions, and $b:\mathcal{E} \to \Psi$ maps transitions to formulas in $\Psi$.
\end{definition}
The semantics of FSPA is defined over finite sequences over $X$. We refer to the elements in $\Psi$ as {\it edge guards}. A transition is enabled if the corresponding Boolean formula is satisfied. Formally, by noting that a Boolean formula is a particular case of a TLTL formula, the FSPA transitions from $q_t$ to $q_{t+1}$ at time $t$ if and only if $\rho(x_{t:t+k}, b(q_t, q_{t+1})) > 0$, where $x_{t:t+k}$ is a sequence of $X$ from $t$ to $t+k$. Note that the robustness is only evaluated at $s_t$ instead of the whole sequence. Thus we abbreviate $\rho(x_{t:t+k}, b(q_t, q_{t+1}))$ to $\rho(x_t, b(q_t, q_{t+1}))$.
A trajectory over $Q$ is accepting if it ends in the final states $F$. A trajectory over $X$ is accepted by $\mathcal A$ if it leads to an accepting trajectory over $Q$.
Each TLTL can be translated into an equivalent FSPA \cite{li2019formal}, in the sense that a trajectory over $X$ satisfies the TLTL specification if and only if the same trajectory is accepted by the FSPA. All trajectories over $X$ that violate the TLTL formula drive the FSPA to the trap states in $T_r$.
\section{Problem Formulation} \label{sec:PF}
The geometry of the world is modeled by an environment graph $G = (V, E)$, where $V$ is a set of vertices and $E \subset V\times V$ is a set of edges. The motions of the agents are restricted by this graph. Assume we have a set of agents $\{i | i \in I\}$ where $I$ is a label set, and a set of service requests $\Sigma$. Let $ l: \Sigma \to 2^V$ be a function indicating the locations at which a request occurs. For a given request $\sigma \in \Sigma$, let $V_\sigma=l(\sigma)\subseteq V$ be the set of vertices where $\sigma$ occurs. The definition of $V_\sigma$ implicitly assumes that one request can occur at multiple vertices, since $V_\sigma\in2^V$. Also, multiple requests can occur at the same vertex, since $V_{\sigma_1}$ and $V_{\sigma_2}$ may have non-empty intersection for $\sigma_1\neq\sigma_2$. We use a distribution $\Delta = \{\Sigma_i \subset \Sigma, i\in I\}$ to model the agents' capabilities for completing different service requests: $\sigma \in \Sigma_i$ means that request $\sigma$ can be serviced by agent $i$. We further define $I_\sigma=\{i\in I|\sigma\in\Sigma_i\}$ as the set of agents that can service request $\sigma$. We consider two types of services: \textit{independent} and \textit{shared}. An independent request is any $\sigma \in \Sigma$ such that $|I_\sigma| = 1$. This type of request can only be serviced by agent $i$ such that $\sigma \in \Sigma_i$. A shared request $\sigma$ is defined by $|I_\sigma| > 1$, which means that servicing it requires the cooperation of all the agents that own $\sigma$.
We model the motion and actions of agent $i$ by using a deterministic finite transition system $\mathcal{T}_i = \langle V_i, {v_0}_i, A_i, \delta_i, \Sigma_i, \models_i\rangle$ where $V_i \subseteq V$ is the set of vertices that can be reached by agent $i$; ${v_0}_i \in V_i$ is the initial location; $A_i = V_i\cup\Sigma_i\cup \{\epsilon\}$ is the discrete action space; $\delta_i:V_i \times A_i \to V_i$ is a deterministic transition function; $\Pi_i = \Sigma_i \cup \{\epsilon\}$ is a proposition set; $\models_i \subseteq V \times \Pi_i$ is the satisfaction relation such that 1) $(v,\epsilon) \in \models_i$ for all $i\in I$, $v\in V_i$, and 2) $(v,\sigma) \in \models_i$, $\sigma \in \Sigma_i$, if and only if $v \in l(\sigma)$. The meaning of taking action $a\in A_i$ at state $v$ is as follows: if $a \in V_i$, then the agent tries to move to vertex $a$ from $v$; if $a \in \Sigma_i$, then agent $i$ tries to conduct service $a$ at $v$; $a = \epsilon$ indicates that the agent stays at the current vertex without conducting any requests. Formally,
\begin{equation}
\delta_i(v,a) = \begin{cases}
v' & \text{if} \; a = v' \in V_i \;\text{and}\; (v,v') \in E \\
v & \text{otherwise} \\
\end{cases}
\end{equation}
For $v, v'\in V_i$ and $a \in A_i$, a transition $v' = \delta_i(v,a)$ is also denoted by $v\xrightarrow{a}_{\mathcal{T}_i} v'$. Now we define the Motion and Service (MS) plan for an agent, which is inspired by \cite{chen2011formal}.
\begin{definition}[Motion and Service Plan] A Motion and Service (MS) plan for an agent $i \in I$ is a sequence of states represented by ordered pairs $z^i_0z^i_1 \ldots z^i_T \in (V \times \Pi_i)^*$ that satisfies the following properties:
\begin{enumerate}
\item $z^i_0 = ({v_0}_i,\epsilon)$.
\item For all $t\geq0$, $z^i_t=(v,\sigma)\in\models_i$.
\item For all $t\geq1$, given $z^i_{t-1} = (v, \sigma)$ and $z^i_t = (v', \sigma')$, if $v \neq v'$ then $\sigma' = \epsilon$.
\item For all $t\geq1$, given $z^i_{t-1} = (v, \sigma)$ and $z^i_t = (v', \sigma')$, then $\exists a\in A_i$ such that $\delta_i(v,a) = v'$.
\end{enumerate}
\end{definition}
We use $z^i_{0:T}$ to denote the MS plan for an agent $i$ from time $0$ to $T$, i.e., $z^i_{0:T} = z^i_0z^i_1\cdots z^i_T$. A Team Motion and Service (TMS) plan is then defined as $z_{0:T} = \times^n_{i=1}z^i_{0:T}$.
We assume that there exists a global discrete clock that synchronizes the motions and the services of requests of all agents. We also assume that the times needed to service requests are all equal to 1. In other words, similar to POSGs, at each time step, an agent either chooses to move to a vertex according to $\delta_i$ or to stay where it is to conduct a particular request in $\Sigma_i$. Before the beginning of the next time step, all motions and requests are completed and the agents are ready to execute the next state in their MS plans. A MS plan thus uniquely defines a sequence of actions for an agent. More specifically, the action derived from a MS plan $z^i$ at $t \geq 1$ is determined by $z^i_{t-1} = (v,\sigma)$ and $z^i_t = (v',\sigma')$. If $\sigma' = \epsilon$ and $v \neq v'$, then the agent moves to vertex $v$ from $v'$ (for the case where $v = v'$, the agent does not conduct any request and waits at $v$). On the other hand, $\sigma \neq \epsilon$ means that the agent should conduct request $\sigma$ and it must have reached vertex $v$ at the previous time step according to property $3$ from the above definition.
In this paper, we use the term \textit{global behavior} to refer to the sequence of requests serviced by the whole team. An independent request $\sigma$ is considered to be finished at time $t$ if and only if $z^i_t = (v,\sigma)$, $i \in I_\sigma$. For a shared request $\sigma$, we assume that all the agents that own this request are capable of communicating with each other at the same vertex where $\sigma$ occurs. A shared service request $\sigma$ is completed at time $t$ if and only if $z^i_t = (v_i,\sigma)$ for all $i \in I_\sigma$ and $v_j = v_i$ for all $i,j \in I_\sigma, i \neq j$. Thus, given individual MS plans of all the agents, one single global behavior is then uniquely determined. This is directly deduced from the assumption of a constant finishing time for all requests. Next we define a term called \textit{Team Trajectory} to describe the results of executing a global TMS plan.
\begin{definition}[Team Trajectory]
Given a team of $n$ agents $\mathcal{T}_i$, a set of service requests $\Sigma$, a distribution $\Delta = \{\Sigma_i\subseteq \Sigma, i\in I\}$, a function $l: \Sigma \to V$ that shows the locations of requests and a TMS plan $z_{0:T}$ for the whole team of agents, the Team Trajectory from executing the TMS plan is a sequence of $T+1$ states $x_{0:T}$, each of which is a $(n+1)$-tuple, with the first $n$ elements equal to the vertices in the corresponding individual MS plan. The last element of the team trajectory at each time $t$ is a set $\Bar{\Sigma}_t \subseteq \Sigma$, such that $\Bar{\Sigma}_t:= \{\sigma\in\Sigma | \sigma \;\text{is finished at time t}\}$.
\end{definition}
Now the problem considered in this paper can be formulated as follows.
\begin{problem}\label{problem formulation}
Given an environment graph $G$, a team of agents $\mathcal{T}_i$ as defined above, $i\in I$, a set of service requests $\Sigma$, a distribution $\Delta = \{\Sigma_i\subseteq \Sigma, i\in I\}$, a function $l: \Sigma \to V$ that shows locations of requests, find a set of MS plans for each agent such that the team trajectory obtained by executing individual MS plans satisfying a given global task specification encoded by a TLTL formula over predicate functions of states of the team trajectory $x_{0:T}$.
\end{problem}
A natural, classical motion planning approach to Problem \ref{problem formulation} would be to design in advance MS plans for each agent. However, this requires the transition function $\delta_i$ for each agent $i$ to be known, which is not always true in realistic applications. Moreover, a priori top-down design that guarantees satisfactory performance can become extremely difficult in complex and time-varying environments \cite{bucsoniu2010multi}. Therefore, we tackle this problem using MARL, in which agents learn how to act by constantly interacting with the environment and other agents, and adjusting their behaviors according to feedback received from the environment. The transition functions $\delta_i$ are assumed to be unknown.
\begin{example} \label{example}
Consider a team of two heterogeneous agents that have to service three requests: $\sigma_1, \sigma_2$ and $\sigma_3$, with an additional requirement that $\sigma_3$ must not be finished until $\sigma_1$ or $\sigma_2$ has been finished. The environment is shown in Fig. \ref{fig:setup} a $5 \times 5$ as a grid world, which can be abstracted by a graph $G = (V,E)$. Each cell represents a vertex $v \in V$ (labeled with values at the bottom right) and each facet forms a reflective, two-way edge in $E$. There are three service requests: $\Sigma = \{\sigma_1, \sigma_2, \sigma_3\}$. There is a team of two robots modeled by transition systems $\mathcal{T}_1 = \langle V, v_4, A_1, \delta_1, \Sigma_1, \models_1\rangle$ and $\mathcal{T}_2 = \langle V, v_{18}, A_2, \delta_2, \Sigma_2, \models_2\rangle$. Assume both agents are able to move through any edges in $E$. Thus the action spaces are $A_1 = V\cup\Sigma_1\cup \{\epsilon\}$ and $A_2 = V\cup\Sigma_2\cup \{\epsilon\}$. The capabilities of conducting services are captured by the distribution $\Delta = \{\Sigma_1, \Sigma_2\}$, $\Sigma_1 = \{\sigma_1, \sigma_3\}$ and $\Sigma_2 = \{\sigma_2, \sigma_3\}$, which means $\sigma_1$ and $\sigma_2$ are independent requests and $\sigma_3$ is a shared request. For instance, to finish request $\sigma_1$, robot $\mathcal{T}_1$ must move to vertex $v_{21} \in l(\sigma_1)$ and then choose to take action $\sigma_1 \in A_1$. This behavior, which can be interpreted as a sub-task towards the success of the global mission, is formulated by a TLTL formula $\phi_1 = go_1(\cdot) \mathcal{T} do_1(\cdot) \land \square \neg (\neg go_1(\cdot)\land do_1(\cdot))$ where $go_1(\cdot)$ and $do_1(\cdot)$ are predicate functions defined over states of the team trajectory as
\begin{equation}
\begin{aligned}\label{eqn:predicates_independSigma}
go_1(x_{t:T}) &= C_1 - \min_{v \in l(\sigma_1)} dist(v^1_t, v) \\
do_1(x_{t:T}) &= \begin{cases}
C_2 \quad \sigma_1 \in \bar{\Sigma}_t \\
-C_2 \quad \text{otherwise}
\end{cases}
\end{aligned}
\end{equation}
where $v^1_t$ is the location of robot $1$ at time $t$ (included in team trajectory state $x_t$), $dist(v^1_t, l(\sigma_1))$ is the distance from $v^1_t$ to the location at which $\sigma_1$ occurs, and $C_1$ and $C_2$ are positive constants. Then the global specification is given by the TLTL formula $\phi = (\lozenge \phi_1 \lor \lozenge \phi_2) \land \lozenge \phi_3 \land \neg \phi_3 \mathcal{U} (\phi_1 \lor \phi_2)$. The FSPA corresponding to $\phi$ can be found in Fig. \ref{fig:FSPA}. The predicate functions $go_2(\cdot)$, $do_2(\cdot)$ and $do_3(\cdot)$ are defined similarly as \eqref{eqn:predicates_independSigma}. Predicates $go_3(\cdot)$ are designed in a slightly different form to encode the cooperative behaviors:
\begin{equation}
\label{eqn:predicates_dependSignma}
go_3(x_{t:T}) = C_3 - \min_{v \in l(\sigma_3)}\left(\max_{i \in I_{\sigma_3}}[dist(v^i_t, v)]\right),
\end{equation}
where $C_3$ is some constant. Note that although formula $\phi$ is constructed hierarchically, the resultant FSPA is actually nonhierarchical.
\end{example}
\begin{figure}
\centering
\begin{minipage}[t]{0.56\textwidth}
\label{fig:setup}
\centering
\includegraphics[height=4cm]{Fig/gridWorld.png}
\caption{\small The grid world of Example \ref{example}. Independent and shared service requests are depicted by yellow and red cells respectively. Cells (vertices) are labeled by the value at the bottom right corner. }
\end{minipage}
\quad
\begin{minipage}[t]{0.40\textwidth}
\label{fig:FSPA}
\centering
\includegraphics[height=4cm]{Fig/FSPA.PNG}
\caption{\small The FSPA corresponding to $\phi$. State $7$ is the trap state and state $6$ is the final state.}
\end{minipage}
\vspace{-10pt}
\end{figure}
\section{Problem Solution Using MARL}\label{sec:solution}
In summary, our method can be divided into three steps. First, in Sec. \ref{sec:POSG formulation}, we formulate an POSG whose solution can be easily transferred to a set of MS plans that solves Problem \ref{problem formulation}. Second, in Sec. \ref{sec:TLRS}, we introduce a reward shaping technique that employs robustness and a heuristic energy-like function to produce an additional reward signal. Finally in Sec. \ref{sec:Solving POSG}, we demonstrate how to solve the POSG with reshaped rewards using algorithms available in the realm of MARL.
\subsection{Formulation of Equivalent POSG}\label{sec:POSG formulation}
It is straight forward to define a POSG whose state is identical with the team trajectory state $x_t$, i.e., $S = V_1 \times \cdots \times V_n \times 2^\Sigma$. However, by doing so, the satisfaction of TLTL specifications can only be determined based on the entire team trajectory. Hence, defining the reward function as the TLTL robustness contradicts the Markovian behaviors of the POSG in the sense that the reward $r$ only depends on current state of the system. Therefore, an extra element must be added to the POSG as a tracer of the whole team in the process of satisfying the global task. The state of the FSPA corresponding to the TLTL formula suits this job perfectly, and the state space of the POSG will be a product of the team trajectory states and the FSPA states. Now we are ready to introduce the following POSG. We use $v_t^i \in V_i$ and $a_t^i \in A_i$ to denote the state and action of agent $i$ at time $t$.
\begin{definition}[FSPA augmented POSG] \label{def:FSPA-POSG}
Given Problem \ref{problem formulation} and an FSPA $\mathcal{A} = \langle Q, q_0, F, T_{r}, \Psi, \mathcal{E},\\ b \rangle$ corresponding to the TLTL specification, an FSPA augmented POSG (FSPA-POSG) is defined as a tuple $\langle I, S, s_0, \{A_i, O_i, B_i\}_{i\in I}, p, \{r_i\}_{i\in I}\rangle$, in which $I=\{1,2,\ldots,n\}$ is the index set for agents, $S = V_1 \times \cdots \times V_n \times 2^\Sigma \times Q$ is the discrete state space. $s_0 = ({v_0}_1, \ldots, {v_0}_n, \bar\Sigma_0, q_0)$ is the initial state. The action space $A_i = V \cup \Sigma_i \cup \epsilon$ of agent $i$ is identical with the action space of transition system $\mathcal T_i$. The observation space $O_i=V_i$ is the state space of $\mathcal T_i$. The observation function $B_i:S\rightarrow V_i$ maps the state of the FSPA-POSG to the state of agent $i$, which means that the policy of agent $i$ only depends on its own state. Let $s_t = (v_t^1, \ldots, v_t^n, \bar\Sigma_t, q_t)$ and $a_t = (a_t^1, \ldots, a_t^n)$ be the state of the FSPA-POSG and the joint action of all agents at time $t$ respectively. Then, the transition probability from $s_t$ to $s_{t+1}$ under action $a_t$ is defined as
\begin{equation*}
p(s_{t+1}| s_t, a_t) = \\
\begin{cases}
1 & \text{if} \; \forall i \in I, v^i_t\xrightarrow{a_t^i}_{\mathcal{T}_i} v_{t+1}^i,\; \bar\Sigma_{t+1}=\{\cup_{i\in I}a_t^i | a_t^i\in\Sigma\}\ \text{and} \;\rho(x_t, \psi_{q_t,q_{t+1}}) > 0\\
0 & \text{otherwise}
\end{cases}
\end{equation*}
where $\psi_{q_t,q_{t+1}} = b(q_t,q_{t+1}) \in \Psi$ is the edge guard of the transition from $q_t$ to $q_{t+1}$ and $x_t$ is a state of the Team Trajectory included in $s_t$. Let the reward function $r_1=\ldots=r_n=r$ be defined as
\begin{equation}
r(s_t,a_t,s_{t+1}) = \\
\begin{cases}
C & \text{if} \; q_{t+1} \in F \; \text{and} \; q_t \not\in F\\
-C & \text{if} \; q_{t+1} \in T_r\; \text{and} \; q_t \not\in T_r \\
0 & \text{otherwise}
\end{cases}
\end{equation}\label{eqn:original reward}
\vspace{-5pt}
where $C>0$ is a constant.
\end{definition}
In this paper, we assume that the FSPA-POSG has a finite horizon $T$. The rationale of this assumption lies in the fact that any accepting or rejecting team trajectory must be finite (the corresponding FSPA reaches one state either in $F$ or in $Tr$). However, $T$ must be large enough to give agents adequate time to complete the task and thus it becomes a design factor of our approach. Clearly, since the team trajectory state is included in the FSPA-POSG state, constructing a team trajectory $x_{0:T}$ from $s_{0:T}$ is very simple, then constructing a TMS is also straightforward.
\subsection{Temporal Logic Reward Shaping}\label{sec:TLRS}
As discussed in Sec. \ref{sec:intro}, we assume that the state of the FSPA-POSG is fully observable during training, while the observation function during deploying is as defined in Def. \ref{def:FSPA-POSG}. The reward function defined above only provides feedback to agents at the very end of each episode, which could cause serious problems when the probability of success under randomly selected actions converges to zero as the mission becomes more and more complicated. Missions described by temporal logics are expected to be complicated. Therefore, reward shaping techniques are introduced to add intermediate rewards to improve the speed and successful rate of learning.
We developed two additional rewards for each agent $i$ to guide learning: $r^i_\rho$ and $r^i_J$. $r^i_\rho$ is defined based on TLTL's quantitative semantics as follows:
\begin{equation}
\label{eqn:reward rho}
r^i_\rho(s_t,a_t,s_{t+1}) = \rho(x_{t+1},D_{q_{t+1}}) - \rho(\hat{x}^i_{t+1}, D_{\hat q_{t+1}}),
\end{equation}
where $D_{q_{t+1}} = \bigvee_{(q_{t+1}, q_{t+2})\in \mathcal{E}, q_{t+2}\not\in T_r} b(q_{t+1}, q_{t+2})$ is the disjunction of all predicates of the outgoing edges from $q_{t+1}$ to other non-trapped states, $\hat{x}^i_{t+1} = (v^i_t \times^n_{j=1, j\neq i} v^j_{t+1}, \hat{\bar{\Sigma}}_{t+1})$ is a predicted FSPA-POSG state that all agents take actions and transition to the next state except agent $i$, $\hat q_{t+1}$ is the corresponding predicted automaton state.
The robustness value $\rho(\cdot,D_{q_{t+1}})$ can be interpreted as a metric that measures how far it is from leaving the FSPA state $q_{t+1}$ given $x_{t+1}$. The individual reward $r^i_\rho$ is designed to be the difference between the robustness value under the whole team's actions and the robustness value with agent $i$ being idle. This is reasonable because of two reasons. First, $\hat{x}^i_{t+1}$ is generated by only using past observations so it does not require the agent model $\delta_i$. Second, since we know $\Sigma_i$, the value of predicate functions $do_i(\cdot)$ can be predicted by removing all the services that requires agent $i$'s action from the set of services that are finished at $t$, which makes $\hat{q}_{t+1}$ available. $r_\rho^i$ is able to distinguish each agent's contribution to the global behavior, and thus provide more informative rewards to agents.
By incorporating $r^i_\rho$, the agents are encouraged to leave the current FSPA state. However, the limitation of $r^i_{\rho}$ is that it cannot distinguish whether a transition is beneficial or not. For example, in Fig. \ref{fig:FSPA}, the bad transitions that we want to avoid are colored by orange. As for state $q_5$, two agents have already reached the location of $\sigma_3$ and the next thing to do is to finish $\sigma_3$ cooperatively. However, if they leave the vertex where $\sigma_3$ lives, $r^i_\rho$ will generate positive rewards because the FSPA has left $q_5$, which clearly violates our intentions. It is true that the learning can still converge to the correct policies, but only through the effect of discount factor $\gamma < 1$, in the sense that transition from $q_5$ to $q_4$ will delay the time of completing the mission and thus yield a lower final reward.
To overcome this issue, we design another reshaped reward signal $r^i_J$, which can be seen as a "potential energy" of the corresponding FSPA state. Before giving the definition of $r^i_J$, we first introduce the concept of path length. Let $H(q,q')$ denote the set of all finite trajectories from $q \in Q$ to $q' \in Q$. The path length L is defined as
\begin{equation}
\vspace{-3pt}
\label{eqn:path length}
L(\textbf{q}) = \sum_{k=1}^{n-1}\omega_{\mathcal{A}}(q_k, q_{k+1}),
\vspace{-1pt}
\end{equation}
where $\textbf{q} = q_1\ldots q_n \in H(q_1, q_n)$ is a finite sequence of FSPA states connected by transitions in $\mathcal{E}$ and $\omega_{\mathcal{A}}: \mathcal{E} \to \mathbb{R}^+$ is a weight function. Then a distance function from $q$ to $q'$ can be defined as
\begin{equation}
\vspace{-2pt}
\label{eqn:distance_q}
d(q,q') = \begin{cases}
\min_{\textbf{q} \in H(q,q')} L(\textbf{q}) \qquad & \text{if}\; H(q,q') \neq \emptyset \\
\infty \qquad & \text{if}\; H(q,q') = \emptyset.
\end{cases}
\vspace{-2pt}
\end{equation}
Now an energy function $J(q)$, $q \in Q$ can be defined as
\begin{equation}
\vspace{-2pt}
\label{eqn:energy function}
J(q) = \begin{cases}
0 \qquad & \text{if}\; q \in F \\
\min_{q'\in F} d(q,q') & \text{if}\; q \not\in F.
\end{cases}
\vspace{-2pt}
\end{equation}
In words, the energy function $J$ of a state $q$ is the minimum weighted sum of transitions it takes to reach the set of final states. Finally, $r^i_J$ is constructed as follows:
\begin{equation}
\vspace{-2pt}
\label{eqn:rJ}
r^1_J = \cdots = r^n_J = J(q) - J(q_{t+1}).
\vspace{-2pt}
\end{equation}
$r^i_J$ ensures that agents will receive an assessment of every FSPA transition immediately. There is no standard procedure for determining the weight function $\omega_{\mathcal{A}}$ and hence it becomes a design factor.
\subsection{Solving the equivalent FSPA-POSG}\label{sec:Solving POSG}
We use the policy gradient method with policy graphs as in \cite{meuleau1999learning} and \cite{peshkin2001learning} to find the optimal policy for each agent in the FSPA-POSG. In the training phase, both the original reward and the two reshaped rewards require the information of the FSPA state, which is determined by all the agents. Hence, a centralized coordinator is necessary. We assume that the communication among all agents during the training phase is available, which is reasonable as stated in Sec. \ref{sec:intro}. After training, each agent obtains a policy that tracks its own history states to decide the action to be taken at each time step, i.e., the policies are distributed.
\begin{comment}
We use soft-max functions as the activation function over parameters $W^i_\psi, w^i_\eta$ and $ W^i_{\eta_0}$ of each agent $i$. For instance, in example \ref{example}, $W^1_\psi$ is a $N_1$ by $7$ matrix in which $N_1$ is the number of internal state of the policy graph of agent $1$. In this case, each parameter is interpreted as a "Q-value":
\begin{align}
\label{eqn:soft-max activation}
\psi^i(n,a) & = \frac{e^{W^i_{\psi}(n,a)/\theta}}{\sum_{a'\in A_i}e^{W^i_{\psi}(n,a')/\theta}} \\
\eta^i(n,o,n') & = \frac{e^{W^i_{\eta}(n,o,n')/\theta}}{\sum_{n''\in N_i}e^{W^i_{\eta}(n,o,n'')/\theta}}\\
\eta_0^i(o,n) & = \frac{e^{W^i_{\eta_0}(o,n)/\theta}}{\sum_{n'\in N_i}e^{W^i_{\eta_0}(o,n')/\theta}}
\end{align}
where $\theta \in \mathbb{R}$ is a temperature parameter. The update rule of each parameter is shown in the following:
\begin{align}
\label{eqn:policy gradient update}
\Delta T_k^\psi(t) & = \frac{\partial}{\partial w_k}\ln \psi(n_{t-1}, a_{t-1}), \\
\Delta T_k^\eta(t) & = \frac{\partial}{\partial w_k}\ln \eta(n^{t-1}, o^t, n^t),\\
\Delta w_k(t) & = -\alpha\left[\frac{\partial}{\partial w_k} r_t + r_t(T_k^\psi(t) + T_k^\eta(t))\right]
\end{align}
where $w_k$ is the parameter to be updated, $r_t$ is the reward received at time t and $\alpha$ is the learning rate. The terms $T^\psi_k$ and $T^\eta_k$ are initialized to be zero. Note that each parameter is updated for each time step in each episode.
\end{comment}
\section{Simulation}\label{sec:simulation}
To verify the efficacy of the method described in Sec. \ref{sec:solution}, we conducted $500$ simulations of training on Example \ref{example} with reshaped rewards $r^i_\rho$, $r^i_\rho + r^i_J$ and $0$ (only original reward), respectively. In each simulation, we trained the policies for $10000$ episodes. We set the discount factor $\gamma$ to $0.995$, the learning step size $\alpha$ to $0.01$, the constant C in Eqn. \eqref{eqn:original reward} to $1$. The number of internal states of the policy graphs are chosen to be 10 for both agents. In the process of training policy graphs, we utilized Boltzmann exploration with a temperature parameter of 1. As a comparison, we also applied decentralized Q-learning \cite{matignon2012independent} with reshaped rewards $r^i_\rho + r^i_J$ and an $\epsilon$-greedy exploration strategy to learn policies without memory.
It was not guaranteed that agents' policies would always converge to the optimal ones. The rates at which the optimal policies were learned within $10000$ episodes are shown in Table \ref{tb:1}. We can see that using $r_J^i+r_\rho^i$ boosted the convergence rate by $12.8\%$ compared to the original reward. Note that memoryless Q-learning completely fails the task. The averaged learning curves in which optimal policies successfully converged are shown in Fig. \ref{fig:L-curves}. We can see that the proposed reward shaping technique can accelerate the speed of learning and significantly reduce the variance.
The behaviors of agents governed by the learned policies are visualized in Fig. \ref{fig:route}. In example \ref{example}, there are two possible routes to finish the TLTL task: finish $\sigma_1$ then $\sigma_3$; finish $\sigma_2$ then $\sigma_3$. It can be seen that the agents are able to figure out the optimal policy with minimum route length.
We also inspected the final policy graphs and their corresponding trajectories in simulations where agents with reshaped rewards failed to satisfy the TLTL specification. It turns out that in the majority of the failures, the agents
would first finish either $\sigma_1$ or $\sigma_2$ and then never reach $\sigma_3$ by wondering among several vertices near $\sigma_1$ or $\sigma_2$. One possible cause is that the gradient ascent took steps that were too large, which cannot be fully handled by simply decreasing $\alpha$ \cite{schulman2017proximal}.
More advanced policy gradient methods such as PPO \cite{schulman2017proximal} or TRPO \cite{schulman2015trust} might solve this issue. We will explore this direction in future research.
\begin{figure}
\centering
\begin{minipage}[t]{0.55\textwidth}
\centering
\includegraphics[height=4cm]{Fig/l-curves.png}
\caption{\small The averaged learning curves of the three rewards. The values on the $y$-axis represent the normalized original reward signal in Eqn. \eqref{eqn:original reward}.
\label{fig:L-curves}
\end{minipage}
\quad
\begin{minipage}[t]{0.41\textwidth}
\centering
\includegraphics[height=4cm]{Fig/route.PNG}
\caption{\small Final trajectories generated by policy graphs after training. }
\label{fig:route}
\end{minipage}
\vspace{-10pt}
\end{figure}
\begin{table}
\vspace{-10pt}
\centering
\setlength{\belowcaptionskip}{10pt}
\caption{Rates of convergence to the optimal policies with different rewards}
\label{tb:1}
\begin{tabular}{|l|c|c|c|c|}
\hline
Reshaped rewards& $r^i_\rho$ & $r^i_\rho+r^i_J$ & $0$ & $r^i_\rho+r^i_J$ (memoryless) \\
\hline
Optimal policy convergence rate & $68.4\%$& $83.0\%$ & $70.2\%$ & $0\%$\\
\hline
\end{tabular}
\vspace{-10pt}
\end{table}
\section{Conclusion} \label{sec:discussion}
In this paper, we applied MARL to synthesize distributed controls for a heterogeneous team of agents from Truncated Linear Temporal Logic (TLTL) specifications. We assumed a partially observable environment, where each agent's policy depends on the history of its own states. We used the TLTL robustness as the reward of the MARL and introduce two additional reshaped rewards. Simulation results demonstrated the efficacy of our framework and showed that reward shaping significantly improves the convergence rate to the optimal policy and the learning speed. Future work includes consideration of non-deterministic SGs and other policy gradient methods.
\acks{This work was supported by NSF under Grant IIS-2024606 and Grant OIA-2020983.}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,607
|
package mcl;
import java.io.*;
import javax.servlet.*;
import javax.servlet.http.*;
import lotus.domino.*;
import sun.misc.BASE64Decoder;
import org.jruby.embed.*;
public class RubyInterpreter extends HttpServlet {
private static final long serialVersionUID = 2229617989934548785L;
private ScriptingContainer container;
public void init(ServletConfig config) throws ServletException {
super.init(config);
container = new ScriptingContainer(LocalContextScope.THREADSAFE);
}
public void doGet(HttpServletRequest req, HttpServletResponse res) throws ServletException, IOException {
PrintWriter out = res.getWriter();
try {
NotesThread.sinitThread();
// Create a session using the currently-authenticated user
Session session = NotesFactory.createSession(null, req);
// This servlet is intended to be used with NSF-based .rb file resources
String dominoURI = "notes://" + req.getRequestURI() + "?OpenFileResource";
Base obj = session.resolve(dominoURI);
if(obj instanceof Form) {
// File resources are presented as Form objects, the worst kind
Form formObj = (Form)obj;
Database database = formObj.getParent();
// Find the Form's UNID via its URL and use that to get the file resource as a document
String universalID = strRightBack(strLeftBack(formObj.getURL(), "?"), "/");
Document fileResource = database.getDocumentByUNID(universalID);
String dxl = fileResource.generateXML();
String rubyCode = new String(new BASE64Decoder().decodeBuffer(strLeft(strRight(dxl, "<filedata>\n"), "\n</filedata>")));
// Add some important environment variables as constants
container.put("$session", session);
container.put("$database", database);
container.put("$request", req);
container.put("$response", res);
// Set the writer to be the HTTP output
container.setWriter(out);
// Execute the code
container.runScriptlet(rubyCode);
}
obj.recycle();
session.recycle();
} catch(Exception ne) {
out.println();
out.println("=======================");
ne.printStackTrace(out);
} finally {
NotesThread.stermThread();
}
}
private String strLeft(String input, String delimiter) {
return input.substring(0, input.indexOf(delimiter));
}
private String strRight(String input, String delimiter) {
return input.substring(input.indexOf(delimiter) + delimiter.length());
}
private String strLeftBack(String input, String delimiter) {
return input.substring(0, input.lastIndexOf(delimiter));
}
private String strRightBack(String input, String delimiter) {
return input.substring(input.lastIndexOf(delimiter) + delimiter.length());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,857
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{"url":"https:\/\/chat.stackexchange.com\/transcript\/11?m=53380855","text":"12:02 AM\n@Someone_Evil chuckle\n\n@Yuuki Wait, Zathura is a Jumanji sequel? Not just a Jumanji ripoff?\n\n@Adeptus Wait, Zathura is something other than a standalone movie?\n\nI had no idea Zathura was even a movie until yesterday so you all are ahead of the game\n\nWait what is Zathura? XP\n\nHello.\n\n12:17 AM\nZathura was a spiritual sequel to Jumanji, by the same author.\n\nHey, if a bug bears attacks with a whip, range is 15ft right?\n\n@RandomDudeWithAKnife (5e) Yup\n\nChris Van Allsberg was a prolific and creative author\/illustrator. He also wrote The Polar Express.\n\nFair enough\n\nHe specialized in quirky and slightly unsettling 32-page high-concept fantastical-realism books filled with beautiful and bizarre graphite illustrations.\nHe's also the author of The Mysteries of Harris Burdick and Bad Day at Riverbend.\n\n12:25 AM\npicks up knife\n\n@BESW whoa TIL\n\n@Rubiksmoose I do recommend the original Polar Express book! And even more, Mysteries of Harris Burdick and Bad Day at Riverbend.\n\n@BESW I'll have to add them to my list :)\n\nIf you like jumanji and zathura you should check out a book I think by william sleator called interstellar pig.\n@RandomDudeWithAKnife narrows eyes slowly picks up wooden pot lid for shield\n\n12:43 AM\nSo I'm about to take the Jeopardy online test lol\nFun random fact\nGood thing I don't know monarchs or state capitals or presidents!\n\nhey there @RandomDudeWithAKnife, welcome to the RPG.SE lair :)\n\n1:03 AM\n7\n\nI like a lot D&D vampires. I even bought Curse of Strahd. But something I always wandered is \"What is running water?\" Rivers are running waters, since it is used as an example in others editions. But how about rain (no answers are accepted on the related question)? How about a bottle? How about...\n\n9\n\nWar Magic PHB pg. 75 states: When you use your action to cast a cantrip, you can make one weapon attack as a bonus action. Counterspell states: You attempt to interrupt a creature in the process of casting a spell. If the creature is casting a spell of 3rd level or lower, its spell fail...\n\n4\n\nI've got a wizard with 11 Dex and I'm trying to figure out what my most efficient use of spell slots is to help me from getting hit. We typically have 2-4 very difficult encounters per day and I want to balance my spell slot uses where I can provide as much firepower as needed. The monsters are...\n\n11\n\nThe Warlock Invocation \"Improved Pact Weapon\" from Xanathar's Guide allows the warlock to summon a ranged weapon: Finally, the weapon you conjure can be a shortbow, longbow, light crossbow, or heavy crossbow. Does this include the associated ammunition (arrows or bolts)? Or will the warloc...\n\n@Rubiksmoose They are singular bowed structures, liquid, gas or solid letters, or treefolk which existed before to the Civilisation games, respectively\nAnd I guess it was time to refresh our HNQ list\n\n1:15 AM\n@Someone_Evil Good thing you told me that it helped a lot ;)\n\nBus update: Well, the union and the region reached a tentative bus deal. So now they vote to ratify in 2 days, and then the regional council meets to consider it if the vote passes, then if all that works, we get a date of when buses will be back.\nStill too many points of failure for my liking but...progress\n\n1:37 AM\nAlso a distressing amount of time for any side involved to change their minds\n\n3 hours later\u2026\n4:08 AM\n14\n\nI recently started playing with a large group of about 7-8 people. I noticed that, in our group, a couple of the people are very active: whenever there is a chance for the group to do things, it only takes them a second or two to come up with an action for their character and narrate it. On top o...\n\n4:18 AM\n@nitsua60 hi pal :-)\n\n5:13 AM\nhmm. Isn't twitter supposed to onebox?\n(never mind... it was here that I saw it first anyway...)\n\n@Adeptus it's supposed to but it stopped about a year or two ago\n\n6:00 AM\nIf a question \/should\/ be a duplicate of another, but the later's answer doesn't actually cover the situation in the first, should votes to close still be cast?\n\nThis question about Arcane Trickster spell slots: https:\/\/rpg.stackexchange.com\/questions\/163807\/how-do-spells-known-work-in-multiclassing-in-5e\n\nwas marked as a duplicate of this one about multiclassing spell slots: https:\/\/rpg.stackexchange.com\/questions\/151000\/if-i-multiclass-into-2-or-more-spellcasting-classes-how-do-i-determine-my-known\n\n6:49 AM\nIs this pinned anywhere? It's pretty useful for sketching out tables. isaurssaurav.github.io\/mathjax-table-generator\n5\n\n1 hour later\u2026\n7:57 AM\n@Himitsu_no_Yami Are you proficient with shields? (If so, how?)\n@BradleyLindsey Oh man, I read that ages ago. I recall it being very weird but good. I think I had a few dreams inspired by it.\n@Ash Good luck!\n@Adeptus It's been broken for ages.\n@jgn ...Neither of those questions is about spell slots. (I should know, I wrote the second question and the answer on it. :P) That's also why I rolled back your edit to my answer.\nThe first question you linked specifically says:\n> I know how spell slots work, to a degree, when multiclassing, but I'd like a better understanding of \"spells known\".\n@Glazius Oooh. I'm sure we have a meta or two about MathJax tables... It'd be good to have that link somewhere more permanent and easily searchable than chat.\nI suppose the original \"MathJax is live\" post isn't exactly a typical Q&A, so this might be suited to an answer on it:\n25\n\nWe have MathJax In this meta we discussed, requested, and gathered evidence for the utility of MathJax in RPG.SE posts. Now it's time to use MathJax For those familiar with LaTeX it will likely suffice to say that \\$... \\$ are our delimiters ($$...$$ for equations centered on their own lin...\n\nAlternately, it might be worth asking and self-answering a meta post specifically about how to format tables using MathJax, maybe including both an explanation of the MathJax syntax relevant to formatting such tables manually and a link to that page for auto-generating such tables.\n(Alongside your description of the linked resource, it might also be worth suggesting manually cleaning up the formatting to make it more readable and easier to maintain - by which I mean things like hitting enter at the end of each row of the table to mark where each row begins.)\n\n8:22 AM\n@V2Blast I read the last part of the question as asking about spell slots \"do I prepare those, and then cast from my Arcane Trickster spell lists as normal and just keep track of which is which\" casting spells is about spell slots to me. Your answer is about 75% about spell slots, so it may be worth covering the remaining 2 classes, especially since the only other question about AT\/EK spell slots isn't very broad in scope.\n\n8:40 AM\nWell, you've misunderstood the Arcane Trickster question, then. It's entirely about spells known. The querent's talking about whether they need to keep track of cantrips and known\/prepared spells separately for each class. At no point does their question relate to spell slots aside from saying it's a thing they already understand.\nAnd my answer to the spells known\/prepared question is also entirely about spells known\/prepared; the only time I mention spell slots are when I reference each class's spellcasting rules (which state that your spells known\/prepared for the class must be of a level for which you have spell slots), and when I quote the Pact Magic multiclassing rule (which is silent about spells known\/prepared).\nThere could be a separate question about determining spell slots for multiclassed casters, but that one ain't it.\n(I don't think the issue of how to determine spell slots has really come up much, so there hasn't been a need for a canonical question addressing that.)\n\n2 hours later\u2026\n10:22 AM\nWould this chat be ok with dnd5e character build questions? (i.e. something like \"are 3 levels of rogue worth it for an archery ranger that wants to increase their damage output?)\n\nAug 7 '18 at 20:02, by doppelgreener\nAt least for this chat, going by its history, the most off-topic thing anyone's probably asked in here was a tabletop RPG question.\n\n:D well then, here it goes :D\n\nWe certainly can, though if well structured that should also work on main which would let you get more complete (\/elaborate) answers\n\nlet's see then, I'll post it here and if it turns out to be site-worthy I'll migrate it\n\nWhat are the other levels and subclasses you had in mind?\n\n10:26 AM\nI'm building a level 4 archery-based character with a spy background. I'm going for 4 levels of Ranger as a start, and will certainly take the 5th level too for the extra attack. At that point, level 6 is notoriously bad so I was thinking to take a dip into Rogue for the sneak attack and cunning action. At that point, the 3rd level would give me access to the assassin subclass and an extra sneak attack dice, so it seems worthy at a glance.\nI have to pick ranger over fighter because the party is very small and having another character able to cast cure wounds is required - I'm willing to change class if I can get a semi-reliable access to cures and fighting primarily with a bow\ntl;dr is a ranger 5\/ Rogue 3 better than a Ranger 8 in terms of damage output?\nif yes, what would be the ideal level progression?\n\nWhich Ranger archetype, if you had one in mind?\n\nhunter\nmostly because I know that we tend to face groups more often than single big baddies, and they tend to group up in front of our front-line cavalier fighter\nbut I'm not set in stone with that\n\nHmm... so which Hunter's features would you be looking at?\n\nHorde Breaker: Once on each of your turns when you make a weapon Attack, you can make another Attack with the same weapon against a different creature that is within 5 feet of the original target and within range of your weapon.\nthat's accessible at third level\n\nHorde breaker is better for hordes (I think), but Giant Killer works really well with sneak attack, but requires big enemies\n7th level doesn't affect damage output so I guess it's not important for the question\n\n10:39 AM\nIt's not, unless something like \"Multiattack Defense: When a creature hits you with an Attack, you gain a +4 bonus to AC against all subsequent attacks made by that creature for the rest of the turn.\" is so major to be a dealbreaker for a character, from your perspective\nI can't guarantee that we're going to face many Large enemies (some, for sure) but I aim not to be attacked in the first place, if I can avoid it, so Giant Killer might be off :)\nalso, as an archer I'd rather not be within 5 feet of a Large+ big bad enemy\n\n10:51 AM\n\n11:02 AM\nimho assassin is not a good class\nat least not for a combat focused character\nthe special abilities depend too heavily on you definitely being able to go first and enemies being surprised, which is difficult to engineer for most parties I have played with\nfor a rogue dip I'd probably suggest taking Scout as the archetype, the Skirmish ability to just nope away from any enemy that gets close to you as a reaction is very good for keeping your distance\n\nbut at that point, is it worth it to get to level 3 rather than say just level 1 for the sneak attack bonus?\n\nwell, level 3 will get you more sneak attack bonus.\n\n11:21 AM\nTrue. But rather than scout I'd probably go Mastermind, to help out the frontline tank, if I don't go for assassin. That would still increase damage output, even if very indirectly.\n\n11:56 AM\nYour goal is an archer with some healing in their back pocket, have you considered just picking up the healer feat?\n\nshould I change something in my campaign if a level 4 monk averages 64 damage\/round\n\nHow are they doing that, and are you sure?\n\nyes, I did the math, and, somehow got it to work\n\n@Someone_Evil nice tip, thanks, i hadn't considered it\n\n*They got it to work\nshould I ban tiefling monks, (thats thier race)\nbtw, its higher when they use flurry of blows\n\n12:03 PM\nIs the race that relevant for the damage? What are they using?\n\nyou're going to have to explain how you worked out that average damage because that number is excessive\nquick back of the envelope, an L4 monk does 39 damage if it has a +5 ability mod and flurry of blows and crits for max damage on all its attacks\n(assuming martial arts damage die, you can eke out a couple more points by making the initial attack with a better weapon)\n\ncough cough greataxe cough cough\n\nMonk subclass?\nBut I still think you should show the whole working out, because I'm still not seeing how you get to 64\n\n12:19 PM\na greataxe is not a monk weapon\na monk who attacks with a greataxe doesn't get to do flurry of blows\n\nnor can they do the unarmed bonus action attack from martial arts\n\nin complete honesty it seems more likely you guys have misunderstood some rules somewhere to get those numbers because they're dramatically higher than we would expect, so if you go through your working we can see if it's right or not.\n\nI concur that a full explanation of how they are doing this would be better\n\nsunsoul\n\n@Someone_Evil When I did this one it was fighter (for Archery and precision BM dice), rogue (for bonus-action Hide and SA dice), and Magic Initiate for healing word.\n\n12:27 PM\n64 is rather low compared to what a wizard can do a first level :(\n\n@RandomDudeWithAKnife You're talking about D&D 5e, right? If so, this does not ring true to me. 64 is insane\n\na 1st-level wizard might be able to do 64 damage with one spell if they manage to drop a burning hands on a perfect cluster of enemies but they're hardly doing that as average damage\nregardless there's not much point in us speculating\n\n@nitsua60 Yeah, I thought about suggesting MI for healing word or cure wounds, the problem is you only have one per long rest, and if you need the second one, you might really need it\n\n12:50 PM\nif I shot off my bow 5 times in combat, and afterwords searched the field for ammo, and I can't half 1\/2 of an arrow, how many arrows are retrieved?\n\n@RandomDudeWithAKnife Unless otherwise stated round down so 2\n\nCan I search for arrows I didn't fire. aka goblin shoots his be 6 times. can I search for his arrows, for a total of 5?\n\nUp to the DM whether any equipment from monsters are usable\n\nAnd arrow recovery is still 1\/2, isn't it?\n\nDM. You see a goblin dying and a spear sticking out of his chest. Rouge. I take the spear out of him and pocket it.\n\n12:57 PM\n@Someone_Evil my dead wizard grabbed MI for healing word .\n\n@RandomDudeWithAKnife Does \"it\" refer to the spear, or the goblin? :p\n\n@Someone_Evil rogues aren't allowed to keep pets lol\nthats rangers\n\n@NautArch I'm by no means saying MI for a healing spell isn't good (and having it as a BA is really good), I'm saying there's an argument for taking healer instead. Comes a bit down to threat modelling, I suppose\n@RandomDudeWithAKnife I'm not sure a dead goblin qualifies as a pet. I think a rock is better for that end\n\n@Someone_Evil yeah, for me it was wanting to bring an ally back into the fight from distance.\n\n2:01 PM\n@STTLCU are you looking by for a high damage Archer build?\n\n@NautArch yes, with some restriction such as being able to heal, and tailored for levels 5-10 as I doubt we'll ever get past that\n\n@STTLCU Being able to heal yourself, or others? Is that a hard requirement?\n(hard as in strict, not as in \"difficult to achieve\". I mean if you play with feats you could always take the Healer feat...)\n\nOthers primarily, and yes. The other party members are a Cavalier Fighter and a Lore bard, so having another source of magical\/special healing is required. Potions aren't extremely common for us (as of yet)\n\nMorning everyone\n\nHm, I don't know what kind of a campaign you're planning for, but the Lore Bard could have Healing Word and that'd be already quite good\n\n2:08 PM\nyes, but their slots are limited and with a reduced party size we tend to be in need of many heals between long rests\n\nHealers are generally not very important in the normal course of action in DnD 5e except as exception-handlers (reviving KO'd characters)\nOf course your mileage may vary, esp. if the campaign has a somewhat less common threat profile. If the bulk of the damage comes from environmental hazards you can't fireball your way through, healing certainly beats offensive magic and bashing :)\n\nIt's mostly a matter of basic redundancy. With a half-healer only in the party, we'd need someone to be able to revive them if they go down for any reason\n\nI can get the redundancy part, but I wouldn't consider a bard a half-healer\n\nthey would rather cast their spells for something else rather than healing, though\nif you bring a bard for healing you'd be better of with a cleric, no?\n\nNot that much?\n\n2:13 PM\nwouldn't you be losing out on all the other interesting spells that a bard has?\nI am no expert of 5th ed, by any means, but that sounds odd at a first glance\n\n@kviiri Neither would I really (jumping in with little context though)\n\n@STTLCU 5e doesn't really support the \"designated healer\" trope where a character spends a lot of their daily resources doing healing. (Of course you can do it, but that's seldom even close to optimal)\n\n@STTLCU hmm, you're going to have to make some compromises, because high ranged weapon damage and good healing may not be something you can totally realize together.\nAnd bards are definitely capable of being full healers. WIth the spell slots, healing word, and cure wounds, they're good at battlefield healing to keep allies up and fighting.\n\n@NautArch I'm aware of that, I'm looking for the best I can do with the hand I'm dealt :)\n\nAnd the short rest song of rest extra hit die is also no joke. Combine with Inspiring Leader feat, and they're a great healer.\n\n2:17 PM\nBut anyway, I actually have a suggestion! Check out the Healer feat. It saved the butts of our entire party in a particularly nasty session.\n\nyes, I got that suggestion and it seems quite great, in fact\n\n@STTLCU well...it's not hte hand you're dealt. It's what you want :) WHat are your stats?\n\nIn particular, it allows a no-roll heal to 1 hp, at the cost of an action but that's already enough to get your main healer back online if needed.\n\n@kviiri i'm actually considering that feat for my wizard at my next ASI\n\nstr 12, dex 18, con 14, int 12, wis 16, cha 13 with a wood elf\nI was planning for an ASI to 20 dex\nbecause it's just too good for the bonuses it gives\n\n2:20 PM\nOr just have everyone be able to off heal.\n\nThe game I'm currently running has a celestial warlock, a ranger, (battle) cleric, paladin, and druid.\n\n@STTLCU check out this build. It's more about ranged damage, but the cleric dip gives you some healing. But the spell slots are really more about bless for yourself to increase your damage chances.\n\nBut yeah, to rephrase what I tried to say earlier: there is really no way for healing magic to keep up with the expected damage outputs of various mobs (even before accounting for the fact that healing consumes daily resources in almost all cases, while competing attacks are often free). Therefore the best places to concentrate healing on are reversing KO's (or preventing imminent ones), although the option to convert slots to HP over a non-combat stretch can be handy at times.\n\nI agree with @kviiri, @STTLCU. Healing in-combat is really a last resort thing to bring people back into the fight to kill the enemy. It's a short-term solution. Almost always, the better choice is offense over healing. Kill the thing that's damaging you and you won't be damaged anymore.\n\nyes that's a valid point\nSo you're all saying I should make a Fighter Archer (I'd assume BattleMaster) with maybe or maybe not a dip into rogue, and maybe keep a potion or two for emergency revives?\n\n@STTLCU for pure archery options, fighter is generally the best. But hunter rangers deal some good consistent damage with some other spell control options, too.\nI personally don't love the rogue dip, i'm not sure it's worth it.\nBut going sharpshooter for the -5\/+10 and having way like bless to mitigate the -5 is the path to high damage.\n\n2:36 PM\ncunning action and an extra damage die per turn seems juicy though\n> i'm not sure it's worth it.\nAlso I'm open to change my mind :D\n\n@STTLCU would you rather have an extra d6, or a +10?\n\nhmmm\nthe d6 would apply when I hit but the +10 might end up missing\n\nnot going to deny cunning action is good, but you also need to dip two levels into rogue for that. delaying your extra attack\ndelaying action surge\n\n@NautArch we all know what the +10 is, but the d6 could be anything! it could be +6!\n\n:D\n\n2:39 PM\n@goodguy5 i'll take that door!\n\nI would miss out on horde breaker though. That seems quite strong in our current setting\n\nI think I'm referencing Family guy, or maybe the Simpsons.\n\nSomehow gives me The Simpsons vibes\n\n@goodguy5 i think it's family guy. it's the boat?\n\n@kviiri There's a lot of thematic overlap\nI think Homer loves his kids more, and I THINK Peter is a little smarter?\n\n2:42 PM\n@STTLCU well, wanting area effect stuff is another lever you don't currently have. Most martials don't get that.\n\nOh right now I remember. In that one episode Mr. Burns' power plant was under threat of closure because of a safety inspection and Burns offered to settle for a washer-dryer or something like that, \"or you could trade it for this mystery box!\"\nTo think i have neurons dedicated to this\n\n@STTLCU And horde breaker works when you fight mobs, but colossus slayer works on every damaged creature.\n\n@goodguy5 That'd probably depend on the rough time frame we're talking about. Homer changed a lot over the years\nAlthough Homer's flanderization hasn't been the fastest or most obvious in the franchise\n\nthat's true\n\nbrb gotta catch a bus\n\n2:51 PM\nHow big is their mitt?\n\n@goodguy5 their mitt is about the size of an ant\n\nMakes catching the bus seem unlikely\n\nTaking the missus out for some delicious Ethiopian food today\n\noooh. hands on!\n\n(As in, a date. Not a trade. Not a trade in the other sense either. And I think that covers all the obvious misinterpretations of that sentence)\n\n3:00 PM\n@goodguy5 good call on offering the group rolling option. I think you should either submit it as an answer or do a related link.\n\n(I guess a purposefully obtuse person could ask if you intended to kill your wife in exchange for ethiopian food)\n\n@goodguy5 I sure hope so, an Ethiopian coworker from a few years back used to complain some Ethiopian restaurant in Helsinki charges extra for the bread so most people ate with a fork and a knife!\n\n@NautArch I asked a comment first, confirming that's acceptable\n\n@goodguy5 I personally think it is, but it's not my question :)\n\nThough, I'm not sure I said that anywhere, I was just thinking it....\n\n3:01 PM\no\/\n\nEthiopian cuisine has this delicious bread called injera, a bit like our sour breads up here but a tad more fruity in aroma and softer in consistency. Very nice stuff\n\nlike a soughdough pita or some such?\nregardless, sounds great\nanyway, either prerecord a bunch of rolls and then just go down the line\n\n@goodguy5 oh, i thought you were referecing something around Nits' group save stuff.\n\noh no\nThough the DMG also offers this:\nd20 Roll Needed Attackers Needed for One to Hit\n1-5 1\n6-12 2\n13-14 3\n15-16 4\n17-18 5\n19-19 10\n20-20 20\n\n@goodguy5 Hm, I guess it does resemble pita a bit yes\n\n3:06 PM\n@goodguy5 where is that?\n\nDMG p.250\noh, and there's a calculator on slyflourish\nhttps:\/\/slyflourish.com\/mob_calculator.html\n\n@goodguy5 doy ou want to put up an answer with this or can i include it in mine?\n\nI'll put one up in a bit. I've got to have this meeting\n\nare you doing things outside of that bit, or is the whole answer it? Was working on an update to mine.\nBut also realizing detailing the whole system may not be appropriate?\n\n@NautArch Hmm... I'm tempted to give my ol' \"figure the expected number of hits +d6 - d6\" answer. But that's a bunch of time I don't really have today =\\\n\n3:18 PM\nbut yeah, that's what I was thinking of.\n\nOh, yes--I see that you-all mention it later on.\nFeel free to cannibalize that if it fits in somewhere--I'd be glad to see it recommended to that GM, and don't feel like doing it myself =)\n\ni think the related link works\n\n@NautArch It was good, thanks. Squeezed in a run, had a double-length department meeting, memorial service, rehearsal, and watched a show with my wife. Solid.\n@NautArch It feels a little like an answer-in-comment, but if you say it's good, I'll leave it.\n\nyeah, i'm not sure, either. I mean, it is related, tho :)\nbut it's borderline, too as it can 100% be an answer.\ninterestingly, @goodguy5, it looks like RyanCThompson was also thinking about handling mods. THey've got a deleted answer with that headline.\n\n3:34 PM\ndone\n\n@goodguy5 That was a pretty fast meeting\n\nit was a 1:1 and it was cut short because of technical difficulties. I'm pretty sure the VPN does a DNS swap around this time of day or something\n\n@goodguy5 aaah! acronyms!!!\n\n@goodguy5 I'm not sure about prerolling my own character's (or summons) rolls. Don't love that suggestion.\nand how do you handle multiple creatures against multiple enemies? How do you pick which ones hit?\nOr is it that you just say 4 enemies against creature X, and then say 2 hit?\n\n@NautArch DM prerolls\nyes, group them by attackee\n\n3:44 PM\n@goodguy5 so you've taken agency of the summons away from the players?\n\nI mean.... they still have the agency, you've just taken rolling away from them\nI clarified in post\nbut, if it's detracting from the game for a player to roll 25 times, then it's completely fair, imo, to remove that aspect of the game\nanyway, need to drive into work. My remote morning is at an end.\n\n3:56 PM\n10\n\nThis is effectively the reverse of How can I end combat quickly when the outcome is inevitable? Lame duck scenarios in game design occur when a player cannot win, but the game isn't over yet. They're obviously undesirable, and good game design would seek to minimize it. I'm interested to see ho...\n\nI wonder if D&D villains would be scarier as acronyms:\n\n- TtDQ\n- DPoD\n- SVZ\n- AtLotN\n- JtFL\n- XtE\n4\n- KtWtW\n10 points to whoever guesses what these are xD\n\n1- Tiamat the Dragon Queen 3- Strahd Von Zarovich. All I got","date":"2020-03-29 13:30:32","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\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3915252983570099, \"perplexity\": 3890.956633625978}, \"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-2020-16\/segments\/1585370494331.42\/warc\/CC-MAIN-20200329105248-20200329135248-00200.warc.gz\"}"}
| null | null |
{"url":"https:\/\/math.meta.stackexchange.com\/questions\/9383\/is-it-possible-to-change-enlarge-font","text":"Is it possible to change\/enlarge font?\n\nLatex is very hard to read especially with complicated expressions that involve powers etc.\n\nIs it possible to change font to Modern Computer and make it larger?\n\nRight now I have to zoom in to read some mathematical expressions even though my monitor has a very dense resolution.\n\n1 Answer\n\n$$e^{\\int_0^1 \\log(1-x^2)\\,dx}$$\n\nRight click on the math, go to Math Settings > Scale All Math... and enter your favorite number bigger than 100.\n\n\u2022 For some reason this isn't working on meta for me but it does work on the main site. I use Chrome. \u2013\u00a0Antonio Vargas May 6 '13 at 14:24\n\u2022 You can also define a Zoom Trigger which only actives the math zoom when the trigger is set. \u2013\u00a0Asaf Karagila May 6 '13 at 14:24","date":"2021-05-12 11:55:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 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\": 0.5676351189613342, \"perplexity\": 1171.9612627444333}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989693.19\/warc\/CC-MAIN-20210512100748-20210512130748-00542.warc.gz\"}"}
| null | null |
<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
<feed><tipo>Rua</tipo><logradouro>Jaguara</logradouro><bairro>Fundão</bairro><cidade>Recife</cidade><uf>PE</uf><cep>52221080</cep></feed>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,149
|
Q: Way to express charge i would like to ask a question regarding electric charge. I see that charge has 3 units
*
*Elementary charge
*Coulomb
*Esu
For example
A proton has a charge of $+1e$ or of $+1.602 * 10^{-19}$ Coulombs..
In terms of the elementary charge unit, I sometimes I see people write
A proton has a charge of $+1$ instead of $+1e$. Is there any name for the kind of expression or it's just about habit's stuff ?
I apologize if i have violated a rule. I also ask this question in chemistry section because I don't know if this explanation is clear in either of the two subjects..
A: The esu (electrostatic unit) is an old unit of charge which preceded the SI system of units.
The system had the centimetre, the gramme and the second as the base units hence the name cigs units.
In this system of units Coulomb's law was $F = \frac{q_1q_2}{r^2}$ with the charge measured in esu or statcoulomb or franklin.
The whole topic of conversion between esu and SI units is a minefield because there was another set of units for magnetism called emu.
There is information about the conversions factors here.
What you have called elementary charge is really relative charge and can be compared with quoting the masses of atoms in atomic mass units.
Masses of atoms are compared to the mass of a twelfth of a Carbon-12 atom.
Charges are compared to the charge on an electron.
So a charge of $+2$ means that the particle is positively charged and has a charge whose magnitude is twice that of the charge on the electron.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,233
|
Q: Increase or View all Stored History in Gedit I would like to be able to see the last ~200 files opened in Gedit. I was able to increase the number it shows to 40 using this command:
gconftool-2 --type int --set /apps/gedit-2/preferences/ui/recents/max_recents 40
but it is only showing me the last 23. I looked in .recently-used.xbel and found that it has only stored the last 23 opened gedit files, as well as files from other applications. Is there a way to increase this? I accept that the last ~200 opened in gedit are no longer stored, but I would like to store them in the future so I can view them. Essentially, can I tell ubuntu to keep ~10X more history in that recently-used.xbel file?
A: Inspired by http://www.richud.com/wiki/Ubuntu_gedit_recent_files_dconf
Make usage of dconf-editor to change the property 'max-recents'.
Under org > gnome > gedit > preferences > ui:
Change the value of max-recents.
It was tested on ubuntu 12.0.4 and 14.04 LTS 64 bits.
In the 2 following screen dumps you can find:
*
*Where it was configured in dconf-editor
*An example that it works perfectly.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,976
|
import { compose, withHandlers, withState } from "recompose"
import { connect } from "react-redux"
import Create from "./create"
import { createShow } from "../../actions/shows"
const minLengthForCreate = 2
export default compose(
withState("name", "updateName", props => props.name),
connect(null, dispatch => ({
create: item => dispatch(createShow(item)),
})),
withHandlers({
updateName: props => event => {
props.updateName(event.target.value)
if (event.key === "Enter" && props.name && props.name.length > minLengthForCreate) {
props.create({
name: props.name,
})
props.updateName("")
}
},
})
)(Create)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,518
|
Romansk indgår i flere betydninger:
Romansk stil – en stilart indenfor arkitekturen
Romanske sprog – en sprogstamme, indeholdende fx fransk, italiensk og rumænsk
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,915
|
New Weapons Against Inflammation & Arthritis
by Aviva Natural Health Solutions November 29, 2006
Neil E. Levin
Why would we need something new in our arsenal against inflammation and arthritis? After all, people use aspirin and other Non-Steroidal Anti-Inflammatory Drugs (NSAIDs) every day, including ibuprofen and acetaminophen.
Are the new class of drugs called COX-2 Inhibitors safe and effective?
Can certain dietary supplements be useful in place of these drugs? What is the comparable effectiveness and safety of NSAIDs, COX-2 Inhibitor drugs and dietary supplements?
There are about 16,000 deaths a year from NSAIDs, and 100,000 people hospitalized with serious complications. NSAIDs are blamed for over half of all liver failures in this country. These serious side effects have caused a demand for COX-2 Inhibitor drugs.
Gideon Bosker, MD, Assistant Clinical Professor at Yale University School of Medicine, reports on the use of NSAIDs for Osteoarthritis (OA) and Rheumatoid Arthritis (RA): "As every primary care practitioner knows, NSAID-associated GI toxicity has become a public health problem, especially among older patients with OA and RA. Gastrointestinal intolerance has been reported in up to 50% of patients on long-term NSAIDs."
Two recent large studies, called CLASS and VIGOR, looked at the relative safety of NSAIDs versus COX-2 drugs. NSAIDs were shown to be associated with significantly more upper G.I. tract complications, including ulcers and bleeding. Partly because of such studies, COX-2 drugs have become a major success story for pharmaceutical companies over the past few years, becoming a multi-billion dollar a year business.
Research published in the British Medical Journal found that 21% of adults with asthma are sensitive to aspirin. Aspirin may trigger a deadly reaction, as may ibuprofen, diclofenac and naproxen. The doctors recommend new warning labels on all products containing these drugs.
COX-2 INHIBITOR DRUGS
The newest darlings of the medical community, COX-2 Inhibitors help to prevent inflammation from developing by blocking the action of a certain chemical called COX-2. These drugs are noted for reducing "risk of clinically important GI events" by some 50-60% versus NSAIDs. However, most of the COX-2 Inhibitor drugs are also associated with an increased risk of cardiovascular problems like MI. And there are still a goodly number of GI complaints in the COX-2 groups.
There are dietary supplements that may help control the inflammation associated with osteoarthritis.
SAMe (S-Adenosylmethionine) has been studied for depression, arthritis, and a host of other ills. Pronounced "Samm-ee", this substance was deemed effective enough to be studied in comparison to the COX-2 Inhibitor drug celecoxib (Celebrex), reportedly the least dangerous COX-2 drug in terms of cardiovascular risks.
In this study 61 patients were enrolled in a randomized, double-blind, cross-over trial over a 4-month period. The researchers found that "SAMe is equivalent in almost all measures to COX-2 inhibitors (celecoxib) in relieving pain and improving function in subjects with osteoarthritis of the knee." Their functional parameters included depression, pain, impairment of physical activity and knee mobility and strength.
The anti-inflammatory effects of aspirin and other drugs can also be achieved with concentrated blends of spices and herbs that have a wide range of benefits. These formulas also block the COX-2 enzyme, which triggers inflammation in tissues as a response to chemical signals. The herbs also serve as antioxidants and anticoagulants (blood thinners).
One formula, Zyflamend, has been shown to be helpful for inflammation and joint health, and also promotes normal cell growth (preventing abnormal growth). Zyflamend is being studied at Columbia University to show how it fights prostate cancer cells. It contains highly concentrated common spices like ginger and turmeric, which have been naturally extracted to contain the full spectrum of therapeutic chemicals in the plants.
The tremendous safety difference between dietary supplements and drugs is staggering: over a hundred thousand deaths a year from drugs versus a handful from all dietary supplements, which are far safer than any other category of food. Our risk of dying from eating dinner is far greater than from taking any dietary supplement.
Recent claims of 155 deaths "linked" to dietary supplements is nonsense, as clearly stated in the conclusions of the Rand Corporation report that was cited. Those are possible links, not proven. Almost all anecdotal reports evaporate quickly under real scrutiny by experts at the FDA, the Centers for Disease Control & Prevention and the American Association of Poison Control Centers.
Health Disclaimer. Content provided by NOW Foods. Copyright ©2006-2018. Published with permission. Neil E. Levin CCCN, DANLA is a certified clinical nutritionist and is a professional member of the International & American Associations of Clinical Nutritionists.
Back to Health Articles
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We proudly ship healthy lifestyle products to customers worldwide.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,816
|
Matwijiwka (; ) ist ein Dorf in der ukrainischen Oblast Saporischschja mit etwa 900 Einwohnern (2001).
Das 1842 von Staatsbauern aus den Gouvernements Kursk und Orel gegründete Dorf liegt auf einer Höhe von 10 km östlich vom Gemeindezentrum Nowouspeniwka (), 21 km östlich vom ehemaligen Rajonzentrum Wessele und etwa 100 km südlich vom Oblastzentrum Saporischschja.
Im Dorf trifft die nach Norden führende Territorialstraße T–08–18 auf die von West nach Ost verlaufende T–08–11.
Am 3. August 2017 wurde das Dorf ein Teil der neugegründeten Landgemeinde Nowouspeniwka, bis dahin bildete es zusammen mit dem Dorf Woschod () die gleichnamige Landratsgemeinde Matwijiwka (Матвіївська сільська рада/Matwijiwska silska rada) im Nordosten des Rajons Wessele.
Seit dem 17. Juli 2020 ist sie ein Teil des Rajons Melitopol.
Weblinks
Webseite der Landgemeinde Nowouspeniwka (ukrainisch)
Webseite des Gemeinderates auf rada.info (ukrainisch)
Einzelnachweise
Ort in der Oblast Saporischschja
Rajon Melitopol
Gegründet 1842
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,981
|
\section{Introduction}
\label{sec:intro}
To facilitate the discussion we introduce first the basic notation and definitions.
\subsection{Basic notation and definitions}
\begin{definition}
Let $c > 1$. By $\mathcal{E}_c$ we denote an interior of an ellipse, such that
the foci of $\mathcal{E}_c$ are located at points $\pm 1$ and the
sum of semi-axes is equal to $c$.
\end{definition}
\begin{definition}
Let $D \subset \mathbb{C}$. We call $D$ a {\em nice domain} if it is an open, connected and simply connected set, symmetric with respect to the real axis (i.e.\ if $z \in D$, then $\bar{z} \in D$).
\end{definition}
\begin{definition}
Let $D \subset \mathbb{C}$ be a nice domain. Let $M \geq 0$.
We will write:
\begin{itemize}
\item[] $\A(D)$ for the set of analytic functions
on $D$ such that $\|f\|=\sup_{z \in D}|f(z)| < \infty$,
\item[] $\A(D,M)$ for the set of analytic functions
on $D$ such that $|f(z)| \leq M$ for $z \in D$,
\item[] $\A_0(D,M)$ for a subset of $\A(D,M)$ consisting of the functions
which are real on the real line.
\end{itemize}
\end{definition}
We denote by $I(f,\alpha)$ the integral
\begin{equation*}
I(f,\alpha)=\int_{-1}^1 f(x) d\alpha(x), \label{eq:int-with-weight}
\end{equation*}
where $\alpha$ is a finite Borel measure on $[-1,1]$ which is absolutely continuous with respect to the Lebesgue measure. Usually we drop $\alpha$
and write $I(f)$, when $\alpha$ is known from the context. Let $\Q(n,\R)$, where $n \in \mathbb{N}$ and $\R = (r_1, \ldots, r_n)$, $r_1, \ldots, r_n\in \NN\backslash \{0\}$, denote the class of all possible (even non-linear) quadratures that use $n$ nodes $z_1,\dots,z_n \in [-1,1]$ and derivatives of an integrand up to the order $r_j - 1$ for each $z_j$. By $\Oq(n, \R)$ we denote a subclass of $\Q(n,\R)$ containing quadratures of the form
\begin{equation}
S_\mathcal{R}(f)=\sum_{j=1}^n \sum_{k=0}^{r_j - 1} b_{kj} f^{(k)}(z_j). \label{eq:quad-with-der}
\end{equation}
Additionally, $|\R|$ denotes the sum $r_1 + \ldots + r_n$ and $\R_2 = (2, \ldots, 2)$.
Following \cite{P98} we introduce the following definitions.
\begin{definition}
\label{def:petras-optimal}
Let $D \subset \mathbb{C}$ be open set, such that $[-1,1] \subset D$.
For a given quadrature $Q \in\Q(n,\R)$ the remainder term is defined as
\begin{equation}
R(f,\alpha)=I(f,\alpha) - Q(f).
\end{equation}
The error constant of $Q$ with respect to $\A(D)$ is given by
\begin{equation}
\rho(Q,\A(D),\alpha)=\sup_{f \in \A(D) \setminus \{0\}} \frac{|R(f, \alpha)|}{\|f\|}
\end{equation}
and the respective optimal error constant by $ \rho_n(\A(D),\alpha)$
\begin{equation}
\rho_n(\A(D),\alpha)=\inf_{Q\in \Q(n,\R_2)} \rho(Q,\A(D),\alpha).
\end{equation}
A quadrature formula is called \emph{optimal} if its error constant attains $\rho_n(\A(D),\alpha)$.
\end{definition}
Let us stress that in the above definition only $\R_2$ is used to define the optimal error constant and optimal quadrature .
To measure the quality of a quadrature formula $Q_n \in \Oq(n,\R_2)$ Petras in \cite{P98} proposed the following definition.
\begin{definition}
\label{def:petras-loss}
\begin{equation}
\mbox{\rm loss}(Q,\A(D),d\alpha)=\frac{\rho(Q,\A(D),\alpha)}{\rho_n(\A(D),\alpha)}.
\end{equation}
The sequence $\{Q_n\}_{n \in \mathbb{N}}$, where $Q_n\in \Oq(n,\R_2)$, is called \emph{near-optimal}, if the sequence of corresponding losses is bounded.
\end{definition}
\subsection{Our motivation and main result}
In the works of Bakhvalov \cite{B} and Petras \cite{P98} there are convincing arguments for the near-optimality of the Gaussian quadrature in the case
when the domain of analyticity of the integrand is an ellipse; for other regions, it will be the Gaussian quadrature transported from the unique
ellipse via the Riemann mapping theorem. In Petras' article \cite{P98} one can find a demonstration of how the Gaussian quadrature fails to be nearly optimal,
when the analyticity region is not an ellipse.
To describe briefly the results of Bakhvalov and Petras we assume that $\alpha$ is the Lebesgue measure and $\R=\R_2$ (results in \cite{B} and \cite{P98} have been established for more general situations discussed in more details in Section~\ref{sec:lower-upper-bnds}). Let $G_n$ denotes the Gauss-Legendre quadrature with $n$ nodes.
The claim of an almost optimal performance of the Gauss-Legendre quadrature formula for ellipses is based on the following estimates
\begin{itemize}
\item there exists a bounded and positive function $\kappa_l:(1,\infty) \to \mathbb{R}_+ $, such for any $c>1$ and for any quadrature $Q_n\in \Oq(n,\R_2)$ there is an $f_0 \in A_0(\mathcal{E}_c,M)$ such that
\begin{equation*}
|I(f_0) - Q_n(f_0)| \geq M \kappa_{l}(c) c^{-2n}, \label{eq:lo-bnd-err-intro}
\end{equation*}
\item there exists a bounded and positive function $\kappa_g:(1,\infty) \to \mathbb{R}_+ $ such that, for any $c>1$ for the Gauss-Legendre quadrature $G_n$ for any $f \in A_0(\mathcal{E}_c,M)$ holds
\begin{equation*}
|I(f) - G_n(f)| \leq M \kappa_{g}(c) c^{-2n}. \label{eq:up-bnd-err-intro}
\end{equation*}
\end{itemize}
Observe that the above estimates lead to asymptotically the same bounds for $n$ needed to get the
quadrature error less than $\eps$. We obtain
\begin{equation}
N_l\left(\frac{M}{\eps},c \right)\leq n \leq N_g\left(\frac{M}{\eps},c \right)
\end{equation}
where
\begin{eqnarray}
N_l\left(\frac{M}{\eps},c \right)=\frac{\ln \frac{M}{\eps} + \ln \kappa_l(c)}{2 \ln c}, \qquad N_g\left(\frac{M}{\eps},c \right)=\frac{\ln \frac{M}{\eps} + \ln \kappa_g(c)}{2 \ln c}. \label{eq:estm-Nl-Ng}
\end{eqnarray}
For $\eps \to 0^+$ we have $N_l \approx N_g \approx \left(\ln \frac{M}{\eps}\right)/(2 \ln c)$, so both lower and upper bounds predict more
or less the same number of nodes.
The motivation for our work comes from the following observation. From the estimates for $\kappa_l(c)$ given in \cite{B,P98} it follows that
\begin{equation}
\lim_{c \to 1^+} \kappa_l(c)=0. \label{eq:kappa-to-0}
\end{equation}
Thus if $c-1$ is small, $N_l<0$ in (\ref{eq:estm-Nl-Ng}) unless $\eps$ is very small, so in fact the lower bound given by (\ref{eq:estm-Nl-Ng}) does not have
any predictive power w.r.t. the number of nodes required to get the error less than $\eps$ for a substantial range of the parameters $c$ and $\eps$.
The main technical result of our paper is a new lower bound for errors of arbitrary quadratures of bounded analytic function using $N$ values of functions or its derivatives at some nodes, which
does not suffer from the bad qualitative behavior exemplified by equation (\ref{eq:kappa-to-0}). This allows to obtain more meaningful lower
bounds on the cost of quadratures in the sprit of IBC approach to the complexity of integration
of bounded analytic functions (see \cite{KWW} and references given there).
Our approach is based on the conformal distance on the domain of analyticity $D$.
The theorem below is an example of our lower bound for the case of the Lebesgue measure.
\begin{theorem} \label{thm:main}
Let $D \subset \mathbb{C}$, $D \neq \mathbb{C}$ and let $D$ be a nice domain
such that $[-1,1] \subset D$.
For any $Q\in {\mathcal Q}(n, \R)$, where $|\R| = N$, and for any $M>0$ there
exists a function $f_0 \in \A_0(D,M)$ such that
\begin{equation}
|I(f_0) - Q(f_0)| \geq \gamma M,\label{eq:from-below}
\end{equation}
where
\begin{equation}\label{eq:gamma}
\gamma = \left((1+1/(2\delta_D))^{2\delta_D}(2\delta_D+1)\right)^{-2N}
\end{equation}
and
\begin{equation*}
\delta_D:=\sup\{\delta_D(x):x\in[-1,1]\}, \quad \delta_D(x):=\inf\{|x-z|:z\in\mathbb{C}\setminus D\}.
\end{equation*}
\end{theorem}
This theorem is proved in Section~\ref{sec:new-lower-bound}.
Corollary \ref{cor:ellipse} therein contains the version of this result for the ellipse~$\mathcal{E}_c$.
This result improves the results of Bakhvalov \cite{B} and Petras \cite{P98} as it allows higher derivatives in the quadrature formula and more general measures $\alpha$. On the other hand, in these works the nodes used in the quadrature are not restricted to the segment $[-1,1]$. However, the most
important qualitative improvement is that our bound does not tend to $0$ for $c\to 1^+$.
To the best of our knowledge, the only similar result, i.e.\ the fact that the lower bound does not go to $0$ when the ellipse shrinks
to $[-1,1]$, has been established by Osipenko~\cite{O95} for a very particular weight function, namely the Chebyshev weight function.
Let us describe briefly the content of the paper. In Section~\ref{sec:lower-upper-bnds} we discuss in detail the results of Bakhvalov and Petras concerning
the lower bounds for the integration error for arbitrary quadrature and the upper bounds for the error of the Gauss-Legendre quadrature, and we compare them.
In Section~\ref{sec:new-lower-bound} we develop a new lower bound for the error of an arbitrary quadrature.
\section{Existing error bounds for quadratures of analytic functions}
\label{sec:lower-upper-bnds}
\subsection{Bakhvalov's lower bound for quadratures of analytic functions}
\label{subsec:bach}
The following theorem has been proven in \cite[Thm. 1]{B} (as an
improvement of a previous result from \cite{S}).
\begin{theorem}
\label{thm:bach1}
Assume that $d \alpha=p(x) dx$ and there exists a polynomial $t(x)$ such that $p(x)/t(x) \geq \eta > 0$ for $x \in [-1,1]$.
Let $z_1,z_2,\dots,z_n \in \mathbb{R}$ ($n \leq N$) and let $z_{n+1},\dots,z_N\in \CC$ be points contained
in upper half-plane ($\im z >0$). Let $\mathcal{E}_c$ be an ellipse which encloses all of these points.
For any quadrature formula of the form
\begin{eqnarray}
Q_N (f)= \sum_{j=1}^n \left(b_{1j} \re f(z_j) + b_{2j} \re f'(z_j) + b_{3j} \im f(z_j) + b_{4j} \im f'(z_j)\right) + \nonumber \\
\sum_{j=n+1}^N \left(b_{1j} \re f(z_j) + b_{2j} \re f'(z_j) + b_{3j} \im f(z_j) + b_{4j} \im f'(z_j)\right),
\end{eqnarray}
any $c > 1$ and $M>0$, there exists a function $f_0 \in \A_0(\mathcal{E}_c,M)$ such that
\begin{equation*}
I(f_0) - Q_N(f_0) \geq \kappa_0 M c^{-2N},
\end{equation*}
where $\kappa_0$ depends on $c$ and the weight function $p(x)$ only.
\end{theorem}
\textbf{Comment:}
\begin{itemize}
\item In terms of the notions introduced earlier, for $N = n$ we have
\begin{equation}
\rho_n(\A(D),d\alpha) \geq \kappa_0 c^{-2n}.
\end{equation}
\item In \cite{B} the following formula for
$\kappa_0$ is given (see page 67)
\begin{equation}
\kappa_0=\pi P_0 (1-c^{-1})c^{-2m}(\sinh h)^m,
\label{eq:kappa0-corrected}
\end{equation}
where $h= \ln c$ (hence $\sinh h = (c - c^{-1})/2$) and
constants $P_0\in \mathbb{R}_+$, $m \in \mathbb{N}$ depend on
the weight function only ($P_0$ appears as $Q_0$ in \cite{B}).
In fact, \cite{B} misprints the formula for $\kappa_0$ as $(1-c^{-1})^{-1}$ instead of
$(1-c^{-1})$.
The constants $m$ and $P_0$ are determined as follows: after the substitution $x=\cos u$ we
have
\begin{equation}
I(f)=\int_0^\pi f(\cos u) q(u) du, \qquad q(u)=p(\cos u) \sin u.
\end{equation}
Under the assumptions of the theorem the following holds true
\begin{equation}
q(u)=P(u) l(\cos u),
\end{equation}
where $l$ is a polynomial of degree $m$ and $P(u)=q(u)/(l(\cos u)) \geq P_0 >0$ for $u \in [0,\pi]$ ($P$~appears as $Q$ in \cite{B}).
Therefore, $m$ is the number of zeros in $q(u)$ counted with multiplicities. It is related to the number of zeros in the
weight function $p(x)$: it is the number of zeros $p(x)$ counted with multiplicities plus two if the zeros at $0$ and $\pi$
introduced in $q(u)$ by the factor $\sin u$ are not canceled by the singular behavior of $p(x)$, when $x \to \pm 1$. Such cancelations happen for the
Chebyshev weight (see below).
\begin{itemize}
\item
For $p(x)\equiv 1$ we have $q(u)=\sin u$. Therefore $l(z)=1-z^2$, $$\frac{q(u)}{l(\cos u)}=\frac{\sin u}{1-\cos^2 u }= \frac{1}{\sin u} \geq P_0=1$$ for
$x \in [0,\pi]$. Hence $m=2$.
Easy computations show that for $m=2$ and $P_0=1$ we obtain
\begin{eqnarray}
\kappa_0 & = & \frac{\pi}{4} (1-c^{-1}) c^{-4} (c-c^{-1})^2
= \frac{\pi}{4}(c-1)^3 \frac{(c+1)^2}{c^7}= \label{eq:ko-bach}\\
& = & \pi (c-1)^3 + O((c-1)^4), \quad \mbox{ for $c\to 1^+$}. \nonumber
\end{eqnarray}
\item For the Chebyshev weight
$p(x)=1/\sqrt{1-x^2}$
we have
\begin{equation*}
q(u) = p(\cos u) \sin u = 1.
\end{equation*}
Hence $m=0$ and $P_0=1$, and consequently
\begin{eqnarray}
\kappa_0=\pi (1-c^{-1}) =\frac{\pi (c-1)}{c}.
\end{eqnarray}
\end{itemize}
We obtain a counter-intuitive statement that when $c-1$ is small (i.e.\ the integrated function is difficult to calculate due to the possible presence of
singularities nearby),
the lower bound for the error is also small. Hence the
quality of the bound is rather poor and can be considerably
improved.
\item $\rho_n(\A(D),d\alpha)$ is estimated as follows. For for any $n$ nodes of polynomial $f_0 \in \A(\mathcal{E}_c,1)$ of degree $2n+m$ is defined, such that its quadrature error is bounded from below by $\kappa_0 c^{-2n}$. For fixed set of nodes this polynomial is the same for all $c>1$ up to a multiplicative constant depending on $c$. Therefore, the functions considered have no singularities outside the ellipse.
\end{itemize}
\subsection{Petras' lower bounds}
\label{subsec:petras}
Petras in \cite{P98} considers the quadrature of the same type as in Theorem~\ref{thm:bach1}, ellipses as analyticity regions
and the Szeg\"o class of weights (measures), which are defined as follows:
$d \alpha(x)=w(x)dx$, where $w$ is a function for which $\int_0^\pi \ln w(\cos x)dx$ exists. It contains the class of weights considered by Bakhvalov.
The reasoning in \cite{P98} goes as follows. Petras proves the following theorem for even more general class of weight measures.
\begin{theorem} \cite[Thm. 2.1]{P98}
\label{thm:PetrasOrtPol}
Assume that the measure $\alpha$ is supported on at least $n+1$ points. Let $D$ be a symmetric domain. Let $p_0,p_1,\ldots,p_n$ be the orthonormal polynomials with respect to the measure~$\alpha$. Then
\begin{equation}
\rho_n(\A(D),d \alpha) \geq k_n(\A(D),d \alpha):= \left(\sum_{\nu=0}^n (\sup_{z \in D}|p_\nu(z)|)^2\right)^{-1}.
\end{equation}
\end{theorem}
For weights in the Szeg\"o class and $D=\mathcal{E}_c$ Petras obtains (see Corollary 3.1 in \cite{P98}) the following result
\begin{equation}
\lim_{n \to \infty} c^{2n} k_n(\A(\mathcal{E}_c), d\alpha)= 2\pi (1-c^{-2}) \cdot \min_{|z|=c} |D(z^{-1})|^2 >0, \label{eq:Petras-Asymptotic}
\end{equation}
where the so-called Szeg\"o function $D(z)$ is given by
\begin{equation}
D(z):= \exp \left( \frac{1}{4 \pi} \int_{-\pi}^\pi \frac{1 + ze^{-it}}{1- z e^{-it}} \ln (w(\cos t)|\sin t|) dt \right).
\end{equation}
Observe that apparently from the above formula one obtains $\rho_n \geq O(c-1)/c^{2n}$, the exact form of lower bound depending
on the term involving function $D(z)$ in (\ref{eq:Petras-Asymptotic}) and for $c \geq R > 1$ one obtains
\begin{equation}
\rho_n(\A(\mathcal{E}_c),d\alpha) \geq \kappa_0 c^{-2n},
\end{equation}
generalizing Theorem~\ref{thm:bach1}.
For several particular weights Petras computes an explicit lower bound for $ k_n(\A(\mathcal{E}_c), d\alpha)$, but it exhibits an incorrect behavior for $c\to 1^+$.
Below we list only the results for
the Lebesgue measure and the Chebyshev weight.
\begin{itemize}
\item From Corollary 3.6 in \cite{P98} it follows that for the weight $w(x)\equiv 1$ it holds
\begin{equation}
\rho_n(\A(\mathcal{E}_c),dx)\geq \pi(1-c^{-2})^2 c^{-2n} \cdot (1 + \eps_n)^{-1},
\end{equation}
where
\begin{equation*}
0 \leq \eps_n \leq \frac{c^4 + 4c^2 + 18}{4n c^2 (c^2-1)} + \frac{(n+2)^{3/2}}{c^{n+2}}.
\end{equation*}
It is clear that for a fixed $n$, this bound is $O((c-1)^3)$ for $c \to 1^+$. To be more precise we have (for fixed $n$
\begin{equation}
\rho_n(\A(\mathcal{E}_c),dx) \geq \frac{\pi}{c^{2n}} \left(\frac{32}{23} (c-1)^3 n c^2+O\left((c-1)^4\right)\right).
\end{equation}
\item From Corollary 3.5 in \cite{P98} it follows that for the Chebyshev weight $d\alpha(x)=$ $w(x)dx=$ $dx/\sqrt{1-x^2}$ we have
\begin{equation}
\rho_n(\A(\mathcal{E}_c),d \alpha) \geq \frac{\pi (1-c^{-2})^3}{2 c^{2n}} \left( 1- \frac{(2n+3)(c^2 -1)+ c^{-2n - 2}}{c^{2n +4}} \right)^{-1} \geq \frac{\pi (1-c^{-2})^3}{2 c^{2n}}.
\end{equation}
For $c\to 1^+$ we obtain the following estimate
\begin{equation}
\rho_n(\A(\mathcal{E}_c),d \alpha) \geq \frac{\pi}{c^{2n}} \left( \frac{3 }{2 n^3+9 n^2+13 n+6}+\frac{6 (c-1) n}{2 n^3+9 n^2+13
n+6}+O\left((c-1)^2\right) \right).
\end{equation}
In this case for fixed $n$ the lower bound for $c\to 1^+$ does not go to zero, however it goes when $n \to \infty$, which turns out to be unsatisfactory.
\end{itemize}
In fact the bound in Theorem~\ref{thm:PetrasOrtPol} obtained by considering polynomials of degree $2n$. Hence the functions producing this bound have no singularities outside the ellipse.
\subsection{Osipenko estimates}
\label{subsec:Osip}
Osipenko in \cite[Thm. 6]{O95} obtained the following explicit estimate for the Chebyshev weight $d\alpha(x)=dx/\sqrt{1-x^2}$
\begin{equation}
\rho_n(\mathcal{A}(\mathcal{E}_c),d\alpha)=\frac{2\pi}{c^{2n}} +O(c^{-6n}) \label{eq:Osip-estm}
\end{equation}
and the limit behavior
\begin{equation}
\lim_{c \to 1^+} \rho_n(\mathcal{A}(\mathcal{E}_c),d\alpha)=2\pi.
\end{equation}
Osipenko uses transformation of an ellipse to an infinite strip, which transforms the problem of integration of bounded analytic functions defined on the ellipse with the Chebyshev weight to the problem of integration of analytic periodic functions with the Lebesgue measure. He uses Blaschke products to find lower estimate for the error, which is natural for this kind of problem. This should be contrasted with the polynomials used to derive lower bounds in \cite{B,P98}.
\subsection{Final comments on Bakhvalov's and Petras' lower bounds}
Both Bakhvalov and Petras mention that the Riemann mapping theorem allows to transport the results for an ellipse to other domains. However, no quantitative statements related to the geometry of the domain $D$ are given.
As it was mentioned in the introduction we have found the behavior of $\kappa_{l}(c)$ for
$c \to 1^+$ obtained by Bakhvalov and by Petras overly pessimistic. In the argument below we will show how bad this
bound is qualitatively. Namely, if $\kappa_{g}(c)$ were of the same order as $\kappa_{l}(c)$, i.e.\ $\lim_{c \to 1^+}\kappa_g(c)=0$, the quadrature would be exact even for $n = 1$. This is formalized in the following remark.
\begin{remark}
\label{rem:wrong-lower-bnd}
Let $Q\in \Q(n,\R)$ and a positive bounded function $\kappa:(1,\infty) \times \NN \to \mathbb{R}_+$ be such that
\begin{equation}
|I(f) - Q(f)| \leq M \kappa(c,n) c^{-2n}, \quad f \in
\A_0(\mathcal{E}_c,M). \label{eq:err-estm-hyp}
\end{equation}
Assume that for each $n \in \NN$ holds
\begin{equation}
\lim_{c \to 1^+}\kappa(c,n)=0. \label{eq:kappa-lim}
\end{equation}
Then for any $M>0$, $c > 1$, $n \in \NN$ and $f \in
\A_0(\mathcal{E}_c,M)$ holds
\begin{equation*}
I(f)=Q(f).
\end{equation*}
\end{remark}
\begin{proof}
Since $\mathcal{E}_c \subset \mathcal{E}_{c_1}$
for $c<c_1$, we have
\begin{equation}
\A_0(\mathcal{E}_{c_1},M) \subset \A_0(\mathcal{E}_{c},M), \quad c <
c_1. \label{eq:ellipse-incl}
\end{equation}
The above inclusion holds in the following sense: for a function
$f \in \A_0(\mathcal{E}_{c_1},M)$ we consider its restriction to
$\mathcal{E}_c$. It is immediate to see that $f_{|\mathcal{E}_c}
\in \A_0(\mathcal{E}_{c},M)$.
Let us fix $n$ and take a function $f \in
\A_0(\mathcal{E}_{c_1},M)$. By (\ref{eq:err-estm-hyp}) and
(\ref{eq:ellipse-incl})
\begin{equation*}
|I(f) - Q(f)| \leq M \kappa(c,n) {c}^{-2n}, \quad 1<c\leq c_1.
\end{equation*}
Passing to the limit $c \to 1$ we obtain
\begin{equation*}
|I(f) - Q(f)| =0.
\end{equation*}
\qed
\end{proof}
\subsection{Upper bounds for Gauss-Legendre quadratures}
\label{secsub:Gauss-quadrature}
We assume that $d\alpha(x)=dx$ and $G_n(f)$ denotes the Gauss-Legendre quadrature with $n$ nodes on $[-1,1]$.
Let us define
\begin{equation}
r_n(c)=\rho(G_n, \A_0(\mathcal{E}_c,1),dx)=\sup_{f \in \A_0(\mathcal{E}_c,1) }|I(f) - G_n(f)|.
\end{equation}
Obviously
\begin{equation}
|I(f) - G_n(f)| \leq M r_n(c) , \quad f \in \A_0(\mathcal{E}_c,M).
\end{equation}
Let us list two estimates for the error of Gauss quadrature known in the literature.
Let us start with the estimates for the error of the Gauss
quadrature due to Rabinowitz \cite[eq.~(18)]{R}, see also
\cite[Thm.~90]{Br} and \cite[Thm.~4.5]{T}
\begin{theorem}
\label{thm:rabinowitz}
\begin{equation}\label{eq:gauss-upper-bnd}
r_n(c) \leq \min \left(4,\frac{64 }{15(1 - c^{-2})}c^{-2n}\right).
\end{equation}
\end{theorem}
The non-constant part of this estimate has an undesirable property. For $c \to 1$ it explodes,
which may lead to non-uniform estimates in some contexts.
The bounds which are much more uniform in $c$ for $c \to 1$ are given by Petras in \cite{P95}.
\begin{theorem} \cite[Thm. 4]{P95}
\label{thm:gauss-petras}
\begin{equation*}
r_n(c) \leq \frac{4}{c^{2n}} \left( 1 + \frac{3}{2n c^2}+ \frac{4}{c^{n+1}}\right).
\end{equation*}
\end{theorem}
In fact \cite[Thm. 4]{P95} contains four estimates for $r_n(c)$, such that their mutual ratios are bounded. Here we chose the one, which appears the easiest to handle.
From Theorem~\ref{thm:gauss-petras} one can easily obtain the following Corollary.
\begin{cor}
\begin{eqnarray}
r_n(c) &\leq& \frac{26}{c^{2n}}, \label{eq:GL-error-26} \\
\forall \eps >0 \quad \exists c_0(\eps) \quad \forall c \geq c_0(\eps) \quad r_n(c) &\leq& \frac{4 + \eps}{c^{2n}}. \label{eq:GL-error-4-eps}
\end{eqnarray}
\end{cor}
\begin{remark}\label{rmk:chebyshev-2n}
In \cite{P95} (in part (b) of a remark just below Theorem 4 there) Petras mentions that taking $f$ to be a suitably scaled ($2n$)-th Chebyshev polynomial of the first kind $T_{2n}$, i.e.\ $f=\frac{2c^{2n}}{c^{4n}+1} T_{2n} \in \A_0(\mathcal{E}_c,1)$ one obtains
\begin{equation}
|I(f) - G_n(f)| \geq \frac{\pi (1 - (4n)^{-1})}{c^{2n}(1 + c^{-4n})}.
\end{equation}
Hence, the bounds given in Theorem~\ref{thm:gauss-petras} are optimal, up to a constant independent of $c$ and $n$.
\end{remark}
Observe that from (\ref{eq:GL-error-26}) it follows that if $M/\eps > 26$, then in order to have the error less than $\eps$ for
functions from $\A_0(\mathcal{E}_c,M)$ it is enough to use $N_g$ nodes, where
\begin{equation}
N_g \geq \frac{\ln \frac{M}{\eps}}{\ln c}. \label{eq:N-GL-nodes}
\end{equation}
\subsection{Comparison of lower and upper bounds}
\label{subsec:lo-upp-bnds}
We are now ready to compare in detail the lower bounds of Bakhalov and Petras with the bounds for the Gauss-Legendre quadrature for the ellipses with the Lebesgue measure as the weight function.
Let $c>1$ and let $\kappa_l(c)$ and $\kappa_g(c)$ be positive numbers such that
\begin{itemize}
\item for any $Q_n\in \Oq(n,\R_2)$ there is an $f_0 \in A_0(\mathcal{E}_c,M)$ such that
\begin{equation}
|I(f_0) - Q_n(f_0)| \geq M \kappa_{l}(c) c^{-2n}, \label{eq:lo-bnd-err}
\end{equation}
\item for the Gauss-Legendre quadrature $G_n$, for any $f \in A_0(\mathcal{E}_c,M) $ we have
\begin{equation}
|I(f) - G_n(f)| \leq M \kappa_g(c) c^{-2n}, \label{eq:up-bnd-err}
\end{equation}
\end{itemize}
where $\kappa_l$ is Bakhvalov's or Petras' lower bound discussed in Sections \ref{subsec:bach} and \ref{subsec:petras} and $$\kappa_g(c)=\sup_{n \geq 1}c^{2n}r_n(c)$$ obtained
from Theorem~\ref{thm:rabinowitz} or Theorem~\ref{thm:gauss-petras}.
From Theorem~\ref{thm:bach1} (with
$\kappa_l=\kappa_0$ given by (\ref{eq:ko-bach})) for $c$ close to $1$ we get
\begin{eqnarray*}
\kappa_{l}(c) & = & (c-1)^3\pi +O((c-1)^4), \\
\kappa_{g}(c) & = & 26.
\end{eqnarray*}
For large values of $c$ (which means that we are considering very regular functions) we have from~(\ref{eq:GL-error-4-eps})
\begin{eqnarray*}
\kappa_{l}(c) & = & \frac{\pi}{4}c^{-2} + O(c^{-4}), \\
\kappa_g(c) &=& 4.1 + O(c^{-2}).
\end{eqnarray*}
Note that in both cases the quotient
$\kappa_{g}/\kappa_{l}\to \infty$.
Both bounds~(\ref{eq:lo-bnd-err}) and~(\ref{eq:up-bnd-err}) are $O(c^{-2n})$ as the
function of $n$ and they give the following estimates for
$n$, needed to obtain the integral with error less
than $\eps$.
For the Gauss-Legendre quadrature it is enough to take $n \geq
N_g$, where
\begin{equation*}
N_g=N_g\left(\frac{M}{\eps},c\right)=\max\left(1,\frac{1}{2 \ln
c}\ln\left(\frac{M}{\eps}\kappa_g(c)\right)\right),
\end{equation*}
while (\ref{eq:lo-bnd-err}) implies that whatever the quadrature
is we cannot take $n$ smaller than
\begin{equation*}
N_l=N_l\left(\frac{M}{\eps},c\right)=\max\left(1,\frac{1}{2 \ln
c}\ln\left(\frac{M}{\eps}\kappa_l(c)\right)\right).
\end{equation*}
For $\eps \to 0$ we have
\begin{equation*}
\frac{N_l}{N_g}\approx
\frac{\ln\left(\frac{M}{\eps}\kappa_l(c)\right)}{\ln\left(\frac{M}{\eps}\kappa_g(c)\right)} \to 1.
\end{equation*}
Apparently both numbers $N_l$ and $N_g$ are of similar magnitude up to
a factor depending on $c$ but not on $n$.
However, if we fix $\eps$ and let $c \to 1$, we have $\kappa_l(c) \to 0$, hence $N_l \to 1$ the lower bound $N_l$ loses its predictive power.
We are not concerned with the behavior of $\kappa_l$ and $\kappa_g$ for $c \to \infty$, because it does not necessarily make sense to increase $c$ while
keeping $M$ constant; the functions in
$\A_0(\mathcal{E}_c,M)$ become very flat for large $c$ and in this limit we obtain $N_l=N_g=1$.
Summing up, the bounds (\ref{eq:lo-bnd-err}) and (\ref{eq:up-bnd-err}) might give completely different estimates $N_l$ and $N_g$ of information needed to bring the error below $\eps$. For `difficult' functions ($c$ close to 1) we obtain the obvious bound $n\ge N_l = 1$ for a significant range of the ratio $M/\eps$.
It appears to us that it makes sense to require the following condition to maintain the optimality of Gauss-Legendre quadratures on ellipses:
there exists $\eta_0$ such that for all $M/\eps \in \mathbb{R}_+$ and $c>1$
\begin{equation}
0 < \eta_0 \le \frac{N_l\left(\frac{M}{\eps},c\right)}{N_g\left(\frac{M}{\eps},c\right)}. \label{eq:optimality-crit}
\end{equation}
Observe that, when compared to Definition~\ref{def:petras-loss}, we now want
the ratio to be bounded also when we change the ellipse.
\section{New lower bounds }\label{sec:new-lower-bound}
In this section we study the problem of estimating from below the quadrature error
in a class of analytic functions with possible singularities outside
a nice domain. In the special
case of ellipses the formulas are given so that they can be directly compared with the
known ones. Since the methods may probably be applicable in a more general class
of domains (not necessarily simply connected) we introduce distances (metrics) that could
be tools for studying them in several complex variables. But we restrict
our consideration to the case of simply connected domains in the complex plane
where the considered (hyperbolic) metric and distance may be described in many
equivalent ways. The question which description could (and should) be applied in
the case of domains being not simply connected remains open.
\subsection{Definitions and description of the problem}
By $\lambda_1(A)$ we denote the Lebesgue measure of the set $A \subset \mathbb{R}$.
We recall that \textit{the Poincar\'e distance $p$ } on the open unit disk $\DD:=\{z\in\mathbb C:|z|<1\}$ is given by the formula
\begin{equation}
p(z,w):=\frac{1}{2}\ln\frac{1+m(w,z)}{1-m(w,z)}=:\arctanh(m(w,z)),\; w,z\in\DD, \label{eq:def-pzw}
\end{equation}
where $m(w,z)=\left|\frac{w-z}{1-\bar wz}\right|$.
The Poincar\'e distance induces the pseudodistance $c_D$ on any domain (i.e.\ connected and open set)
$D\subset\CC$ by the following formula
\begin{equation}
c_D(w,z):=\sup\{p(F(w),F(z)):F\in\OO(D,\DD)\}, w,z\in D,
\end{equation}
where $\OO(D,\DD)$ denotes the set of holomorphic (analytic) functions $D$ to $\DD$.
We also put
\begin{equation}
c_D^*(w,z):=\tanh c_D(w,z).
\end{equation}
We remind the following property of $c_D$ (called \textit{the holomorphic contractibility of $c$}):
$c_G(F(w),F(z))\leq c_D(w,z)$ for any $F\in\OO(D,G)$, $w,z\in D$.
In the case of simply connected domains the function $c_D$ coincides with the distance induced by the metric $\gamma_D$ (often called \textit{hyperbolic metric} for planar domains) defined by the formula
\begin{equation}
\gamma_D(z;X):=\sup\{|F^{\prime}(z)X|/(1-|F(z)|^2):F\in\OO(D,\DD)\},\; z\in D,\; X\in\CC. \label{eq:gammaD}
\end{equation}
It is well-known that $\gamma_{\mathbb D}(z;X)=|X|/(1-|z|^2)$, $z\in\mathbb D$, $X\in\mathbb C$ (we call the function $\gamma_{\mathbb D}$ \textit{the Poincar\'e metric}).
Similarly as before we get a version of holomorphic contractibility of $\gamma$, namely the inequality
\begin{equation}
\gamma_G(F(w);F^{\prime}(w)X)\leq\gamma_D(w;X),\; w\in D;X\in\mathbb{C}, \label{eq:gamma-ineq}
\end{equation}
for any $F\in\OO(D,G)$. For domains $D\subset G$ in $\mathbb C$ we may use the holomorphic contractibility for the inclusion function $\iota:D\mapsto G$ where $D\subset G\subset\mathbb C$ which gives, among others, the inequality $\gamma_D(z;1)\geq\gamma_G(z;1)$, $z\in D$.
Note that although we defined
the functions $c_D$ and $\gamma_D$ in a very general situation we shall consider them in the very special case of $D$ being a simply connected domain.
The geometry induced by the Poincar\'e distance is an example of a
non-Euclidean geometry. Recall that the lines (geodesics) in this
geometry are diameters and the arcs of circles lying in $\DD$ and
being orthogonal to the unit circle $\partial\DD$. In particular,
for three consecutive points $x,y,z$ on such geodesics one has the
equality $p(x,z)=p(x,y)+p(y,z)$. Note also that the biholomorphic
mappings transform geodesics into geodesics, and the geodesics in the domain $D$ satisfy the equality $c_D(x,z)=c_D(x,y)+c_D(y,z)$ for three consecutive points lying in the geodesic.
The distance of two points $w,z$ from the simply connected domain $D$ lying in a geodesic may be given with the help of the function $\gamma_D$ as follows. If $\alpha:[0,1]\to D$
is a parametrization of the part of the geodesic joining $w$ and $z$ lying between $w$ and $z$; then
\begin{equation*}
c_D(w,z)=\int\sb{0}\sp{1}|\alpha^{\prime}(t)|\gamma_D(\alpha(t);1)dt.
\end{equation*}
We should also keep in mind that the Poincar\'e distance on $\mathbb D$ (as well as the Poincar\'e metric) are invariant under holomorphic automorphisms of the unit disk ($\operatorname{Aut}(\mathbb D)$). Recall that
\begin{equation}
\operatorname{Aut}(\mathbb D)=\left\{e^{i\theta}m_{\eta}:\theta\in\mathbb R,\eta\in\mathbb D\right\},
\end{equation}
where $m_{\eta}(z):=(\eta-z)/(1-\bar{\eta} z)$, $z\in\mathbb D$.
A special role in our considerations will be played by \textit{the finite Blaschke products}.
Some of basic properties of the finite Blaschke products are that they extend holomorphically to a neighborhood of $\overline{\mathbb D}$ (they are rational with poles lying outside of the closed unit disk. The finite Blaschke product $B$ is a proper holomorphic mapping of $\mathbb D$ onto $\mathbb D$. Moreover, $|B(z)|=1$, $|z|=1$.
We refer the reader to any of the textbooks
\cite{Rud 1966}, \cite{Con 1978},
\cite{Con 1995} and
\cite{Jar-Pfl 1993}. In the last reference the theory of
holomorphically invariant metrics and distances in several complex
variables is presented.
In higher-dimensional case the metric $\gamma_D$ depends
on points $z\in D$ and the vectors $X$ from the tangent space to
$D$; that is the reason why the value of the differential at
vector $X\in\mathbb{C}$ (generally $\mathbb{C}^n$) is studied.
However, the facts that we use are standard in the theory of one
complex variable and may be found in many textbooks on the theory
of complex variable. As to the theory of (bounded) holomorphic
functions, except for the above mentioned textbooks, we refer the reader to \cite{Dur 1970}, \cite{Gar 1981} (where one may also
see how the Blaschke products appear naturally when considering some extremal problems in the theory of analytic functions). Out of many
possible references for the properties of the Carath\'eodory
distance (induced by the hyperbolic metric) we recommend the paper
\cite{Ban-Car 2008} and the references therein concerning
estimates for the hyperbolic metric in the ellipses.
Note that the hyperbolic density $\sigma_D$ considered in \cite{Ban-Car 2008}
is related to $\gamma_D$ by the relation
$\gamma_D(z;X)=|X|\sigma_D(z)$. The paper \cite{Ban-Car 2008}
could also possibly be applied to sharpen some of the results
presented in the paper in the case of ellipses.
In this section, unless otherwise stated, the domain $D\subset\mathbb C$ contains $[-1,1]$ is simply connected, $D\neq\CC$ and $D$ is
symmetric with respect to the $x$-axis, i.e.\ $z\in D$ iff $\bar z\in
D$. Let $\alpha$ be a finite, positive, Borel measure on $[-1,1]$ absolutely continuous with respect to the Lebesgue measure.
Let $f_D:D\to\DD$ be a conformal mapping (i.e.\ biholomorphic) such that $f_D(0)=0$,
$f_D([0,1])\subset[0,1)$ (the latter is possible because of the
symmetry of $D$). Note also that the function $f_D$ is defined is actually unique
(it follows from the uniqueness part of the Riemann mapping
theorem). The set $\mathbb R\cap D$ is a geodesic. We shall often make use of the identity
\begin{equation*}
c_D^*(w,z)=m(f_D(w),f_D(z)),\;w,z\in D. \label{eq:cd-wzor}
\end{equation*}
Given an integer $k$ let $r(k)$ be the least even integer bigger than or equal to $k$.
Certainly, $r(k)$ is either $k$ or $k+1$.
For the sequence of $n$ distinct points $X:=(x_1,\ldots,x_n)$
where $-1\leq x_1<\ldots<x_n\leq 1$, the sequence of $n$ positive integers
$\mathcal K=(k_1,\ldots,k_n)$ we define
\begin{equation*}
\mathcal F(D;X;\mathcal K):=\{f\in\OO(D,\DD):f^{(l)}(x_j)=0:l=0,\ldots,k_j-1;\;j=1,\ldots,n\},
\end{equation*}
\begin{equation*}
\mathcal F_r(D;X;\mathcal K):=\{f\in\mathcal F(D;X;\mathcal K):f(D\cap\mathbb R)\subset\mathbb R\},
\end{equation*}
\begin{equation*}
\mathcal F_+(D;X;\mathcal K):=\{f\in\mathcal F_r(D;X;\mathcal K):f\geq 0 \text{ on } D\cap\mathbb R\}
\end{equation*}
and
\begin{equation*}
J_a(D;X;\mathcal K):=\sup\left\{\left|\int\sb{-1}\sp{1}g(x) d\alpha(x) \right|:g\in \mathcal F_a(D;X;\mathcal K)\right\},
\end{equation*}
where $a$ is $+$, $r$ or empty sign.
We are now in a position to prove the following lemma.
\begin{lemma}\label{lem:unique-extremal}
Let $D$, $f_D$, $\alpha$, $X$ and $\mathcal K$ be defined as above. Then there is exactly
one $f\in \mathcal F_+(D;X;\mathcal K)$ such that
\begin{equation*}
\int\sb{-1}\sp{1}f(x)d\alpha(x)=J_+(D;X;\mathcal K).
\end{equation*}
Moreover, $f$ is given by the formula
\begin{equation}
f(z)=\prod\sb{j=1}\sp{n}\left(\frac{f_D(z)-f_D(x_j)}{1-f_D(x_j)f_D(z)}\right)^{r(k_j)}, \; z\in D
\label{eq:func-extremalna}
\end{equation}
and
\begin{equation*}
J_+(D;X;\mathcal K) = \int\sb{-1}\sp{1}\left(\prod\sb{j=1}\sp{n}(c_D^*(x,x_j))^{r(k_j)}\right)d\alpha= \int\sb{-1}\sp{1}\left(\prod_{j=1}^n
m(f_D(x),f_D(x_j))^{r(k_j)}\right)d\alpha.
\end{equation*}
\end{lemma}
\begin{proof}
Let $g\in \mathcal F_+(D;X;\mathcal K)$.
The non-negativity of $g$ together
with the vanishing of derivatives at $x_j$ implies that the multiplicity
of $g$ at $x_j$ is at least $r(k_j)$. Let $f$ be the function given by the formula (\ref{eq:func-extremalna}). Then the function $h:=\frac{g}{f}$
extends to a well-defined holomorphic function on $D$.
Moreover, the function $f$ is the composition of the finite Blaschke product with the conformal function $f_D$ so $\lim\sb{z\to\partial D}|f(z)|=1$ and thus $\limsup\sb{z\to\partial D}|h(z)|\leq 1$.
This together with the maximum principle for holomorphic functions implies that $|h(z)|\leq 1$, $z\in D$.
Additionally, the maximum principle gives that the equality at one point $z\in D$ holds iff $h$ is constant. And the
non-negativity of $f$ and $g$ on $[-1,1]$ implies that this constant is one.
Consequently, either $g(z)=f(z)$, $z\in D$ or $|g(z)|<|f(z)|$,
$z\in D\setminus \{x_1,\ldots,x_n\}$, which completes the proof.
\qed
\end{proof}
\begin{remark} It is obvious that
\begin{equation}
J_+(D;X;\mathcal K)\leq J_r(D;X;\mathcal K)\leq J(D;X;\mathcal K).
\end{equation}
Moreover, the second inequality above is actually the equality. To see this take any $g\in\mathcal F(D;X;\mathcal K)$. Let $|\omega|=1$ be such that $\omega\int\sb{-1}\sp{1}g(x)dx=\left|\int\sb{-1}\sp{1}g(x)dx\right|$. Define
$h(\lambda):=(\omega g(\lambda)+\overline{\omega g(\overline{\lambda})})/2$, $\lambda\in D$.
Then $h\in\mathcal F_r(D;X;\mathcal K)$ and $h(x)=\re (\omega g(x))$, $x\in[-1,1]$. Consequently,
\begin{equation}
\left|\int\sb{-1}\sp{1}g(x)dx\right|=\re\left(\omega\int\sb{-1}\sp{1}g(x)dx\right)=\int\sb{-1}\sp{1}h(x)dx,
\end{equation}
which implies the inequality $ J(D;X;\mathcal K)\leq J_r(D;X;\mathcal K)$.
On the other hand $J_+(D;X;\mathcal K)$ is, in general, less than $J_r(D;X;\mathcal
K)$. It can already be seen when considering $n=1$, $k_1=1$, $d\alpha(x)=dx$ and
$x_1$ close to $-1$. In fact, first note that for $x_1=-1$ we get the inequalities
\begin{equation}
1>\frac{f_D(x)-f_D(x_1)}{1-f_D(x_1)f_D(x)}>\left(\frac{f_D(x)-f_D(x_1)}{1-f_D(x_1)f_D(x)}\right)^2>0,\;x\in(-1,1]
\end{equation}
so the inequality
\begin{equation}
\int\sb{-1}\sp{1}\left(\frac{f_D(x)-f_D(x_1)}{1-f_D(x_1)f_D(x)}\right)dx >
\int\sb{-1}\sp{1}\left(\frac{f_D(x)-f_D(x_1)}{1-f_D(x_1)f_D(x)}\right)^2dx
\end{equation}
holds for $x_1\geq -1$ sufficiently close to $-1$.
\end{remark}
\begin{remark}
Recall that the finite Blaschke products are extremal in many problems which involve bounded holomorphic functions on the unit disk. In the context
of the optimal quadrature formula the Blaschke products have been used by Osipenko \cite{O95} and Bojanov \cite{Bo73,Bo74} for the analytic functions on the unit circle. Therefore, it is very natural that the function for which the supremum in Lemma~\ref{lem:unique-extremal} is attained is, up to a conformal mapping $f_D$, a finite Blaschke product.
\end{remark}
In the next subsection
we shall estimate from below the number
\begin{eqnarray*}
J_+(D;N) & := & \inf\{J_+(D;(x_1,\ldots,x_n);(k_1,\ldots,k_n)):\\
& & \mbox{}\hspace*{3cm} n\in\mathbb N, -1\leq x_1<\ldots<x_n\leq 1,k_1+\ldots+k_n=N\}.
\end{eqnarray*}
\subsection{Lower estimate}
First we recall the classical Koebe one-quarter theorem.
\begin{theorem}
\label{thm:Koebe} \rm{ (see e.g. \cite{Con 1995}, Thm 14. 7. 8)}
The image of an injective holomorphic function $f: \DD \to \mathbb{C}$ contains the disk centered at $f(0)$ with radius $|f'(0)|/4$.
\end{theorem}
Before we proceed further with estimates for nice domains we present a result on a more general class of domains. First we remind that for any domain $D\subset\mathbb C$, $D\neq \mathbb C$ we define
$\delta_D(x):=\inf\{|x-z|:z\in\CC\setminus D\}$, $x\in D$.
\begin{lemma}\label{lemma:distance} Let $D$ be a simply connected domain in $\mathbb C$, $D\neq\mathbb C$ (we do not assume the symmetry of $D$!). Let $z_0\in D$. Then
$\gamma_D(z_0;1)\geq\frac{L}{\delta_D(z_0)}$ where $L=1/4$. If $D$ is additionally convex then we may take in the inequality $L=1/2$.
\end{lemma}
\begin{proof} Let $g:\mathbb D\to D$ be the conformal mapping such that $g(0)=z_0$. Applying Theorem~\ref{thm:Koebe} to
$g$ we get that $\delta_D(z_0)\geq|g^{\prime}(0)|/4$. But then $\gamma_D(z_0;1)\geq \left|\left(g^{-1}\right)^{\prime}(z_0)\right|=1/|g^{\prime}(0)|$ which finishes the proof in the general case.
Assume now additionally that $D$ is convex.
Then after translating and rotating the set $D$, we can assume that $D\subset H:=\{\operatorname{Re}z>0\}$ and $z_0=\delta_D(z_0)$.
Define the biholomorphism $F:H \to \mathbb{D}$, $F(z) = (z-1)/(1+z)$. From (\ref{eq:gamma-ineq}) and (\ref{eq:gammaD}) if follows that
\begin{eqnarray*}
\gamma_D(z_0;1)\geq\gamma_H(z_0;1)= \frac{|F'(z_0)|}{1 - |F(z_0)|^2}.
\end{eqnarray*}
Taking into account that $z_0 =\delta_D(z_0)>0$ we obtain the following estimate
\begin{equation*}
\gamma_D(z_0;1) \geq \frac{1}{2z_0}=\frac{1}{2\delta_D(z_0)}.
\end{equation*}
\qed
\end{proof}
Recall now that we assume that $D$ is a simply connected domain, symmetric with respect to the real axis and such that $[-1,1]\subset D\subset\mathbb C$, $D\neq\mathbb C$. We remind that in such a case we define
\begin{equation*}
\delta_D:=\sup\{\delta_D(x):x\in[-1,1]\}.
\end{equation*}
Observe that $\delta_D$ is the radius of the largest disk with the center in $[-1,1]$, which is contained in $D$.
\begin{lemma} For all $w,z\in[-1,1]$ the following inequality holds $c_D(w,z)\geq\frac{L}{\delta_D}|w-z|$, where
$L=1/4$. Moreover, in the case $D$ is additionally convex we may
take $L=1/2$. Consequently,
\begin{equation}\label{eq:koebe}
m(f_D(w),f_D(z))=c_D^*(w,z)=\tanh c_D(w,z)\geq \frac{\exp\left(\frac{2L|w-z|}{\delta_D}\right)-1}{\exp\left(\frac{2L|w-z|}{\delta_D}\right)+1},\;w,z\in [-1,1].
\end{equation}
\end{lemma}
\begin{proof}
Due to the simple fact that $\mathbb R\cap D$ is a geodesic and applying Lemma~\ref{lemma:distance} we get
\begin{equation*}
c_D(w,z)=\int\sb{0}\sp{1}|w-z|\gamma_D(tw+(1-t)z;1)dt\geq \frac{L|w-z|}{\delta_D}.
\end{equation*}
As to the last inequality in (\ref{eq:koebe}), recall that $\tanh$ is an increasing function so we
obtain
\begin{equation*}
c_D^*(w,z)= \tanh c_D(w,z)\geq \tanh
\frac{L}{\delta_D}|w-z| =
\frac{\exp\left(\frac{2L|w-z|}{\delta_D}\right)-1}{\exp\left(\frac{2L|w-z|}{\delta_D}\right)+1},\;w,z\in
[-1,1].
\end{equation*}
\qed
\end{proof}
Let us prove the general estimate for $J_+$.
\begin{theorem}\label{thm:estimates}
Given a positive number $N\in\mathbb N$ the following inequality holds
\begin{equation}
J_+(D;N)\geq \sup_{\eps >0}\left\{\left(\frac{\exp\left(\frac{2L\eps}{\delta_D}\right)-1}{\exp\left(\frac{2L\eps}{\delta_D}\right)+1}\right)^{2N}\left(\alpha([-1,1])-\omega(2N\eps,\alpha)\right)\right\}.
\end{equation}
where $\omega(\delta,\alpha):=\sup\left\{\alpha(A):A\subset [-1,1]\text{ is a Borel subset, } \lambda_1(A)\leq \delta\right\}$.
Moreover,
\begin{equation}
\lim\sb{\delta_D\to 0}J_+(D;N)=\alpha([-1,1]).
\end{equation}
\end{theorem}
\begin{proof} Fix
$\eps>0$. For any compact set $K$ denote $K^{\eps}:=\{z\in\CC:|z-x|<\eps
\text{ for some $x\in K$}\}$. Denote also $r:=r(k_1)+\ldots+r(k_n)$.
By decreasing the set of integration, applying Lemma~\ref{lem:unique-extremal} and the estimate (\ref{eq:koebe}), keeping in mind that the integrands take the values in the interval $[0,1)$ we get the following inequality
\begin{equation*}
J_+(D;(x_1,\ldots,x_n),(k_1,\ldots,k_n))\geq\int\sb{[-1,1]\setminus \{x_1,\ldots,x_n\}^{\eps}}\left(\frac{\exp\left(\frac{2L\eps}{\delta_D}\right)-1}{\exp\left(\frac{2L\eps}{\delta_D}\right)+1}\right)^{r}d\alpha.
\end{equation*}
Since $n\leq N$, we get that $r\leq 2N$ so
\begin{equation*}
J_+(D;(x_1,\ldots,x_n),(k_1,\ldots,k_n))\geq\left(\frac{\exp\left(\frac{2L\eps}{\delta_D}\right)-1}{\exp\left(\frac{2L\eps}{\delta_D}\right)+1}\right)^{2N}\int\sb{[-1,1]\setminus \{x_1,\ldots,x_n\}^{\eps}}d \alpha.
\end{equation*}
Since $\lambda_1(\{x_1,\ldots,x_n\}^{\eps})\leq 2n\eps\leq 2N\eps$ we conclude the proof of the proposition.
\qed
\end{proof}
Note that Theorem~\ref{thm:estimates} gives essential improvement of the
estimates in \cite{B}, \cite{P98}, $J(D;N)$ is estimated from below by a function
tending to $0$ as $\delta_D\to 0$. Moreover, the estimate in
\cite{B}, \cite{P98} are studied in detail for ellipses only.
\begin{theorem}\label{thm:lower-estimate}
Let $D\subset\mathbb C$ be a domain as above (i. e. simply connected, symmetric with respect to the $x$-axis, $[-1,1]\subset D$, $D\neq\mathbb C$) and let $\alpha=\lambda_1$. Then for any positive integer $N$ we get the following estimate (recall that in general case $L=1/4$ and in the case of $D$ convex $L=1/2$)
\begin{equation*}
J_+(D;N)\geq 2 L^{2N}\frac{\delta_D^{(2N\delta_D)/L}}{(\delta_D+L)^{(2N/L)(\delta_D+L)}}.
\end{equation*}
In the case $D$ is convex the above inequality gives
\begin{equation*}
J_+(D;N)\geq
2\left((1+1/(2\delta_D))^{2\delta_D}(2\delta_D+1)\right)^{-2N}\geq 2 \exp(-2N)(2\delta_D+1)^{-2N}.
\end{equation*}
\end{theorem}
\begin{proof}
Since for $ t\geq 0$
$$\frac{\exp(t)-1}{\exp(t)+1}\geq \frac{t}{2+t}$$ by (\ref{eq:koebe}) and Lemma~\ref{lem:unique-extremal} we get
\begin{multline*}
J_+(D;N) \geq \\
\inf\left\{\int\sb{-1}\sp{1}\prod\sb{j=1}\sp{n}\left(\frac{L|x-x_j|}{\delta_D+L|x-x_j|}\right)^{r(k_j)}dx:
n\in\mathbb N,\; -1\leq x_1<\ldots<x_n\leq 1,\;k_1+\ldots+k_n=N\right\}.
\end{multline*}
The Jensen inequality now implies that $J_+(D;N)$ is not less than the infimum of
\begin{equation}\label{eq:jensen2}
2\exp\left(1/2\sum\sb{j=1}\sp{n}r(k_j) \int\sb{-1}\sp{1}\left(\ln(L|x-x_j|)-\ln(\delta_D+L|x-x_j|)\right)dx\right).
\end{equation}
taken over all sequences $-1\leq x_1<\ldots<x_n\leq 1$, $k_1+\ldots+k_n=N$.
The integral in (\ref{eq:jensen2}) equals
\begin{multline*}
I_j =
2\ln L+(1-x_j)\ln (1-x_j)+(1+x_j)\ln (1+x_j) + \\
-\left(1/L\right)(L(1-x_j)+\delta_D)\ln (\delta_D+L(1-x_j)) +\\
-\left(1/L\right)(L(1+x_j)+\delta_D)\ln (\delta_D+L(1+x_j))+(2/L)\delta_D\ln \delta_D.
\end{multline*}
We now rewrite it in the form
\begin{equation*}
I_j=g(x_j) + g(-x_j) + 2\ln L + (2/L)\delta_D\ln \delta_D,
\end{equation*}
where
\begin{equation*}
g(t)=(1+t) \ln(1+t) -\frac{1}{L}(L(1+t)+ \delta_D) \ln (L(1+t)+ \delta_D), \; t\in[-1,1].
\end{equation*}
By setting $h(t):=g(t)+g(-t)$, $t\in[-1,1]$
we get
\begin{eqnarray*}
h'(t) & = & g'(t) - g'(-t)\\
& = & \ln \frac{1+t}{1-t} - \ln\frac{L(1+t)+ \delta_D}{L(1-t)+ \delta_D}\\
& = & \ln \frac{(1+t)(L(1-t)+ \delta_D)}{(1-t)(L(1+t)+ \delta_D)}.
\end{eqnarray*}
It is clear that $h$ is even and $h'(0)=0$. Moreover, $h'(t)>0$ for $t\in(0,1)$.
Indeed $h'(t)>0$ iff $(1+t)(L(1-t)+ \delta_D) > (1-t)(L(1+t)+ \delta_D)$. This condition is equivalent to
\begin{displaymath}
L(1-t^2) + (1+t) \delta_D > L(1-t^2) + (1-t) \delta_D,
\end{displaymath}
which is satisfied for $t>0$.
The above calculations show that the function defined by the formula
\begin{multline*}
(1+t)\ln(1+t)+(1-t)\ln(1-t)-\left(1/L\right)(L(1+t)+\delta_D)\ln(\delta_D+L(1+t))+\\
- \left(1/L\right)(L(1-t)+\delta_D)\ln(\delta_D+L(1-t))
\end{multline*}
attains its minimum on the interval $[-1,1]$ at $t=0$. Since $r=\sum r(k_j) \leq 2N$ we get
\begin{equation*}
\ln\left(J_+(D;N)/2\right) \geq 2N\left(\ln L-\left(1/L\right)(L+\delta_D)\ln (L+\delta_D)+\left(1/L\right)\delta_D\ln\delta_D\right)
\end{equation*}
and consequently
\begin{equation*}
J_+(D;N)\geq 2 L^{2N}\frac{\delta_D^{(2N\delta_D)/L}}{(\delta_D+L)^{(2N/L)(\delta_D+L)}}.
\end{equation*}
Note that the last expression tends to $2$ as $\delta_D\to 0$ (compare Theorem~\ref{thm:estimates}).
On the other hand, in the case when $D$ is convex, we have
\begin{eqnarray*}
J_+(D;N) & \geq & 2 L^{2N}\frac{\delta_D^{(2N\delta_D)/L}}{(\delta_D+L)^{(2N/L)(\delta_D+L)}}\\
& = & 2\left((1+1/(2\delta_D))^{2\delta_D}(2\delta_D+1)\right)^{-2N}\\
& > & 2 \exp(-2N)(2\delta_D+1)^{-2N}\label{eqn:low-exp},
\end{eqnarray*}
in view of the inequality $\left(1+1/x\right)^x < e$ for $x >0$.
\qed
\end{proof}
\textbf{Proof of Theorem~\ref{thm:main}:} Without loss of generality we may assume that $M=1$. Fix also the nodes $x_j$ and integers $k_j$, $j=1,\ldots,n$. Let $f$ be the unique function for which the supremum in the definition of $J_+(D;X;\mathcal K)$ is attained (compare Lemma~\ref{lem:unique-extremal}). Since the function $f$ belongs to the class $\mathcal F(D;X;\mathcal K)$, we get $f^{(l)}(z_j)=0$ for $l=0, \ldots, k_j-1;\;j=1,\ldots,n$ and consequently it gives the quadrature $Q$ the same information as does the function $g\equiv 0$. Therefore,
\begin{equation}
Q(f)=Q(g).
\end{equation}
From Theorem~\ref{thm:lower-estimate} it follows that
\begin{equation}
I(f) \geq 2 \gamma,
\end{equation}
where $\gamma$ is defined by (\ref{eq:gamma})
Since $Q(f)=0$, we immediately get
\begin{equation}
|I(f) - Q(f)| \geq \gamma.
\end{equation}
\qed
\subsection{The case of ellipses}
In the case when $D$ is an ellipse
\begin{equation*}
\mathcal{E}_c:=\{(x,y)\in\mathbb R^2:x^2/a^2+y^2/b^2<1\},
\end{equation*}
$a^2-b^2=1$, $c:=a+b$, $a,b>0$, simple computations lead to the relations $a=(c^2+1)/(2c)$, $b=(c^2-1)/(2c)$ and the formula\\
$$
\delta_{E_C}(x) = \left\{
\begin{array}{ll}
\sqrt{a^2-1}\sqrt{1-x^2},& x\in [-1/a,1/a],\\
\min\{|x\pm a|\}, & x\in[-1,1]\setminus(-1/a,1/a).
\end{array}
\right.
$$
Consequently, $\delta_{\mathcal{E}_c}=\sqrt{a^2-1}=(c^2-1)/(2c)$.
Therefore, as an immediate consequence of Theorem~\ref{thm:lower-estimate}, we get the following lower bound in the case of the ellipse and $\alpha$ being the Lebesgue measure.
\begin{cor}
\label{cor:ellipse}
Let $c>1$. Then
\begin{equation}
J_+(\mathcal{E}_c;N)\geq 2\left(\left(\frac{c^2-1+c}{c^2-1}\right)^{(c^2-1)/c}\left(\frac{c^2-1+c}{c}\right)\right)^{-2N}. \label{eq:low-bound-ell}
\end{equation}
\end{cor}
\begin{theorem}
\label{thm:ell-num-nodes-nbd}
Let $Q\in \Oq(n, \R)$, such that $|\R|= N$
be a quadrature on $\mathcal{E}_c$. Then for $c$ close to $1$, in order to have the error of $Q$ smaller than $\eps$, $N$ has to be greater than
\begin{equation}
N_l\left(\frac{M}{\eps},c\right)= \frac{-\ln \frac{M}{\eps}}{4 (c-1) \ln(c-1)} \left(1 + O\left(\left|\frac{1}{\ln(c-1)}\right|\right)\right).
\end{equation}
\end{theorem}
\proof
Let us remind the reader that for all functions appearing in the definition of $J_+(\mathcal{E}_c;N)$ we have had a bound $|f(z)| \leq 1$ for $z \in \mathcal{E}_c$.
Therefore from (\ref{eq:low-bound-ell}) (see also the proof of Thm.~\ref{thm:main}) it follows there exists a function $f_0 \in \A_0(\mathcal{E}_c,M)$ such that
\begin{equation}
\left|I(f_0) - Q(f_0)\right|\geq M \left(\left(\frac{c^2-1+c}{c^2-1}\right)^{(c^2-1)/c}\left(\frac{c^2-1+c}{c}\right)\right)^{-2N}.
\end{equation}
Therefore, to have an error less than $\eps$ we need to take $N \geq N_l$, where
\begin{eqnarray}\label{eqn:lower-N}
N_l= \frac{1}{2} \left(\ln \frac{M}{\eps}\right) \left(\ln\left( \left(\frac{c^2-1+c}{c^2-1}\right)^{(c^2-1)/c}\left(\frac{c^2-1+c}{c}\right)\right)\right)^{-1}.
\end{eqnarray}
Let us denote $\Delta=c-1$. Then for $c \to 1$ we obtain
\begin{eqnarray*}
D & :=& \ln\left( \left(\frac{c^2-1+c}{c^2-1}\right)^{(c^2-1)/c}\left(\frac{c^2-1+c}{c}\right)\right)\\
& = & \frac{c^2-1}{c}\left( \ln(c^2-1+c) - \ln(c-1)- \ln(c+1) \right) + \ln\left(1+\frac{c^2-1}{c}\right)\\
& = & (2\Delta + O(\Delta^2))\left(\ln (1+O(\Delta)) - \ln \Delta - \ln (2+\Delta)\right) + \ln(1+O(\Delta)) \\
& = & (2\Delta + O(\Delta^2))\left(O(\Delta) - \ln \Delta + O(1) \right) + O(\Delta)\\
& = & -2\Delta \ln \Delta + O(\Delta).
\end{eqnarray*}
Therefore
\begin{eqnarray*}
D^{-1}&=& \frac{-1}{2 \Delta \ln \Delta} \left(1 + O\left(\left|\frac{1}{\ln \Delta}\right|\right)\right),
\end{eqnarray*}
and from~(\ref{eqn:lower-N}) we obtain
\begin{eqnarray*}
N_l&=&\frac{-1}{4} \left(\ln \frac{M}{\eps}\right) \frac{1}{ \Delta \ln \Delta} \left(1 + O\left(\left|\frac{1}{\ln \Delta}\right|\right)\right).
\end{eqnarray*}
\qed
\section{Conclusions}
For an ellipse $\mathcal{E}_c$ and $\alpha$ being the Lebesgue measure, let us compare $N_l$, the lower bound for the pieces of information required, with $N_g$, the estimate of the number of nodes in the Gauss-Legendre quadrature
needed to obtain an error less than $\eps$, for $f \in \A_0(\mathcal{E}_c,M)$. From (\ref{eq:N-GL-nodes}) we obtain for $c \to 1^+$
\begin{eqnarray*}
\frac{N_l}{N_g} & = & \frac{-\ln \frac{M}{\eps}}{4 (c-1) \ln(c-1)} \left(1 + O\left(\left|\frac{1}{\ln(c-1)}\right|\right)\right) \cdot \left(\frac{\ln \frac{M}{\eps} }{\ln c}\right)^{-1} \\
& = & \frac{\ln c}{4 (c-1) \ln(c-1)}\left(1 + O\left(\left|\frac{1}{\ln(c-1)}\right|\right)\right)\\
& = & \frac{1+ O(c-1)}{4\ln(c-1)}\left(1 + O\left(\left|\frac{1}{\ln(c-1)}\right|\right)\right)\\
& \approx & \frac{1}{4 \ln (c-1)}.
\end{eqnarray*}
We see that $ N_l/N_g \to 0$ for $c \to 1^+$, hence we have not obtained (\ref{eq:optimality-crit}). It will be interesting to see whether the lower bound can be improved to obtain a positive lower bound for this ratio not dependent on $c$. By Remark~\ref{rmk:chebyshev-2n} the estimate for error for the Gauss-Legendre
quadrature is optimal and the improvement should be sought through better estimation of $J_+(\mathcal{E}_c,N)$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,664
|
Steam is a minimal and customizable theme for bloggers and was developed by [Tommaso Barbato](//github.com/epistrephein). He created it as a slightly adapted version of the [Vapor](//github.com/sethlilly/Vapor) Ghost theme by [Seth Lilly](//github.com/sethlilly). Noteworthy features of this Hugo port are the integration of a comment-system either powered by Disqus or Google Plus, the customizable appearance by changing theme colors, support for RSS feeds, syntax highlighting for source code and the optional use of Google Analytics. Enough to read. Let's take the first steps to get started.
## Contents
- [Installation](#installation)
- [The config file](#the-config-file)
- [Add links to the navigation](#add-links-to-the-navigation)
- [Customize theme colors](#customize-theme-colors)
- [Comments](#comments)
- [Nearly finished](#nearly-finished)
- [Contributing](#contributing)
- [License](#license)
- [Annotations](#annotations)
## Installation
Inside the folder of your Hugo site run:
$ mkdir themes
$ cd themes
$ git clone https://github.com/digitalcraftsman/hugo-steam-theme.git
For more information read the official [setup guide](//gohugo.io/overview/installing/) of Hugo.
### The config file
Take a look inside the [`exampleSite`](//github.com/digitalcraftsman/hugo-steam-theme/blob/dev/exampleSite/) folder of this theme. You'll find a file called [`config.toml`](//github.com/digitalcraftsman/hugo-steam-theme/blob/dev/exampleSite/config.toml).
To use it, copy the [`config.toml`](//github.com/digitalcraftsman/hugo-steam-theme/blob/dev/exampleSite/config.toml) in the root folder of your Hugo site. Feel free to change strings as you like to customize your website.
## Add links to the navigation
You can add custom pages like this by adding `menu = "main"` in the frontmatter:
```toml
+++
date = "2015-08-22"
title = "About me"
menu = "main"
+++
```
If no document contains menu = "main" in the frontmatter than the navigation will not be shown
## Customize theme colors
This theme features four different theme colors (green as default, blue, red and orange) that change the appearance of you Hugo site slightly. Just set the `themeColor` variable to the color you like.
Furthermore you can create your own theme. Under [`layouts/partials/themes`](//github.com/digitalcraftsman/hugo-steam-theme/tree/dev/layouts/partials/themes) you'll find a stylesheet template called [`custom-theme.html`](//github.com/digitalcraftsman/hugo-steam-theme/blob/dev/layouts/partials/themes/custom-theme.html). Customize the colors as you like and save the new theme with the schema `<myNewColor>-theme.html` within the same folder. As you can see, the color is the prefix of the stylesheet template. Therefore you just need to set `themeColor` in the [`configs`](//github.com/digitalcraftsman/hugo-steam-theme/blob/dev/exampleSite/config.toml)) to that self-defined prefix.
## Comments
This theme features a comment system that's either powered by Disqus or Google Plus. Enable one of those services by setting the `comments` variable in the the [`config.toml`](//github.com/digitalcraftsman/hugo-steam-theme/blob/dev/exampleSite/config.toml) to `disqus` or `googleplus`. In order to use Disqus you need to enter your shortname in disqusShortname at the top of the configuration file too.
## Nearly finished
In order to see your site in action, run Hugo's built-in local server.
$ hugo server -w
Now enter [`localhost:1313`](http://localhost:1313) in the address bar of your browser.
## Contributing
Did you found a bug or got an idea for a new feature? Feel free to use the [issue tracker](//github.com/digitalcraftsman/hugo-steam-theme/issues) to let me know. Or make directly a [pull request](//github.com/digitalcraftsman/hugo-steam-theme/pulls).
## License
This theme is released under the MIT license. For more information read the [License](//github.com/digitalcraftsman/hugo-steam-theme/blob/master/LICENSE.md).
## Annotations
Thanks to [Steve Francia](//github.com/spf13) for creating Hugo and the awesome community around the project.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,089
|
{"url":"https:\/\/stacks.math.columbia.edu\/tag\/05Z4","text":"Lemma 66.53.1. Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is a universal homeomorphism (in the sense of Section 66.3) if and only if for every morphism of algebraic spaces $Z \\to Y$ the base change map $Z \\times _ Y X \\to Z$ induces a homeomorphism $|Z \\times _ Y X| \\to |Z|$.\n\nProof. If for every morphism of algebraic spaces $Z \\to Y$ the base change map $Z \\times _ Y X \\to Z$ induces a homeomorphism $|Z \\times _ Y X| \\to |Z|$, then the same is true whenever $Z$ is a scheme, which formally implies that $f$ is a universal homeomorphism in the sense of Section 66.3. Conversely, if $f$ is a universal homeomorphism in the sense of Section 66.3 then $X \\to Y$ is integral, universally injective and surjective (by Spaces, Lemma 64.5.8 and Morphisms, Lemma 29.45.5). Hence $f$ is universally closed, see Lemma 66.45.7 and universally injective and (universally) surjective, i.e., $f$ is a universal homeomorphism. $\\square$\n\nThere are also:\n\n\u2022 1 comment(s) on Section 66.53: Universal homeomorphisms\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).","date":"2022-11-27 15:07:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9977553486824036, \"perplexity\": 197.03154108639615}, \"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-2022-49\/segments\/1669446710409.16\/warc\/CC-MAIN-20221127141808-20221127171808-00118.warc.gz\"}"}
| null | null |
\section{Introduction}
This paper studies automorphisms of hyperbolic and relatively hyperbolic groups in relation with their splittings.
The first result in this direction is due to Paulin \cite{Pau_arboreal}: if $G$ is a hyperbolic group with $\Out(G)$ infinite,
then $G$ has an action on an $\bbR$-tree with virtually cyclic (possibly finite) arc stabilizers.
Rips theory then implies that $G$ splits over a virtually cyclic group.
Understanding the global structure of $\Out(G)$ requires techniques which depend on the number of ends of $G$.
If $G$ is one-ended, there is an $\Out(G)$-invariant JSJ decomposition, and its study leads to Sela's description of $\Out(G)$
as a virtual extension of a direct product of mapping class groups
by a virtually abelian group \cite{Sela_structure,Lev_automorphisms}.
If $G$ has infinitely many ends, one does not get such a precise description because there is no $\Out(G)$-invariant splitting. One may study $\Out(G)$ by letting it act on a suitable space of splittings, the most famous being Culler-Vogtmann's outer space for $\Out(F_n)$.
Before moving on to relatively hyperbolic groups, here is a basic problem about which we get new results in the context of hyperbolic, and even free, groups.
Given a finitely generated subgroup $P$ of a group $G$,
consider the group $\Out(P \mathop{\nnearrow} G)\subset\Out(P)$ consisting of outer automorphisms of $P$ which extend to automorphisms of $G$.
What can one say about this group ?
For instance, is it finitely generated? finitely presented?
This question was asked by D.\ Calegari for automorphisms of free groups, and we answer it when $P$ is malnormal.
The answer is related to splittings through the following simple remark:
if $P$ is a vertex group in a splitting of $G$, say $G=A*_{C_1}P*_{C_2} B$,
then any automorphism of $P$ which acts \emph{trivially} (\ie as conjugation by some element $p_i\in P$)
on each incident edge group $C_i$,
extends to $G$
(this is the algebraic translation of the fact that any self-homeomorphism of a closed subset $Y\subset X$ which is the identity
on the frontier of $Y$ extends by the identity to a homeomorphism of $X$).
The following theorem says that this phenomenon accounts for almost all of $\Out(P \mathop{\nnearrow} G)$.
\begin{thm}[see Corollary \ref{cor_induit}]
\label{calhyp}
Let $P$ be a quasiconvex malnormal subgroup of a hyperbolic group $G$. If $\Out(P \mathop{\nnearrow} G)$ is infinite, then $P$ is a vertex group in a splitting of $G$, and the group of outer automorphisms of $P$ acting trivially
on incident edge groups has finite index in $\Out(P \mathop{\nnearrow} G)$.
\end{thm}
We call the group of outer automorphisms of $P$ acting trivially on a family of subgroups a \emph{McCool group of $P$}, because of McCool's paper about subgroups of $\Out(F_n)$ fixing a finite set of conjugacy classes \cite{McCool_fp}.
In this language,
Theorem \ref{calhyp}
says that $\Out(P\mathop{\nnearrow} G)$ is virtually a McCool group of $P$.
It is a theme of this paper
that
many groups of automorphisms may be understood in terms of McCool groups, and that many results concerning the full group $\Out(G)$
also apply to McCool groups (see also \cite{GL_McCool}).
The groups considered by McCool are finitely presented \cite{McCool_fp}.
In fact, they have a finite index subgroup with a finite classifying space \cite{CuVo_moduli}.
In \cite{GL_McCool} we extend these results to all McCool groups of torsion-free hyperbolic groups (and more generally of toral relatively hyperbolic groups). From this, one deduces that $\Out(P \mathop{\nnearrow} G)$ has a finite index subgroup with a finite classifying space when $G$ and $P$ are as in Theorem \ref{calhyp}, with $G$ torsion-free.
Our hypotheses for Theorem \ref{calhyp}, namely quasiconvexity and malnormality of $P$,
imply that $G$ is hyperbolic relative to $P$ (see \cite{Bow_relhyp}).
In fact, Theorem \ref{calhyp} is just a special case of a result describing
$\Out(P \mathop{\nnearrow} G)$ as a virtual McCool group when $G$ is relatively hyperbolic
and $P$ is a maximal parabolic subgroup (see Theorem \ref{caleg_general} for a precise statement).
\\
This paper also addresses the question of whether $\Out(G)$ is finite or infinite.
It turns out that the answer is much simpler when $G$ is torsion-free, owing to the fact that $\Out(G)$ is then infinite whenever $G$ has infinitely many ends (see Lemma \ref{freep}).
Things are more complicated when torsion is allowed. For instance, characterizing virtually free groups with $\Out(G)$ infinite is a non-trivial problem which was solved by Pettet \cite{Pettet_virtually}. The following theorem gives a different characterization.
We say that a
subgroup of $G$ is $\calz_{\mathrm{max}}$ if it is maximal for inclusion among virtually cyclic subgroups
with infinite center.
\begin{thm} [see Theorems \ref{thm_twist_hyp} and \ref{thm_MC_infini}] \label{twinf}
Let $G$ be a hyperbolic group.
Then $\Out(G)$ is infinite if and only
$G$ splits over a $\calz_{\mathrm{max}}$ subgroup $C$; in this case, any element $c\in C$ of infinite order defines a Dehn twist which has infinite order in $\Out(G)$.
Moreover, one may decide algorithmically whether $\Out(G)$ is finite or infinite.
\end{thm}
The first assertion
answers a question asked by D.\ Groves.
See \cite{Carette_automorphism} for a related result proved
independently by M.\ Carette, and
\cites{Lev_automorphisms, DG2} for the one-ended case.
Let us now consider (in)finiteness of $\Out(G)$ when $G$ is relatively hyperbolic.
Suppose that $G$ is hyperbolic with respect to a finite family $\calp$ of finitely generated subgroups $P_i$.
Since automorphisms of $G$ need not respect $\calp$ (for instance, $G$ may be free and $\calp$ may consist of any finitely generated malnormal subgroup), we consider the group $\Out(G;\calp)$ consisting of automorphisms mapping each $P_i$ to a conjugate (in an arbitrary way).
Note that $\Out(G;\calp)$ has finite index in the full group $\Out(G)$
when the groups $P_i$ are small but not virtually cyclic, more generally when they are not themselves relatively hyperbolic in a nontrivial way \cite{MiOs_fixed}.
Given a splitting of $G$, we have already pointed out that any automorphism of a vertex group
acting trivially on incident edge groups extends to an automorphism of $G$.
\emph{Twists} around edges of the splitting also provide automorphisms of $G$.
For instance, if $G=A*_C B$, and $a\in A$ centralizes $C$, there is an automorphism of
$G$ equal to conjugation by $a$ on $A$ and to the identity on $B$.
Note that we do not require that $C$ be virtually cyclic or that $a\in C$
(see Subsection \ref{arb}).
The following result says that
infiniteness of $\Out(G;\calp)$ comes from twists or from a McCool group of a parabolic group.
\begin{thm}[see Corollary \ref{cor_outu_infini}] \label{twinfr}
Let
$G$ be hyperbolic relative to a family $\calp=\{P_1,\dots,P_n\}$ of finitely generated subgroups.
Then $\Out(G;\calp)$ is infinite if and only if
$G$ has a splitting over virtually cyclic or parabolic subgroups, with each $P_i$ contained in a conjugate of a vertex group, such that one of the following holds:
\begin{itemize}
\item
the group of twists of the splitting is infinite;
\item
some $P_i$ is a vertex group and there are infinitely many outer automorphisms of $P_i$ acting trivially
on incident edge groups.
\end{itemize}
\end{thm}
As mentioned above, one can get similar results characterizing the infiniteness of McCool groups of $G$.
We refer to Section \ref{outinfi}, in particular Theorem \ref{thm_marked} and Corollary \ref{cor_outu_infini},
for more detailed statements.
\\
Let us now discuss the techniques that we use. We assume that $G$ is hyperbolic relative to $\calp$, and we distinguish two cases according to the number of ends
(technically, we
consider relative one-endedness, but we will ignore this in the introduction).
When $G$ is one-ended, we use a canonical $\Out(G;\calp)$-invariant decomposition $\Gamma_{\mathrm{can}}$ of $G$, namely (see Subsection \ref{cano}) the JSJ decomposition over \emph{elementary} (i.e.\ parabolic or virtually cyclic) subgroups relative to parabolic subgroups (i.e.\ parabolic subgroups have to be contained in conjugates of vertex groups).
One may thus generalize the description of $\Out(G)$ given by Sela for $G$ hyperbolic, and
express $\Out(G;\calp)$ in terms of mapping class groups, McCool groups of maximal parabolic subgroups, and a group of twists $\calt$. For simplicity we restrict to a special case here (see Section \ref{gen} for a general statement).
\begin{thm}[see Corollary \ref{limgp}]\label{thm_limgp_intro}
Let $G$ be toral relatively hyperbolic and one-ended.
Then some finite index subgroup $\Out^1(G)$ of $\Out(G)$ fits in an exact sequence
\begin{equation*}
1\ra \calt}%{\mathrm{Tw} \ra \Out^1(G) \ra \prod_{i=1}^p MCG^0(\Sigma_i) \times \prod_{k=1}^m
GL_{r_k,n_k}(\bbZ)
\ra 1
\end{equation*}
where $\calt}%{\mathrm{Tw}$ is finitely generated free abelian, $MCG^0(\Sigma_i)$ is the group of isotopy classes of homeomorphisms of a compact surface $\Sigma_i$ mapping each boundary component to itself in an orientation-preserving way, and $GL_{r,n}(\bbZ) $
is the group of automorphisms of $\bbZ^{r+n}$ fixing the first $n$ generators.
\end{thm}
More generally, McCool groups of a one-ended toral relatively hyperbolic group $G$
have a similar description
(see Corollary \ref{cor_MCtoral}).
A more general statement (without restriction on the parabolic subgroups) is given in Theorems \ref{thm_struct_m} and \ref{thm_struct_m_rel}.
We also show that the \emph{modular group} of $G$, introduced by Sela \cite{Sela_hopf,Sela_diophantine1}
and usually defined by considering all suitable splittings of $G$, may be seen on the single splitting $\Gamma_{\mathrm{can}}$. We refer to Section \ref{modul} for details.
To prove Theorem \ref{calhyp} when $G$ is one-ended,
one applies the previous analysis, viewing $G$ as hyperbolic relative to $P$. Note that we use a JSJ decomposition which is relative (to $P$), and over subgroups which are not small (any subgroup of $P$ is allowed).
Another example of the usefulness of relative JSJ decompositions is to prove the Scott conjecture about fixed subgroups of automorphisms of free groups. The proof that we give in Section \ref{fixed}, though not really new, is simplified by the use of the cyclic JSJ decomposition relative to the fixed subgroup.
We therefore work consistently in a relative context. We fix another family of finitely generated subgroups $\calh=\{H_1,\dots,H_{q}\}$, and we define $\Out (G;\calp,\mk\calh)$ as the group of automorphisms mapping $P_i$ to a conjugate (in an arbitrary way) and acting trivially on $H_j$ (\ie
as conjugation by an element $g_j$ of $G$).
In order to understand the structure of the automorphism group of a one-ended relatively hyperbolic group from its
canonical JSJ decomposition,
one needs to control automorphisms of rigid vertex groups. This is made possible by
the following generalisation of Paulin's theorem mentioned above:
\begin{thm}[see Theorem \ref{thm_sci}] \label{thm_sci3}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots,P_n\}$, with $P_i$ finitely
generated and $P_i\neq G$.
Let $\calh=\{H_1,\dots,H_q\}$ be another family of
finitely generated subgroups.
If $\Out(G;\calp,\mk\calh)$ is infinite, then $G$ splits
over an elementary (virtually cyclic or parabolic) subgroup relative to $\calp\cup\calh$.
\end{thm}
Note that there is no quasiconvexity or malnormality assumption on groups in $\calh$, but the automorphisms that we consider have to act
trivially on them (see also Remark \ref{lent}).
The theorem is proved using the Bestvina-Paulin method (extended to relatively hyperbolic groups in \cite{BeSz_endomorphisms})
to get an action on an $\R$-tree\ $T$, and then applying Rips theory as developed in \cite{BF_stable} to get a splitting.
There are technical difficulties in the second step because $G$ may fail to be finitely presented (the $P_i$'s are not required to be finitely presented),
and the action on $T$ may fail to be stable if the $P_i$'s are not
slender; it only satisfies a weaker property which we call \emph{hypostability},
and in the last section we generalize \cite{BF_stable} to hypostable actions of relatively finitely presented groups.
Theorem \ref {thm_sci3} explains why McCool groups appear in Theorems \ref{twinfr} and \ref{thm_limgp_intro}.
Indeed, given a rigid vertex group $G_v$ in a JSJ decomposition of a one-ended group,
Theorem \ref{thm_sci3} implies that only finitely many outer automorphisms of $G_v$ extend to automorphisms of $G$.
In turn, this implies that, after passing to a finite index subgroup, automorphisms of $G$ act
trivially on edge groups of the JSJ decomposition. See Subsection \ref{autar}
for details.
When $G$ is not one-ended, one has to consider splittings over finite groups.
We do not have an exact sequence as
in
Theorem \ref{thm_limgp_intro}
because there is no $ \Out (G;\calp)$-invariant splitting. In order to prove Theorems \ref{twinf} and \ref{twinfr}, we use the tree of cylinders introduced in \cite{GL4} to obtain a non-trivial splitting over finite groups which is invariant or has an infinite group of twists (Corollary \ref{propc_induct}).
The paper is organized as follows. Section \ref{prelim} consists of preliminaries (JSJ decompositions, automorphisms of a tree, trees of cylinders). Section \ref{relhyp} contains generalities about relatively hyperbolic groups. We point out that vertex groups of a splitting over relatively quasiconvex subgroups are relatively quasiconvex, and that the canonical JSJ decomposition $\Gamma_{\mathrm{can}}$ has finitely generated edge groups. In Section \ref{gen} we study the structure of the automorphism group of a one-ended relatively hyperbolic group.
Section \ref{modul} is devoted to the modular group. In Section \ref{sec_induced} we study extendable automorphisms; Theorem \ref{calhyp} is a special case of Theorem \ref{caleg_general}. Section \ref{outinfi} is devoted to the question of whether groups like $\Out (G;\calp,\mk\calh)$ and $\Out (G; \mk\calp,\mk\calh)$ are finite or infinite. Section \ref{fixed} contains a proof of the Scott conjecture, and a partial generalization to relatively hyperbolic groups. Theorem \ref{thm_sci3} is proved in Section \ref{sec_Rips}.
Acknowledgments. We thank D.\ Groves and D.\ Calegari for asking stimulating questions, and the organizers of the 2007 Geometric Group Theory program at MSRI where this research was started. We also thank the referee for helpful comments.
This work was partly supported by
ANR-07-BLAN-0141, ANR-2010-BLAN-116-01,
ANR-06-JCJC-0099-01,
ANR-11-BS01-013-01.
\tableofcontents
\section{Preliminaries}\label{prelim}
Unless mentioned otherwise, $G$ will always be a finitely generated group.
Given a group $A$ and a subgroup $B$, we denote by $Z(A)$ the center of $A$, by $Z_A(B)$ the centralizer of $B$ in $A$, and by $N_A(B)$ the normalizer of $B$ in $A$.
We write $B^g$ for $gBg\m$.
A subgroup $B\subset A$ is \emph{malnormal} if $B^g\cap B$ is trivial for all $g\notin B$, \emph{almost malnormal} if $B^g\cap B$ is finite for all $g\notin B$.
A group is \emph{virtually cyclic} if it has a cyclic subgroup of finite index; it may be finite or infinite. Its outer automorphism group is finite.
A group $G$ is \emph{slender} if $G$ and all its subgroups are finitely generated. We say that $G$ is \emph{small} if it contains no non-abelian free group (see \cite{BF_bounding} for a slightly weaker definition).
Let $\calp$ be a family of subgroups $P_i$.
In most cases, $\calp$ will be a finite collection of finitely generated groups $\calp=\{P_1,\dots,P_n\}$.
The group $G$ is \emph{finitely presented relative to $\calp=\{P_1,\dots, P_n\} $}
if it is the quotient of $P_1*\dots* P_n*F $ by the normal closure of a finite subset, with $F$ a finitely generated free group. If $G$ is finitely presented relative to $\calp=\{P_1,\dots, P_n\} $, and if $g_1,\dots,g_n\in G$,
then $G$ is also finitely presented relative to $\{P_1^{g_1},\dots,P_n^{g_n}\}$.
Let $H$ be a finitely generated subgroup. If $G$ is finitely presented relative to $\calp$, then $G$ is finitely presented relative to $\calp\cup\{H\}$; the converse is not true in general.
\subsection{Relative automorphisms}\label{sec_relaut}
Given $G$ and $\calp$, we denote by
$\Aut (G;\calp)$ the subgroup of $\Aut(G)$ consisting of automorphisms mapping each $P_i$ to a conjugate, and by $\Out (G;\calp)$ its image in $\Out(G)$. If $\calp=\{P_1,\dots, P_n\}$, we use the notations $\Aut (G;P_1,\dots, P_n)$ and $\Out (G;P_1,\dots, P_n)$. We also write $\Out (G; \calp, \calq)$ for $\Out (G; {\calp\cup\calq})$.
We define
$\Out (G;\mk\calp)\subset\Out (G;\calp)$ by restricting to automorphisms whose restriction to each $P_i$ agrees with conjugation by some element $g_i$ of $G$ (we call them \emph{marked automorphisms},
or \emph{automorphisms acting trivially on $\calp$}). An outer automorphism $\Phi$ belongs to $\Out (G;\mk\calp)$ if and only if, for each $i$, it has a representative $\alpha_i\in \Aut(G)$ equal to the identity on $P_i$.
The group $\Out (G;\mk\calp)$ is denoted by PMCG in \cite{Lev_automorphisms}, by $\Out_m(G;\calp)$ in \cite{DG2}, and is called a (generalized) McCool group in \cite{GL_McCool}.
Note that $\Out(G;\mk G)$ is trivial, and that $\Out (G;\mk\calp)$ has finite index in $\Out (G;\calp)$ if $\calp$ is a finite collection of finite groups (more generally, of groups $P$ with $\Out(P)$ finite).
If $\calh=\{H_1,\dots,H_{q}\}$
is another family of subgroups,
we define
$\Out (G;\calp,\mk\calh)=\Out(G;\calp)\cap\Out(G;\mk\calh)$. We allow $\calp$ or $\calh$ to be empty,
in which case we do not write it.
The groups defined above do not change if we replace each $P_i$ or $H_j$ by a conjugate,
or if we add conjugates of the $P_i$'s to $\calp$ or conjugates of the $H_j$'s to $\calh$.
\subsection{Splittings}
A \emph{splitting} of a group $G$ is an isomorphism between $G$ and the fundamental group of a graph of groups $\Gamma$. Equivalently, using Bass-Serre theory, we view a splitting of $G$ as an action of $G$ on a simplicial tree $T$, with $T/G=\Gamma$. This tree is well-defined up to equivariant isomorphism,
and two splittings are considered equal if there is an equivariant isomorphism between the corresponding Bass-Serre trees.
The group $\Out(G)$ acts on the set of splittings of $G$ (by changing the isomorphism between $G$ and $\pi_1(\Gamma)$, or precomposing the action on $T$).
Trees will always be simplicial trees with an action of $G$ without inversion. We usually assume that the splitting is \emph{minimal} (there is no proper $G$-invariant subtree). Since $G$ is assumed to be finitely generated, this implies that $\Gamma$ is a finite graph.
A splitting is \emph{trivial} if $G$ fixes a point in $T$ (minimality then implies that $T$ is a point).
A splitting is \emph{relative to $\calp$} if every $P_i$ is conjugate to a subgroup of a vertex group, or equivalently if $P_i$ is \emph{elliptic} (i.e.\ fixes a point) in $T$. If $\calp=\{P_1,\dots,P_{n}\}$, we also say that the splitting is relative to $P_1,\dots, P_n$.
The group $G$ \emph{splits over a subgroup $A$} (relative to $\calp$) if there is a non-trivial
minimal splitting (relative to $\calp$) such that
$A$ is an edge group. The group is \emph{one-ended relative to $\calp$} if it does not split over a finite group relative to $\calp$.
If $\Gamma$ is a graph of groups, we denote by $V$ its set of vertices, and by $E$ its set of oriented edges. The origin of an edge $e\in E$ is denoted by $o(e)$. A vertex $v$ or an edge $e$ carries a group $G_v$ or $G_e$, and there is an inclusion $i_e:G_e\to G_{o(e)}$
\subsection{Trees and deformation spaces \cite{For_deformation,GL2}}\label{defsp}
In this subsection we consider trees rather than graphs of groups. We denote by $G_v$ or $G_e$ the stabilizer of a vertex or an edge.
We often restrict edge stabilizers of $T$ by requiring that they belong to a family $\cala$ of subgroups of $G$ which is stable under taking subgroups and under conjugation. We then say that $T$ is an \emph{$\cala$-tree}. For instance, $\cala$ may be the set of finite subgroups, of cyclic subgroups, of abelian subgroups, of elementary subgroups of a relatively hyperbolic group (see Section \ref{relhyp}). We then speak of cyclic, abelian, elementary trees (or splittings).
Besides restricting edge stabilizers, we also often restrict to trees $T$ \emph{relative to $\calp$}: every $P_i$ is elliptic in $T$. We then say that $T$ is an \emph{$(\cala,\calp)$-tree}.
A tree $T'$ is a \emph{collapse} of $T$ if it is obtained from $T$ by collapsing
each edge in a certain $G$-invariant
collection
to a point; conversely, we say that $T$ \emph{refines} $T'$.
In terms of graphs of groups, one passes from $\Gamma=T/G$ to $\Gamma'=T'/G$ by collapsing edges;
for each vertex $v'\in\Gamma'$,
the vertex group $G_{v'}$ is the fundamental group of the graph of groups occuring as the preimage of $v'$ in $\Gamma$.
Given two trees $T$ and $T'$, we say that $T$ \emph{dominates} $T'$ if there is a $G$-equivariant map $f:T\to T'$, or equivalently if every subgroup which is elliptic in $T$ is also elliptic in $T'$. In particular, $T$ dominates any collapse $T'$.
Two trees belong to the same \emph{deformation space} if they dominate each other. In other words, a deformation space $\cald$ is the set of all trees having a given family of subgroups as their elliptic subgroups. We denote by $\cald(T)$ the deformation space containing a tree $T$, and by $\Out(\cald)\subset\Out(G)$ the group of automorphisms leaving $\cald$ invariant. The set of $\cala$-trees contained in a deformation space is called a deformation space over $\cala$ (and usually denoted by $\cald$ also).
A tree is \emph{reduced} if $G_e\ne G_v,G_w$ whenever an edge $e$ has its endpoints $v,w$ in different $G$-orbits. Equivalently, no tree obtained from $T$ by collapsing the orbit of an edge belongs to the same deformation space as $T$. If $T$ is not reduced, one may collapse edges so as to obtain a reduced tree in the same deformation space.
Any two reduced trees in a
deformation space over finite groups may be joined by slide moves (see \cite{For_deformation,GL2} for definitions).
In particular, they have the same set of edge and vertex stabilizers.
\subsection{Induced structures} \label{ind}
\begin{dfn} [Incident edge groups $\mathrm{Inc}_v$] \label{iv}
Given a vertex $v$ of a graph of groups $\Gamma$, we denote by $\mathrm{Inc}_v$ the collection of all subgroups $i_e(G_e)$ of $G_v$, for $e$ an edge with origin $v$. We call $\mathrm{Inc}_v$ the set of \emph{incident edge groups.} We also use the notation $\mathrm{Inc}_{G_v}$.
\end{dfn}
Similarly, if $v$ is a vertex of a (minimal) tree,
there are finitely many $G_v$-orbits of incident edges,
and
$\mathrm{Inc}_v$ is the family of stabilizers of some representatives of these orbits.
This is a finite collection of subgroups of $G_v$, each well-defined up to conjugacy.
Any splitting $\Gamma_v$ of $G_v$ relative to $\mathrm{Inc}_v$ extends (non-uniquely) to a splitting $\Lambda$ of $G$, whose edges are those of $\Gamma_v$ together with those of $\Gamma$; an edge $e$ of $\Gamma$ incident to $v$ is attached to a vertex of $\Gamma_v$ whose group contains $G_e$ (up to conjugacy). We call this \emph{refining} $\Gamma $ at $v$ using $\Gamma_v$. One recovers $\Gamma$ from $\Lambda$ by collapsing edges of
$\Gamma_v$.
\begin{lem} \label{mar} Consider subgroups $H\subset K\subset G$ such that, if $g\in G$ and $H^g\subset K$, then $ g\in K$
(this holds in particular when $K$ is a vertex stabilizer of a tree $T$, and $H$ is a subgroup which fixes no edge of $T$).
\begin{itemize*}
\item
If $H'\subset K$ is conjugate to $H$ in $G$, it is conjugate to $H$ in $K$.
\item
If $\alpha\in \Aut(G)$ leaves $K$ invariant and
maps $H$ to $gHg\m$, then $g\in K$. \qed
\end{itemize*}
\end{lem}
This lemma is trivial, but very useful.
Given a vertex stabilizer $G_v$ of a tree $T$,
it allows us to define a family
$\calp_{ | G_v}$ as follows (like $\mathrm{Inc}_v$, it
is a
finite set of subgroups of $G_v$, each well-defined up to conjugacy).
\begin{dfn}[Induced structure $\calp_{|G_v}$] \label{indu}
Let $\calp=\{P_i\}$ be a collection of subgroups of $G$, and let $G_v$ be a vertex stabilizer in a
tree $T$ relative to $\calp$.
For each $i$ such that $P_i$ fixes a point in the orbit of $v$, but fixes no edge of $T$, let $\Tilde P_i\subset G_v$ be a conjugate of $P_i$.
When defined, $\Tilde P_i$ is unique up to conjugacy in $G_v$ by Lemma \ref{mar}.
We define $\calp_{|G_v}$ as this collection of subgroups $\Tilde P_i\subset G_v$;
we define $\calp_{|G_v}$ similarly if $G_v$ is a vertex group of $\Gamma=T/G$.
\end{dfn}
\begin{rem} \label{tric}
Given $v$ and $i$, one of the following always holds: $P_i$ fixes a vertex of $T$ not in the orbit of $v$,
or some conjugate of $P_i$ fixes $v$ and an edge incident to $v$,
or $P_i$ is conjugate to a group in $\calp_{ | G_v}$.
\end{rem}
\begin{rem}
In this definition, $\calp_{|G_v}$ depends not only on $\calp$ and $G_v$, but also
on the incident edge groups of $G_v$.
In practice, we will not work with $\calp_{|G_v}$ alone, but with $\calq_v=\calp_{|G_v}\cup\mathrm{Inc}_v$. This is
the case for instance in the following lemma.
\end{rem}
\begin{lem} \label{extens}
If $\Gamma_v$ is a splitting of $G_v$ relative to $\calq_v=\mathrm{Inc}_v\cup\calp_{ | G_v}$,
refining $\Gamma$ at $v$
using $\Gamma_v$ yields a splitting $\Lambda$ of $G$ which is relative to $\calp$.
\end{lem}
\begin{proof} The refinement is possible because $\Gamma_v$ is relative to $\mathrm{Inc}_v$.
Each $P_i$ is elliptic in $\Lambda$ by Remark \ref{tric}.
\end{proof}
\subsection{JSJ decompositions \cite{GL3a}}\label{JSJ}
Fix $\cala$ as in Subsection \ref{defsp}, and a (possibly empty) family $\calp$
of subgroups. All trees considered here are $(\cala,\calp)$-trees.
A subgroup $H$ is \emph{universally elliptic} if it is elliptic (fixes a point) in every tree. A tree is universally elliptic if its edge stabilizers are.
A tree $T$ is a \emph{JSJ tree} over $\cala$ relative to $\calp$ if it is universally elliptic and dominates every universally elliptic tree (see Section 4 of \cite{GL3a}). JSJ trees exist if $G$ is finitely presented relative to $\calp$.
The set of JSJ trees, if non-empty, is a deformation space called the \emph{JSJ deformation space} over $\cala$ relative to $\calp$.
When $\cala$ is the set of cyclic (abelian, elementary...) subgroups, we refer to the cyclic (abelian, elementary...) JSJ.
When $\cala$ is the set of finite subgroups, and $\calp=\emptyset$,
the JSJ deformation space is the \emph{Stallings-Dunwoody deformation space}, characterized by the property that its trees have vertex stabilizers with at most one end (see Section 6 of \cite{GL3a}).
Because it is a deformation space over finite groups, all its reduced trees have the same edge and vertex stabilizers (see Subsection \ref{defsp}).
This space exists if and only if $G$ is accessible, in particular when $G$ is finitely presented \cite{Dun_accessibility}.
If $G$ is torsion-free, the Stallings-Dunwoody deformation space is
the \emph{Grushko deformation space}; edge stabilizers are trivial, vertex stabilizers are freely indecomposable and non-cyclic.
More generally, the JSJ deformation space over finite subgroups relative to $\calp$ will be called the Stallings-Dunwoody deformation space relative to $\calp$.
We will also consider JSJ spaces over finite subgroups of cardinality bounded by some $k$; these exist whenever $G$ is finitely generated by Linnell's accessibility \cite{Linnell}.
If $T $ is a tree (in particular, if it is a JSJ tree), a vertex stabilizer $G_v$ of $T $ not belonging to $\cala$ (or $v$ itself) is \emph{rigid} if it is universally elliptic.
Otherwise, $G_v$ (or $v$) is \emph{flexible}.
In many situations, flexible vertex stabilizers $G_v$ of JSJ trees are {quadratically hanging} subgroups (see Section 7 of \cite{GL3a}).
\begin{dfn} [QH vertex] \label{dqh}
A vertex stabilizer $G_v$ (or $v$) is \emph{quadratically hanging}, or \emph{QH},
(relative to $\calp$) if
there is a normal subgroup $F\normal G_v$ (called the \emph{fiber} of $G_v$)
such that $G_v/F$ is isomorphic to the fundamental group $\pi_1(\Sigma)$ of a hyperbolic $2$-orbifold $\Sigma$ (usually with boundary); moreover, if $H\subset G_v$ is an incident edge stabilizer, or is the intersection of $G_v$ with a conjugate of a group in $\calp$, then the image of $H$
in $\pi_1(\Sigma)$ is finite or contained in a \emph{boundary subgroup}
(a subgroup conjugate to the fundamental group of a boundary component).
\end{dfn}
\begin{dfn}[Full boundary subgroups]
\label{bv} Let $G_v$ be a QH vertex stabilizer. For each boundary component of $\Sigma$, we select a representative for the conjugacy class of its fundamental group in $\pi_1(\Sigma)$,
and we consider its full preimage in $G_v$.
This defines a finite family $\calb_v$ of subgroups of $G_v$.
\end{dfn}
If $G_v$ is QH with finite fiber,
every infinite incident edge stabilizer
is virtually cyclic and (up to conjugacy in $G_v$) contained with finite index in a group of $\calb_v$.
If $G$ is one-ended relative to $\calp$, then every incident edge stabilizer is infinite, so
is contained in a group of $\calb_v$.
\begin{rem}\label{rem_UEQH}
If $G_v$ is a \emph{flexible} QH vertex stabilizer
with finite fiber, and $H\subset G_v$ is universally elliptic,
then the image of $ H$ in $\pi_1(\Sigma)$ is finite or contained in a boundary subgroup (see Proposition 7.6 of \cite{GL3a}; this requires a technical assumption on $\cala$, which holds in all cases considered in the present paper).
In particular, $H$ is virtually cyclic.
\end{rem}
\subsection{The automorphism group of a tree \cite{Lev_automorphisms}}\label{arb}
Let $T$ be a tree with a minimal action of $G$. We assume that $T$ is not a line with $G$ acting by translations.
We denote by $\Aut(T)\subset\Aut(G)$ the group
of automorphisms $\alpha$ \emph{leaving $T$ invariant}:
there exists an isomorphism
$f_\alpha:T\ra T$ which is $\alpha$-equivariant in the sense that $f_\alpha(gx)=\alpha(g)f_\alpha(x)$ for $g\in G$ and $x\in T$.
Following \cite{Lev_automorphisms},
we describe
the image $\Out(T)$ of $\Aut(T)$ in $\Out(G)$ in terms of the graph of groups $\Gamma=T/G$. Our assumptions on $T$ imply that $\Gamma$ is minimal, and is not a mapping torus (as defined in \cite{Lev_automorphisms}).
The group $\Out(T)$ acts on the finite graph $\Gamma=T/G$,
and we define $\Out^0(T)$ as the finite index subgroup acting trivially.
We use the notations $\Out(T;\calp,\mk\calh)=\Out(T)\cap\Out(G;\calp,\mk\calh)$,
and $\Out^0(T;\calp,\mk\calh)=\Out^0(T)\cap\Out(G;\calp,\mk\calh)$ (see Section \ref{sec_relaut}).
\paragraph{Action on vertex groups.}
If $v\in V$ is a vertex of $\Gamma$, there is a natural map $\rho_v:\Out^0(T)\to \Out(G _v)$ defined as follows. Let $\Phi\in\Out^0(T)$. When $N_G(G_v) $ acts on $G_v$ by inner automorphisms, in particular when $G_v$ fixes a unique point in $T$ (in this case $N_G(G_v)=G_v$), one defines $\rho_v(\Phi)$ simply by choosing any representative $\alpha\in\Aut(T)$ of $\Phi$ leaving $G_v$ invariant and considering its restriction to $G_v$. In general, one has to choose $\alpha$ more carefully: one fixes a vertex $\tilde v$ of $T$ mapping to $v$ such that the stabilizer of $\Tilde v$ is $G_v$, one chooses $\alpha$ so that $f_\alpha$ fixes $\tilde v$, and
$\rho_v(\Phi)$ is represented by the restriction $\alpha_{|G_{v}}$.
If $e$ is an edge of $\Gamma$, one may define $\rho_e:\Out^0(T)\to \Out(G _e)$ similarly.
Let $\rho :\Out^0(T)\to \prod_{v\in V}\Out(G _v)$ be the product map.
As pointed out
in Subsection 2.3 of \cite{Lev_automorphisms}, the image $\rho (\Out^0(T))$ contains $\prod_{v\in V}\Out (G_v;\mk\mathrm{Inc}_v)$
and is contained in $\prod_{v\in V}\Out (G_v;\mathrm{Inc}_v)$. More precisely:
\begin{lem}\label{automind} Let $\calp,\calh$ be two
families of
subgroups of $G$.
Let $T$ be a tree relative to $\calp\cup \calh$.
Then
\begin{equation}\label{eq_incl}
\Out(G_v;\mk\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})
\subset \rho_v (\Out^0(T;\calp,\mk \calh))
\subset
\Out(G_v;\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v}) \end{equation}
for every $v\in V$, and
$$\prod_{v\in V}
\Out(G_v;\mk\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})
\subset \rho (\Out^0(T;\calp,\mk \calh))
\subset \prod_{v\in V}
\Out(G_v;\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v}).$$
If $\Out(G_e)$ is finite for all edges (resp.\ for all edges $e$ incident to $v$),
all inclusions (resp.\ all inclusions in (\ref{eq_incl})) have images of finite index.
\end{lem}
Recall that $\calp_{|G_v}$ was defined in Definition \ref{indu}.
\begin{proof}
The inclusion $ \prod_{v\in V}\Out(G_v;\mk\mathrm{Inc}_v)\subset \rho (\Out^0(T))$ is proved in \cite{Lev_automorphisms} by extending any
$\Phi_v\in \Out (G_v;\mk\mathrm{Inc}_v)$
``by the identity'' to get
$\Phi\in\Out^0(T)$, with $\Phi_v=\rho_v(\Phi)$, acting as a conjugation on each edge group
and on each $G_w$ for $w\ne v$. The left hand side inclusions in the lemma
follow from Remark \ref{tric}.
The inclusion $\rho_v(\Out^0(T))\subset \Out(G_v;\mathrm{Inc}_v)$
follows from the fact that, given an edge $e$ of $T$ containing the lift $\tilde v$ of $v$ used to define $\rho_v$,
any $\Phi\in \Out^0(T)$ has a representative $\alpha$ such that $f_\alpha$ fixes $e$; this representative induces an automorphism of $G_{\tilde v}$ leaving $G_e$ invariant. To prove the right hand side inclusions, apply Lemma \ref{mar} with $K=G_{\tilde v}$, recalling that groups in
$\calh_{|G_v}$ or $\calp_{|G_v}$ fix a unique point in $T$.
If $\Out(G_e)$ is finite for all incident edge groups,
$\Out(G_v;\mk\mathrm{Inc}_v)$ has finite index in $\Out(G_v;\mathrm{Inc}_v)$ (see Proposition 2.3 in \cite{Lev_automorphisms}).
Intersecting with $\Out(G_v;\calp_{|G_v},\mk\calh_{|G_v})$,
we get that $\Out(G_v;\mk\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})$
has finite index in
$ \Out(G_v;\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})$.
This conludes the proof.
\end{proof}
\begin{rem}\label{automind2}
We can also consider automorphisms which do not leave $T$ invariant, but
only leave some vertex stabilizer $G_v$ invariant.
Assuming that $G_v$ equals its normalizer, there is a natural map $\rho_v:\Out(G;G_v)\to\Out(G_v)$ and
$$
\Out(G_v;\mk\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})
\subset \rho_v (\Out(G;G_v,\calp,\mk \calh))
\subset
\Out(G_v; \calp_{|G_v},\mk\calh_{|G_v}).$$
\end{rem}
\paragraph{Twists.}
As in Subsection 2.5 of \cite{Lev_automorphisms},
we now consider the kernel of the product map
$\rho :\Out^0(T)\to \prod_{v\in V}\Out(G _v)$.
It consists of automorphisms in $\Out^0(T)$ having, for each $v$,
a representative in $\Aut(T)$ whose restriction to $G_v$ is the identity.
If $T$ is relative to $\calp$, the group $\ker\rho$ is contained in
$\Out(T;\mk\calp
)
$.
To study $\ker\rho$, we need to introduce the \emph{group of twists} $\calt$ associated to $T$ or equivalently to $\Gamma$ (we write $\calt}%{\mathrm{Tw}(T)$ or $\calt}%{\mathrm{Tw}(\Gamma)$ if there is a risk of confusion).
Let $e $ be
a separating edge of $\Gamma$ with origin $v$ and endpoint $w$. Then $G=A*_{G_e}B$ with $G_v\subset A$ and $G_w\subset B$. Given $g\in Z_{G_v}(G_e)$, one defines the
\emph{twist by $g$ around $e$ near $v$} as the (image in $\Out(G)$ of the) automorphism of $G$ equal to the identity on $B$ and to conjugation by $g$ on $A$. There is a similar definition in the case of an HNN extension $G=A*_C=\grp{ A,t\mid tct\m=\phi(c),c
\in C}$: given $g\in Z_A(C)$,
the twist by $g$ is the identity on $A$ and sends $t$ to $tg$.
The group $\calt$ is the subgroup of $\Out(G)$ generated by all twists.
It is a quotient of $\prod_{e\in E}Z_{G_{o(e)}}(G_e)$ and is contained in $\ker\rho$.
The following facts follow directly from Section 2 of \cite{Lev_automorphisms}.
\begin{lem} \label{lem_virt_nobitwist}
\begin{enumerate}
\item
If every $\Out(G_e)$ is finite, then $\calt$ has finite index in $\ker \rho$.
\item Assume that every non-oriented
edge $e$ of $\Gamma$ has an endpoint $v$ such that $N_{G_v}(G_e)$ acts on $G_e$ by inner automorphisms (this holds in particular if $G_v$ is abelian, or if $G_e$ is malnormal in $G_v$, or if $G_e$ is infinite and almost malnormal in $G_v$). Then $\calt=\ker\rho$. \qed
\end{enumerate}
\end{lem}
The kernel of the epimorphism from $\prod_{e\in E}Z_{G_{o(e)}}(G_e)$ to $\calt$ is the image of a natural map $$j:\prod_{v\in V}Z(G_v)\times\prod_{e\in\cale}Z(G_e)\to\prod_{e\in E}Z_{G_{o(e)}}(G_e)$$
where $\cale$ is the set of non-oriented edges of $\Gamma$ (see Proposition 3.1 of \cite{Lev_automorphisms}). The image of an element of $Z(G_v)$ is called a \emph{vertex relation} at $v$, the image of an element of $Z(G_e)$ is an \emph{edge relation}.
For instance, if $\Gamma$ is a non-trivial amalgam $G=A*_C B$, then $\calt}%{\mathrm{Tw}$ is the image of the map $p:Z_A(C)\times Z_B(C)\to\Out (G)$ sending $(a,b)$ to the class of the automorphism acting on $A$ as conjugation by $a$ and on $B$ as conjugation by $b$. The kernel of $p$ is generated by the elements $(a,1)$ with $a\in Z(A)$ and $(1,b)$ with $b\in Z(B)$ (vertex relations), together with the elements
$(c,c)$ with $c\in Z(C)$ (edge relations).
\begin{lem}\label{lem_twist0}
Let $e$ be an edge of $\Gamma$ with origin $v$. If $Z(G_e)$ and $Z(G_v)$ are finite, but
$Z_{G_v}(G_e)$ is infinite,
then the image of $Z_{G_v}(G_e)$ in $\calt}%{\mathrm{Tw} $ is infinite.
In particular, $\calt}%{\mathrm{Tw} $ is infinite.
\end{lem}
Note that $Z_{G_v}(G_e)$ is infinite if $N_{G_v}(G_e)$ is infinite and $G_e$ is finite.
\begin{proof}
It is pointed out in \cite[Lemma 3.2]{Lev_GBS} that the image of $Z_{G_v}(G_e)$ in $\calt}%{\mathrm{Tw}$ maps onto the quotient $Z_{G_v}(G_e)/\langle Z(G_v), Z(G_e)\rangle$. Since $Z(G_v)$ and $Z(G_e)$ are commuting finite subgroups, this quotient is infinite.
\end{proof}
Let $\Gamma$ be a graph of groups with fundamental group $G$, and $\Gamma_0\subset\Gamma$ a connected subgraph. We view $\Gamma_0$ as a graph of groups, with fundamental group $G_0\subset G$ and associated group of twists $\calt}%{\mathrm{Tw}(\Gamma_0)\subset\Out(G_0)$.
\begin{lem}\label{lem_extension_twist}
If $\calt}%{\mathrm{Tw} (\Gamma_0)$ is infinite, then so is $\calt}%{\mathrm{Tw}(\Gamma)$.
\end{lem}
\begin{proof}
Let $E_0$ be the set of oriented edges of $\Gamma_0$. The projection from $\prod_{e\in E}Z_{G_{o(e)}}(G_e)$ to $\prod_{e\in E_0}Z_{G_{o(e)}}(G_e)$ obtained by keeping only the factors with $e\subset \Gamma_0$ is compatible with the vertex and edge relations, so induces
an epimorphism from $\calt}%{\mathrm{Tw} (\Gamma)$ to $\calt}%{\mathrm{Tw}(\Gamma_0)$.
\end{proof}
\begin{lem}\label{lem_collapse_twist}
If $\Gamma$ is a graph of groups with $\calt}%{\mathrm{Tw} (\Gamma)$ infinite, there is an edge $e$ such that the graph of groups $\Gamma_e$ obtained by collapsing every edge except $e$ has $\calt}%{\mathrm{Tw} (\Gamma_e)$ infinite.
\end{lem}
\begin{proof}
There is an edge $e$ such that $Z_{G_{o(e)}}(G_e)$ has infinite image in $\Out(G)$. Twists of $\Gamma$ around $e$ are also twists of $\Gamma_e$.
\end{proof}
\subsection{Trees of cylinders \cite{GL4}}\label{defcyl}
We recall some basic properties of the tree of cylinders (see Section 4 of \cite{GL4} for details).
Besides $\cala$ (and possibly $\calp$),
we have to fix
a conjugacy-invariant subfamily $\cale\subset\cala$ and an admissible equivalence relation $\sim$ on $\cale$. Rather than giving a general definition, we describe the examples that will be used in this paper (with $\cala$ consisting of all finite, elementary, or abelian subgroups respectively):
\begin{enumerate}
\item $\cale$ consists of all finite subgroups of a fixed order $k$, and $\sim$ is equality.
\item $G$ is relatively hyperbolic (see Section \ref{relhyp}), $\cale$ consists of all infinite elementary subgroups (parabolic or loxodromic), and $\sim$ is co-elementarity: $A\sim B$ if and only if $\langle A,B\rangle$ is elementary.
\item $G$ is a torsion-free CSA group, $\cale$ consists of all infinite abelian subgroups, and $\sim$ is commutation: $A\sim B$ if and only if $\langle A,B\rangle$ is abelian (recall that $G$ is CSA if centralizers of non-trivial elements are abelian and malnormal).
\end{enumerate}
Let $T$ be a tree with edge stabilizers in $\cale$. We declare two (non-oriented) edges $e,f$ to be equivalent if $G_e\sim G_f$.
The union of all edges in an equivalence class is a subtree $Y$, called a \emph{cylinder} of $T$. Two distinct cylinders meet in at most one point.
The \emph{tree of cylinders} $T_c$ of $T$ is the bipartite tree such that $V_0(T_c)$ is the set of vertices $x$ of $T$ which belong to at least two cylinders,
$V_1(T_c)$ is the set of cylinders $Y$ of $T$, and there is an edge $\varepsilon=(x,Y)$ between $x$ and $Y$ in $T_c$ if and only if $x\in Y$. In other words, one obtains $T_c$ from $T$ by replacing each cylinder $Y$ by the cone on its boundary (defined as the set of vertices of $Y$ belonging to some other cylinder).
Note that $T_c$ may be trivial even if $T$ is not.
The tree $T_c$ is dominated by $T$ (in particular, it is relative to $\calp$ if $T$ is). It only depends on the deformation space $\cald$ containing $T$ (we sometimes say that it is the tree of cylinders of $\cald$). In particular, $T_c$ is invariant under any automorphism of $G$ leaving $\cald$ invariant.
The stabilizer of a vertex $x\in V_0(T_c)$ is the stabilizer of $x$, viewed as a vertex of $T$.
The stabilizer $G_Y$ of a vertex $Y\in V_1(T_c)$ is the stabilizer $G_\calc$ of the equivalence class
$\calc\in\cale/\sim$ containing stabilizers of edges in $Y$, for the action of $G$ on $\cale$ by conjugation (see Subsection 5.1 of \cite{GL4}). It is the normalizer of a finite subgroup in case 1, a maximal elementary (resp.\ abelian) subgroup in case 2 (resp.\ 3). Note that $A\subset G_\calc$ if
$A\in\cale$ and $\calc$ is its equivalence class.
The stabilizer of an edge $\eps=(x,Y)$ of $T_c$ is $G_\varepsilon=G_x\cap G_Y$; it is elliptic in $T$. In cases 2 and 3, $G_\varepsilon$ belongs to $\cala$.
But, in case 1, it may happen that edge stabilizers of $T_c$ are not in $\cala$, so
we also consider
the \emph{collapsed tree of cylinders} $T_c^*$
obtained from $T_c$ by collapsing each edge whose stabilizer does not belong to $\cala$ (see Subsection 5.2 of \cite{GL4}).
It is an $(\cala,\calp)$-tree if $T$ is, and $(T_c^*)_c^*=T_c^*$.
\section{Relatively hyperbolic groups}\label{relhyp}
In this section we assume that $G$ is hyperbolic relative to a finite family $\calp=\{P_1,\dots,P_n\}$ of finitely generated subgroups; we say that $\calp$ is the \emph{parabolic structure}.
The group $G$ is finitely generated. It is not necessarily finitely presented, but it is finitely presented relative to $\calp$ \cite{Osin_relatively}, so JSJ decompositions relative to $\calp$ exist. In particular, there is a deformation space of relative Stallings-Dunwoody decompositions (see Subsection \ref{JSJ}).
\subsection{Generalities}\label{gene}
We refer to \cite{Hruska_quasiconvexity} for equivalent definitions of relative hyperbolicity. In particular, $G$ acts properly discontinuously on a proper geodesic $\delta$-hyperbolic space $X$
(which may be taken to be a graph \cite{GroMan_dehn}), the action is cocompact in the complement of a $G$-invariant union $\calb$ of disjoint horoballs, and the $P_i$'s are representatives of conjugacy classes of stabilizers of horoballs.
Any horoball $B\in\calb$ has a unique point at infinity $\xi$, and the stabilizer of $\xi$ (for the action of $G$ on $\partial X$) coincides with the stabilizer of $B$.
For each constant $M>0$, one can change the system of horoballs so that any two distinct horoballs are at distance at least $M$.
Indeed, for each horoball $B$ with stabilizer $P$ defined by a horofunction $h$, the function $h'(x)=\sup_{g\in P} h(gx)$
is another (well-defined) horofunction which is $P$-equivariant; then $B'=h'{}\m([R,\infty))$ is a new $P$-invariant horoball such that $d(B',X\setminus B)\geq M$ for $R$ large enough.
Doing this for a chosen horoball in each orbit, and extending by equivariance, one gets a system of horoballs
at distance at least $M$ from each other.
A subgroup of $G$ is \emph{parabolic} if it is contained in a conjugate of some $P_i$, \emph{loxodromic} if it is infinite, virtually cyclic, and not parabolic, \emph{elementary} if it is parabolic or virtually cyclic (finite or loxodromic). Any small subgroup is elementary.
The group $G$ itself is elementary if it is virtually cyclic or equal to a $P_i$. We say that $A$ is an elementary subgroup of $B$ if it is elementary and contained in $B$.
One may remove any virtually cyclic
subgroup from $\calp$, without destroying relative hyperbolicity (see e.g.\ \cite[Cor 1.14]{DrSa_tree-graded}). Conversely, one may add to $\calp$ a finite subgroup or a maximal loxodromic subgroup (see e.g.\ \cite{Osin_elementary}). These operations do not change the set of elementary (or relatively quasiconvex, as defined below) subgroups, and it is sometimes convenient (as in \cite{Hruska_quasiconvexity}) to assume that every $P_i$ is infinite. Any infinite $P_i$ is a maximal elementary subgroup.
The following lemma is folklore, but we have not found it in the literature.
\begin{lem}\label{borne}
Given a relatively hyperbolic group $G$, there exists a number $M$ such that any
elementary subgroup $H\subset G$ of cardinality $>M$ is contained in a unique maximal elementary subgroup $E(H)$. There are finitely many conjugacy classes of non-parabolic finite subgroups.
\end{lem}
\begin{proof} We may assume that every $P_i$ is infinite.
Let $H$ be elementary.
The existence of $E(H)$ is well-known if $H$ is infinite (see for instance \cite{Osin_relatively}),
so assume $H$ is finite.
We also assume that the distance between any two distinct horoballs in $\calb$ is bigger than $6\delta$.
Given $r>0$, let $X_r$ be the set of points of $X$ which are moved less than $r$ by $H$.
It follows from Lemma 3.3 p.\ 460 of \cite{BH_metric} (existence of quasi-centres) that
$X_{5\delta}$ is nonempty.
Arguing as in the proof of Lemma 1.15 page 407 and Lemma 3.3 page 428 in \cite{BH_metric},
one sees that any geodesic joining two points of $X_{5\delta}$, or a point of $X_{5\delta}$ to a fixed point of $H$ in $\partial X$,
is contained in $X_{9\delta}$.
If $X_{9\delta}$ meets $X\setminus \calb$,
properness of the action of $G$ on $X\setminus \calb$ implies that there are only finitely many possibilities for $H$ up to conjugacy, so we can choose $M $ to ensure $X_{9\delta}\subset \calb$.
This implies that $X_{5\delta}$ is contained in a unique horoball $B_0$ of $\calb$. This horoball is $H$-invariant since horoballs are $6\delta$-apart, so $H$ fixes the point at infinity $\xi_0 $ of $B_0$ and is contained in the maximal parabolic subgroup $E(H)=\Stab (B_0)$. In particular, $H$ is parabolic.
There remains to prove uniqueness of $E(H)$. It suffices to check that $H$ cannot fix any point $\xi \ne \xi _0$ in $\partial X$. If it did, a geodesic joining $\xi $ to a point of $X_{5\delta}$ would be contained in $X_{9\delta}$ and meet $X\setminus \calb$. This contradicts our choice of $M$.
\end{proof}
Since maximal elementary subgroups are equal to their normalizer, we get:
\begin{cor}[{\cite[Lemma 4.20]{DrSa_groups}}]\label{almar}
Maximal elementary subgroups $E$ are \emph{uniformly almost malnormal}: if $E\cap gEg\m$ has cardinality $>M$, then $g\in E$. \qed
\end{cor}
\subsection{Quasiconvexity}\label{qconv}
\begin{dfn}[Hruska \cite{Hruska_quasiconvexity}] Let $X$ and $\calb$ be as above, and $C>0$.
A subspace $Y\subset X$ is relatively $C$-quasiconvex if, given $y,y'\in Y$, any geodesic $[y,y']\subset X$
has the property that $[y,y']\setminus \calb$ lies in the $C$-neighbourhood of $Y$. The space $Y$ is relatively quasiconvex if it is relatively $C$-quasiconvex for some $C$.
A subgroup $H<G$ is relatively quasiconvex if some (equivalently, every) $H$-orbit is relatively quasiconvex in $X$.
\end{dfn}
\begin{prop}\label{gvqc}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots, P_n\}$, with $P_i$ finitely generated.
If $G$ acts on a simplicial tree $T$ relative to $\calp$
with relatively quasiconvex edge stabilizers, then vertex stabilizers are relatively quasiconvex.
\end{prop}
The proposition applies in particular if edge stabilizers are elementary, since elementary subgroups are relatively quasiconvex. It was proved by Bowditch
\cite[Proposition 1.2]{Bo_cut} and Kapovich \cite[Lemma 3.5]{Kapovich_quasiconvexity} for $G$ hyperbolic. We generalize Bowditch's argument.
\begin{proof}
As usual, we assume that $G$ acts on $T$ minimally and without inversion.
Since $G$ is finitely generated,
the graph $T/G$ is finite. We also assume that $X$ is a connected graph,
and edges of $T$ have length 1.
Since $P_i$ is elliptic in $T$, and stabilizers of points of $X$ are finite, hence elliptic in $T$,
there exists an equivariant map $f:X\ra T$ sending vertices to vertices, mapping each edge linearly to an edge path,
and
constant on each horoball of $\calb$.
For each edge $e$ of $T$, let $m_e$ be the midpoint of $e$, and $Q_e=f\m(m_e)$.
Let $v$ be a vertex of $T$,
and let $E_v$ be the set of edges of $T$ with origin $v$.
Let $Q_v\subset X$ be the preimage under $f$ of the closed ball of radius $\frac12$ around $v$ in $T$.
Note that $Q_e\subset Q_v$ for all $e\in E_v$ and $Q_e\cap\calb=\emptyset$.
Also note that $Q_e\neq\emptyset$ by minimality of $T$ and connectedness of $X$.
If $f(x)=f(hx)$ for $x\in X$ and $h\in G$, then $h$ fixes $ f(x)$. Since $G
$ acts cocompactly
on $X\setminus\calb$, it follows that $G_v$ acts cocompactly on $Q_v\setminus \calb $ and $G_e$ acts cocompactly on $Q_e=Q_e\setminus \calb $.
In particular, $Q_e$ is the $G_e$-orbit of a finite set.
Relative quasiconvexity of $G_e$ implies that $Q_e$ is relatively quasiconvex.
Since $T/G$
is a finite graph, there exists a common constant $C$ such that all subsets $Q_e$ are relatively $C$-quasiconvex.
We now fix a vertex $v$, and we show that $G_v$ is relatively quasiconvex. Choose $x\in Q_v\setminus\calb$.
Since $G_v$ acts cocompactly on $Q_v\setminus \calb $, the Hausdorff distance between $Q_v\setminus \calb$ and the $G_v$-orbit of $x$ is finite, so it suffices to prove that $Q_v$ is relatively quasiconvex. Let $\gamma$ be a geodesic of $X$ joining two points of $Q_v$, and let $\gamma_0$ be a maximal subgeodesic contained in $\gamma\setminus Q_v$. Considering the image of $\gamma$ in $T$, we see that
both endpoints of $\gamma_0$ belong to the same $Q_e$, for some $e\in E_v$. Thus $\gamma_0\setminus\calb$ is $C$-close to $Q_e$, hence to $Q_v$. This shows that $Q_v$ is relatively $C$-quasiconvex.
\end{proof}
A relatively quasiconvex subgroup is relatively hyperbolic in a natural way (\cite[Theorem 9.1]{Hruska_quasiconvexity}).
In particular:
\begin{lem} \label{Hrufi}
If $ G_v$ is an infinite vertex stabilizer of a tree with finite edge stabilizers, it is hyperbolic relative to the family $\calp_{ | G_v}$ of Definition \ref{indu}.
\end{lem}
\begin{proof} This follows from Theorem 9.1 of \cite{Hruska_quasiconvexity}, adding finite groups belonging to $\calp_{ | G_v}$ to the
parabolic structure if needed.
\end{proof}
\subsection{The canonical JSJ decomposition}\label{cano}
In this section we recall the description of the canonical relative JSJ decomposition.
The content of the word canonical is that the JSJ tree (not just the JSJ deformation space) is invariant under automorphisms.
Let $G$ be hyperbolic relative to $\calp$, and denote by $\cala$ the family of elementary subgroups.
In this subsection we fix another (possibly empty) family of finitely generated subgroups $\calh=\{H_1,\dots, H_q\}$
and we assume that $G$ is one-ended relative to $\calp\cup\calh$.
We consider the canonical $\Out(G;\calp\cup\calh)$-invariant JSJ tree $T_{\mathrm{can}}$ defined (under the name $T_c^*$) in Theorem 13.1 of \cite{GL3b} (see also Theorem 7.5 of \cite{GL4}).
It is the tree of cylinders (see Subsection \ref{defcyl}) of the JSJ deformation space $\cald$ over elementary subgroups relative to $\calp\cup\calh$, and it belongs to $\cald$. It is $\Out(G;\calp\cup\calh)$-invariant because
the JSJ deformation space $\cald$ is.
Being a tree of cylinders, $T_{\mathrm{can}}$ is bipartite, with vertices
$x\in V_0(T_{\mathrm{can}})$ and $Y\in V_1(T_{\mathrm{can}})$.
Stabilizers of vertices in $V_0(T_{\mathrm{can}})$ are non-elementary, and stabilizers of vertices in $V_1(T_{\mathrm{can}})$ are maximal elementary subgroups. Non-elementary vertex stabilizers may be rigid or flexible (see Subsection \ref{JSJ}), and flexible vertex stabilizers are QH with finite fiber (see Theorem 13.1 of \cite{GL3b}). Elementary vertex stabilizers are infinite by one-endedness, they may be parabolic or loxodromic. Thus there are exactly four possibilities for a vertex $v\in T_{\mathrm{can}}$:
\begin{enumerate}
\item[0.a.] \emph{rigid}: $G_v$ is non-elementary and is elliptic in every $(\cala, \calp\cup\calh)$-tree.
\item[0.b.] \emph{(flexible) QH}: $G_v$ is non-elementary and not universally elliptic. Then $v$ is a flexible QH vertex with finite fiber
as in Subsection \ref{JSJ}.
\item[1.a.] \emph{maximal parabolic}: $G_v$ is conjugate to a $P_i$.
\item[1.b.] \emph{maximal loxodromic}: $G_v$ is a maximal virtually cyclic subgroup of $G$, and $G_v$ is not parabolic.
\end{enumerate}
\begin{rem}
A QH vertex $v\in V_0(T_{\mathrm{can}})$ is flexible, except in a
few cases; for instance, if $G$ is torsion-free, the only exceptional case is when the underlying surface is a pair of pants (thrice punctured sphere). In these cases we view $v$ as rigid rather than QH. This should not cause confusion. In particular, Propositions \ref {imro} and \ref{imro_rel} would remain valid with $v$ viewed as QH.
\end{rem}
If $\eps=(x,Y)$ is an edge, then $G_\eps=G_x\cap G_Y$ is an infinite maximal elementary subgroup of $G_x$ (but $G_\varepsilon$ may fail to be maximal elementary in $G$ and $G_Y$).
In particular, $G_\varepsilon$ is always almost malnormal in $G_x$, so that Assertion 2 of Lemma \ref{lem_virt_nobitwist} applies to $T_{\mathrm{can}}$, showing $\calt=\ker\rho$.
Let now $v$ be a QH vertex. We claim that, \emph{if $\calh=\emptyset$ and every $P_i$ is infinite, then $\calb_v=\mathrm{Inc}_v\cup \calp_{|G_v} $, and $\calb_v=\mathrm{Inc}_v$ if no $P_i$ is virtually cyclic} (see Definitions \ref {iv}, \ref{indu} and \ref{bv}).
Groups in $\mathrm{Inc}_v\cup \calp_{|G_v} $ are infinite maximal elementary subgroups of $G_v$, and groups in $\calb_v$ are virtually cyclic, so $\mathrm{Inc}_v\cup \calp_{|G_v}\subset \calb_v $ by Definition \ref{dqh}. Conversely, because of one-endedness,
every boundary component of $\Sigma$ is \emph{used} by $\mathrm{Inc}_v\cup \calp_{|G_v} $ \cite[Subsection 2.5 and Theorem 13.1]{GL3b}:
for any full boundary subgroup $B\in\calb_v$, there exists a subgroup $H\in \mathrm{Inc}_v\cup \calp_{|G_v} $
such that some $G_v$-conjugate of $H$ is a finite index subgroup of $B$ (hence equals $B$). This proves the converse. Since groups in $\calb_v $ are virtually cyclic, $\calb_v=\mathrm{Inc}_v$ if no $P_i$ is virtually cyclic. This proves the claim.
When $\calh\ne\emptyset$, we still have $\mathrm{Inc}_v\cup \calp_{|G_v}\subset \calb_v $. If $H_j$ is infinite, the intersection of any of its conjugates with $G_x$
is contained in a full boundary subgroup,
in particular is virtually cyclic. Conversely, a group of $\calb_v$ belongs to $\mathrm{Inc}_v\cup \calp_{|G_v} $
or contains with finite index a $G_v$-conjugate of a group $H\in \calh_{|G_v} $.
This analysis implies that
a group $P_i $ which is not virtually cyclic is contained in a rigid $G_x$
or is equal to some $G_Y$ (which may be contained in a rigid $G_x$). A group $H_j$ which is not virtually cyclic is contained in a rigid $G_x$ or in a $G_Y$.
\begin{lem}\label{fingen}
$T_{\mathrm{can}}$ has finitely generated edge (hence vertex) stabilizers.
\end{lem}
We do not assume that the $P_i$'s are slender, so there may exist infinitely generated elementary subgroups.
\begin{proof}
Since $G$ is finitely presented relative to $\calp\cup\calh$,
there is an elementary JSJ tree $T_J$
relative to $\calp\cup\calh$ having finitely generated edge stabilizers (\cite{GL3a}, Theorem 5.1).
The tree $T_{\mathrm{can}}$ is the tree of cylinders of $T_J$.
Consider an edge $\eps=(x,Y)$ of $T_{\mathrm{can}}$, and $G_\eps=G_x\cap G_Y$. We view $Y$ as a subtree of $T_J$ containing $x$.
If $G_Y$ is virtually cyclic, then $G_\eps$ is obviously finitely generated, so we can assume that $G_Y$ is a
maximal parabolic group $P_i$.
Since $G_Y=P_i$ is elliptic in $T_J$ and leaves $Y$ invariant, it fixes a vertex $y\in Y$.
If $y=x$, then $G_Y\subset G_x$, so $G_\eps=G_Y=P_i$ is finitely generated.
If $y\neq x$, let $e$ be the initial edge of the segment $[x,y]$ in $T_J$.
It is contained in $Y$, so $G_e\subset G_Y$, hence $G_e\subset G_x\cap G_Y
\subset G_x\cap G_y \subset G_e$.
It follows that $G_\eps=G_x\cap G_Y=G_e$ is finitely generated.
\end{proof}
By Proposition \ref{gvqc}, vertex groups of $T_{\mathrm{can}}$ are relatively quasiconvex, hence relatively hyperbolic. We make the parabolic structure explicit.
\begin{lem} \label{rh}
If $x\in V_0(T_{\mathrm{can}})$, the group
$G_x$ is hyperbolic relative to the finite family of finitely generated subgroups $\calq_x=\mathrm{Inc}_x\cup \calp_{ | G_x}$, where $\mathrm{Inc}_x$ is a set of representatives of conjugacy classes of incident edge stabilizers and $\calp_{ | G_x}$ is the induced structure (see Definitions \ref{iv} and \ref{indu}).
\end{lem}
\begin{proof} By Theorem 9.1 of \cite{Hruska_quasiconvexity}, we have to consider infinite groups of the form $G_x\cap gP_ig\m$. Recall that stabilizers of vertices adjacent to $x$ are maximal elementary subgroups, and distinct maximal elementary subgroups have finite intersection.
Thus $gP_ig\m$ must be the stabilizer of an adjacent vertex, or have $x$ as unique fixed point, so $G_x\cap gP_ig\m$ is conjugate in $G_x$ to a group in $\mathrm{Inc}_x\cup \calp_{ | G_x}$. Conversely, a group in $\mathrm{Inc}_x\cup \calp_{ | G_x}$ which is not an infinite group of the form $G_x\cap gP_ig\m$ is finite or is a loxodromic maximal virtually cyclic subgroup of $G_x$. Such groups may be added to the parabolic structure.
\end{proof}
\subsection{Rigid groups have finitely many automorphisms} \label{rig}
\begin{SauveCompteurs}{thm_sci}
\begin{thm} \label{thm_sci}
Let $G$ be hyperbolic relative to finitely generated subgroups $\calp=\{P_1,\dots,P_n\}$,
with $P_i\ne G$.
Let $\calh=\{H_1,\dots,H_q\}$ be another family of finitely generated subgroups. If $\Out(G;\calp,\mk\calh)$ is infinite, then $G$ splits over an elementary subgroup relative to $\calp\cup\calh$.
\end{thm}
\end{SauveCompteurs}
The hypothesis means that there are infinitely many (classes of) automorphisms which map each $P_i$ to a conjugate (in an arbitrary way) and act on each $H_j$ as conjugation by an element of $G$.
The proof of the theorem has two steps. First, using the Bestvina-Paulin method (see \cite{Pau_arboreal}),
extended by Belegradek-Szczepa\'nski \cite{BeSz_endomorphisms} to relatively hyperbolic groups, one constructs an action of $G$ on an $\R$-tree $T$. Rips theory then yields a splitting.
This is fairly standard but there are technical difficulties, in particular because the action on $T$ is not necessarily stable if the $P_i$'s are not assumed to be slender. Details are in Section \ref{sec_Rips}.
\section{Automorphisms of one-ended relatively hyperbolic groups} \label{gen}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots,P_n\}$, with $P_i$ infinite and finitely generated.
We assume that $G$ is one-ended relative to $\calp $. In Subsection \ref{autar} we study $\Out(G;\calp)$ through its action on the canonical JSJ tree. This leads to the main results of this section, which are stated in Subsection \ref{aut_1bout}.
In Subsection \ref{rela} we study automorphisms which
act trivially on another family $\calh$.
\subsection{Automorphisms of the canonical JSJ splitting}\label{autar}
Let $T_{\mathrm{can}}$ be the canonical $\Out(G;\calp)$-invariant JSJ tree as in Subsection \ref{cano} (with $\calh=\emptyset$),
and $\Gamma_{\mathrm{can}}=T_{\mathrm{can}}/G$. Edge stabilizers are infinite elementary subgroups. Vertex stabilizers may be
rigid (non-elementary), QH with finite fiber, maximal parabolic (conjugate to a $P_i$), or maximal loxodromic (virtually cyclic). A rigid or QH stabilizer $G_x$ fixes a unique point in $T_{\mathrm{can}}$, hence is equal to its normalizer; incident edge stabilizers are maximal elementary subgroups of $G_x$.
We study $\Out(G;
\calp)$ through its action on $T_{\mathrm{can}}$ as in Subsection \ref{arb}.
In general, $\Out(G; \calp)$ is a proper subgroup of $\Out(T_{\mathrm{can}})$.
We define finite index subgroups $\Out^0 (G;\calp )$ and $\Out ^0(G;\mk\calp )$ by taking the intersection of
$\Out (G;\calp )$ and $\Out (G;\mk\calp )$ with the group $\Out^0(T_{\mathrm{can}})$
consisting of automorphisms acting trivially on the graph $\Gamma_{\mathrm{can}}=T_{\mathrm{can}}/G$.
By the second assertion of Lemma \ref{lem_virt_nobitwist}, the kernel of $\rho:\Out^0(T_{\mathrm{can}})\to \prod _{v\in V} \Out(G _v)$ is the group of twists $\calt$. Note that $\calt\subset \Out ^0(G;\mk\calp)
\subset \Out ^0(G;\calp)
$ since every $P_i $ is elliptic in $T_{\mathrm{can}}$ and a twist acts as a conjugation on any vertex stabilizer.
The group $\calt}%{\mathrm{Tw} $ is the image in $\Out(G)$ of
a finite direct product $\prod _{e\in E }
Z_{G_{o(e)}}(G_e)$. Each factor is virtually cyclic or contained in a conjugate of some $P_i$.
We now consider the image of $\Out^0 (G;\calp )$ and $\Out ^0(G;\mk\calp )$ by $\rho_v:\Out^0(T_{\mathrm{can}})\to \Out(G _v)$, for $v$ a vertex of $\Gamma_{\mathrm{can}}$ (viewed as a vertex of $T_{\mathrm{can}}$ with stabilizer $G_v$).
Using Theorem \ref{thm_sci}, we shall show that both images are finite if $G_v$ is rigid. If $G_v=P_i$, the index of $\Out(G_v;\mk\mathrm{Inc}_v)$ in the image of
$\Out^0 (G;\calp )$ is finite because $w$ is rigid whenever $e=vw$ is an edge with $\Out(G_e)$ infinite. More precisely:
\begin{prop} \label{imro} The images of $\Out^0 (G;\calp )$ and $\Out ^0(G;\mk\calp )$ by $\rho_v:\Out^0(T_{\mathrm{can}})\to \Out(G _v)$ may be described as follows:
\begin{itemize*}
\item If $G_v$ is virtually cyclic or rigid, both images are finite.
\item
If $G_v$ is a QH vertex stabilizer,
both images contain $\Out(G_v;\mk\calb_v)$
with finite index. \item
If $G_v$ is (conjugate to) $P_i$, the image of $\Out^0(G;\mk\calp)$ is trivial.
The image of $\Out^0(G;\calp)$
contains $\Out(G_v;\mk\mathrm{Inc}_v) $ with finite index.
\end{itemize*}
If $e$ is any edge of $T_{\mathrm{can}}$,
the images of $\Out^0 (G;\calp )$ and $\Out ^0(G;\mk\calp )$ by $\rho_e:\Out^0(T_{\mathrm{can}})\to \Out(G _e)$ are finite.
\end{prop}
See Definition \ref{bv} for the definition of the family of full boundary subgroups $\calb_v$, and recall that $\calb_v= \mathrm{Inc}_v$ if no $P_i$ is virtually cyclic.
\begin{proof}
If $G_v$ is virtually cyclic, $\Out(G_v)$ is finite.
If $G_v$ is rigid, finiteness follows from Theorem \ref{thm_sci}, as we now explain. We have seen (Lemma \ref{rh}) that $G_v$ is hyperbolic relative to a finite family
$\calq_v=\mathrm{Inc}_v\cup\calp_{|G_v}$ consisting of incident edge groups and conjugates of the $P_i$'s having $v$ as unique fixed point. These groups are finitely generated by Lemma \ref{fingen}.
By Lemma \ref{automind}, the group $\rho_v(\Out^0 (G;\calp))$ is contained in $\Out(G_v;\calq_v)$.
If it is infinite,
$G_v$ splits over an elementary subgroup relative to $\calq_v$ by Theorem \ref{thm_sci}
(applied with $\calp=\calq_v$ and $\calh=\emptyset$).
By Lemma \ref{extens}, this splitting may be used to refine $T_{\mathrm{can}}$, yielding an elementary splitting of $G$ relative to $\calp$ in which $G_v$ is not elliptic. This contradicts rigidity.
If $G_v$ is QH,
first note that
$\Out(G_v;\mk\calb_v)=\Out(G_v;\mk\mathrm{Inc}_v, \mk\calp_{|G_v})$ and
$\Out(G_v; \calb_v)=\Out(G_v; \mathrm{Inc}_v, \calp_{|G_v})$ because $\calb_v=\mathrm{Inc}_v\cup \calp_{|G_v}$
(see Subsection \ref{cano}, recalling that groups in $\calp$ are assumed to be infinite).
Lemma \ref{automind} then yields
$$\Out(G_v;\mk\calb_v)\subset
\rho_v(\Out^0 (G;\mk\calp))
\subset \rho_v(\Out^0 (G;\calp))
\subset\Out(G_v;\calb_v).$$
We conclude by observing that $\Out(G_v;\mk\calb_v)$ has finite index in $\Out(G_v;\calb_v)$
because groups in $\calb_v$ are virtually cyclic, hence have finite outer automorphism group.
If $G_v$ is (conjugate to) $P_i$, the image of $\Out^0(G;\mk\calp)$ is clearly trivial.
Since $\calp_{|G_v}$ equals $\{G_v\}$ or is empty, the image of $\Out^0(G;\calp)$
contains $\Out(G_v;\mk\mathrm{Inc}_v)=\Out(G_v; \mk\mathrm{Inc}_v, \calp_{|G_v})$, and we have to show that the index is finite.
If $\Out(G_\varepsilon)$ is finite for every incident edge $\varepsilon$,
this follows from Lemma \ref{automind}.
In general, we have to control the action of automorphisms on $G_\varepsilon$ for incident edges $\varepsilon$ with $\Out(G_\varepsilon)$ infinite. Note that there is no natural map from $\Out(G_v; \mathrm{Inc}_v)$ to $\Out(G_\varepsilon)$ if $N_{G_v}(G_\varepsilon)$ acts non-trivially on $G_\varepsilon$.
Infiniteness of $\Out(G_\varepsilon)$ implies that the other endpoint of $\varepsilon$ is a rigid vertex $x$: it cannot be QH since $G_\varepsilon$ would then be virtually cyclic. As explained above, the image of $\Out^0(G;\calp)$ in $\Out(G_x)$ is finite. Any $\Phi\in\Out^0(G;\calp)$ has a representative $\alpha$ leaving $G_x$ and $G_v$ invariant (the associated map $f_\alpha$ fixes $\varepsilon$). Replacing $ \Out^0(G;\calp)$ by a finite index subgroup, we may suppose that
$\alpha$ acts on $G_x$ as conjugation by some element $g$. This $g$ must be in $G_x$ since $G_x$ equals its normalizer, and in fact in $G_\varepsilon$ because $G_\varepsilon$ is almost malnormal in $G_x$. This shows that $
\Phi$
maps into $\Out(G_v;\mk G_\varepsilon)$. Arguing in this way for each incident edge proves the result for the image of $\rho_v$.
Since any edge of $T_{\mathrm{can}}$ has a vertex $v$ with $G_v$ virtually cyclic or conjugate to a $P_i$, the previous argument also shows finiteness for images by $\rho_e$.
\end{proof}
\begin{dfn} Let $\Sigma$ be a compact 2-dimensional hyperbolic orbifold.
The \emph{extended mapping class group} $MCG^*(\Sigma)$ is the group of outer automorphisms of $\pi_1(\Sigma )$ preserving the set of boundary subgroups.
If $v$ is
a QH vertex of $T_{\mathrm{can}}$ with underlying orbifold $\Sigma$ and finite fiber $F$,
we define $MCG^0_{T_{\mathrm{can}}}(\Sigma )$ as the group
$\Out(G_v;\mk\calb_v)$.
This group
maps with finite kernel onto a finite index subgroup of $MCG^*(\Sigma)$ \cite[pp.\ 240 and 268]{DG2}.
\end{dfn}
As noted in \cite{Fujiwara_outer,DG2},
one can understand the extended mapping class group of an orbifold with mirrors
in terms of the extended mapping class group of a suborbifold without mirrors.
If $G$ is torsion free, $\Sigma $ is a surface, $MCG^*(\Sigma)$ is the group of isotopy classes of homeomorphisms, and $MCG^0_{T_{\mathrm{can}}}(\Sigma )$ is the group of isotopy classes of homeomorphisms which map each boundary component to itself in an orientation-preserving way (in this case it only depends on $\Sigma$ since the fiber is trivial).
Mapping class groups are usually infinite, but there are exceptions. In the torsion-free case, the exceptions are
the pair of pants and the twice punctured projective plane \cite[Cor 4.6]{Korkmaz_MCG};
all other hyperbolic surfaces contain
an essential 2-sided simple closed curve not bounding a M\"obius band, so there is a Dehn twist of infinite order.
As a QH vertex, a pair of pants is rigid; every simple closed curve is homotopically trivial or boundary parallel. A twice punctured projective plane is flexible, but every 2-sided simple closed curve is homotopically trivial, boundary parallel, or bounds a M\"obius band, so there is no non-trivial Dehn twist.
\subsection{Automorphisms of $G$}\label{aut_1bout}
Motivated by the previous subsection, we define a subgroup $\Out^1(G;\mk\calp)\subset \Out^0(G;\mk\calp)$ as the set of $\Phi$ such that $\rho_v(\Phi)$ is trivial if $G_v$ is virtually cyclic, rigid, or conjugate to a $P_i$,
and $\rho_v(\Phi)\in\Out(G_v;\mk\calb_v)=MCG^0_{T_{\mathrm{can}}}(\Sigma )$ if $G_v$ is QH. Proposition \ref{imro} shows that this subgroup has finite index.
We define a finite index subgroup
$\Out^1(G; \calp)\subset \Out^0(G; \calp)$ similarly, allowing $\rho_v(\Phi)\in \Out(G_v;\mk\mathrm{Inc}_v)$
if $G_v$ is conjugate to a $P_i$.
We may now sum up the discussion in Subsection \ref{autar} as:
\begin{thm}\label{thm_struct_m}
Let $G$ be hyperbolic relative to
$\calp=\{P_1,\dots,P_n\}$, with $P_i$ infinite and finitely
generated. Assume that $G$ is one-ended relative to $\calp$.
Then $\Out(G;\mk\calp)$ and $\Out(G;\calp)$ have finite index subgroups
$\Out^1(G;\mk\calp)$ and $\Out^1(G;\calp)$ which fit in exact sequences
$$1\ra \calt}%{\mathrm{Tw} \ra \Out^1(G;\mk\calp) \ra \prod_{i=1}^p MCG^0_{T_{\mathrm{can}}}(\Sigma_i)\ra 1$$
$$1\ra \calt}%{\mathrm{Tw} \ra \Out^1(G;\calp) \ra \prod_{i=1}^p MCG^0_{T_{\mathrm{can}}}(\Sigma_i) \times \prod _j \Out(P_j;\mk\mathrm{Inc}_{P_j})\ra 1$$
where:
\begin{itemize*}
\item $\calt}%{\mathrm{Tw} $ is the group of twists of the canonical elementary JSJ decomposition $T_{\mathrm{can}}$ relative to $\calp$; it is a quotient of a finite direct product where each factor is a subgroup of $G$ which is virtually cyclic or contained in a $P_i$;
\item
$\Sigma_1,\dots,\Sigma_p$ are the 2-orbifolds occuring in flexible QH vertices $v$ of $T_{\mathrm{can}}$, and $MCG^0_{T_{\mathrm{can}}}(\Sigma_i)$
maps with finite kernel onto a finite index subgroup of the extended mapping class group $MCG^*(\Sigma_i)$;
\item
the last product is taken only over those $P_j$'s which occur as vertex stabilizers of $T_{\mathrm{can}}$, and $\mathrm{Inc}_{P_j}$ is the set of incident edge groups. \qed
\end{itemize*}
\end{thm}
Note that $\calt}%{\mathrm{Tw}$ is slender (resp.\ small, virtually solvable, virtually nilpotent, virtually abelian) if the $P_i$'s are. Also note that $\Out(G;\calp)$ has finite index in $\Out(G)$ if the $P_i$'s are small but not virtually cyclic, since they may be characterized up to conjugacy as maximal among the subgroups of $G$ which are small but not virtually cyclic. More generally, this holds if no $P_i$ is relatively hyperbolic \cite[Lemma 3.2]{MiOs_fixed}.
Recall that $G$ is \emph{toral relatively hyperbolic} if it is torsion-free and hyperbolic relative to a finite family $\calp$ of finitely generated abelian subgroups.
Limit groups, and more generally groups acting freely on $\R^n$-trees, are toral relatively hyperbolic \cite{Dah_combination,Gui_limit}.
\begin{cor} \label{limgp}
Let $G$ be toral relatively hyperbolic and one-ended.
Then some finite index subgroup $\Out^1(G)$ of $\Out(G)$ fits in an exact sequence
$$1\ra \calt}%{\mathrm{Tw} \ra \Out^1(G) \ra \prod_{i=1}^p MCG^0(\Sigma_i) \times \prod_{k=1}^m
GL_{r_k,n_k}(\bbZ)
\ra 1$$
where $\calt}%{\mathrm{Tw}$ is finitely generated free abelian, $MCG^0(\Sigma_i)$ is the group of isotopy classes of homeomorphisms of a compact surface $\Sigma_i$ mapping each boundary component to itself in an orientation-preserving way, and $GL_{r,n}(\bbZ) =M_{r,n}(\bbZ)\rtimes GL_{r}(\bbZ)$
is the group of automorphisms of $\bbZ^{n+r}$ fixing the first $n$ generators.
\end{cor}
See Theorem 5.3 of \cite{BuKhMi_isomorphism} and Theorem 6.5 of \cite {GL1} for the case of limit groups, based on results from \cite{KhMy_effective} and \cite{BuKhMi_isomorphism}.
\begin{proof} We may assume that no $P_i\in\calp$ is cyclic, so $\Out(G;\calp)$ has finite index in $\Out(G)$.
If $P_j$ is isomorphic to ${\mathbb {Z}}^{a}$, then $\Out(P_j;\mk\mathrm{Inc}_{P_j})$ is isomorphic to some $GL_{r,n}(\bbZ)
$ with $r+n=a$, so the exact sequence follows from Theorem \ref{thm_struct_m}. We know that the group of twists $\calt}%{\mathrm{Tw}$ of $T_{\mathrm{can}}$ is finitely generated and abelian. There remains to check that it is torsion-free.
Recall from Subsection \ref{arb} that $\calt}%{\mathrm{Tw}$ is generated by the product of all $Z_{G_{o(e)}}(G_e)$, subject to edge and vertex relations. Denoting an edge of $\Gamma_{\mathrm{can}}=T_{\mathrm{can}}/G$ by
$ \varepsilon=(x,Y)$ with $x\in V_0(\Gamma_{\mathrm{can}})$ and $Y\in V_1(\Gamma_{\mathrm{can}})$,
there is no relation at the vertex $x$ since $Z(G_x)$ is trivial.
Moreover, $Z_{G_x}(G_\varepsilon)=G_\varepsilon$, so the edge relation identifies the twists around $\varepsilon$ near $x$ with twists near $Y$. Thus $\calt}%{\mathrm{Tw}$ is the direct product, over vertices $v$ of
$ \Gamma_{\mathrm{can}}$ carrying an abelian group, of
$(\prod_{e\in E_v} Z_{G_v}(G_e))/Z(G_v)$ where $E_v$ is the set of oriented edges with origin $v$ and
$Z(G_v)$ is embedded diagonally.
Since $G_v$ is abelian, $Z_{G_v}(G_e)= Z(G_v)=G_v$, so $\calt}%{\mathrm{Tw}$ is isomorphic to a finite direct product $\prod (G_v)^{ | E_v | -1}$
of abelian vertex groups. It is therefore torsion-free.
\end{proof}
\begin{cor} \label{VFL}
If $G$ is a toral relatively hyperbolic group,
$\Out(G)$ is virtually torsion-free and has a finite index subgroup with a finite classifying space.
\end{cor}
\begin{proof} This follows from Corollary \ref{limgp} if $G$ is one-ended. In general, we write
$G=G_1*\dots*G_q*F$ with $G_\ell$ one-ended and $F$ free. All groups $G_\ell$ and $G_\ell/Z(G_\ell)$ have a finite classifying space \cite{Dah_classifying}, so
we
can apply Theorem 5.2 of
\cite{GL1}. (We mention here that the arguments given in \cite{GL1} are insufficient to get a finite classifying space: there should exist finite classifying spaces for the groups $\Out^S(G)$ themselves (rather than for finite index subgroups); this is achieved by restricting to some finite index subgroup of $\Out(G)$, see \cite {GL_McCool} for details.)
\end{proof}
\subsection{The relative case}\label{rela}
We generalize the analysis of Subsection \ref{autar}
to a relative situation.
Let $G,\calp$ be as above,
and fix another
family of finitely generated subgroups $\calh=\{H_1,\dots, H_q\}$.
Assume that $G$ is one-ended relative to $\calp\cup\calh$,
and
let now $T_{\mathrm{can}}$ be
the canonical elementary JSJ tree relative to $\calp\cup\calh$.
\begin{thm}\label{thm_struct_m_rel}
Under these hypotheses,
$\Out(G;\mk\calp,\mk\calh)$ and $\Out(G;\calp,\mk\calh)$ have finite index subgroups
$\Out^1(G;\mk\calp,\mk\calh)$ and $\Out^1(G;\calp,\mk\calh)$ which fit in exact sequences
$$1\ra \calt}%{\mathrm{Tw} \ra \Out^1(G;\mk\calp,\mk\calh) \ra \prod_{i=1}^p MCG^1_{T_{\mathrm{can}}}(\Sigma_i)\ra 1$$
$$1\ra \calt}%{\mathrm{Tw} \ra \Out^1(G;\calp,\mk\calh) \ra \prod_{i=1}^p MCG^1_{T_{\mathrm{can}}}(\Sigma_i) \times \prod_j \Out(P_j;\mk\mathrm{Inc}_{P_j},\mk \calh_{|P_j})\ra 1$$
as in Theorem \ref{thm_struct_m}.
The group $MCG^1_{T_{\mathrm{can}}}(\Sigma_i)$ equals $\Out(G_v;\mk\calb_v, \mk\calf_v)$,
where $\calf_v$ is a set
of representatives of conjugacy classes of finite subgroups in $G_v$;
it is a finite index subgroup of $MCG^0_{T_{\mathrm{can}}}(\Sigma_i)=\Out(G_v;\mk\calb_v)$.
\end{thm}
Note that $\calf_v$ is a finite set since the QH vertex group $G_v$ maps onto a 2-orbifold group with finite kernel. The family $\calf_v$ is not needed if all groups in $\calh$ are infinite (see the proof below).
The theorem is proved as in the absolute case, replacing Proposition \ref{imro} by the following result.
\begin{prop} \label{imro_rel} The images of $\Out^0 (G;\calp,\mk\calh )$ and $\Out ^0(G;\mk\calp,\mk\calh )$
by $\rho_v:\Out^0(T_{\mathrm{can}})\to \Out(G _v)$ may be described as follows:
\begin{itemize*}
\item If $G_v$ is virtually cyclic or rigid, both images are finite.
\item If $G_v$ is a QH vertex stabilizer,
both images contain $\Out(G_v;\mk\calb_v,\mk \calf_v)$
with finite index (where $\calf_v$ is as in Theorem \ref{thm_struct_m_rel}
and $\calb_v$ is as in
Definition \ref{bv}).
\item
If $G_v$ is (conjugate to) $P_i$, the image of $\Out^0(G;\mk\calp,\mk\calh)$ is trivial.
The image of $\Out^0(G;\calp,\mk\calh)$
contains $\Out(G_v;\mk\mathrm{Inc}_v,\mk\calh_{|G_v}) $ with finite index.
\end{itemize*}
\end{prop}
\begin{proof}
We only mention the differences with the proof of Proposition \ref{imro}.
If $v$ is a (non-elementary) rigid vertex,
the images of $\Out^0 (G;\calp,\mk\calh )$ and $\Out ^0(G;\mk\calp,\mk\calh )$
by $\rho_v$ are contained in
in $\Out(G_v;\calq_v,\mk\calh_{|G_v})$ by Lemma \ref{automind}.
It follows that the images are finite since, otherwise,
Theorem \ref{thm_sci} would yield a splitting relative to
$\calq_v\cup\calh_{|G_v}$, which extends to a splitting of $G$ relative to $\calp\cup\calh$
by Lemma \ref{extens}.
When $G_v$ is conjugate to a parabolic group $P_j$, Lemma \ref{automind}
says that the image of
$\Out^0(G;\calp,\mk\calh)$ contains $\Out(G_v;\mk\mathrm{Inc}_ v, \mk \calh_{|G_v})$,
and the index is finite for the same reason as before.
When $G_v$ is QH,
we write
$$\Out(G_v;\mk\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})
\subset \rho_v (\Out^0(T;\calp,\mk \calh))
\subset
\Out(G_v;\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})$$
using Lemma \ref{automind}.
The proof in the non-relative case relied on the equality $\calb_v=\mathrm{Inc}_v\cup \calp_{|G_v}$.
Here (see Subsection \ref{cano}) we have $\mathrm{Inc}_v\cup \calp_{|G_v}\subset \calb_v$, and a group $B\in\calb_v$ not in $\mathrm{Inc}_v\cup \calp_{|G_v} $ contains with finite index a group $H'$ conjugate to some $H\in \calh_{|G_v} $. Since $B$ is the only maximal elementary subgroup of $G_v$ containing $H'$, any automorphism preserving $H'$ preserves $B$, so $\Out(G_v;\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})\subset \Out(G_v;\calb_v)$.
If all groups in $\calh$ are infinite, the intersection of any conjugate of $H_j$ with $G_v$ is contained in a full boundary subgroup, so $\Out(G_v;\mk\calb_v )\subset \Out(G_v;\mk\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})$.
Otherwise
$\calh_{|G_v}$ may contain finite groups (fixing $v$ but no other vertex of $T_{\mathrm{can}}$) and we can only write $\Out(G_v;\mk\calb_v,\mk \calf_v )\subset \Out(G_v;\mk\mathrm{Inc}_v,\calp_{|G_v},\mk\calh_{|G_v})$.
The proposition follows because the index of $\Out(G_v;\mk\calb_v,\mk \calf_v) $ in $ \Out(G_v;\calb_v)$ is finite.
\end{proof}
\begin{rem} \label{rips}
Because we use Theorem \ref{thm_sci} to control automorphisms of rigid groups,
we do not have a similar result concerning $\Out(G;\mk \calp\cup\calh)$ or
$\Out(G;\calp\cup\calh)$:
we have to impose that automorphisms act trivially on $\calh$. We also need finite generation of groups in $\calh$.
\end{rem}
Arguing as in the previous subsection, one gets:
\begin{cor}\label{cor_MCtoral}
Let $G$ be toral relatively hyperbolic, one-ended relative to a family $\calh=\{H_1,\dots, H_q\}$ of finitely generated subgroups. Then
$ \Out(G; \mk\calh) $ has a finite index subgroup $ \Out^1(G;\mk\calh)$ fitting in an exact sequence
as in Corollary \ref{limgp}. \qed
\end{cor}
\section{The modular group}\label{modul}
The goal of this section is to show that the modular group, usually defined by considering all suitable splittings of a group $G$, may be seen on a single splitting, namely the canonical JSJ decomposition.
\subsection{Definitions and examples}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots, P_n\}$, where each $P_i$ is
finitely generated. Without loss of generality, we assume that no $P_i$ is
virtually cyclic (in particular, $P_i$ is infinite). Let $\calh$ be another finite family of finitely generated subgroups $H_j$ such that every $P_i$ which contains a free group $F_2$ is contained in a group of $\calh$.
In particular, we may take $\calh=\calp$, or $\calh=\emptyset$ if every $P_i$ is small.
We will assume that $G$ is one-ended relative to $ \calh$ (equivalently, relative to $\calp\cup\calh$ since every $P_i$ is one-ended or contained in a group of $\calh$).
We consider trees $T$
with elementary edge stabilizers,
which are relative to $\calh$ (universal ellipticity will be with respect to these trees, unless indicated otherwise). They are not necessarily relative to $P_i$ if $P_i$ is small, but our assumption on $\calh$ implies that
elementary subgroups which are not small
have a conjugate contained in a group in $\calh$,
so are universally elliptic.
We shall associate a modular group $\mo(T)\subset\Out(T)\subset\Out(G )$ to such a tree $T$.
\begin{lem}\label{lem_elemQH}
If $v$ is a flexible QH vertex with finite fiber, then any elementary subgroup of $G_v$ is virtually cyclic.
\end{lem}
Recall (Definition \ref{dqh}) that $G_v$ maps onto
$\pi_1(\Sigma)$ with finite kernel $F$; flexibility of $G_v$ is equivalent to the 2-orbifold $\Sigma$ containing an essential 1-suborbifold. Since $T$ is only assumed to be relative to $\calh$, Definition \ref{dqh} only restricts intersections of $G_v$ with conjugates of groups in $\calh$.
\begin{proof}
Assume on the contrary that some subgroup $E<G_v$ is elementary, but not virtually cyclic.
Then $E$ contains $F_2$ and is parabolic. As pointed out before, our assumption on $\calh$ implies that
$E$ is universally elliptic. This contradicts Remark \ref{rem_UEQH}
saying that such a group has to be virtually cyclic.
\end{proof}
\begin{rem*} If $G_v$ is a flexible QH vertex stabilizer with elementary fiber $F$, then $F$ is finite. Indeed, if $F $ is infinite, then $G_v$ is elementary by almost malnormality of maximal elementary subgroups (Corollary \ref{almar}). Since it contains $F_2$, it is universally elliptic, contradicting flexibility.
\end{rem*}
\begin{dfn} We say that a vertex $v$ of $T$ (or of $\Gamma=T/G$) is \emph{modular} if $G_v$ is flexible and QH (relative to $\calh$) with finite fiber,
or $G_v$ is elementary. Note that $G_v$ cannot be both.
\end{dfn}
Recall (Subsection \ref{arb}) the maps $\rho_v: \Out^0(T)\to \Out(G_v)$ defined on the finite index subgroup of $\Out(T)$ consisting of automorphisms acting trivially on $\Gamma=T/G$.
\begin{dfn}
We define $\mo(T)$ by saying that $\Phi\in\Out^0(T)$ belongs to $\mo(T)$
if it satisfies the following conditions:
\begin{itemize}
\item If $v$ is not modular, $\rho_v(\Phi)$ is trivial.
\item If $G_v$ is elementary, $\rho_v(\Phi)\in\Out(G_v;\mk\mathrm{Inc}_v)$; in other words, $\rho_v(\Phi)$ acts on each incident edge group as a conjugation.
\item If $G_v$ is QH with finite fiber, and flexible, then $\rho_v(\Phi)\in\Out(G_v;\mk\calb_v)$, with $\calb_v$ consisting of full preimages of boundary subgroups of $\pi_1(\Sigma)$ as in Definition \ref{bv}.
\end{itemize}
\end{dfn}
Note that $\mo(T)$ contains the group of twists $\calt}%{\mathrm{Tw}(T)$, and that automorphisms in $\Mod(T)$ need
not preserve $\calh$.
We have assumed that $G$ is one-ended relative to $\calh$,
so we can consider
the canonical elementary JSJ
tree $T_{\mathrm{can}}$ relative to $\calp\cup\calh$ as in Subsection \ref{cano}.
Note that $\Mod(T_{\mathrm{can}})$ has finite index in $\Out(G; \calp)$ when $\calh=\calp$
(it contains the group $\Out^1(G;\calp)$ defined in
Subsection \ref{aut_1bout}, possibly strictly because of vertices with $G_v$ virtually cyclic).
\begin{thm} \label{thm_mod}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots, P_n\}$, with each $P_i$
finitely generated, not virtually cyclic. Let $\calh$ be a family of subgroups such that every $P_i$ which contains $F_2$ is contained in a group of $\calh$, and $G$ is one-ended relative to $\calh$.
If $T$ is any elementary splitting of $G$ relative to $\calh$, then $\mo(T)\subset\mo(T_{\mathrm{can}})$, where $T_{\mathrm{can}}$ is the canonical elementary JSJ tree relative to $\calp\cup\calh$.
\end{thm}
This applies in particular if $G$ is one-ended and no $P_i$ contains $F_2$ (taking $\calh=\emptyset$), or if $G$ is one-ended relative to an arbitrary $\calp$ and we restrict to splittings relative to $\calp$ (taking $\calh=\calp$).
\begin{rem} Rather than defining $\mo(T)$ by imposing conditions on the action on vertex groups, as we just did, one could define it by giving generators: twists around edges, and certain automorphisms of vertex groups. This would yield a slightly smaller group $\mo'(T)$: its intersection with $\ker\rho$ is $\calt}%{\mathrm{Tw} $, whereas $\mo(T)$ contains all of $\ker \rho$. Theorem \ref{thm_mod} (and Theorem \ref{thm_modCSA} below) also hold with this more restrictive definition, since $\mo'(T_{\mathrm{can}})=\mo(T_{\mathrm{can}})$ by Assertion 2 of Lemma \ref{lem_virt_nobitwist}.
\end{rem}
There is a similar statement for torsion-free CSA groups (recall that $G$ is CSA if centralizers of non-trivial elements are abelian and malnormal). We now consider abelian splittings of $G$. A vertex $v$ is modular if $G_v$ is either abelian or QH as above (in this case $F$ is trivial and $\Sigma$ is a surface). The definition of $\mo(T)$ is the same (with elementary replaced by abelian). The tree $T_{\mathrm{can}} $ is the
canonical abelian JSJ tree relative to non-cyclic abelian subgroups; it is also the tree of cylinders of the (non-relative) abelian JSJ deformation space (see Theorem 11.1 of \cite{GL3b}).
\begin{thm} \label{thm_modCSA} Let $G$ be a finitely generated, torsion-free, one-ended, CSA group. If $T$ is any splitting of $G$ over abelian groups, then $\mo(T)\subset\mo(T_{\mathrm{can}})$, where $T_{\mathrm{can}}$ is the tree of cylinders of the abelian JSJ deformation space.
\end{thm}
\begin{example}
Let $G$ be the Baumslag-Solitar group $BS(2,4)=\grp{a,b\mid ba^2b\m=a^4}$.
Any splitting of $G$ as a graph of infinite cyclic groups is a cyclic JSJ decomposition of $G$
\cite{For_uniqueness,GL3a}.
Its modular group coincides with its group of twists, and is a finite abelian group (see \cite{Lev_GBS}).
But the JSJ deformation space of $BS(2,4)$ is quite large \cite{Clay_deformation},
and JSJ splittings of $BS(2,4)$ may have modular groups of arbitrarily large order. In particular,
there is no splitting whose modular group contains all others.
\end{example}
\begin{example}
Even if $G$ is as in Theorem \ref{thm_mod},
one cannot replace $T_{\mathrm{can}}$ by an arbitrary tree in its deformation space:
there exists such trees whose modular group is not maximal (even up to finite index).
Indeed, let $G=A_1*_{C_1}B*_{C_2} A_2$, where:
\begin{itemize*}
\item $A_1$ and $A_2$ are torsion-free hyperbolic groups with no cyclic splitting;
\item $C_i$ is a maximal infinite cyclic subgroup of $A_i$;
\item $B$ is torsion-free, hyperbolic relative to a subgroup $\Hat C= C_1\oplus C_2\oplus \bbZ\simeq{\mathbb {Z}}^3$, and does not split over an abelian group.
\end{itemize*}
The group $G$ is hyperbolic relative to $\Hat C$ by \cite{Dah_combination}.
The graph of groups
$$\xymatrix{ & B \ar@{-}[d]^{C_1\oplus C_2}\\
A_1\ar@{-}[r]_{C_1\hspace{0.4cm} }&C_1\oplus C_2 \ar@{-}[r]_{\hspace{0.4cm} C_2}&A_2}$$
is an elementary JSJ decomposition of $G$ (both absolute and relative to $\Hat C$) because its vertex groups are universally elliptic (see Lemma 4.7 of \cite{GL3a}). Given any $z\in \Hat C\setminus (C_1\oplus C_2)$, the automorphism $\tau$ defined as the identity on $A_1$ and $B$ and as conjugation by $z$
on $A_2$ is not an automorphism of this graph of groups. But $\tau$ is a twist of $T_{\mathrm{can}}$, which is the Bass-Serre tree of the graph of groups below.
$$\xymatrix{ & B \ar@{-}[d]^{\Hat C}\\
A_1\ar@{-}[r]_{C_1
}&\Hat C \ar@{-}[r]_{
C_2}&A_2}$$
\end{example}
\subsection{Proof of Theorem \ref{thm_mod}}
We prove Theorem \ref{thm_mod}.
The proof of Theorem \ref{thm_modCSA} is similar and left to the reader.
The main difference in the context of a general CSA group is that we have no version of Theorem
\ref{thm_sci} saying that a rigid vertex group only has finitely many outer automorphisms.
But the proof given below does not use
Theorem \ref{thm_sci}.
By Theorem 13.1 of \cite{GL3b}, the trees $T_{\mathrm{can}} $ and $T$ are compatible: they have a common refinement $\hat T$ (as defined in Subsection \ref{defsp}). We may assume that no edge of $\hat T$ is collapsed in both $T_{\mathrm{can}} $ and $T$ (so $\hat T$ is the lcm of $T_{\mathrm{can}} $ and $T$ as defined in Section 3 of \cite{GL3b}). The tree $\hat T$ has elementary edge stabilizers and is relative to $\calh$ since $T_{\mathrm{can}} $ and $T$ are (Proposition 3.22 of \cite{GL3b}).
We first claim that $T_{\mathrm{can}} $ is $\mo(T)$-invariant. To see this, it suffices to show that the image of an infinite group $J\in \calp\cup\calh$ by a modular automorphism $\Phi$ has a finite index subgroup which is contained (up to conjugacy) in a group
belonging to $\calp\cup\calh$.
If $J$ is a small $P_i$, its image is elementary and not virtually cyclic, so is parabolic. If $J $ is a $P_i$ containing $F_2$, it is contained in a group of $\calh$, so we only have to consider groups $J\in\calh$. Such a group fixes a vertex $v$ in $T$, and $G_v$ is $\Phi$-invariant.
We distinguish three cases.
If $v$ is non-modular, $\Phi$ acts trivially on $J$.
If $v$ is a flexible QH vertex, then $J$ is contained in a group of $\calb_v$, hence $\Phi$-invariant. Now suppose that $J$ is contained in an elementary $G_v$. If $G_v$ contains $F_2$, it is a $\Phi$-invariant group contained in a group of $\calh$, so the image of $J$ is contained in a group of $\calh$. The case when $G_v$ is small but not virtually cyclic has been dealt with before. If $G_v$ is virtually cyclic, $\Phi(J)$ has a finite index subgroup contained in $J$.
This completes the proof of the claim.
Let $\Phi\in\mo(T)$.
The heart of the proof of the theorem is to study the action of $\Phi$
on a non-elementary vertex stabilizer $G_v$ of $T_{\mathrm{can}}$ (it is rigid or QH).
In particular, given an edge $e=vw$ in $T_{\mathrm{can}}$, we show that $\Phi$ has a representative $\alpha$ leaving $G_v$ invariant and equal to the identity on $G_e$.
We have defined $T_{\mathrm{can}}$ as a JSJ tree relative to $\calp\cup\calh$. When $T_{\mathrm{can}}$ is viewed as relative to $ \calh$ only, a flexible QH vertex remains flexible QH.
It follows from Sections 8 and 13 of \cite{GL3b} that a rigid (non-elementary) vertex stabilizer $G_v$ of $T_{\mathrm{can}}$ remains universally elliptic relative to $\calh$. The argument goes as follows:
if $T$ is an elementary splitting relative to $\calh$,
then $G_v$ is elliptic in its tree of cylinders $T_c$ because $T_c$ is relative to $\calh\cup\calp$ and $G_v$ is rigid relative to $\calh\cup\calp$; since
groups elliptic in $T_c$ but not in $T$ are elementary (see Subsection \ref{defcyl}), and
$G_v$ is not,
this implies that $G_v$ is elliptic in $T$.
We distinguish two cases.
Case 1: $v$ is rigid.
Then $G_v$ is universally elliptic, so fixes a point $u$ in $T$,
which is unique because edge stabilizers are elementary and $G_v$ is not.
The group $G_u$ cannot be elementary. By Remark \ref{rem_UEQH},
it cannot be flexible QH because its subgroup $G_v$ is universally elliptic and non-elementary.
Thus $u$ is not modular and $\Phi$ has a representative $\alpha$ equal to the identity on $G_u$, hence on $G_v$.
Case 2: $v$ is a flexible QH vertex
of $T_{\mathrm{can}}$.
Let $e $ be an adjacent edge and $\hat e$ its lift to $\hat T$. Recall that $G_e=G_{\hat e}$ is a maximal elementary subgroup of $G_v$. We define a point $\hat v\in \hat T$ as follows. If $G_v$ is elliptic in $\hat T$, we call $\hat v$ its unique fixed point. If it is not elliptic, its action on its minimal subtree in $\hat T$ is dual to a family of 1-suborbifolds of $\Sigma$ (see Lemma 7.4 of \cite{GL3a}). We let $\hat v$ be the point of that subtree closest to $\hat e$ (possibly an endpoint of $\hat e$).
The stabilizer of $\hat v$ is QH, associated to a suborbifold $\hat \Sigma$ of $\Sigma$ (if $\hat \Sigma$ contains no essential $1$-suborbifold, $\hat v$ is a rigid vertex of $\hat T$).
Note that $G_{\Hat v}$ is non-elementary by Lemma \ref{lem_elemQH}.
The stabilizer of $\hat e$, and also of edges between $\hat e$ and $\hat v$ if any,
is contained in $G_{\hat v}$, in fact in the preimage of a boundary subgroup of $\pi_1(\Sigma)$
and $\pi_1(\hat\Sigma)$.
Let $u$ be the image of $\hat v$ in $T$.
Subcase 2a: $u$ is not modular. Then $\Phi$ has a representative $\alpha$ equal to the identity on $G_{u}$,
hence on $G_{\hat v}$ and on $G_e=G_{\hat e}$.
Note that $\alpha$ leaves $G_v$ invariant because $\alpha$ is an automorphism of $T_{\mathrm{can}}$
and $v$ is the only vertex of $T_{\mathrm{can}}$ fixed by $G_{\hat v}$.
Subcase 2b: $G_{u}$ is QH with finite fiber.
Then $G_{u}$ is elliptic in $T_{\mathrm{can}}$ (see Proposition 7.13 of \cite{GL3a}, which is valid in a relative setting),
hence in $\hat T$ \cite[Proposition 3.22]{GL3b}, so $G_{u}=G_{\hat v}$.
Unfortunately, this argument says nothing about incident edge groups at $u$.
First suppose that $\hat v$ is an endpoint of $\hat e$.
Choose an edge path with origin $\hat v$, starting with $\hat e$, such that all edges except the last one get collapsed to $ u$ in $T$ (this path consists of the single edge $\hat e$ if $\hat e$
is not collapsed to a point in $T$). Call this last edge $\hat e'$, and its initial vertex $a$. We have $G_{\hat e'}\subset G_a\subset G_{u}=G_{\hat v}$, so $G_{\hat e'}\subset G_{\hat e}$.
The group $G_{u}$ is QH with finite fiber, and
$G_{\hat e'}$ is an incident edge group. It is infinite by one-endedness, so is contained in a unique maximal elementary subgroup $C$ of $G_{u}$ (the preimage of a boundary subgroup of the underlying 2-orbifold). Since
$G_{\hat e'}\subset G_{\hat e}$, we have $G_{\hat e}\subset C$. By definition of $\mo(T)$, there is a representative $\alpha$ of $\Phi$ leaving $G_{u}$ invariant and equal to the identity on $C$, hence on $G_e=G_{\hat e}$. As above, $\alpha$ leaves $G_v$ invariant.
If there are edges between $\hat v$ and $\hat e$, call $\hat e'$ the edge that contains $\hat v$. It is not collapsed to a point in $T$, since it is collapsed in $T_{\mathrm{can}}$, so $G_{\hat e'}$ is an incident edge group of $G_{u}$. We now have $G_{\hat e }\subset G_{\hat e'}\subset C$ and we argue as in the previous case. This completes the analysis of the action of $\Phi$ on $G_e$ in case 2.
Still in case 2 (\ie assuming that $v$ is a flexible QH vertex stabilizer of $T_{\mathrm{can}}$),
we also need to understand the action of $\Phi$ on an element $B\in\calb_v$ which is not an incident edge stabilizer.
Such a $B$ contains a conjugate of an $H_j$ with finite index.
By minimality of $\hat T$, the group $B$ fixes a QH vertex $\hat v\in\hat T$. We then argue as above.
In subcase 2b, we have $B\in\calb_u$ (up to conjugacy) because $B$ contains a conjugate of $H_j$, so we can find $\alpha$ leaving $G_{u}$ invariant and equal to the identity on $B $ since $\Phi\in \mo(T)$.
This finishes case 2.
We can now conclude. Consider $\Gamma_{\mathrm{can}}=T_{\mathrm{can}}/G$, and recall that $T_{\mathrm{can}}$ is a tree of cylinders, so $\Gamma_{\mathrm{can}}$ is bipartite, with edges joining a vertex $x\in V_0(\Gamma_{\mathrm{can}})$ carrying a non-elementary group to a vertex $Y\in V_1(\Gamma_{\mathrm{can}})$ carrying an elementary group. We know that $\Phi$ fixes each vertex $x\in V_0(\Gamma_{\mathrm{can}})$,
and its action on $G_x$ is trivial if $x$ is not modular,
in $\Out(G_x;\mk\calb_x)$ if $G_x$ is QH.
Since $T_{\mathrm{can}}$ is a tree of cylinders, distinct edges of $\Gamma_{\mathrm{can}}$ with origin $x$ in $V_0(\Gamma_{\mathrm{can}})$ carry groups which are not conjugate in $G_x$. As $\rho_x(\Phi)\in\Out(G_x; \mk\mathrm{Inc}_x)$, we deduce that $\Phi$ acts as the identity on edges of $\Gamma_{\mathrm{can}}$ with origin $x$, hence on the whole of $\Gamma_{\mathrm{can}}$. Thus $\Phi\in\Out^0(T_{\mathrm{can}})$.
There remains to check that
$\rho_Y(\Phi)\in\Out(G_Y;\mk\mathrm{Inc}_Y)$ for $Y\in V_1(\Gamma_{\mathrm{can}})$. If $\varepsilon=(x,Y)$ is an adjacent edge, we have seen that $\Phi$ has a representative $\alpha$ equal to the identity on $G_\varepsilon$. Since $G_\varepsilon$ is infinite, $G_Y$ is the unique maximal elementary subgroup containing it, so $\alpha$ leaves $G_Y$ invariant. This completes the proof.
\section{Induced automorphisms} \label{sec_induced}
In this section $G$ is hyperbolic relative to $\calp=\{P_1,\dots, P_n\}$, and we study automorphisms of $P_1$ which are induced by automorphisms of $G$. We then apply this to the case when $G$ is hyperbolic and $H$ is a malnormal quasiconvex subgroup, viewing $G$ as hyperbolic relative to $H$.
\begin{dfn} Given families of subgroups $\calp$ and $\calh$, and a subgroup $Q$,
we say that $\alpha\in\Aut(Q)$ is \emph{extendable to $G$ relative to
$\calp$ and $\mk\calh$} if it is the restriction to $Q$ of an automorphism of $G$ representing an element of
$\Out(G;\calp,\mk\calh)$.
Being extendable only depends on the image of $\alpha$ in $\Out(Q)$,
so we define the \emph{group of extendable automorphisms} $\Out(Q \mathop{\nnearrow} (G;\calp, \mk\calh))\subset\Out(Q )$. We write $\Out(Q \mathop{\nnearrow} (G;\calp ))$ when $\calh=\emptyset$, and $\Out(Q \mathop{\nnearrow} G)$ for $\Out(Q \mathop{\nnearrow} (G;\emptyset))=\Out(Q \mathop{\nnearrow} (G;\{Q \} )) $.
If $Q$ equals its normalizer (for instance if $Q$ is an infinite maximal parabolic subgroup), there is a map $ \Out(G;Q)\to\Out(Q)$, and $\Out(Q \mathop{\nnearrow} G)$ is its image.
\end{dfn}
Suppose that
$P_1=G_v$ is a vertex group of a splitting of $G$ relative to $\calp=\{P_1,\dots, P_n\}$,
and $P_1$ contains no conjugate of $P_i$ for $i>1$ (this is automatic if no $P_i$ is finite).
Then $\Out(P_1\mathop{\nnearrow} (G;\calp))$ contains $\Out (G_v;\mk\mathrm{Inc}_v)$ (see Lemma \ref{automind}).
The following theorem says that virtually all extendable automorphisms occur in this fashion.
\begin{thm}\label{caleg_general}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots, P_n\}$, with $P_i$ finitely generated and infinite,
and $P_i\ne G$. Let $\calh$ be a finite family of finitely generated subgroups of $G$.
If $\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))$ is infinite, then:
\begin{enumerate}
\item
$P_1$ is a vertex group $G_v$ in
an elementary JSJ decomposition $\Gamma$ relative to $\calp\cup\calh$. Edge groups of $\Gamma$ are finitely generated.
\item
The group $\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))\subset\Out(P_1)$ has a finite index subgroup equal to
$\Out(P_1;\mk\calk)$, where $\calk=\mathrm{Inc}_v\cup\calh_{|G_v}$ is a finite family of finitely generated
subgroups of $P_1$
(the family of incident edge groups $\mathrm{Inc}_v$, and $\calh_{|G_v}$, are defined in Subsection \ref{ind}).
\end{enumerate}
\end{thm}
Since we do not assume that $G$ is one-ended relative to $\calp\cup\calh$, there is no \emph{canonical} JSJ decomposition.
\begin{proof}
$\bullet$
First assume that $G$ is one-ended relative to $ \calp\cup \calh$.
Consider the canonical elementary JSJ tree $T_{\mathrm{can}}$ relative to $ \calp\cup\calh$ as in Subsection \ref{cano}.
Let $G_v$ be a vertex stabilizer containing $P_1$.
It cannot be flexible QH because $P_1$ is not virtually cyclic (see Remark \ref{rem_UEQH}).
If it is rigid (non-elementary), we have seen in Subsection \ref{rela}
that the image of $\Out^0(G; \calp,\mk\calh)$ in $\Out(G_v)$ is finite
(recall that $\Out^0(G; \calp,\mk\calh)$ is the finite index subgroup of $\Out(G;\calp,\mk\calh)$
acting trivially on $T_{\mathrm{can}}/G$).
Since $P_1$ equals its normalizer, this implies that $\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))$ is finite, a contradiction.
Thus $G_v$ is elementary, so $G_v=P_1$. This proves Assertion 1 in the one-ended case (edge stabilizers of $T_{\mathrm{can}}$ are finitely generated by Lemma \ref{fingen}).
Assertion 2 is also clear since
$\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))$ is virtually
$\Out(P_1;\mk\mathrm{Inc}_{v},\mk\calh_{|G_v})$ by Proposition \ref{imro_rel}.
$\bullet$
We now consider the general case, first assuming that $G$ is torsion-free.
Let $F=G_u$ be the vertex stabilizer containing $P_1$ in a Grushko decomposition $S$ relative to $\calp\cup\calh$
(see Subsection \ref{JSJ}), and let $\calp_{ | F},\calh_{|F} $ be the induced structures (see Definition \ref{indu});
if $\calp\cup\calh=\{P_1\}$, then $F$ is simply the smallest free factor containing $P_1$.
Since $F$ is hyperbolic relative to $\calp_{ | F} $ by Lemma \ref{Hrufi},
and $\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))=\Out(P_1\mathop{\nnearrow} (F;\calp_{|F},\mk\calh_{|F}))$ because $F$ is a free factor (or by Remark \ref
{automind2}),
the results of the previous case apply.
The group $P_1$ is a vertex group $G_v$ of a splitting $\Gamma_F$ of $F$, which may be used to refine $S$ to an elementary JSJ decomposition $\Gamma$ of $G$ having $G_v$ as a vertex group
(see Subsection 8.1 of \cite{GL3a}). The families $\mathrm{Inc}_v$ and $\calh_{|G_v}$ are the same for $\Gamma_F$ and $\Gamma$.
$\bullet$
If $G$ has torsion, we define $F=G_u$ and $\calp_{ | F},\calh_{|F} $ as above, using a
Stallings-Dunwoody tree $S$ relative to $\calp\cup\calh$.
The proof is technically more complicated because we cannot neglect the incident edge groups $\mathrm{Inc}_u$.
All Stallings-Dunwoody trees $S$ have a unique vertex stabilizer $G_{u(S)}$ equal to $F$, but the incident edge groups may vary. This was studied in Section 4 of \cite{GL2}, where we defined a ``peripheral structure'' for $F$.
To state the relevant result, we choose a Stallings-Dunwoody tree $S$ for which the valence of $u(S)$
in the quotient graph of groups $S/G$ is minimal.
Since no edge stabilizer is properly contained in a conjugate,
it follows from Proposition 4.9 of \cite{GL2} that the incident structure $\mathrm{Inc}_{u(S)}$
does not depend on the choice of such an $S$ (in trees with non-minimal valence, there may be more incident edge groups;
such a group is contained in a group belonging to $\mathrm{Inc}_{u(S)}$).
We fix $S$, and from now on we write $u$ rather than $u(S)$, so $F=G_u$.
Any automorphism representing an element of $\Out(G;\calp,\mk\calh)$ and leaving $P_1$ invariant
also leaves $F$ invariant.
Since $P_1$ and $F$ are equal to their normalizers, $\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))$ is the image of the map $p:\Out(G;\calp,\mk\calh)\to\Out(P_1)$, and $p$ factors through $\rho_u:\Out(G;\calp,\mk\calh)\to\Out(F)$.
By Remark \ref{automind2}, the image of $\rho_u$ contains $\Out(F;\mk \mathrm{Inc}_u,\calp_{ | F},\mk\calh_{|F})$
and is contained in $\Out(F;\calp_{ | F},\mk\calh_{|F})$.
Our choice of $S$ implies that automorphisms in the image of $\rho_u$ preserve $\mathrm{Inc}_{u}$ globally. Since
$\mathrm{Inc}_{u}$ consists of finitely many finite subgroups of $F$ (well-defined up to conjugacy),
the index of $\Out(F;\mk \mathrm{Inc}_u,\calp_{ | F},\mk\calh_{|F})$
in $\Out(F;\calp_{ | F},\mk\calh_{|F})$ is finite. It therefore suffices to
study the image of
$q:\Out(F;\mk \mathrm{Inc}_u,\calp_{ | F},\mk\calh_{|F})\to\Out(P_1)$, and to show that it is virtually $\Out(P_1;\mk\calk)$.
The group $F=G_u$ is hyperbolic relative to the family $\calp_{ | F}$ (see Lemma \ref{Hrufi}),
and one-ended relative to $\calp_{ | F}\cup\calh_{|F}$. Since the image of $q$ is infinite, we have seen that $P_1$ is a vertex group $G_v$ in the canonical elementary JSJ decomposition $\Gamma_{\mathrm{can}}$ of $F$ relative to $\calp_{ | F}\cup\calh_{|F}$. One obtains an elementary JSJ tree $T$ of $G$ relative to $\calp\cup\calh$ by refining $S$
using
JSJ decompositions of vertex groups (see Subsection 8.1 of \cite{GL3a}), so Assertion 1 is proved.
Moreover, $\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))$ is virtually $\Out(P_1;\mk{\calk'})$, where $\calk'$ is the union of $\mathrm{Inc}_v$ (the incident edge groups of $P_1$ in $T_{\mathrm{can}}$) and $(\mathrm{Inc}_u\cup\calh_{|F})_{ | P_1}$.
We now show that $\calk'=\calk$ if we construct $T$ carefully.
When $S$ is refined to yield $T$, the vertex $u$ is replaced by $T_{\mathrm{can}}$. There is some freedom in the
way edges of $S$ containing $u$ are attached to $T_{\mathrm{can}}$: an edge $e$ may be attached to any vertex of $T_{\mathrm{can}}$ which is fixed by $G_e$.
We may therefore assume that,
if $e$ is an edge of $T\setminusT_{\mathrm{can}}$ attached to $v$,
then $v$ is the only fixed point of $G_e$ in $T_{\mathrm{can}}$.
The family
$(\calh_{|F})_{ | P_1}$ is defined viewing $F$ as a vertex group of $S$, and then $P_1$ as a vertex group of $T_{\mathrm{can}}$. Since groups in $\calh$ are infinite and edge stabilizers of $S$ are finite, $(\calh_{|F})_{ | P_1}$ equals $\calh _{ | P_1}$, defined viewing $P_1$ as a vertex group of $T$. We complete the proof by showing that $\mathrm{Inc}_v\cup(\mathrm{Inc}_u )_{ | P_1}$ is the family of incident edge groups in $P_1=G_v$ viewed as a vertex stabilizer of $T$.
There are two types of incident edge groups of $G_v$
in $T$. Those fixing edges in $T_{\mathrm{can}}$ are precisely those in $\mathrm{Inc}_v$. Because of the way we contructed $T$, those fixing edges in $T\setminus T_{\mathrm{can}}$ have $v$ as unique fixed point in $T_{\mathrm{can}}$, they are the groups in $(\mathrm{Inc}_u )_{ | P_1}$ (see Definition \ref{indu}).
\end{proof}
If $G$ is (absolutely) hyperbolic, and $P$ is a subgroup,
then $G$ is hyperbolic relative to $\{P\}$
if (and only if) $P$ is quasiconvex and almost malnormal,
see \cite[Theorem 7.11]{Bow_relhyp} or \cite{Osin_elementary}.
If so, Theorem \ref{caleg_general} applies and describes $\Out (P\mathop{\nnearrow} G)$, the automorphisms of $P$ which extend to $G$.
\begin{cor}\label{cor_induit}
Let $P$ be a
quasiconvex, almost malnormal subgroup of a hyperbolic group $G$, with $P\ne G$.
\begin{itemize}
\item
If $\Out (P\mathop{\nnearrow} G)$ is infinite, then $P$ is a vertex group in a splitting of $G$
with finitely generated edge groups, and
$\Out (P\mathop{\nnearrow} G)$
is virtually
$\Out(P;\mk\calk)$ with $\calk$
the family of incident edge groups
(a finite family of finitely generated subgroups of $P$).
\item
If $P$ is torsion-free, then $\Out (P\mathop{\nnearrow} G)$ has a finite index subgroup with a finite classifying space.
\end{itemize}
\end{cor}
\begin{proof}
The first assertion follows from Theorem \ref{caleg_general}.
Being quasiconvex, $P$ is a hyperbolic group.
It is proved in \cite{GL_McCool} that, if $P$ is a torsion-free hyperbolic group and $\calk$ is an arbitrary family of subgroups, then $\Out(P;\mk\calk)$ has a finite index subgroup with a finite classifying space.
\end{proof}
If $G=F_n$, every finitely generated subgroup is quasiconvex (it is a virtual retract by \cite{Hall_1949}),
so we get:
\begin{cor} If $P\subset F_n$ is finitely generated and malnormal,
then $\Out (P\mathop{\nnearrow} F_n)$ is virtually $\Out(P;\mk\calk)$ for some finite family $\calk$ of finitely generated subgroups of $P$. It has a finite index subgroup with a finite classifying space. \qed
\end{cor}
This is a partial answer to a question that was asked by D.\ Calegari. Note that the proof uses JSJ decompositions over groups which are not small.
\begin{example} Let $P\subset F_n$ be a characteristic subgroup of finite index, with $n\ge3$.
Then $\Out (P\mathop{\nnearrow} G)$ is not virtually of the form $\Out(P;\mk\calk)$ because there exist automorphisms of $F_n$ with no nontrivial periodic conjugacy class.
There are similar exemples with $P$ of infinite index.
\end{example}
\section{Groups with infinitely many automorphisms}\label{outinfi}
In this section, we characterize those relatively hyperbolic groups whose automorphism group is infinite.
In the first subsection, we point out that determining whether $\Out(G)$ is infinite or not is relatively easy when $G$ is torsion-free or one-ended. In particular, we give a complete answer for toral relatively hyperbolic groups.
The most interesting case is thus when $G$ has torsion and splits over a finite group. For instance, virtually free groups with $\Out$ finite were
determined by M.\ Pettet \cite{Pettet_virtually}. We will give a different characterization (see Example \ref{Pett}).
If $G$ is hyperbolic relative to $\calp$, we will show in Subsection \ref{sec_infini_marked} that
the group $\Out(G;\mk \calp)$ of automorphisms which act trivially on each parabolic subgroup is infinite if and only if $G$ has an elementary splitting relative to $\calp$ whose group of twists is infinite.
In Subsection \ref{sec_infini_unmarked}, we get a characterization for the full group $\Out(G;\calp)$ being infinite:
this happens if and only if $G$ has an elementary splitting relative to $\calp$ whose group of twists is infinite,
or in which a maximal parabolic subgroup $P$ occurs as a vertex group and $P$ has infinitely many outer automorphisms acting trivially on incident edge groups (such automorphisms extend to $G$).
When $G$ is hyperbolic, we show in Subsection \ref{hyp} that $\Out(G)$ being infinite is equivalent to $G$ having a splitting over a maximal virtually cyclic group with infinite center;
this is decidable algorithmically.
\subsection{Torsion free groups}
We first note:
\begin{lem} \label{freep}
If a torsion-free, finitely generated, group $G$ is a non-trivial free product,
then $\Out(G)$ is infinite.
\end{lem}
\begin{proof} Write $G=A*B$. If $a\in A$ is not central, the automorphism of $G$ equal to conjugation by $a$ on $A$ and to the identity on $B$ has infinite order in $\Out(G)$. Assuming that $\Out(G)$ is finite, we deduce that $Z(A)$ has finite index in $A$, so $A$ is abelian because $[A,A]$ is finite by a result due to Schur
\cite[10.1.4]{Robinson_course}. Similarly, $B$ is abelian. Moreover, $\Out(A)$ and $\Out(B)$ are finite, so $A=B={\mathbb {Z}}$. This is a contradiction since $\Out({\mathbb {Z}}*{\mathbb {Z}})$ is infinite.
\end{proof}
Thus, for torsion-free groups, infiniteness of $\Out(G)$ is only interesting for one-ended groups.
One can get a similar result in a relative setting.
\begin{prop}\label{prop_freep_rel}
Let $G$ be a finitely generated, non-cyclic, torsion-free group, and $ \calh$ a finite collection of finitely generated subgroups.
If the Grushko decomposition of $G$ relative to $ \calh$ is non-trivial, and not an amalgam $G=A_1*A_2$ with $A_1, A_2$ abelian,
then $\Out(G;\mk\calh)$ is infinite.
\end{prop}
\begin{rem} \label{pab}
If the
Grushko decomposition $\Gamma$ relative to $ \calh$ is $G=A_1*A_2$ with $A_1, A_2$ abelian, then $\Out(G; \mk\calh)$ is finite if and only if, for $i=1,2$, the subgroup of $A_i$ generated by subgroups conjugate to a group in $\calh$ has finite index. This is because $\Gamma$ is $\Out(G; \mk\calh)$-invariant by \cite{For_deformation} (its Bass-Serre tree is the unique reduced tree
in its deformation space). Twists are trivial because $A_1$ and $A_2$ are abelian, so $\Out(G; \mk\calh)$ is infinite if and only if $A_1$ or $A_2$ has infinitely many automorphisms acting trivially on $\calh_{|A_i}$.
\end{rem}
\begin{proof}
Assume that $\Out(G; \mk\calh)$ is finite, and
let $\Gamma$ be a reduced Grushko decomposition of $G$ relative to $ \calh$. We assume that $\Gamma$ is non-trivial and we show that it is an amalgam as in the proposition.
We first note that $G$ cannot split relative to $ \calh$ as an HNN extension $G=A*_{\{1\}}$ over the trivial
group. Indeed,
the group of twists of this HNN extension is isomorphic to $(A\times A)/Z(A)$,
with $Z(A)$ embedded diagonally, so contains
the infinite group $A$, a contradiction.
It follows that $\Gamma$ is a tree of groups.
The proof of Lemma \ref{freep} shows that, whenever $G$ splits as a free product $A*B$ relative to $ \calh$,
then $A$ and $B$ are abelian: otherwise the group of twists of the splitting is infinite.
Since $\Gamma$ is reduced, it follows that it is an amalgam $G=A_1*A_2$ with $A_1, A_2$ abelian: if $\Gamma$ has more than one edge, collapsing an edge provides a decomposition
with a non-abelian vertex group.
\end{proof}
Let now $G$ be hyperbolic relative to
$\calp=\{P_1,\dots,P_n\}$, with $P_i$ finitely
generated, not virtually cyclic.
If $G$ is one-ended relative to $\calp$, one can read infiniteness of
$\Out(G;\calp)$ from
the JSJ decomposition thanks to
Theorem \ref{thm_struct_m}:
$\Out(G;\calp)$ is finite if and only if the canonical elementary JSJ decomposition relative to $\calp$ has no flexible QH vertex with infinite mapping class group, the parabolic subgroups $P_j$ appearing as vertex stabilizers have
$\Out(P_j;\mk\mathrm{Inc}_{P_j})$ finite, and the group of twists is finite.
We may be more specific under additional conditions on the parabolic subgroups.
\begin{prop} \label{rht} Let $G$ be a non-abelian toral relatively hyperbolic group. The following are equivalent:
\begin{enumerate}
\item $\Out(G)$ is finite.
\item Every non-trivial abelian one-edge splitting of $G$ is an amalgam $A*_CB$ with $C$ of finite index in $A$ or $B$.
\item $G$ has no non-trivial splitting over an abelian subgroup stable under taking roots.
\item $G$ is freely indecomposable and its canonical abelian JSJ decomposition $\Gamma_{\mathrm{can}}$ relative to non-cyclic abelian subgroups satisfies the following:
\begin{itemize*}
\item $\Gamma_{\mathrm{can}}$ consists of a central vertex, possibly connected to terminal vertices carrying an abelian group;
\item the central vertex is rigid, or QH with underlying surface $\Sigma$ homeomorphic to a pair of pants or a twice punctured projective plane;
\item at each terminal vertex, the incident edge group has finite index in the vertex group.
\end{itemize*}
\end{enumerate}
\end{prop}
A subgroup $A$ is stable under taking roots if $g\in A$ whenever $g^n\in A$ for some $n\ge2$ (this is also called pure, or isolated).
The pair of pants and the twice punctured projective plane appear in this statement because they are the only compact hyperbolic surfaces with finite mapping class group (see the end of Subsection \ref{autar}). The fundamental group of a pair of pants is rigid.
The fundamental group of a twice punctured projective plane has two (incompatible) cyclic splittings
relative to the boundary
(it is flexible), but none over a maximal cyclic subgroup.
Automorphisms of toral relatively hyperbolic groups were considered in \cite{DaGr_isomorphism}, and some of the equivalences in Proposition \ref{rht}
follow from their results (note for instance that a splitting as in (3) is an essential splitting in the sense of their Definition 3.30).
\begin{proof}
If $G$ is a free product, (2) and (3) are false, and so is (1) by
Lemma \ref{freep}. We therefore assume that $G$ is freely indecomposable.
We prove $(1)\imp (2)$ by assuming that (2) does not hold, and showing that $\Out(G)$ is infinite.
If $G$ is an HNN extension over an abelian group, or an amalgam with $A$ and $B$ non-abelian, the group of twists of the splitting is infinite.
If $G=A*_CB$ with $A$ abelian containing $C$ with infinite index, $\Out(G)$ is infinite because $A$ has nontrivial automorphisms equal to the identity on $C$.
It is clear that (2) implies (3). To prove that (3) implies (2), first assume that $G=A*_C B$
with $C$ abelian of infinite index in both $A$ and $B$.
Let $\Hat C$ be the set of all roots of elements of $C$, an abelian subgroup containing $C$ with finite index (recall that all abelian subgroups of $G$ are cyclic or parabolic, hence finitely generated).
Since $\hat C$ is elliptic in the amalgam, up to exchanging the role of $A$ and $B$, we can assume that $\Hat C<A$.
Then $G=A*_{\hat C}\grp{B,\hat C}$ is a decomposition contradicting (3).
The case of an HNN extension is similar, but we do not need the hypothesis that $C$ has infinite index in $A$.
If (4) does not hold, we construct a splitting contradicting (2).
If $\Gamma_{\mathrm{can}} $ has a flexible QH vertex, and if the underlying surface is not a twice punctured projective plane,
then
one
simply considers the cyclic splitting dual to a 2-sided essential simple closed curve not bounding a M\"obius band.
The other possibility is that $\Gamma_{\mathrm{can}} $
has a vertex $v$ carrying an abelian group such that either $v$ has valence $\ge2$, or $v$ is terminal with the edge group of infinite index in $G_v$. One gets the required splitting by collapsing edges of $\Gamma_{\mathrm{can}}$.
We have proved $(1)\imp (2)\imp (4)$ and $(2)\Leftrightarrow(3)$ .
We conclude by deducing from the exact sequence of Corollary \ref{limgp} that $\Out(G)$ is finite if (4) holds.
The groups $GL_{r_k,n_k}({\mathbb {Z}})$ are trivial because $r_k=0$ by the finiteness assumption at terminal vertices.
The mapping class group of $\Sigma$ is finite.
Twists are trivial because terminal vertices carry an abelian group.
\end{proof}
\begin{prop} Let $G$ be torsion-free, hyperbolic relative to nilpotent subgroups. Assume that $G$ is not nilpotent. If $\Out(G)$ is finite, then $G$ is freely indecomposable and
its canonical JSJ decomposition $\Gamma_{\mathrm{can}}$ over nilpotent groups relative to non-cyclic nilpotent subgroups consists of a central vertex
which is
rigid, or QH with underlying surface $\Sigma$ homeomorphic to a pair of pants or a twice punctured projective plane,
possibly connected to terminal vertices carrying a nilpotent group.
\end{prop}
\begin{proof}
We may assume that no $P_i$ is cyclic, so $\Out(G)=\Out(G;\calp)$. As above, $G$ is freely indecomposable and there is no QH vertex
other than those mentioned.
Recall that an infinite torsion-free nilpotent group has infinite center. As in the proof of Corollary \ref{limgp}, the group of twists of $\Gamma_{\mathrm{can}}$ contains the direct product $\prod (Z(G_v))^{ | E_v | -1}$ taken over vertices carrying a nilpotent group, so is infinite as soon as there is a vertex with valence $ | E_v |\ge2$.
\end{proof}
\subsection{Infinity of marked automorphisms}\label{sec_infini_marked}
\begin{thm}\label{thm_marked}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots, P_n\}$, with each $P_i$
finitely generated.
Then $\Out(G;\mk\calp)$ is infinite if and only if
there is a splitting of $G$ over finitely generated elementary subgroups, relative to $\calp$,
with an infinite group of twists $\calt$ (see Subsection \ref{arb}).
More generally, if $\calq$ is a finite family of finitely generated subgroups with $\calp\subset\calq$,
the same characterization holds for $\Out(G; \mk\calq)$, with a splitting relative to $ \calq$.
\end{thm}
By Lemma \ref{lem_collapse_twist}, the splittings may be assumed to
have only one edge. The proof will be given at the end of the subsection.
\begin{example} \label{Pett}
Consider the virtually free group
$G= D_4*_CD_6 *_C D_4$,
where $D_n$ is the dihedral group of order $n$ and $C$ has order 2 (note that $C$ is central in $D_4$, but equal to its centralizer in $D_6$). For this two-edge splitting $\Gamma$,
and any splitting obtained by collapsing an edge, the group of twists is finite.
The one-edge splitting $\Delta$ given by the amalgam $G=D_6*_C \left(D_4*_C D_4\right)$, however,
has a twist of infinite order.
The Bass-Serre tree of $\Delta$ is the tree of cylinders of the Bass-Serre tree $T$ of $\Gamma$, and
Assertion 1 of Proposition \ref{prop_induct} holds in this case. Compare \cite{Pettet_virtually}.
\end{example}
When $G$ is hyperbolic, the group of twists of the splitting provided by Theorem \ref{thm_marked} contains an element of infinite order
(see also Theorem \ref{thm_twist_hyp}).
The following example shows that this does not hold for general relatively hyperbolic groups.
\begin{example}\label{ex2}
Let $G=B_1*B_2$ be the free product of
two infinite torsion groups with trivial center.
It is hyperbolic relative to $\calp=\{B_1, B_2\}$, and $\Out(G;\mk\calp)$ is infinite. But no splitting over elementary subgroups has a twist of infinite order, as we now show. By way of contradiction, suppose that some $Z_{G_{o(e)}}(G_e)$ contains an element of infinite order.
The group $G_e$ is trivial, or contains a torsion element $g\ne1$, or is isomorphic to ${\mathbb {Z}}$. It cannot be trivial since $G_{o(e)}$ would then be a torsion group. The existence of a torsion element
$g$ also leads to a contradiction since the centralizer of such a $g$ is a torsion group. We conclude by observing that $G$
does not split over ${\mathbb {Z}}$: if it does, an equivariant map from the Bass-Serre tree of the amalgam $B_1*B_2$ to that of the splitting maps vertices to vertices and must be locally injective, hence globally injective, a contradiction.
\end{example}
Before proving
Theorem \ref{thm_marked}, we note the following fact, which follows from the presentation of $\calt}%{\mathrm{Tw}$ recalled in Subsection \ref{arb}.
\begin{lem} \label{ti} Let $G$ be a relatively hyperbolic group. Let $\Gamma$ be
a one-edge splitting of $G$ over a virtually cyclic group $G_e$ with infinite center, with $G_e$ not parabolic.
Any element of infinite order in $Z(G_e)$ defines a twist which has infinite order in $\Out(G)$, unless $\Gamma$ is an amalgam and one of the vertex groups is virtually cyclic with infinite center. \qed
\end{lem}
The following result is a key step in the proof of Theorem \ref{thm_marked}.
\begin{prop}\label{prop_induct}
Let $ T$ be a non-trivial tree with edge stabilizers of constant finite cardinality $k$.
Let $T_c$ be its tree of cylinders for the equality equivalence relation (see Subsection \ref{defcyl}).
Then at least one of the following holds:
\begin{enumerate}
\item $T_c$ is nontrivial and its edge stabilizers are finite;
\item $T$ has a collapse $T'$ which is nontrivial and has an infinite group of twists $\calt}%{\mathrm{Tw}(T')$;
\item $T$ has a collapse $T'$ which is nontrivial and invariant under $\Out(\cald(T))$.\end{enumerate}
\end{prop}
$\cald(T)$ denotes the deformation space of $T$ over groups of cardinality $\le k$. All reduced trees in $\cald(T)$ have edge stabilizers of order $k$ (see Subsection \ref{defsp}). Also note that $T_c$ is invariant under $\Out(\cald(T))$, and dominated by $T$, so we get:
\begin{cor} \label{propc_induct}
$T$ has a collapse which is nontrivial and has an infinite group of twists, or $T$ dominates a nontrivial tree $T'$ with finite edge stabilizers which is invariant under $\Out(\cald(T))$. \qed
\end{cor}
\begin{proof}[Proof of Proposition \ref{prop_induct}]
We can assume that $T$ is reduced.
We first consider the case when $T_c$ is trivial.
Since edges of $T$ belong to the same cylinder if and only if they have the same stabilizer, all edges of $T$ have the same stabilizer, a finite normal subgroup $A$.
If there is only one orbit of edges, $T $ is the unique reduced tree in $\cald(T)$ because no edge stabilizer may be properly contained in another
\cite{Lev_rigid}.
It follows that $T'= T$ is invariant under $\Out(\cald(T))$.
Assume that there is more than one orbit of edges.
Since $A$ is normal and finite, its centralizer in $G$ has finite index, so for any vertex $v$ in an arbitrary collapse $T'$ of $T$ (including $T$ itself),
and any edge $e$ of $T'$ incident to $v$, the group $Z_{G_v}(G_e)$ is infinite as soon as $G_v$ is infinite.
By Lemma \ref{lem_twist0}, if Assertion 2 does not hold, then all vertex stabilizers of nontrivial collapses $T'$
are either finite or infinite with infinite center.
Since $T$ is reduced, this is possible only if $T$ has one orbit of vertices,
all vertex stabilizers equal to $A$,
and only two orbits of edges
(this is respectively because a non-trivial amalgam $H_1*_A H_2$,
an HNN extension $H_1*_A$ with $H_1\neq A$, and
a double HNN extension $(A*_A)*_A$, have finite center).
In particular, $G/A$ is free of rank $2$.
One easily checks that collapsing the orbit of any edge gives a tree $T'$ with $\calt}%{\mathrm{Tw} (T')$ infinite,
though Lemma \ref{lem_twist0} does not apply.
This concludes the case when $T_c$ is trivial.
From now on, we assume that $T_c$ is nontrivial and some edge $\eps=(x,Y)$ of $T_c$ has infinite stabilizer.
View the cylinder $Y$ as a subtree of $T$ containing $x$, and consider an edge $e\subset Y$ with origin $x$. Then $G_Y=N_G(G_e)$ and $G_\eps=G_Y\cap G_x=N_{G_x}(G_e)$. We know that $Z_{G_x}(G_e)$ is infinite, but we cannot apply Lemma \ref{lem_twist0} since we do not know that $Z(G_x)$ is finite.
Assume for a moment that $Y$ contains edges from at least two $G$-orbits. Consider $ T'$ obtained from $T$ by collapsing all edges in the orbit of $e$,
and denote by $ x'$ the image of $x$ in $T'$. Since $T$ is reduced, $G_x\subsetneq G_{ x'}$.
By the assumption on $Y$, there is an edge $e'$ of $ T'$ incident to $ x'$ with $G_{e'}=G_e$.
Since $Z_{G_{ x'}}(G_{e'})$ is infinite, we are done if $Z(G_{ {x'}})$ is finite. We now show that $Z(G_{ {x'}})$ being infinite leads to a contradiction.
Edge stabilizers of $T$ being finite, $x$ is the unique point of $T$ fixed by $G_\eps$.
Since $G_\varepsilon\subset G_x\subset G_{ x'}$, the group $Z(G_{ {x'}})$ normalizes $G_\eps$, hence also fixes $x$,
and only $x$ since it is infinite.
This implies that $G_{ x'}$ fixes $x$, a contradiction to $G_x\subsetneq G_{ x'}$.
Returning to the general case, there is a $G$-invariant partition of the set of cylinders: those for which there is an edge $(x,Y)$ of $T_c$ with infinite stabilizer, and the others. Thanks to the previous argument, we may assume that all edges contained in a given cylinder of the first type belong to the same $G$-orbit.
Let now $T'$ be the tree obtained from $T$ by collapsing all edges in cylinders of the second type. It is nontrivial (but $ T'=T$ is possible).
We show that $T'$ does not change if we replace $T$ by another reduced tree $T_1$ in $\cald(T)$. This implies that $T'$ is invariant under $\Out(\cald(T))$.
One may join $T$ and $T_1$ by slide moves (see Subsection \ref{defsp}). In a slide move, an edge $e$ slides over an edge $f$ belonging to a different orbit, with $G_e\subset G_f$. Here one must have $G_e=G_f$, so $e$ and $f$ belong to the same cylinder, necessarily of the second type. The slide move does not change $T'$ since the cylinder gets collapsed.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm_marked}]
We prove the ``only if '' direction (the other direction is clear since twists act by conjugations on vertex groups).
We may assume that all groups in $ \calq$ are infinite.
We first suppose that $G$ is one-ended relative to $ \calq$.
Let $T$ be the canonical elementary JSJ tree relative to $ \calq=\calp\cup\calh$ as in Subsection \ref{cano}. Its edge stabilizers are finitely generated by
Lemma \ref{fingen}.
By Theorem
\ref{thm_struct_m_rel},
if $\Out(G; \mk\calq)$ is infinite, then either $\calt}%{\mathrm{Tw}(T)$ is infinite and we are done,
or $T$ has at least a non-rigid QH vertex $v$ with $\Out(G_v;\calb_v)$ infinite.
The underlying orbifold group $\pi_1(\Sigma)$ splits, relative to its boundary subgroups, over a maximal virtually cyclic subgroup with infinite center (see \cite{DG2}, Proposition 3.1). This induces an elementary splitting of $G_v$ which extends to a splitting of $G$.
By Lemma \ref {extens}, this splitting of $G$ is relative to $ \calq$ (because the intersection of $G_v$ with a conjugate of a group in $ \calq$ projects into a boundary subgroup in $\pi_1(\Sigma)$), and has an infinite group of twists by Lemmas \ref{ti} and \ref{lem_extension_twist}.
This proves the theorem if $G$ is one-ended relative to $\calq$.
In the general case,
let $k\in \bbN$ be the smallest number such that $G$ splits relative to $ \calq$ over a group of cardinality $k$.
Let $T$ be a reduced JSJ tree relative to $ \calq$ over subgroups of cardinality $k$ (see Subsection \ref{JSJ}).
Its deformation space is invariant under $\Out(G; \mk\calq)$.
By Corollary \ref{propc_induct}, either some collapse of $T$ has an infinite group of twists (and we are done),
or $T$ dominates a nontrivial tree $T'$ with finite edge stabilizers
which is invariant under
$ \Out(G;\mk\calq)$.
Note that the groups in $ \calq$ are elliptic in $T'$, since they are elliptic in $T$ and $T$ dominates $T'$.
We may assume that $\calt}%{\mathrm{Tw}(T')$ is finite. Let $ A_0\subset \Out(G; \mk\calq)$ be the finite index subgroup acting trivially on the graph $T'/G$.
By Assertion 1 of Lemma \ref{lem_virt_nobitwist}, there exists a vertex group $G_v$ of $T'$ such that
$\rho_v(A_0)\subset \Out(G_v)$ is infinite. In particular, $G_v$ is infinite.
The group $G_v$ is hyperbolic relative to the family $\calp_{|G_v}$ (see Lemma \ref{Hrufi}).
By Lemma \ref{automind},
we have $\rho_v(A_0)\subset \Out(G_v; \mk\calq_{|G_v})$.
If the theorem holds for $G_v$,
we get a graph of groups decomposition $\Gamma_0$ of $G_v$ relative to $ \calq_{|G_v}$ having an infinite group of twists. Since $T'$ has finite edge stabilizers, $\Gamma_0$ is relative to $\mathrm{Inc}_v$ and one may refine $T'/G$
to
a graph of groups $\Lambda$ by using $\Gamma_0$.
By Lemma \ref{lem_extension_twist}, the splitting $\Lambda$ has an infinite group of twists. It is relative to $ \calq$ by Lemma \ref{extens}.
If the theorem does not hold for $G_v$, we repeat the construction. If the process stops after finitely many steps, we get a splitting $\Lambda$ as in the previous case. If it does not stop, we get an infinite sequence of trees $T_i$ relative to $\calq$ with finite edge stabilizers, with $T_{i+1}$ strictly dominating $T_i$. Since $G$ is finitely presented relative to $\calq$, there is a Stallings-Dunwoody decomposition relative to $\calq$ (see Subsection \ref{JSJ}), and we reach a contradiction (see \cite[p.\ 130]{DicksDunwoody_groups}).
\end{proof}
\subsection{Infinity of unmarked automorphisms} \label{sec_infini_unmarked}
Using Theorem \ref{caleg_general}, we now
characterize relatively hyperbolic groups for which $\Out(G;\calp)$
is infinite.
We first note the following consequence of Theorem \ref{thm_marked}.
\begin{cor} \label{cor_ker}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots,P_n\}$, where each $P_i$ is infinite and finitely generated, and let $\calh$ be a finite family of finitely generated subgroups.
If $\Out(G;\calp,\mk\calh )$ is infinite, there is an elementary splitting relative to $\calp\cup\calh$
with an infinite group of twists, or there is an $i$ such that the natural map $\Out(G;\calp,\mk\calh )\to\Out(P_i)$ (defined because $P_i$ equals its normalizer) has infinite image.
\end{cor}
\begin{proof}
If all maps $\Out(G;\calp,\mk\calh )\to\Out(P_i)$ have finite image, then
the intersection of their kernels, namely $\Out(G;\mk\calp,\mk\calh)$, is infinite.
We can now apply Theorem \ref{thm_marked}.
\end{proof}
\begin{cor}
\label{cor_outu_infini}
Assume furthermore that $G$ is non-elementary (\ie
$P_i\ne G$).
Then
$\Out(G;\calp,\mk\calh)$ is infinite if and only if $G$ has an
elementary splitting as a graph of groups $\Lambda$ relative to $ \calp\cup\calh$
such that one of the following holds:
\begin{itemize}
\item
the group of twists of $\Lambda$ is infinite,
\item
or $\Lambda$ has a vertex $v$ such that $G_v=P_i$ is a maximal parabolic subgroup and
$\Out(G_v;\mk\mathrm{Inc}_v,\mk\calh_{|G_v}) $ is infinite.
\end{itemize}
\end{cor}
\begin{proof} As in Theorem \ref{thm_marked}, the ``if'' direction is clear, so assume that $\Out(G;\calp,\mk\calh)$ is infinite.
By Corollary \ref{cor_ker}, and up to renumbering, we can assume that
$\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))$ is infinite.
By Theorem \ref{caleg_general}, $P_1$ is a vertex group in some elementary decomposition
of $G$ relative to $\calp\cup\calh$
and $\Out(P_1;\mk\mathrm{Inc}_v,\mk\calh_{|G_v})$ has finite index in
$\Out(P_1\mathop{\nnearrow} (G;\calp,\mk\calh))$.
In particular, $\Out(P_1;\mk\mathrm{Inc}_v,\mk\calh_{|G_v})$ is infinite.
\end{proof}
\subsection{Hyperbolic groups} \label{hyp}
We apply the results of the previous subsections to the case when $G$ is a hyperbolic group.
We say that a
subgroup of $G$ is $\calz_{\mathrm{max}}$ if it is maximal for inclusion among virtually cyclic subgroups
with infinite center. Note that any virtually cyclic subgroup $C$ with infinite center is contained in a unique $\calz_{\mathrm{max}}$ subgroup $\hat C$ (the pointwise stabilizer of $\partial C$).
Given any splitting of a hyperbolic group over a $\calz_{\mathrm{max}}$ subgroup $C$, any central element $c\in C$ of infinite order defines
a twist of infinite order in $\Out(G)$.
\begin{thm}\label{thm_twist_hyp}
If $G$ is hyperbolic, $\Out(G)$ is infinite if and only if there is a non-trivial splitting of $G$
over a $\calz_{\mathrm{max}}$ subgroup (such a splitting always has a twist of infinite order).
If $\calh$ is a finite family of finitely generated subgroups, $\Out (G;\mk\calh)$ is infinite if and only if there is a non-trivial
splitting over a $\calz_{\mathrm{max}}$ subgroup
relative to $\calh$.
\end{thm}
It was proved independently by M.\ Carette \cite{Carette_automorphism} that $\Out(G)$ is infinite if and only if
$G$ has a splitting over
a finite group or a (maybe non maximal)
virtually cyclic group with infinite center,
with a twist of infinite order
(see \cite{Lev_automorphisms,DG2} for the one-ended case).
\begin{proof} The ``if'' direction is clear, so we assume that $\Out(G;\mk\calh)$ is infinite. All splittings considered in this proof will be relative to $\calh$.
Theorem \ref{thm_marked} and Lemma \ref{lem_collapse_twist} say that $G$ has a one-edge splitting over a (possibly finite) virtually cyclic group $C$, whose group of twists is infinite. We assume that this splitting is an amalgam $G=A*_C B$; the case of an HNN extension is similar.
We first explain how to replace this amalgam over $C$ by one over a (possibly non-maximal) virtually cyclic subgroup $C'$ with infinite center.
If $C$ is infinite with finite center, its centralizer in $G$ is finite and this forces the group of twists to be finite.
So assume that $C$ is finite.
If both $Z_A(C)/Z(A)$ and $Z_B(C)/Z(B)$ are finite, the group of twists is finite, so assume for instance that $Z_A(C)/Z(A)$ is infinite.
Note that $Z(A)$ has to be finite, since otherwise $A$ would be virtually cyclic,
and $Z_A(C)/Z(A)$ would be finite.
Consider an element of infinite order $t\in Z_A(C)$, and perform a fold to get $G=A*_{\grp{C,t}} \grp{B,t}$. This is a splitting over a virtually cyclic subgroup $C'$ with infinite center, and it is relative to $\calh$.
The twist defined by $t$
has infinite order in $\Out(G)$ by Lemma \ref{ti}.
Now suppose that an amalgam $G=A'*_{C'} B'$ has an infinite group of twists, with $C'$ virtually cyclic with infinite center.
Then $A',B'$ have finite center. The $\calz_{\mathrm{max}}$ subgroup $\Hat C'$ containing $C'$
is elliptic in the amalgam, and one can perform a fold to get an amalgam over $\Hat C'$. This splitting is non-trivial because $\Hat C'$
is not conjugate to $A'$ or $B'$ since they have finite center. The group of twists of the new splitting is clearly infinite.
\end{proof}
\begin{thm} \label{thm_MC_infini}
There is an algorithm which, given a hyperbolic group $G$, decides whether $\Out(G)$ is infinite or not. More generally, if $\calh$ is a finite family of finitely generated subgroups, one may decide whether
$\Out(G; \mk\calh)$ is infinite.
\end{thm}
\begin{proof} We start with the first assertion.
We first construct an algorithm that stops if and only if $\Out(G)$ is finite.
By Theorem 8.1 of \cite{DG2}, one can compute a finite generating set of $\Out(G)$.
Moreover, one can solve the word problem in $\Out(G)$ as this amounts to solving (uniformly)
the simultaneous
conjugacy problem in $G$ (see \cite{BriHow_conjugacy} for a solution).
Thus, for each $R>0$, one can determine the ball $B_R$ of radius $R$ in the Cayley graph of $\Out(G)$. Checking whether $B_R=B_{R+1}$ for some $R$ gives the required algorithm.
It now suffices to construct an algorithm that stops if $\Out(G)$ is infinite. By \cite[Lemma 2.8]{DG2},
one can decide whether a subgroup of $G$ (given by generators) is $\calz_{\mathrm{max}}$ or not.
One can therefore
enumerate all decompositions of $G$ as an amalgam or HNN extension over
$\calz_{\mathrm{max}}$ subgroups.
By Corollary \ref{thm_twist_hyp}, this provides an algorithm that stops if $\Out(G)$ is infinite.
The argument to decide whether
$\Out(G; \mk\calh)$ is infinite is similar.
The first algorithm is the same since Theorem 8.1 of \cite{DG2} provides generators for $\Out(G;\mk\calh)$.
For the second algorithm, one has to restrict to splittings relative to $\calh$, so one needs an algorithm that, given a splitting, stops if the splitting is relative to $\calh$. This is done by choosing a generating set $S_i$ for each $H_i\in\calh$, enumerating all conjugates of $S_i$, and comparing them with words written using the generators of a vertex group.
\end{proof}
In general, we do not know how to decide whether
$\Out(G; \calh)$ is infinite (see Remark \ref{rips}). The following is an answer when $G$ is hyperbolic relative to $\calh$.
\begin{prop} There is an algorithm which, given
a torsion-free hyperbolic group $G$, a finite family $\calp$
of finitely generated locally quasiconvex subgroups $P_i$ such that $G$ is hyperbolic relative to $\calp$,
and a finite family $\calh$ of finitely generated subgroups,
decides whether $\Out(G;\calp,\mk\calh)$ is infinite.
\end{prop}
Since $G$ is assumed to be hyperbolic relative to $\calp$, each $P_i$ is quasiconvex in $G$.
In particular, $P_i$ is itself a hyperbolic group. Local quasiconvexity of $P_i$ means that its finitely generated subgroups are quasiconvex (in $P_i$, hence also in the hyperbolic group $G$).
\begin{proof}
First, using Touikan's algorithm \cite[Theorem A]{Touikan_finding},
one can decide whether $G$ splits as a free product relative to $\calp\cup\calh$.
If it does, it is easy to decide whether $\Out(G;\calp,\mk\calh)$ is infinite using Proposition \ref{prop_freep_rel} and Remark \ref{pab}.
So assume that $G$ is one-ended relative to $\calp\cup\calh$. We may also assume that no $P_i$ is cyclic.
We use \cite[Theorem C]{Touikan_finding} to decide whether $G$ splits in a suitable way.
For this, we need our parabolic groups $P_i$ to be algorithmically tractable
in the sense of \cite[Definition 1.13]{Touikan_finding}.
Since $P_i$ is locally quasiconvex, it is hyperbolic and the conjugacy problem is solvable in $P_i$.
Moreover, local quasiconvexity of $P_i$ implies that
one can decide whether a finite subset $S\subset P_i$ generates $P_i$
or not, by checking whether a given generating set of $P_i$ lies in the quasiconvex subgroup $\grp{S}$ \cite{Kapovich_detecting}.
This says that $P_i$ is algorithmically tractable.
Applying \cite[Theorem C]{Touikan_finding}, one can decide whether there exists
an elementary splitting of $G$ (viewed as a relatively hyperbolic group) relative
to $\calp\cup\calh$ with finitely generated edge groups, and if so find one.
By local quasiconvexity of $P_i$, edge groups of the splitting are quasiconvex in the hyperbolic group $G$,
and so are vertex groups (see Subsection \ref{qconv}).
Iterating this process, one can compute a maximal elementary splitting $\Gamma$ of $G$ relative
to $\calp\cup\calh$ (\ie a splitting that cannot be refined non-trivially into an elementary
splitting relative to $\calp\cup\calh$).
Arguing as in \cite[Section 6]{DaGr_isomorphism}
and \cite[Lemma 2.34]{DG2}, one may then
recognize the QH subgroups in $\Gamma$, and
find the canonical elementary JSJ decomposition of $G$ relative to $\calp\cup\calh$
(see also \cite[Theorem 3.12]{DaTo_isomorphism}).
By Theorem \ref{thm_struct_m_rel}, $\Out(G;\calp,\mk\calh)$ is infinite if and only if
the group of twists $\calt$ is infinite, or there is a vertex $v$ such that $\Out(G_v;\mk\mathrm{Inc}_v,\mk\calh_{|G_v})$ is infinite.
Recall that $\calt$ is isomorphic to a quotient of $\prod_{e\in E} Z_{G_{o(e)}}(G_e)$ by edge and vertex relations.
Each group $Z_{G_{o(e)}}(G_e)$ is either trivial or infinite cyclic, and is computable.
Edge relations and vertex relations are generated by embeddings of groups $Z(G_e)$ and $Z(G_v)$ in this product.
Since one can compute the corresponding subgroups of the abelian group $\prod_{e\in E} Z_{G_{o(e)}}(G_e)$,
one can decide whether the group of twists is infinite or not.
To decide whether $\Out(G_v;\mk\mathrm{Inc}_v,\mk\calh_{|G_v})$ is infinite,
we can apply \cite{DaGr_isomorphism} (or more explicitly \cite[Corollary 3.4]{DG2}),
or Theorem \ref{thm_MC_infini},
but we need to determine $\calh_{|G_v}$ (see Definition \ref{indu}).
Given $H\in \calh$ generated by a finite set $S$, one can decide whether there is $g\in G$ such that $S^g\subset G_v$
in the same way as above because $G_v$ is quasiconvex.
One can similarly decide whether $H$ fixes an edge, which allows to compute $\calh_{|G_v}$.
\end{proof}
\section{Fixed subgroups} \label{fixed}
In this section, we use JSJ decompositions to study fixed subgroups of automorphisms.
This is inspired by arguments due to Sela \cite{Sela_Nielsen}. The proof of the Scott conjecture (Theorem \ref{scott}) given below is not really new, but using the relative JSJ decomposition makes the argument more direct. Using Theorem \ref{somrig}, we will prove in \cite{GL_McCool} that, given a toral relatively hyperbolic group $G$, there are only finitely many possibilities for fixed subgroups of automorphisms of $G$, up to isomorphism. This was proved by Shor \cite{Shor_Scott} for $G$ torsion-free hyperbolic.
\begin{thm}[\cite{BH_tt}]\label{scott} Let $\alpha$ be an automorphism of a free group $F_n$. Its fixed subgroup $\mathop{\mathrm{Fix}}\, \alpha$ has rank at most $n$.
\end{thm}
\begin{proof} The smallest free factor containing $\mathop{\mathrm{Fix}}\, \alpha$ is $\alpha$-invariant. Replacing $G$ by this free factor, we may assume that $F_n$ is one-ended (freely indecomposable) relative to $\mathop{\mathrm{Fix}}\, \alpha$. We also assume that $\mathop{\mathrm{Fix}}\, \alpha$ is not cyclic.
Let $T$ be the canonical cyclic JSJ tree relative to $\mathop{\mathrm{Fix}}\, \alpha$ (see Subsection \ref{cano}), and let $G_v$ be the vertex stabilizer containing $\mathop{\mathrm{Fix}}\, \alpha$. It is $\alpha$-invariant because $T$ is invariant and $v$ is the only vertex fixed by $\mathop{\mathrm{Fix}}\alpha$, and abelianizing shows that it has rank $\le n$.
By Remark \ref{rem_UEQH} it cannot be flexible QH because $\mathop{\mathrm{Fix}}\, \alpha$ is not cyclic, so it is rigid. By standard arguments due to Paulin and Rips,
$\alpha$ has finite order in $\Out(G_v)$: otherwise, applying Theorem \ref{thm_sci} with $\calp $ consisting of incident edge groups and $\calh$ consisting of $\mathop{\mathrm{Fix}}\, \alpha$ (which is finitely generated) yields a cyclic splitting of $G_v$ which contradicts rigidity.
By Dyer-Scott \cite{DySc_periodic}, $\mathop{\mathrm{Fix}}\, \alpha$ is a free factor of $G_v$ so has rank $\le n$.
If we do not wish to use Gersten's result that $\mathop{\mathrm{Fix}}\, \alpha$ is finitely generated \cite{Gersten_fixed_adv},
we argue by contradiction as follows. Let $H$ be a finitely generated free factor of $\mathop{\mathrm{Fix}}\, \alpha$ of rank $>n$.
We claim that $F_n$ is one-ended
relative to $H$ (see \cite{Perin_elementary}
and Lemma 7.6 of \cite{GL3b} for more general statements).
Otherwise, let $\Hat H$ be the smallest free factor of $F_n$ containing $H$.
Then $\Hat H$ is $\alpha$-invariant, and $\mathop{\mathrm{Fix}}\alpha\cap\Hat H$ has rank at most $n-1$
(assuming that the theorem holds in $\Hat H$ by induction on $n$), a contradiction since
$\mathop{\mathrm{Fix}}\alpha\cap\Hat H$ retracts onto $H$.
Define $G_v$ as above, using the cyclic JSJ splitting of $F_n$ relative to $H$. The fixed subgroup of $\alpha_{ | G_v}$ is a subgroup of $\mathop{\mathrm{Fix}}\, \alpha$ which has rank $\le n$ and contains $H$. This is a contradiction since $H$ is a retract of $\mathop{\mathrm{Fix}}\, \alpha$.
\end{proof}
This proof uses \cite{DySc_periodic}, which is specific to free groups. Applying the same argument to (relatively) hyperbolic groups only yields:
\begin{thm} \label{somrig}
Let $G$ be hyperbolic relative to a finite family $\calp$ of slender subgroups.
Consider $\alpha\in \Aut(G;\calp)$
such that $\mathop{\mathrm{Fix}}\alpha$ is not elementary (\ie not
virtually cyclic or parabolic).
Then $\mathop{\mathrm{Fix}}\alpha$ is contained in an $\alpha$-invariant vertex group $G_v $ of a splitting of $G$ over elementary subgroups relative
to $\calp$, and $\alpha_{|G_v}$ has finite order in $\Out(G_v)$.
\end{thm}
\begin{proof}
The proof is similar to the one above. Note that $\mathop{\mathrm{Fix}}\alpha$ is finitely generated by \cite[Cor.\ 9.2]{Hruska_quasiconvexity}
because it is relatively quasiconvex \cite{MiOs_fixed} and groups in $\calp$ are slender.
First assume that $G$ is one-ended relative to $\calp\cup\{\mathop{\mathrm{Fix}}\alpha\}$.
Let $T_{\mathrm{can}}$ be the canonical elementary JSJ decomposition
of $G$ relative to $\calp\cup\{\mathop{\mathrm{Fix}}\alpha\}$.
Let $v$ be the unique vertex of $T_{\mathrm{can}}$ fixed by $\mathop{\mathrm{Fix}}\alpha$. As above, $G_v$ is $\alpha$-invariant;
it cannot be QH because it contains the universally elliptic subgroup $\mathop{\mathrm{Fix}}\alpha$
which is not virtually cyclic,
so $G_v$ is rigid: it has no elementary splitting relative to
$\mathrm{Inc}_v\cup\calp_{|G_v}\cup\{\mathop{\mathrm{Fix}}\alpha\}$.
By Lemma \ref{rh}, $G_v$ is hyperbolic relative to $\mathrm{Inc}_v\cup\calp_{|G_v}$.
By Theorem \ref{thm_sci}, the group $\Out(G_v;\mathrm{Inc}_v,\calp_{|G_v}, \mk{\{\mathop{\mathrm{Fix}} \alpha\}})$ is finite.
It contains the class of $\alpha_{|G_v}$ by Lemma \ref{automind}, so
$\alpha_{|G_v}$ has finite order in $\Out(G_v)$.
If $G$ is not relatively one-ended, we consider a reduced Stallings-Dunwoody tree $S$ relative to $\calp\cup\{\mathop{\mathrm{Fix}}\alpha\}$ (see Subsection \ref{JSJ}).
Since $\mathop{\mathrm{Fix}}\alpha$ is infinite, it fixes a unique vertex $u\in S$.
The deformation space of $S$ is $\alpha$-invariant, so $\alpha(G_u)$ fixes a vertex $u'\in S$.
Since $\mathop{\mathrm{Fix}}\alpha\subset \alpha(G_u)$ fixes only $u$, we have $u'=u$ and $\alpha(G_u)=G_u$.
We now apply the previous analysis to the restriction of $\alpha$ to $G_u$, which is hyperbolic relative to $\calp_{|G_u}$ by Lemma \ref{Hrufi}. We get a splitting $\Lambda$ of $G_u$ relative to $\calp_{|G_u}$, and
we obtain the desired splitting of $G$ by refining $S/G$ using $\Lambda$ (see Lemma \ref{extens}).
\end{proof}
\section{Rigid groups have finitely many automorphisms}
\label{sec_Rips}
The goal of this section is to prove Theorem \ref{thm_sci}.
Let us first recall its statement.
\begin{UtiliseCompteurs}{thm_sci}
\begin{thm} Let $G$ be hyperbolic relative to finitely generated subgroups $\calp=\{P_1,\dots,P_n\}$,
with $P_i\ne G$.
Let $\calh=\{H_1,\dots,H_q\}$ be another family of finitely generated subgroups. If $\Out(G;\calp,\mk\calh)$ is infinite, then $G$ splits over an elementary subgroup relative to $\calp\cup\calh$.
\end{thm}
\end{UtiliseCompteurs}
As mentioned earlier, the proof uses $\R$-tree s. All actions on $\R$-tree s considered here are by isometries. An arc is a subset isometric to an interval $[a,b]\subset \R$ with $a\ne b$.
As in the simplicial case, an action on an $\R$-tree\ $T$ is \emph{relative} to subgroups $H_i$ if each $H_i$ is elliptic (fixes a point) in $T$.
Because the parabolic groups are not assumed to be slender, we
will need to analyze actions on $\bbR$-trees which are not quite stable.
\subsection{Constructing an $\R$-tree}
\begin{thm}[\cite{BeSz_endomorphisms}]\label{th_BS} Let $G,\calp,\calh$ be as in Theorem \ref{thm_sci}.
If $\Out(G;\calp,\mk\calh)$ is infinite, then $G$ has a non-trivial action on an $\bbR$-tree $T$ relative to $\calp\cup\calh$ such that arc stabilizers are elementary.
\end{thm}
The proof is essentially in \cite{BeSz_endomorphisms}, noting that a locally elementary subgroup is elementary by Lemma \ref{borne}.
We also add the remark that the groups $P_i,H_j$ are elliptic in $T$.
\begin{proof}
Let $\varphi_k$ be automorphisms representing distinct elements of $\Out(G;\calp,\mk\calh)$.
Let $X $ be a $\delta$-hyperbolic space on which $G$ acts as in Subsection \ref{gene}.
Consider a finite generating set $S$ of $G$, and the minimal displacement
$d_k=\inf_{x\in X}\max_{s\in S}d_X(x,\phi_k(s).x)$.
Choose a point $x_k\in X$ where $\max_{s\in S}d_X(x_k,\phi_k(s).x_k)\leq d_k+\frac 1k$.
Using the Bestvina-Paulin method, it is shown in \cite{BeSz_endomorphisms} that $d_k$ goes to infinity, the rescaled pointed metric spaces $X_k =(\frac{1}{d_k}X,x_k)$ converge to an $\R$-tree\ $T$ (after taking a subsequence), and the action of $G$ on $X_k$ twisted by $\varphi_k$ converges to a non-trivial isometric action of $G$ on $T$ with locally elementary (hence elementary)
arc stabilizers.
We now prove that the action is relative to $\calp\cup\calh$. Since groups in $\calp\cup\calh$ are finitely generated, it suffices to show that any element $g$ belonging to $P_i$ or $H_j$ is elliptic in $T$.
Suppose $g $ acts hyperbolically in $T$. Then there exists $a\in T$ such that $d_T(a,g^2a)=2d_T(a,ga)>0$.
If $a_k$ is an approximation point of $a$ in $X_k =\frac{1}{d_k}X$, then
$$\displaystyle \frac{d_X(a_k,\phi_k(g^2)a_k)- d_X(a_k,\phi_k(g)a_k)} {d_k}$$ converges to $d_T(a,g^2a)-d_T(a,ga)=d_T(a,ga)>0$,
so for $k$ large enough $$d_X(a_k,\phi_k(g^2)a_k)- d_X(a_k,\phi_k(g)a_k)> 2\delta+ \frac{d_k}{2}d_T(a,ga).
$$
Lemme 9.2.2 of \cite{CDP} implies that $\phi_k(g)$ acts loxodromically on $X$, with translation length going to infinity since $d_k\to\infty$. This is a contradiction if $g\in P_i$, since every $\phi_k(g)$ is parabolic in this case.
If $g\in H_j$, all elements $\varphi_k(g)$ are conjugate so have the same translation length in $X$, also a contradiction.
\end{proof}
\begin{rem} One can show that a group $H\in\calh $ is elliptic in $T$, even if it is not assumed to be finitely generated. We know that every $h\in H$ is elliptic. If $H$ is not elliptic, it fixes an end of $T$, so every finitely generated subgroup of $H$ fixes a ray. This implies that finitely generated subgroups of $H$ are parabolic, so $H $ is parabolic and therefore elliptic in $T$.
On the other hand, Theorem \ref{bf} below requires finite generation.
\end{rem}
\begin{rem} \label{lent} The hypothesis that automorphisms act trivially on $H_j$ may be weakened. It is sufficient to assume that their growth under iteration is slower on $H_j$ than on $G$.
\end{rem}
\subsection{Hypostability}
To deduce a splitting as in Theorem \ref{thm_sci} from the action on the $\bbR$-tree of Theorem \ref{th_BS},
we will generalize
the following basic fact (see Theorem \ref{bf2}):
\begin{thm}[{\cite[Thm 9.6]{BF_stable}}] \label{bf}
Let $G$ be a finitely presented group, and let $\calq$
be a finite family of finitely generated subgroups.
Assume that $G$ has a non-trivial stable action on an $\bbR$-tree $T$ relative to $\calq$.
Then $G$ splits relative to $\calq$ over a group $K$ which is an extension $1\ra A\ra K \ra \bbZ^k\ra 1$, where $A$ fixes an arc of $T$ and $k\geq 0$.
\end{thm}
Recall that an arc $J$ is stable if any subarc of $J$ has the same stabilizer as $J$. An action is \emph{stable} if every arc $I$ contains a stable subarc $J$.
\begin{cor} \label{po}
Theorem \ref{thm_sci} holds if every $P_i$ is slender and $G$ is finitely presented.
\end{cor}
\begin{proof} Assume that every $P_i$ is slender. In this case a subgroup of $G$ is elementary if and only if it is slender. In particular, elementary subgroups satisfy the ascending chain condition, so the action on the $\bbR$-tree
$T$ provided by Theorem \ref{th_BS} is stable.
If furthermore $G$ is finitely presented, Theorem \ref{bf} (applied with $\calq=\calp\cup\calh$) gives a splitting
that satisfies the conclusion of Theorem \ref{thm_sci} (note that $K$ is slender because $A$ is).
\end{proof}
In general, however, $G$ is only finitely presented relative to $\calp$, and the action on $T$ only satisfies a weaker property than stability, which we call hypostability
(see \cite{Kapovich_sequences} for a different property called semistability).
\begin{dfn}\label{hs}
Let $G$ be a group acting on an $\bbR$-tree $T$.
The action is \emph{hypostable}
if, for each arc $I\subset T$, there exists a subarc $J\subset I$
satisfying the following {hypostability condition}: if $g\in G$ acts hyperbolically in $T$ and $gJ\cap J$
is an arc, then $\Stab J=\Stab (gJ)$ (equivalently, $g$ normalizes $\Stab J$).
\end{dfn}
Hypostability is weaker than stability, because any stable arc $J$ satisfies the hypostability condition: if $gJ\cap J$ is an arc, $\grp{\Stab(J),\Stab(gJ)}$ is contained in
the stabilizer of $gJ\cap J$, which coincides with $\Stab(J)$ and $\Stab(gJ)$
by stability of $J$ and $gJ$.
\begin{lem}\label{lem_elem_hypostable} Let $G$ be hyperbolic relative to finitely generated subgroups $\calp=\{P_1,\dots,P_n\}$. Any action of $G$ on an $\R$-tree\ $T$
relative to $\calp$ with elementary arc stabilizers
is hypostable.
\end{lem}
\begin{proof} Let $C$ be such that any elementary subgroup $H$ of $G$ of cardinality $>C$ is contained in a unique maximal elementary subgroup $E(H)$ (see Lemma \ref{borne}).
Let $I\subset T$ be an arc. If the stabilizer of every subarc has cardinality at most $C$, then
a subarc $J\subset I$ whose stabilizer has the greatest cardinality is stable and we are done.
Otherwise, consider $J\subset I$ whose stabilizer $H=\Stab(J)$ has cardinality $>C$.
The stabilizer of every subarc of $J$ is elementary, so is contained in $E(H)$.
If subgroups of $E(H)$ satisfy the ascending chain condition, in particular if $E(H)$ is virtually cyclic, then $J$ contains a stable subarc.
Thus we can assume that $E(H)$ is parabolic.
We prove hypostability by showing that
any $g$ such that $gJ\cap J$ contains an arc is elliptic in $T$. Indeed, $\grp{H,H^g}$ fixes an arc in $T$, so is elementary.
It follows that $\grp{H,H^g}\subset E(H)$, so $E(H)=E(H^g)=E(H)^g$.
Since $E(H)$ is its own normalizer, we get $g\in E(H)$. But $T$ is relative to $\calp$, so $g$ is elliptic.
\end{proof}
\begin{example}
We sketch the construction of an action as in
Lemma \ref{lem_elem_hypostable} which is not stable.
Let $G$ be
the free product $G=P*{\mathbb {Z}}$, with $P$ a (non-slender) finitely generated group
containing a copy of the free abelian group on a countable basis $\bbZ^{(\bbQ)}$. Note that $G$ is hyperbolic relative to $\{P,{\mathbb {Z}}\}$. Informally, identifying the edge of the free product with $[0,1]$,
one can produce an $\R$-tree\ from the Bass-Serre tree of this splitting
by folding the group $\bbZ^{([0,\frac pq]\cap\bbQ)}$ on a length $\frac pq$ for all $0<\frac pq<1$.
The stabilizer of an arc $[a,b]\subset [0,1]$ is then $\bbZ^{([0,b]\cap\bbQ)}$. This action is hypostable but unstable.
\end{example}
\begin{thm} \label{bf2}
Theorem \ref{bf} holds if the action on the $\bbR$-tree is only assumed to be hypostable,
and the group $G$ is only assumed to be finitely presented relative to $\calq$.
\end{thm}
The proof will be given in the next subsection.
The following corollary is an immediate consequence of Lemma \ref{lem_elem_hypostable} and Theorem \ref{bf2}.
\begin{cor}\label{cor_RipsRH}
Let $G$ be a relatively hyperbolic group,
with $\calp,\calh$ as in Theorem \ref{thm_sci}. If $G$ acts non-trivially on an $\R$-tree\ $T$
relative to $\calp\cup\calh$ with elementary arc stabilizers,
then $G$ splits over an elementary subgroup relative to $\calp\cup\calh$. \qed
\end{cor}
Theorem \ref{thm_sci} follows immediately
from this corollary, using the $\bbR$-tree provided by Theorem \ref{th_BS}.
A refinement of Corollary \ref{cor_RipsRH} will be given in Subsection \ref{zm}.
\begin{rem} Let $G$ and $T$ be as in Theorem \ref{bf}. If $T$ is not a line, one can get a splitting over a group $K$ which is an extension of $\bbZ$ or $\bbZ/2\bbZ$ by
a group $A$ fixing an arc in $T$ (see \cite{BF_stable}).
One can also approximate $T$ (in the equivariant Gromov topology) by simplicial trees with controlled edge stabilizers, as in \cite{Gui_approximation}. The same facts are true under the assumptions of Theorem \ref{bf2}.
\end{rem}
\subsection{Proof of Theorem \ref{bf2}}
Recall that a subtree $Y\subset T$ is \emph{indecomposable} \cite{Gui_actions} if, given arcs $I,L\subset Y$, there exist $g_1,\dots,g_n\in G$ such that $L\subset g_1I\cup\dots\cup g_nI$,
and every $g_iI\cap g_{i+1}I$ is an arc. We call $g_1,\dots,g_n $ an \emph{$I$-covering} of $L$.
\begin{lem}\label{lem_superpos}
Assume that $Y\subset T$ is an indecomposable subtree.
Given
two arcs $I,L\subset Y$ with $I\subset L$,
there exists an $I$-covering $ g_1,\dots,g_r $ of $L$ such that $g_1=1$ and
every $ g_{i}g_{i-1}\m$ is hyperbolic in $T$.
\end{lem}
\begin{proof}
Given any $I$-covering of $L$, there exists $i$ such that $I\cap g_i I$ is an arc, and therefore $1,g_i,g_{i-1}, \dots, g_1,g_2,\dots, g_n$ is an $I$-covering starting with 1. From now on, we only consider coverings starting with 1.
The interval covered will always be $L$.
We fix an orientation of $I$. It induces an orientation of $gI$ for any $g\in G$. If $I\cap gI$ is an arc, the orientations of $I$ and $gI$ may agree or disagree on this arc.
We first claim that there exists an $I$-covering $ 1=a_1,\dots,a_p $ of $L$
such that, for each $i$,
the orientation of $a_iI$ agrees with that of $a_{i+1}I$ on their
intersection (we say that such an $I$-covering is orientation-preserving).
If not, we can find arcs $gI,g'I$ whose intersection is an arc on which the orientations disagree.
Define $g_0=g'^{-1}g$ and $J=I\cap g_0 I$; the orientations of $I$ and $g_0I$ disagree on $J$.
Now consider a $J$-covering $1=b_1,\dots,b_p $ of $L$.
It is also an $I$-covering of $L$. Since $J= I\cap g_0 I$, we get another $I$-covering of $L$
if we replace some of the $b_i$'s by $b_ig_0$. Starting with $a_1=b_1=1$, we then define $a_i$ inductively as either $b_i$ or $b_ig_0$,
making sure that orientations agree. This proves the claim.
Next note that there exists a hyperbolic element $h\in G$ mapping an arc $J'\subset I$ to a different arc $h(J')\subset I$ in an orientation-preserving way. To see this, first choose $h_1$ mapping an arc $J_1\subset I$ to a disjoint arc $J_2\subset I$. If orientation is reversed, choose $h_2$ mapping an arc $J_3\subset J_2$ to an arc $J_4\subset I$ different from $ J_1$ and $J_3$. Then take $h$ equal to $h_2$ or $h_2h_1$.
We can now conclude.
Let $1=a_1,\dots,a_r$ be an orientation-preserving $J'$-covering of $L$.
Since $J'\subset I\cap h\m I$,
we get another orientation-preserving $I$-covering if we replace some of the $a_i$'s by $a_ih\m$.
If $a_{i}a_{i-1}\m $ is not hyperbolic, it is the identity on $a_{i-1}J'\cap a_{i} J'$.
We therefore get
the required $I$-covering of $L$ by defining $g_i$ inductively as $a_i$ or $a_ih\m$ so that $g_{i}g_{i-1}\m $ is not the identity on $a_{i-1}J'\cap a_{i} J'$.
\end{proof}
\begin{cor}\label{cor_hypostable}
Let $T$ be hypostable, and let $Y\subset T$ be an indecomposable subtree.
Any element $g\in G$ fixing an arc in $Y$ fixes the whole of $Y$.
In particular, any arc in $Y$ is stable.
\end{cor}
\begin{proof}
Assume that $g$ fixes an arc $I\subset Y$. Given $x\in Y$, we aim to prove that $g$ fixes $x$.
After making $I$ smaller, we can assume that there is an arc $L$ containing $x$ and $I$.
Definition \ref{hs} provides a subarc $J\subset I$ satisfying the hypostability condition. Consider a $J$-covering
$1= g_1,\dots,g_r $ of $L$ as in
Lemma \ref{lem_superpos}. Since $g_{i+1}g_i\m $, hence also $g_i\m g_{i+1}$, is hyperbolic, hypostability of $J$ implies that all arcs $g_iJ$ have the same stabilizer. The element $g$ fixes $g_1J=J$, so it fixes every $g_iJ$ and therefore $L$. In particular, $g$ fixes $x$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{bf2}]
We explain how to adapt the arguments in \cite{BF_stable,Gui_approximation}.
Let $\calq=\{Q_1,\dots,Q_q\}$.
Let $ \grp{S_i\mid\calr_i}$ be a presentation of $Q_i$, with
$S_i$ a finite generating set and $\calr_i$ a possibly infinite set of relators.
Let $\grp{S\mid\calr}$ be a presentation of $G$ such that $S$ is a finite generating set of $G$ containing each $S_i$,
and $\calr$ is the union of $\calr_1\cup\dots\cup\calr_q$ with finitely many additional relators.
Consider a finite subtree $K\subset T$, \ie the convex hull of finitely many points.
We explain how to choose $K$ large enough so as to yield a resolution of $T$
as in \cite[Definition 2.2]{Gui_approximation}, even though $G$ is only relatively finitely presented.
For each $s\in S$, we consider $K_s=K\cap s\m K$ and the restriction $\phi_s:K_s \ra sK_s$ of $s$ (we may assume that no $K_s$ is empty). We then define the suspension $\Sigma$ as the foliated $2$-complex obtained by gluing
foliated bands $K_s\times [0,1]$ to $K$, where we glue $(x,0)$ to $x$ and $(x,1)$ to $\phi_s(x)$.
Note that $\pi_1(\Sigma)$ is naturally
identified with the free group on $S$.
Next, we need all relators of $\calr$ to be represented by
loops contained in leaves of $\Sigma$. Since each $Q_i$ fixes a point $p_i$ in $T$, and $S$ contains $S_i$, requiring that
$K$ contains $p_1,\dots,p_q$ takes care of $\calr_1\cup\dots\cup\calr_q$.
There remain finitely many other relators,
and, as in \cite{BF_stable,Gui_approximation}, one can choose $K$ so that
they also are represented by loops contained in leaves.
The complex $\Sigma$ provides a resolution of $T$ in the sense of Definition 2.2 of \cite{Gui_approximation}
(as pointed out in \cite{Gui_approximation}, the set $\calc$ of curves contained in leaves mentioned in Definition 2.2 is not assumed to be finite).
Obtaining a resolution is the only place where finite presentation is used in \cite{Gui_approximation}.
As for stability, it is used only in Proposition 4.3 of \cite{Gui_approximation}
to prove that, if an element fixes an arc
in the subtree $T_{\Gamma_v}\subset T$ corresponding to a
minimal component of $\Sigma$, then it fixes the whole of $T_{\Gamma_v}$.
By \cite[Proposition 1.25]{Gui_actions}, the geometric $\bbR$-tree dual to a minimal component of $\Sigma$ is indecomposable,
and by Lemma 1.19(1) of \cite{Gui_actions} its image $T_{\Gamma_v}\subset T$
is an indecomposable subtree of $T$.
Corollary \ref{cor_hypostable} then replaces Proposition 4.3 of \cite{Gui_approximation}
under our hypostability assumption.
The rest of the argument of \cite{Gui_approximation} applies without modification.
As in Proposition 4.1 of \cite{Gui_approximation},
the tree dual to $\Sigma$ is a graph of actions on $\bbR$-trees $T(\calg')$,
such that arc stabilizer of the vertex actions lie in the kernel
of these actions. Applying Propositions 5.2, 7.2 and 8.1 of \cite{Gui_approximation},
one can replace these vertex actions by actions on simplicial trees
whose stabilizers are abelian modulo the kernel.
By Bass-Serre theory, this provides a splitting of $G$ over the extension
of an abelian group $A$ by the kernel $K$ of a vertex action.
This splitting is relative to $\calq$ as in the Reduction Lemma in \cite[\S 4]{Gui_approximation}.
\end{proof}
\subsection{$\calz_{\mathrm{max}}$ splittings} \label{zm}
Say that a subgroup of a relatively hyperbolic group is $\calz_{\mathrm{max}}$
if it is maximal for inclusion among non-parabolic virtually cyclic subgroups with infinite center.
\begin{thm}\label{thm_RipsZmax}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots,P_n\}$, with $P_i$ finitely generated.
Let $\calh=\{H_1,\dots H_q\}$ be a (possibly empty) family of finitely generated subgroups.
Assume that $G$ acts non-trivially on an $\R$-tree\ $T$
relative to $\calp\cup\calh$, and that arc stabilizers are either finite, parabolic or $\calz_{\mathrm{max}}$.
Then $G$ splits relative to $\calp\cup\calh$ over a finite, parabolic or $\calz_{\mathrm{max}}$ subgroup.
\end{thm}
Corollary \ref{cor_RipsRH} provides a splitting over an elementary group $A$. Here we assume that every loxodromic arc stabilizer of $T$ is $\calz_{\mathrm{max}}$, and we claim that the same is true for $A$: if it is loxodromic, then it is $\calz_{\mathrm{max}}$.
\begin{proof}
$\bullet$ Assume first that the foliated $2$-complex $\Sigma$ constructed in the proof of Theorem \ref{bf2}
has a minimal component $\Sigma_v$. Let $G_v$ be the image of its fundamental group in $G$,
and let $T_{G_v}\subset T$ be the corresponding subtree of $T$.
In particular, $G_v$ is not elliptic in $T$.
By \cite[Theorem 5.13]{BF_stable} or \cite[Theorem 3.1]{Gui_approximation},
$G_v$ is a vertex group in a decomposition of $G$ as a graph of groups $\Gamma$ relative to $\calp\cup\calh$.
All arcs in $T_{G_v}$ have the same stabilizer $F$,
a normal
subgroup of $G_v$.
We claim that $F$ is finite. Otherwise, there are two cases.
If $F$ is non-parabolic, hence virtually cyclic,
it has finite index in its normalizer and therefore in $G_v$, so $G_v$ is elliptic, a contradiction.
If $F$ is infinite and contained in a maximal parabolic group $P$, then
almost malnormality of $P$ implies that the normalizer of $F$ is contained in $P$,
so $G_v\subset P$. Since $P$ is elliptic in $T$, this is also a contradiction.
Thus $F$ is finite.
We now distinguish several cases, depending on the nature of the minimal component $\Sigma_v$.
First, $\Sigma_v$ cannot be a homogeneous (axial, toral) component since $G_v$ would then be
virtually $\bbZ^k$ for some $k\geq 2$ (\cite[Theorem 9.4(2)]{BF_stable} or \cite[section 5.1]{Gui_approximation}),
hence parabolic, contradicting ellipticity of parabolic groups in $T$.
If $\Sigma_v$ is an exotic (Levitt, thin) minimal component,
one obtains a splitting of $G$ over $F$, and we are done (\cite[Proposition 7.2]{Gui_approximation}, \cite[Theorem 9.4(3)]{BF_stable}).
If $\Sigma_v$ is a surface (IET) component,
then by \cite[Theorem 9.4(1)]{BF_stable} or \cite[section 8]{Gui_approximation},
after performing some moves, one can assume that $\Sigma_v$ is a surface with boundary, and $G_v/F$
is the fundamental group of a $2$-orbifold with conical singularities supporting a measured foliation with dense leaves.
Moreover,
$G_v$ is a QH vertex group (with fiber $F$) of $\Gamma$.
Let $A\subset G_v$ be the preimage of the fundamental group of
an essential two-sided simple closed curve not bounding a M\"obius band, and not boundary parallel. Then $G$ splits over $A$
relative to $\calp\cup\calh$. We check that $A$ is $\calz_{\mathrm{max}}$.
Clearly, $A$ is virtually cyclic with infinite center, and is maximal among virtually cyclic subgroups of $G_v$.
Since $G_v$ is QH and $A$ is not conjugate into a boundary subgroup, it easily follows that
$A$ is $\calz_{\mathrm{max}}$ in $G$.
$\bullet$
The remaining case is when $\Sigma$ has no minimal component.
In this case, all leaves of $\Sigma$ are finite, and the dual tree $T_\Sigma$ is simplicial.
Its edge stabilizers fix an arc in $T$, so we are done if one of these edge stabilizers
is finite or parabolic.
Otherwise, arc stabilizers of $T_\Sigma$ are virtually cyclic with infinite center but may fail to be $\calz_{\mathrm{max}}$.
If this happens, we have to enlarge the finite tree $K$ used to construct $\Sigma$.
By \cite{LP}, we can find an exhaustion of $T$ by an increasing sequence of finite subtrees $K_k$
such that the corresponding dual trees $T_{\Sigma_k}$ strongly converge to $T$. We refer to \cite{LP} for
the definition of strong convergence; we will only use the fact that, if $A$ is a finitely generated
group fixing an arc in $T$, then $A$ fixes an edge in $T_{\Sigma_k}$ for large enough $k$.
We can assume that all dual trees $T_{\Sigma_k}$ are simplicial, and that their edge stabilizers are
infinite and not parabolic. Let $A_0$ be an edge stabilizer of $T_{\Sigma_0}$.
Then $A_0$ fixes an arc $I$ in $T$. By the hypothesis on arc stabilizers of $T$, the stabilizer
of $I$ is a $\calz_{\mathrm{max}}$ subgroup $A\supset A_0$,
which fixes an edge $e$ of $T_{\Sigma_k}$ for $k$ large enough. Since $G_e$ contains $ A$ and fixes an arc in $T$, it is equal to $A$, so
$T_{\Sigma_k}$ provides the desired splitting.
\end{proof}
A similar proof yields the following results.
\begin{thm}\label{thm_Rips_parab}
Let $G$ be hyperbolic relative to $\calp=\{P_1,\dots,P_n\}$, with $P_i$ slender,
and let $\calh=\{H_1,\dots H_q\}$ be a family of finitely generated subgroups.
Assume that
$G$ acts non-trivially on an $\R$-tree\ $T$
relative to $\calp\cup\calh$, with elementary arc stabilizers.
Then $G$ splits relative to $\calp\cup\calh$ over a $\calz_{\mathrm{max}}$ subgroup or over the stabilizer of an arc of $T$. \qed
\end{thm}
\begin{cor} Let $G$ be a toral
relatively hyperbolic group. Consider a non-trivial action of $G$
on an $\bbR$-tree relative to non-cyclic abelian subgroups. If arc stabilizers are abelian and stable under taking roots, then $G$ splits (relative to non-cyclic abelian subgroups) over an abelian subgroup stable under taking roots. \qed
\end{cor}
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title={On accessibility of groups},
date={1983},
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journal={J. Pure Appl. Algebra},
volume={30},
number={1},
pages={39\ndash 46},
review={\MR {85c:20021}},
}
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author={McCool, James},
title={Some finitely presented subgroups of the automorphism group of a free group},
date={1975},
issn={0021-8693},
journal={J. Algebra},
volume={35},
pages={205\ndash 213},
review={\MR {MR0396764 (53 \#624)}},
}
\bib{MiOs_fixed}{unpublished}{
author={Minasyan, Ashot},
author={Osin, Denis},
title={Fixed subgroups of automorphisms of relatively hyperbolic groups},
note={arXiv:1007.2361v2 [math.GR]},
}
\bib{Osin_elementary}{article}{
author={Osin, Denis~V.},
title={Elementary subgroups of relatively hyperbolic groups and bounded generation},
date={2006},
issn={0218-1967},
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volume={16},
number={1},
pages={99\ndash 118},
url={http://dx.doi.org/10.1142/S0218196706002901},
review={\MR {2217644 (2007b:20092)}},
}
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author={Osin, Denis~V.},
title={Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems},
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volume={179},
number={843},
pages={vi+100},
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author={Paulin, Fr{\'e}d{\'e}ric},
title={Outer automorphisms of hyperbolic groups and small actions on {$\mathbb {R}$}-trees},
date={1991},
booktitle={Arboreal group theory (berkeley, ca, 1988)},
publisher={Springer},
address={New York},
pages={331\ndash 343},
}
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author={Perin, Chlo{\'e}},
title={Elementary embeddings in torsion-free hyperbolic groups},
date={2011},
issn={0012-9593},
journal={Ann. Sci. \'Ec. Norm. Sup\'er. (4)},
volume={44},
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pages={631\ndash 681},
review={\MR {2919979}},
}
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author={Pettet, Martin~R.},
title={Virtually free groups with finitely many outer automorphisms},
date={1997},
issn={0002-9947},
journal={Trans. Amer. Math. Soc.},
volume={349},
number={11},
pages={4565\ndash 4587},
review={\MR {MR1370649 (98b:20056)}},
}
\bib{Robinson_course}{book}{
author={Robinson, Derek J.~S.},
title={A course in the theory of groups},
edition={Second},
series={Graduate Texts in Mathematics},
publisher={Springer-Verlag},
address={New York},
date={1996},
volume={80},
isbn={0-387-94461-3},
url={http://dx.doi.org/10.1007/978-1-4419-8594-1},
review={\MR {1357169 (96f:20001)}},
}
\bib{Sela_Nielsen}{article}{
author={Sela, Z.},
title={The {N}ielsen-{T}hurston classification and automorphisms of a free group. {I}},
date={1996},
issn={0012-7094},
journal={Duke Math. J.},
volume={84},
number={2},
pages={379\ndash 397},
}
\bib{Sela_structure}{article}{
author={Sela, Z.},
title={Structure and rigidity in ({G}romov) hyperbolic groups and discrete groups in rank $1$ {L}ie groups. {I}{I}},
date={1997},
issn={1016-443X},
journal={Geom. Funct. Anal.},
volume={7},
number={3},
pages={561\ndash 593},
}
\bib{Sela_hopf}{article}{
author={Sela, Z.},
title={Endomorphisms of hyperbolic groups. {I}. {T}he {H}opf property},
date={1999},
issn={0040-9383},
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pages={301\ndash 321},
}
\bib{Sela_diophantine1}{article}{
author={Sela, Zlil},
title={Diophantine geometry over groups. {I}. {M}akanin-{R}azborov diagrams},
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\end{biblist}
\end{bibdiv}
\begin{flushleft}
Vincent Guirardel\\
Institut de Recherche Math\'ematique de Rennes\\
Universit\'e de Rennes 1 et CNRS (UMR 6625)\\
263 avenue du G\'en\'eral Leclerc, CS 74205\\
F-35042 RENNES C\'edex\\
\emph{e-mail:}\texttt{vincent.guirardel@univ-rennes1.fr}\\[8mm]
Gilbert Levitt\\
Laboratoire de Math\'ematiques Nicolas Oresme\\
Universit\'e de Caen et CNRS (UMR 6139)\\
BP 5186\\
CS 14032\\
14032 CAEN cedex 5\\
France\\
\emph{e-mail:}\texttt{levitt@unicaen.fr}\\
\end{flushleft}
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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\begin{document}
\title{On Schur 2-groups}
\author{Mikhail Muzychuk}
\address{Netanya Academic College, Netanya, Israel}
\email{muzy@netanya.ac.il}
\author{Ilya Ponomarenko}
\address{Steklov Institute of Mathematics at St. Petersburg, Russia}
\email{inp@pdmi.ras.ru}
\thanks{The work of the second author was partially supported by the RFBR Grant 14-01-00156 А}
\date{}
\begin{abstract}
A finite group $G$ is called a Schur group, if any Schur ring over~$G$ is the transitivity module of a point stabilizer
in a subgroup of $\sym(G)$ that contains all right translations. We complete a classification of abelian $2$-groups by proving that the
group ${\mathbb Z}_2\times{\mathbb Z}_{2^n}$ is Schur. We also prove that any non-abelian Schur $2$-group of order larger than $32$ is dihedral
(the Schur $2$-groups of smaller orders are known). Finally, in the dihedral case, we study Schur rings of rank at most $5$, and show that
the unique obstacle here is a hypothetical S-ring of rank $5$ associated with a divisible difference set.
\end{abstract}
\maketitle
\section{Introduction}
Following R.P\"oschel \cite{Poe74}, a finite group $G$ is called a {\it Schur group}, if any S-ring over~$G$ is the transitivity module
of a point stabilizer in a subgroup of $\sym(G)$ that contains all right translations (for the exact definitions, we refer to
Section~\ref{120813a}). He proved there that if $p\ge 5$ is a prime, then a finite $p$-group is Schur if and only if
it is cyclic. For $p=2$ or $3$, a cyclic $p$-group is still Schur, but P\"oschel's
theorem is not true: a straightforward computation shows that an elementary abelian group
of order $4$ or $9$ is Schur. In this paper, we are interested in Schur 2-groups.\medskip
Recently in \cite{EKP14}, it was proved that every finite abelian Schur group belongs to one
of several explicitly given families. In particular, from Lemma~5.1 of that paper it
follows that all abelian Schur $2$-groups are known except for the groups
${\mathbb Z}_{2^{}}\times {\mathbb Z}_{2^n}$, where $n\ge 5$. We prove that all these groups
are Schur (Theorem~\ref{090514b}). As a by-product we can complete the classification of
abelian Schur $2$-groups.
\thrml{170814a}
An abelian $2$-group $G$ is Schur if and only if $G$ is cyclic, or elementary abelian of order at most $32$, or
is isomorphic to ${\mathbb Z}_{2^{}}\times {\mathbb Z}_{2^n}$ for some~$n\ge 1$.
\end{theorem}
Non-abelian Schur groups have been studied in~\cite{PV} where it was proved that they are
metabelian. In particular, from Theorem~4.2 of that paper
it follows that non-abelian Schur $2$-groups are known except for dihedral groups and groups
\qtnl{170914a}
M_{2^n}=\grp{a,b:\ a^{2^{n-1}}=b^2=1,\ bab=a^{1+2^{n-2}}},
\end{equation}
where $n\ge 4$. In this paper we prove that the latter groups are not Schur (Theorem~\ref{071113a}).
As a by-product we obtain the following statement.
\thrml{170814b}
A non-abelian Schur $2$-group of order at least $32$ is dihedral.
\end{theorem}
We do not know whether or not a dihedral $2$-group of order more than $32$ is Schur. A standard technique based
on Wielandt's paper~\cite{W49} enables us to describe S-rings of rank at most~$5$ as follows (see Subsection~\ref{231014w} for a connection between S-rings and divisible difference sets).
\thrml{170814c}
Let be ${\cal A}$ an S-ring over a dihedral $2$-group. Suppose that
$\rk({\cal A})\le 5$. Then one of the following statements is true:
\begin{enumerate}
\tm{1} ${\cal A}$ is isomorphic to an S-ring over ${\mathbb Z}_{2^{}}\times {\mathbb Z}_{2^n}$,
\tm{2} ${\cal A}$ is a proper dot or wreath product,
\tm{3} $\rk({\cal A})=5$ and ${\cal A}$ is associated with a divisible difference set in ${\mathbb Z}_{2^n}$.
\end{enumerate}
\end{theorem}
S-rings in statement (1) of this theorem are schurian by Theorem~\ref{170814a}. By
induction, this implies that all S-rings in statement~(2) are also schurian. Thus, by
Theorem~\ref{081014w} we obtain the following corollary.
\crllrl{031014a}
Under the hypothesis of Theorem~\ref{170814c}, the S-ring ${\cal A}$ is
not schurian only if ${\cal A}$ is associated with a divisible difference set in a cyclic $2$-group.
\end{corollary}
In fact, we do not know whether there exists a non-trivial divisible difference set in a cyclic $2$-group that produces an S-ring ${\cal A}$
in part~(3) of Theorem~\ref{170814c}. If such a set does exist, then the corresponding dihedral $2$-group is not Schur. On the other hand,
in Subsection~\ref{231014w}, we show that using a relative difference set (which is a special case of a divisible one), one can construct
an S-ring of rank~$6$ (over a dihedral $2$-group). These difference sets, and, therefore, S-rings, do exist, but are relatively rare. The only known example is the classical $(q+1,2,q,(q-1)/2)$-difference set
where $q$ is a Mersenne prime. Thus, the question whether a dihedral $2$-group is Schur,
remains open.\medskip
The paper consists of fourteen sections. In Sections~\ref{120813a}, \ref{111014a}
and \ref{231014v}, we give a background of S-rings, Cayley schemes\footnote{Here, we assume
some knowledge of association scheme theory.} and cite
some basic facts on S-rings over cyclic and dihedral groups. In Sections~\ref{231014a1}--
\ref{231014a3}, we develop a theory of S-rings over ${\mathbb Z}_2\times{\mathbb Z}_{2^n}$ that is
culminated in Theorem~\ref{090514b} stating that this group is Schur.
In Section~\ref{231014a4}, we show that the group $M_{2^n}$ is not Schur
for all $n\ge 4$ (Theorem~\ref{071113a}).
In Sections~\ref{231014a6}--\ref{231014a7}, we study S-rings over a dihedral $2$-group:
here, we start with constructions based on cyclic divisible difference sets, and then
complete the proof of Theorem~\ref{170814c}.\medskip
{\bf Notation.}
As usual, by ${\mathbb Z}$ we denote the ring of rational integers.
The identity of a group $D$ is denoted by $e$; the set of non-identity elements in $D$ is denoted by $D^\#$.
Let $X\subseteq D$. The subgroup of $D$ generated by $X$ is denoted by $\grp{X}$;
we also set $\rad(X)=\{g\in D:\ gX=Xg=X\}$. The element $\sum_{x\in X}x$ of the group
ring ${\mathbb Z} D$ is denoted by $\und{X}$. The set $X$ is called {\it regular} if the order
$|x|$ of an element $x\in X$ does not depend on the choice of~$x$.
For a group $H\trianglelefteq D$, the quotient epimorphism from $D$ onto $D/H$ is denoted by $\pi_{D/H}$.
The group of all permutations of $D$ is denoted by $\sym(D)$. The set of orbits of a group
$G\le\sym(D)$ is denoted by $\orb(G)=\orb(G,D)$. We write $G\approx_{\scriptscriptstyle 2}} %{\underset{\scriptscriptstyle 2}{\approx} G'$ if the groups $G,G'\le\sym(D)$
are {\it $2$-equivalent}, i.e. have the same orbits in the coordinate-wise action on~$D\times D$.
Given two subgroups $L\trianglelefteq U\leq D$, the quotient group $U/L$ is called the {\it section} of $D$. For a
set $\Delta\subseteq\sym(D)$ and a section $S = U/L$ of $D$ we set
$$
\Delta^S=\{f^S:\ f\in \Delta,\ S^f=S\},
$$
where $S^f = S$ means that $f$ permutes the right $L$-cosets in $U$ and $f^S$ denotes the bijection of
$S$ induced by~$f$.
The cyclic group of order $n$ is denoted by ${\mathbb Z}_n$.
\section{A background on S-ring theory}\label{120813a}
In what follows, we use the notation and terminology of~\cite{EP12}.\medskip
Let $D$ be a finite group. A subring~${\cal A}$ of the group ring~${\mathbb Z} D$ is called a {\it Schur
ring} ({\it S-ring}, for short) over~$D$ if there exists a partition ${\cal S}={\cal S}({\cal A})$ of~$D$
such that
\begin{enumerate}
\tm{S1} $\{e\}\in{\cal S}$,
\tm{S2} $X\in{\cal S}\ \Rightarrow\ X^{-1}\in{\cal S}$,
\tm{S3} ${\cal A}=\Span\{\und{X}:\ X\in{\cal S}\}$.
\end{enumerate}
When ${\cal S}=\orb(K,D)$ where $K\le\aut(D)$, the S-ring ${\cal A}$ is called {\it cyclotomic} and
denoted by $\cyc(K,D)$. A group isomorphism $f:D\to D'$ is called a {\it Cayley isomorphism} from an S-ring ${\cal A}$ over
$D$ to an S-ring ${\cal A}'$ over $D'$ if ${\cal S}({\cal A})^f={\cal S}({\cal A}')$.\medskip
It follows from (S3) that given $X,Y\in{\cal S}({\cal A})$ there exist non-negative integers
$c_{X Y}^Z$, $Z\in{\cal S}({\cal A})$, such that
$$
\und{X}\,\und{Y}=\sum_{Z\in{\cal S}({\cal A})}c_{X Y}^Z\und{Z}.
$$
One can see that $c_{XY}^Z$ equals the number of different representations $z=xy$ with
$(x,y)\in X\times Y$ for a fixed (and hence for all) $z\in Z$. It is a well-known fact that
$$
c_{Y^{-1}X^{-1}}^{Z^{-1}}=c_{X^{}Y^{}}^{Z^{}}\quad\text{and}\quad |Z|c_{XY}^{Z^{-1}}=|X|c_{YZ}^{X^{-1}}=|Y|c_{ZX}^{Y^{-1}}
$$
for all $X,Y,Z$. A ring isomorphism $\varphi:{\cal A}\to {\cal A}'$ is said to be {\it algebraic} if for any $X\in{\cal S}({\cal A})$
there exists $X'\in{\cal S}({\cal A}')$ such that $\varphi(\und{X})=\und{X'}$.\medskip
The classes of the partition ${\cal S}$ and the number $\rk({\cal A})=|{\cal S}|$ are called the {\it basic
sets} and the {\it rank} of the S-ring~${\cal A}$, respectively. Any union of basic sets
is called an {\it ${\cal A}$-subset of~$D$} or {\it ${\cal A}$-set}. The set of all of them
is closed with respect to taking inverse and product. Given an ${\cal A}$-set $X$, we denote by ${\cal A}_X$
the submodule of~${\cal A}$ spanned by the elements $\und{Y}$, where $Y$ belongs to the set
$$
{\cal S}({\cal A})_X=\{Y\in{\cal S}({\cal A}):\ Y\subseteq X\}.
$$
Any subgroup of $D$ that is an ${\cal A}$-set, is called an {\it ${\cal A}$-subgroup} of~$D$ or
{\it ${\cal A}$-group}. With each ${\cal A}$-set $X$, one can naturally associate two ${\cal A}$-groups,
namely $\grp{X}$ and $\rad(X)$ (see Notation). The following useful lemma was proved
in \cite[p.21]{EvdP09}.\footnote{Apparently, for the first time this lemma was proved in~\cite[Proposition~4.5]{M87}.}
\lmml{090608a}
Let ${\cal A}$ be an S-ring over a group $D$ and $H\le D$ an ${\cal A}$-group. Then
given $X\in{\cal S}({\cal A})$, the cardinality of the set $X\cap xH$ does not depend on $x\in X$.
\end{lemma}
A section $S=U/L$ of the group $D$ is called an {\it ${\cal A}$-section}, if both $U$ and
$L$ are ${\cal A}$-groups. In this case, the module
$$
{\cal A}_S=\Span \{\pi_S(X):\ X\in{\cal S}({\cal A})_U\}
$$
is an S-ring over the group~$S$, the basic sets of which are exactly the sets $\pi_S(X)$ from the
right-hand side of the formula.\medskip
The S-ring ${\cal A}$ is called {\it primitive} if the only ${\cal A}$-groups are $e$ and $D$,
otherwise this ring is called {\it imprimitive}. One can see that if $H$
is a minimal ${\cal A}$-group, then the S-ring ${\cal A}_H$ is primitive. The classical results
on primitive S-rings over abelian and dihedral groups were obtained in papers~\cite{S33,K37,W49}.
A careful analysis of the proofs shows that the schurity assumption there
was superfluous. Therefore, in the first part of the following statement, we formulate the
corresponding results in slightly more general form.
\thrml{030414a}
Let $D$ be a $2$-group which is cyclic, dihedral, or isomorphic to the group ${\mathbb Z}_2\times{\mathbb Z}_{2^n}$. Then any primitive S-ring over $D$
is of rank~$2$. In particular, if ${\cal A}$ is an S-ring over $D$ and $H\le D$ is a minimal ${\cal A}$-group,
then $H^\#$ is a basic set of~${\cal A}$.
\end{theorem}
Let $S=U/L$ be an ${\cal A}$-section of the group $D$. The S-ring ${\cal A}$ is called the {\it generalized $S$-wreath product}
\footnote{In \cite{LM98}, the term {\it wedge product} was used.} if the group $L$ is
normal in~$D$ and $L\le\rad(X)$ for all basic sets $X$ outside~$U$; in this case we write
\qtnl{050813a}
{\cal A}={\cal A}_U\wr_S{\cal A}_{D/L},
\end{equation}
and omit $S$ if $U=L$. When the explicit indication of~$S$ is not important,
we use the term {\it generalized wreath product}. The generalized $S$-wreath product is
{\it proper} if $L\ne e$ and $U\ne D$. When $U=L$, the generalized $S$-wreath product coincides with the ordinary
wreath product.\medskip
Let $D=D_1D_2$, where $D_1$ and $D_2$ are trivially intersecting subgroups of $D$.
If ${\cal A}_1$ and ${\cal A}_2$ are S-rings over the groups $D_1$ and $D_2$ respectively, then the
module
$$
{\cal A}=\Span\{\und{X_1\cdot X_2}: X_1\in{\cal S}({\cal A}_1),\ X_2\in{\cal S}({\cal A}_2)\}
$$
is an S-ring over the group $D$ whenever ${\cal A}_1$ and ${\cal A}_2$ are commute with each other.
In this case, ${\cal A}$ is called the {\it dot product} of ${\cal A}_1$ and
${\cal A}_2$, and denoted by ${\cal A}_1\cdot{\cal A}_2$ \cite{LM98}. When $D=D_1\times D_2$, the
dot product coincides with the {\it tensor product} ${\cal A}_1\otimes {\cal A}_2$. The following
statement was proved in \cite{EKP14}.
\lmml{050813b}
Let ${\cal A}$ be an S-ring over an abelian group $D=D_1\times D_2$. Suppose that $D_1$ and $D_2$
are ${\cal A}$-groups. Then ${\cal A}={\cal A}_{D_1}\otimes {\cal A}_{D_2}$ whenever ${\cal A}_{D_1}={\mathbb Z} D_1$
or ${\cal A}_{D_2}={\mathbb Z} D_2$.
\end{lemma}
The following two important theorems go back to Schur and Wielandt (see \cite[Ch.~IV]{Wie64}).
The first of them is known as the Schur theorem on multipliers, see~\cite{EvdP09}.
\thrml{261009b}
Let ${\cal A}$ be an S-ring over an abelian group $D$. Then given an integer~$m$ coprime
to $|D|$, the mapping $X\mapsto X^{(m)}$, $X\in{\cal S}({\cal A})$, where
\qtnl{141014a}
X^{(m)}=\{x^m:\ x\in X\},
\end{equation}
is a bijection. Moreover, $x\mapsto x^m$, $x\in D$, is a Cayley automorphism of~${\cal A}$.
\end{theorem}
Given a subset $X$ of an abelian group $D$, denote by $\tr(X)$ the {\it trace} of $X$, i.e. the union
of all~$X^{(m)}$ over the integers $m$ coprime to~$|D|$. We say that
$X$ is {\it rational} if $X=\tr(X)$. When $\tr(X)=\tr(Y)$ for some $Y\subseteq D$, the sets
$X$ and $Y$ are called {\it rationally conjugate}. For an S-ring ${\cal A}$ over $D$, the module
$$
\tr({\cal A})=\Span\{\tr(X):\ X\in{\cal S}({\cal A}\}
$$
is also an S-ring; it is called the {\it rational closure} of~${\cal A}$. Finally, the S-ring is
{\it rational} if it coincides with its rational closure, or equivalently, if each of its
basic sets is rational.\medskip
In general, Theorem~\ref{261009b} is not true when $m$ is not coprime to the order of $D$. However, the following
weaker statement holds.
\thrml{261009w}
Let ${\cal A}$ be an S-ring over an abelian group $D$. Then given a prime divisor~$p$ of~$|D|$,
the mapping $X\mapsto X^{[p]}$, $X\in 2^ D$, where
\qtnl{030713a}
X^{[p]}=\{x^p:\ x\in X,\ \nmmod{|X\cap Hx|}{0}{p}\}
\end{equation}
with $H=\{g\in D:\ g^p=1\}$, takes an ${\cal A}$-set to an ${\cal A}$-set.
\end{theorem}
We complete the section by the theorem on separating subgroup that was proved
in~\cite{EP05}.
\thrml{t100703}
Let ${\cal A}$ be an S-ring over a group $D$. Suppose that $X\in{\cal S}({\cal A})$ and $H\le D$ are such that
$$
X\cap H\ne\emptyset\quad\text{and}\quad X\setminus H\ne\emptyset\quad\text{and}\quad
\grp{X\cap H}\le\rad(X\setminus H).
$$
Then $X=\grp{X}\setminus\rad(X)$ and $\rad(X)\le H\le\grp{X}$.
\end{theorem}
\section{S-rings and Cayley schemes}\label{111014a}
In this section, we freely use the language of association scheme theory; in our exposition,
we follow \cite{EP09,MP09}.
\sbsnt{The 1-1 correspondence.}
For a group $D$, denote by $R(D)$ the set of all binary relations on $D$ that are
invariant with respect to the group $D_{right}$ (consisting of the permutations of the set~$D$ induced
by the right multiplications in the group~$D$). Then the mapping
\qtnl{171014a}
2^D\to R(D),\quad X\mapsto R_D(X)
\end{equation}
where $R_D(X)=\{(g,xg):\ g\in D,x\in X\}$, is a bijection. If ${\cal A}$ is an S-ring
over the group $D$, then the pair
\qtnl{210215a}
{\cal X}=(D,S),
\end{equation}
where $S=R_D({\cal S}({\cal A}))$, is
an association scheme. Moreover, it is a {\it Cayley scheme} over $D$, i.e.
$D_{right}\le\aut({\cal X})$. Each basis relation $s\in S$ of this scheme, is a Cayley digraph
over~$D$ the connection set of which is equal to $es=\{g\in D:\ (e,g)\in s\}$. Conversely,
given a Cayley scheme~\eqref{210215a} the module
$$
{\cal A}=\Span\{\und{es}:\ s\in S\}
$$
is an S-ring over~$D$.
\thrml{100909a}{\rm \cite{K85}}
The mappings ${\cal A}\mapsto{\cal X}$, ${\cal X}\mapsto{\cal A}$ form a 1-1 correspondence
between the S-rings and Cayley schemes over the group~$D$.
\end{theorem}
It should be mentioned that the above correspondence preserves the inclusion. Moreover,
the mapping~\eqref{171014a} induces a ring isomorphism from ${\cal A}$ onto the adjacency algebra
of the Cayley scheme ${\cal X}$ associated with ${\cal A}$. It follows that
$c_{XY}^Z=c_{rs}^t$ for all $X,Y,Z\in{\cal S}({\cal A})$, where $r=R_D(X)$, $s=R_D(Y)$
and $t=R_D(Z)$. In particular, the number $|X|$ is equal to the valency $n_r$ of the relation $r$, and
the S-ring~${\cal A}$ is commutative if and only if so is the Cayley scheme~${\cal X}$.
\sbsnt{Isomorphisms and schurity.}
We say that S-rings ${\cal A}$ and ${\cal A}'$ are (combinatorial) {\it isomorphic} if
the Cayley schemes associated with ${\cal A}$ and ${\cal A}'$ are isomorphic. Any isomorphism
between these schemes is called the {\it isomorphism} of ${\cal A}$ and ${\cal A}'$. The group
$\iso({\cal A})$ of all isomorphisms from ${\cal A}$ to itself has a normal subgroup
$$
\aut({\cal A})=\{f\in\iso({\cal A}):\ R_D(X)^f=R_D(X)\ \text{for all}\; X\in{\cal S}({\cal A})\};
$$
any such $f$ is called a (combinatorial) {\it automorphism} of the S-ring~${\cal A}$.
In particular, if ${\cal A}={\mathbb Z} D$ (resp. $\rk({\cal A})=2$), then $\aut({\cal A})=D_{right}$
(resp. $\aut({\cal A})=\sym(D)$).\medskip
The S-ring ${\cal A}$ is called {\it schurian} (resp. {\it normal}) if so is the Cayley scheme
associated with ${\cal A}$. Thus, ${\cal A}$ is schurian if and only if ${\cal S}({\cal A})=\orb(\aut({\cal A})_e,D)$,
and normal if and only if $D_{right}\trianglelefteq\aut({\cal A})$.\medskip
From our definitions, it follows that ${\cal A}={\cal A}_1\wr{\cal A}_2$ if and only if ${\cal X}={\cal X}_1\wr{\cal X}_2$
where ${\cal X}$, ${\cal X}_1$ and ${\cal X}_2$ are the Cayley schemes associated with the S-rings~${\cal A}$, ${\cal A}_1$ and
${\cal A}_2$ respectively. Similarly, ${\cal A}={\cal A}_1\cdot{\cal A}_2$ if and only if ${\cal X}={\cal X}_1\otimes{\cal X}_2$.
On the other hand, the tensor and wreath product of association schemes (and permutation groups) are special cases
of the crested product introduced and studied in~\cite{BC05}. Thus, Theorem~\ref{211014a} below immediately follows from
Remark~23 of that paper and Theorems~21 and~22 proved there.
\thrml{211014a}
Let ${\cal A}={\cal A}_1\ast{\cal A}_2$, where $\ast\in\{\wr,\cdot\}$. Then ${\cal A}$ is schurian if and
only if so are ${\cal A}_1$ and ${\cal A}_2$. Moreover,
$$
\aut({\cal A}_1\wr{\cal A}_2)=\aut({\cal A}_1)\wr\aut({\cal A}_2)\quad\text{and}\quad
\aut({\cal A}_1\cdot{\cal A}_2)=\aut({\cal A}_1)\times\aut({\cal A}_2).
$$
\end{theorem}
The following simple statement is an obvious consequence of the definition of wreath product
and Theorem~\ref{211014a}.
\crllrl{021014a}
Let ${\cal A}$ be an S-ring over a group $D$ and $H$ an ${\cal A}$-group such that
$\rk({\cal A})=\rk({\cal A}_H)+1$. Then ${\cal A}$ is isomorphic to the wreath product of ${\cal A}_H$
by an S-ring of rank $2$ over the group ${\mathbb Z}_{[D:H]}$.
Moreover, ${\cal A}$ is schurian if and only if so is ${\cal A}_H$.
\end{corollary}
\sbsnt{Quasi-thin S-rings.} An S-ring ${\cal A}$ is called {\it quasi-thin} if any of its basic sets consists of at most
two elements. Thus, ${\cal A}$ is quasi-thin if and only if
the Cayley scheme associated with ${\cal A}$ is quasi-thin (the latter means that the valency
of any its basic relation is at most~$2$).
\lmml{100914a}
Let ${\cal A}$ be an S-ring over an abelian group $D$. Suppose that $X\in{\cal S}({\cal A})$ is such
that $|X|=2$ and $\grp{X}=D$. Then ${\cal A}$ is quasi-thin.
\end{lemma}
\noindent{\bf Proof}.\ The Cayley scheme~${\cal X}$ associated with ${\cal A}$ is commutative,
because the group $D$ is abelian. Moreover, the relation $r=R_D(X)$ corresponding
to the set $X$, is of valency $|X|=2$. Thus, the equality $\grp{X}=D$ implies that ${\cal X}$ is a $2$-cyclic
scheme generated by the tightly attached relation $r$ in the sense of~\cite{HM}. So, by Proposition~3.11 of that paper,
${\cal X}$ is a quasi-thin scheme. Therefore, the S-ring~${\cal A}$ is also quasi-thin.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
Following the theory of quasi-thin schemes in \cite{MP}, we say that a basic set $X\ne\{e\}$
of a quasi-thin S-ring~${\cal A}$ is an {\it orthogonal} if $X\subseteq Y\, Y^{-1}$ for some
$Y\in{\cal S}({\cal A})$.
\lmml{100914y}
Any commutative quasi-thin S-ring ${\cal A}$ is schurian. Moreover, if
it has at least two orthogonals, then the group $\aut({\cal A})_e$ has a faithful regular orbit.
\end{lemma}
\noindent{\bf Proof}.\ The first statement immediately follows from ~\cite[Theorem~1.2]{MP}. To prove
the second one, denote by ${\cal X}$ the Cayley scheme associated with the S-ring~${\cal A}$. Then
from \cite[Corollary~6.4]{MP} it follows that the group $\aut({\cal X})_{e,x}$ is trivial for
some $x\in D$. This means that $x^{\aut({\cal X})_e}$ is a faithful regular orbit of the group $\aut({\cal A})_e$.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\section{S-rings over cyclic and dihedral groups}\label{231014v}
\sbsnt{Cyclic groups.}\label{141114b}
Let $C$ be a cyclic group of order $2^n$, $n\ge 1$. Then the group $\aut(C)$
consists of permutations $\sigma_m:x\mapsto x^m$, $x\in C$, where $m$ is an odd integer. In what
follows, $c_1$ denotes the unique involution in $C$.
\lmml{211113a}
Let $X\in\orb(K,C)$, where $K\le\aut(C)$. Then
\begin{enumerate}
\tm{1} $\rad(X)=e$ if and only if $X$ is a singleton, or $n\ge 3$ and $X=\{x,\varepsilon x^{-1}\}$
where $x\in X$ and $\varepsilon\in \{e,c_1\}$,
\tm{2} if $K\ge\{\sigma_m:\ \mmod{m}{1}{2^{n-k}}\}$, then $2^k$ divides $|\rad(X)|$.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof}.\ Statement (1) follows from \cite[Lemma~5.1]{EP02}, whereas statement~(2) is straightforward.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
Let ${\cal A}$ be an S-ring over the group $C$. By the Schur theorem on multipliers,
the group $\rad(X)$ does not depend on a set $X\in{\cal S}({\cal A})$ that contains a generator of $C$.
This group is called the {\it radical} of ${\cal A}$ and denoted by $\rad({\cal A})$. Since $C$ is a $2$-group, from~\cite[Lemma~6.4]{EP02}
it follows that if $\rad({\cal A})=e$, then either $n\ge 2$ and $\rk({\cal A})=2$, or ${\cal A}=\cyc(K,C)$, where $K\le\aut(C)$ is the group
generated by the automorphism taking a generator $x$ of $C$ to an element in $\{x,x^{-1},c_1x^{-1}\}$
(see also statement~(1) of Lemma~\ref{211113a}). In any case, ${\cal A}$ is, obviously, schurian. In fact, the latter statement holds
for any S-ring over a cyclic $p$-group \cite{EKP}.\medskip
For any basic set $X$ of the S-ring~${\cal A}$, one can form an ${\cal A}$-section $S=\grp{X}/\rad(X)$. Then the radical
of the S-ring~${\cal A}_S$ is trivial. Since in our case, $|S|$ is a $2$-group, the result in previous paragraph shows
that either the S-ring ${\cal A}_S$ is cyclotomic or $|S|$ is a composite number and $\rk({\cal A}_S)=2$. In the
former case, $X$ is an orbit of an automorphism group of $C$, whereas in the latter case, $X=\grp{X}\setminus\rad(X)$. Thus, any
basic set of ${\cal A}$ is either regular or equals the set difference of two distinct ${\cal A}$-groups.
\lmml{021109a}
Let ${\cal A}$ be a cyclotomic S-ring over a cyclic $2$-group.
Suppose that $\rad({\cal A})=e$. Then $\rad({\cal A}_S)=e$ for any ${\cal A}$-section $S$ such that $|S|\ne 4$.
\end{lemma}
\noindent{\bf Proof}.\ Follows from \cite[Theorem~7.3]{EKP}.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
The following auxiliary lemma will be used in Section~\ref{140914b}.
\lmml{150414a}
Let $C$ be a cyclic $2$-group, and let $X$ and $Y$ be orbits of some subgroups of $\aut(C)$. Suppose
that $\grp{X}\ne\grp{Y}$ and $\rad(X)=\rad(Y)=e$. Then the product $X\,Y$ contains
no coset of $\grp{c_1}$.
\end{lemma}
\noindent{\bf Proof}.\ Without loss of generality, we can assume that $\grp{Y}$ is a proper subgroup of~$\grp{X}$.
Then from statement (1) of Lemma~\ref{211113a}, it follows that
$X=\{x\}$ or $\{x,\varepsilon x^{-1}\}$, and $Y=\{y\}$ or $\{y,\varepsilon' y^{-1}\}$
where $|x|>|y|$. Therefore, the required statement trivially holds whenever $X$ or $Y$ is a singleton. Thus,
we can assume that $|X|=|Y|=2$, and hence $|x|>|y|\ge 8$. Furthermore,
$$
X\,Y=\{xy,\quad \varepsilon'xy^{-1},\quad \varepsilon x^{-1}y,\quad \varepsilon''x^{-1}y^{-1}\},
$$
where $\varepsilon''=\varepsilon\varepsilon'$. Suppose on the contrary, that this product contains a coset
of $\grp{c_1}$. Then, obviously, $c_1xy=\varepsilon'xy^{-1}$ or
$c_1\varepsilon x^{-1}y=\varepsilon''x^{-1}y^{-1}$.
In any case, $y^2\in\{e,c_1\}$ and hence $|y|\le 4$. Contradiction.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\sbsnt{Dihedral groups.}\label{011014b}
Throughout this subsection, $D$ is a dihedral group and $C$ is the cyclic subgroup of $D$ such that all the
elements in $D\setminus C$ are involutions. A set $X\subseteq D$ is called {\it mixed} if
the sets $X_0=X\cap C$ and $X\setminus X_0$ are not empty. For an
element $s\in D\setminus C$, we denote by $X_1=X_{1,s}$ the subset of
$C$ for which $X=X_0\,\cup\,X_1s$. The following statement (in the other notation) was proved in~\cite{W49}.
\lmml{240913a}
Let ${\cal A}$ be an S-ring over the dihedral group $D$ and $X$ a mixed basic set of ${\cal A}$. Then
\begin{enumerate}
\tm{1} the sets $X_0$, $X_1s$ and $X$ are symmetric, and $X_0$ commutes with $X_1s$,
\tm{2} given an integer $m$ coprime to $|D|$, there exists a unique $Y\in{\cal S}({\cal A})$ such that $(X_0)^{(m)}=Y_0$.
\end{enumerate}
\end{lemma}
When it does not lead to confusion, the set $Y$ from statement~(2) of Lemma~\ref{240913a} will be also denoted by $X^{(m)}$;
for $X\subseteq C$, this notation is consistent with~\eqref{141014a}. For any ${\cal A}$-set $X$ such that $X_0\ne\emptyset$, one can define
its trace $\tr(X)$ to be the union of sets $X^{(m)}$, where $m$ runs over all integers coprime to $|D|$. The following statement was
also proved in~\cite{W49}.
\lmml{290514b}
Let ${\cal A}$ be an S-ring over the dihedral group $D$. Suppose that $X_0\ne\emptyset$ for all $X\in{\cal S}({\cal A})$. Then
given an integer $m$ coprime to $|D|$, the mapping $X\mapsto X^{(m)}$ induces an algebraic isomorphism of~${\cal A}$; in particular,
$|X^{(m)}|=|X|$.
\end{lemma}
The algebraic fusion of the S-ring ${\cal A}$ from Lemma~\ref{290514b} with respect to the group of all algebraic isomorphisms
defined in this lemma, is an S-ring any basic set of which is of the form $\tr(X)$ where $X\in{\cal S}({\cal A})$. This S-ring is called the
{\it rational closure} of ${\cal A}$ and denoted by $\tr({\cal A})$. It should be stressed that this notation has sense only
if the hypothesis of Lemma~\ref{290514b} is satisfied.
\section{S-rings over $D={\mathbb Z}_2\times{\mathbb Z}_{2^n}$: basic sets containing involutions}\label{231014a1}
In what follows, $C\le D$ is a cyclic group of order $2^n$ and $E$ is the Klein subgroup of~$D$. The non-identity elements of
this subgroup are the involution $c_1\in C$ and the other two involutions $s\in D\setminus C$ and $c_1s$.
For an S-ring over~$D$ and an element $t\in E$, we denote by $X_t$ the basic set that contains~$t$. Then
by the Schur theorem on multipliers (Theorem~\ref{261009b}), this set is rational. In this section we completely
describe the sets~$X_t$'s.
\thrml{250814a}
Let ${\cal A}$ be an S-ring over the group $D$. Then the set $H=\bigcup_{t\in E}X_t$
is an ${\cal A}$-group and for a suitable choice of $s$ one of the following statements holds
with $U=\grp{X_{c_1}}$:
\begin{enumerate}
\tm{1} $X_{c_1}=X_s=X_{sc_1}=U\setminus e$,
\tm{2} $X_{c_1}=U\setminus e$ and $X_s=X_{sc_1}=H\setminus U$,
\tm{3} $X_{c_1}=X_{sc_1}=U\setminus\grp{s}$ and $X_s=\{s\}$,
\tm{4} $X_{c_1}=U\setminus e$, $X_s=\{s\}$ and $X_{sc_1}=sX_{c_1}$.
\end{enumerate}
\end{theorem}
The proof of Theorem~\ref{250814a} will be given later. The following
auxiliary statement is, in fact, a consequence of the Schur theorem on multipliers. Below, we fix an S-ring~${\cal A}$ over
the group~$D$.
\lmml{290714c}
Suppose that $X\in{\cal S}({\cal A})$ contains two elements $x$ and $y$ such that $|x| > |y|\geq 2$. Then $x\{e,c_1\}\subseteq X$.
\end{lemma}
\noindent{\bf Proof}.\ Set $m=1 + |x|/2$. Then by Schur's theorem on multipliers, $Y:=X^{(m)}$ is a basic set of ${\cal A}$. On the other hand,
since $|x| > |y|$, we have $y^m=y$. Thus, $y^m\in X$. This implies that $X=Y$, and hence $x^m\in X$. Since $|x|>2$ and
$x^m=xc_1$, we conclude that $x\{e,c_1\}\subseteq X$ as required.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip\medskip
In the following lemma, we keep the notation of Theorem~\ref{250814a}.
\lmml{290714b}
Either $X_{c_1}=U\setminus e$, or $H$ is an ${\cal A}$-group and statement~(3) of Theorem~\ref{250814a} holds.
\end{lemma}
\noindent{\bf Proof}.\
The statement is trivial if the set $X:=X_{c_1}$ is contained in~$E$. So, we can assume that $X$ contains at least one element
of order greater than two. Then $x\{e,c_1\}\subseteq X$ for each $x\in X$ with $|x|>2$ (Lemma~\ref{290714c}).
Thus, $c_1\in\rad(X\setminus E)$. We observe that $X\cap E$ is equal to one of the following sets:
$$
\{c_1\}\quad\text{or}\quad \{c_1,s,sc_1\}\quad\text{or}\quad \{c_1,t\},
$$
where $t\in\{s,sc_1\}$.
In the first two cases, $c_1\in\rad(X\setminus\{c_1\})$. So by Theorem~\ref{t100703} with $H=\grp{c_1}$, we
conclude that $X=\grp{X}\setminus\rad(X)$ and $c_1\not\in\rad(X)$. However, the only non-trivial subgroups of $D$
not containing $c_1$, are $\grp{s}$ and $\grp{sc_1}$. Since none of them equals $\rad(X)$,
we conclude that $\rad(X)=e$ and $X=U\setminus e$.\medskip
In the remaining case, $X\cap E=\{c_1,t\}$, and hence $|c_1 X\cap X|=|X|-2$. Since $c_1\in X$, this implies that the latter number
equals~$c_{XX}^X$ (see Section~\ref{120813a}).
Therefore, $|x^{-1}X\cap X|=|X|-2$ for each $x\in X$. On the other hand, the set $\{c_1,t\}t=\{c_1t,e\}$ does not
intersect $X$. Thus, $t(X\setminus E)=X\setminus E$, and hence
$$
\rad(X\setminus E)=E.
$$
By Theorem~\ref{t100703} with
$H=E$, we conclude that $X=\grp{X}\setminus\rad(X)$ and $E\setminus\rad(X)$ is contained in~$X$. Therefore, $\rad(X)=\grp{t'}$,
where $t'$ is the element of $\{s,sc_1\}$ other than $t$. Thus, $X=U\setminus\grp{t'}$, $H=U$ is an ${\cal A}$-group and we are done.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip\medskip
{\bf Proof of Theorem~\ref{250814a}.} By Lemma~\ref{290714b}, we can assume that $X:=X_{c_1}=U\setminus e$.
If, in addition, $X$ contains $s$ or $sc_1$, then $H=U$ is an ${\cal A}$-group and statement~(1) of the theorem holds. Thus, we can
also assume that
$X\ne X_t$ for each $t\in \{s,sc_1\}$. Suppose first that $X_s=X_{sc_1}$; denote this set by $Y$. Then $Y\cap E=\{s,sc_1\}$, and hence
$c_1\in\rad(Y)$ (Lemma~\ref{290714c}). Since $X=U\setminus e$, this implies that $U\le\rad(Y)$.
Thus, $H=\grp{Y}$ is an ${\cal A}$-group, $X_s=X_{sc_1}=Y=H\setminus U$ and statement~(2) holds.\medskip
Let now $X_s\ne X_{sc_1}$ and $t\in \{s,sc_1\}$. Then $Y\cap E=\{t\}$ where $Y=X_t$. It follows that $|c_1 Y\cap Y|=|Y|-1$, and hence
\qtnl{280814a}
\und{Y}^2 = |Y|e+(|Y|-1) \und{X} + \cdots,
\end{equation}
where the omitted terms in the right-hand side contain neither $e$ nor elements of $X$ with non-zero coefficients. However,
$|X|c_{YY}^X=|Y|c_{YX}^Y$ because $X=X^{-1}$ and $Y=Y^{-1}$. Since $c_{YY}^X=|Y|-1$, this implies
that $|X|$ is divided by $|Y|$. If, in addition, $|Y|=1$, then we have $\{X_s,X_{sc_1}\}=\{\{t\},tX_{c_1}\}$ and $H=U\cup tU$ is an ${\cal A}$-group. So statement~(4) holds. If $|Y|\ne 1$, equality~\eqref{280814a}
implies that $|Y|=|X|=|U|-1$ and $\und{Y}^2 = |Y|e+(|Y|-1) \und{X}$. It follows that $c_{YX}^Y=|X|-1$. Therefore,
$$
|tU\cap Y|=|X|-1=|U|-2.
$$
This is true for $t=c_1$ and $t=sc_1$. On the other hand, $sU=sc_1U$ and the sets $X_s$ and $X_{sc_1}$ are disjoint. Thus,
$$
|U|=|tU|\ge |sU\cap X_s|+|sc_1U\cap X_{sc_1}|=2(|U|-2).
$$
It follows that $|U|=2$ or $|U|=4$. In the former case, $H=E$ and statement~(4) trivially holds. In the
latter one, $|X|=|X_s|=3$ and $|sU\cap Y|=2$. Therefore, there exists a unique $x\in X_s$ outside $sU$. It
follows that $|xU\cap X_s|=1$ which is impossible by Lemma~\ref{090608a}.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip\medskip
\section{S-rings over $D={\mathbb Z}_2\times{\mathbb Z}_{2^n}$: non-regular case}\label{140914a}
A set $X\subseteq D$ is said to be {\it highest} (in $D$) if it contains an element of order $2^n$.
Given an S-ring ${\cal A}$ over $D$, denote by $\rad({\cal A})$ the group generated by
the groups $\rad(X)$, where $X$ runs over the highest basic sets of ${\cal A}$.
Clearly, $\rad({\cal A})$ is an ${\cal A}$-group, and it is equal to $e$ if and only if each highest basic set of ${\cal A}$ has trivial radical.
In what follows, we say that ${\cal A}$ is {\it regular}, if each highest basic set of ${\cal A}$ is regular. Now, the
main result of this section can be formulated as follows.
\thrml{170414a}
Let ${\cal A}$ be an S-ring over the group $D$. Suppose that $\rad({\cal A})=e$. Then ${\cal A}$ is either regular
or rational. Moreover, in the latter case, ${\cal A}={\cal A}_H\otimes{\cal A}_L$, where $\rk({\cal A}_H)=2$ and
$|L|\le 2\le |H|$; in particular, ${\cal A}$ is schurian.
\end{theorem}
The proof of Theorem~\ref{170414a} will be given in the end of the section. The key point of the proof is the
following statement.
\thrml{290714a}
Let ${\cal A}$ be an S-ring over the group $D$. Then any non-regular basic set of ${\cal A}$ either intersects $E$,
or has a non-trivial radical.
\end{theorem}
\noindent{\bf Proof}.\
Let $X$ be a non-regular basic set of ${\cal A}$ that does not intersect $E$. Then the minimal order of an element in $X$
equals $2^m$ for some $m\ge 2$. Denote by $X_m$ the set of all elements in~$X$ of order $2^m$. Clearly,
each of the sets $X\setminus X_m$ and $X_m$ is non-empty. Suppose, towards a contradiction, that $\rad(X)=e$. Then $c_1\not\in\rad(X)$, and $c_1\in\rad(X\setminus X_m)$ (Lemma~\ref{290714c}). It follows that
\qtnl{300814u}
c_1\not\in{\cal A},
\end{equation}
because otherwise $c_1X$ is a basic set other than $X$ that intersects $X$.\medskip
Denote by $K$ the setwise stabilizer of $X$ in the group $G\cong{\mathbb Z}_{2^n}^*$ of all permutations $x\mapsto x^m$,
$x\in D$, where $m$ is an odd integer. Then
by Schur's theorem on multipliers, $X_m$ is the union of at most two $K$-orbits (one inside $C$ and the other outside). The
radicals of these orbits must be trivial, because
$\rad(X)=e$ and $c_1\in\rad(X\setminus X_m)$. Thus, by statement~(1) of Lemma~\ref{211113a}, we have
\qtnl{300814a}
X_m=\{x\}\quad\text{or}\quad \{x,x^{-1}\varepsilon\}\quad\text{or}\quad\{x,ys\}\quad\text{or}\quad\{x,x^{-1}\varepsilon,ys,y^{-1}\varepsilon s\},
\end{equation}
where $x,y\in X_m$ are such that $\grp{x}=\grp{y}$, and $\varepsilon\in\{e,c_1\}$. It should be mentioned that $x\ne \varepsilon y$,
for otherwise $\varepsilon s\in\rad(X)$.\medskip
Let us define ${\cal A}$-groups $U$ and $H$ as in Theorem~\ref{250814a}. Then
$X\subseteq D\setminus H$, because $X$ does not intersect $E$. By the definition of $H$, this implies that it does not contain elements
of order $2^m$. Since $U\le H$, we conclude that
$xU\cap X\subseteq X_m$ for each $x\in X_m$. Therefore, $X_m$ is a disjoint union
of some sets $xU\cap X$ with such $x$. However, by Lemma~\ref{090608a} the number $\lambda:=|xU\cap X|$ doesn't depend
on a choice of $x\in X$. Thus, $\lambda$ divides $|X_m|$.
By~\eqref{300814a} this implies that
$\lambda\in\{1,2,4\}$. Moreover, setting $Y$ to be the basic set
containing $c_1$, we have
\qtnl{141114a}
c_{X^{} Y^{}}^X=\lambda-1,
\end{equation}
because $U=Y\setminus e$ or $U=Y\setminus \grp{\varepsilon s}$,
and $x\ne \varepsilon y$.\medskip
Denote by $\alpha$ the number of $z\in X_m$ for which $c_1z\not\in X_m$.
If $\alpha=1$, then from Theorem~\ref{261009w}, it follows that
${\cal S}({\cal A})$ contains $X^{[2]}=\{z^2\}$ for an appropriate $z\in X_m$.
Since $z\not\in E$, this implies that $c_1\in{\cal A}$ in contrast
to~\eqref{300814u}. Thus, $\alpha\ne 1$. Therefore, $\alpha$ is an
even number less or equal than $|X_m|\le 4$ (see~\eqref{300814a}).
Moreover, it is not zero, because otherwise $c_1\in\rad(X)$. Besides,
from~\eqref{141114a} it follows that
\qtnl{310714a}
|X|\,(\lambda - 1)=|X|c_{X^{} Y^{}}^X =
|Y|\,c_{X^{} X^{-1}}^Y = |Y|\,|c_1X\cap X| = |Y|\,(|X|-\alpha).
\end{equation}
Since $|X|>|X_m|\ge\alpha$, this implies that the right-hand side of
the equality is not zero. Thus, $\lambda\ne 1$, and finally
\qtnl{010914a}
\lambda,\alpha\in\{2,4\}.
\end{equation}
\lmml{310714b}
In the above notation $|X|\geq 2|X_m|$, and the equality holds only if
\begin{enumerate}
\tm{1} $X_m$ is a union of two $K$-orbits and $X\setminus X_m$ is a $K$-orbit,
\tm{2} any element in $X\setminus X_m$ is of order $2^{m+1}$.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof}.\ By the Schur theorem on multipliers, the stabilizers of
an element $x\in X$ in the groups $K$ and $G$, coincide. However,
the stabilizer in $G$ consists of raising to power $1+i|x|$, where
$i=0,1,\ldots,2^n/|x|-1$. Therefore, $|K_x|=2^n/|x|$. For $x\in X_m$ and
$y\in X\setminus X_m$, this implies that
$|K_x|\ge 2|K_y|$, and hence
$$
|x^K|=\frac{|G|}{|K_x|}\le \frac{|G|}{2|K_y|}=\frac{|y^K|}{2}.
$$
Taking into account that $X_m$ is a disjoint union of
at most two $K$-orbits, we obtain that
$$
|X|-|X_m|\geq |y^K|\geq 2|x^K|\geq |X_m|
$$
as required. Since the equality holds only if the second and third
inequalities in the above formula are equalities, we are done.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip\medskip
We observe that $|Y|\ne\lambda-1$: indeed, for $\lambda=2$, this follows from~\eqref{300814u} whereas
for $\lambda=4$, the assumption $|Y|=\lambda-1$ implies by \eqref{310714a} an impossible equality $|X|=|X|-\alpha$.
Thus, by~\eqref{310714a} and Lemma~\ref{310714b}
we have
$$
\frac{\alpha\,|Y|}{|Y|-(\lambda-1)}=|X|\ge 2|X_m|\ge 2\alpha.
$$
Furthermore, if $\lambda=2$, then $|Y|=2$ and $|X|=2\alpha$. On the other hand, if $\lambda=4$, then $\lambda-1<|Y|\le 6$ and
$|Y|\in \{2^a-1,2^a-2\}$ for some~$a$ (Theorem~\ref {250814a}); but then $|Y|=6$ and $|T|=2\alpha$.
Thus, by~\eqref{010914a} there are exactly four possibilities:
\begin{enumerate}
\tm{1} $\alpha =2, \lambda=2, |Y|=2, |X|=4, |X_m|=2$,
\tm{2} $\alpha =4, \lambda=2, |Y|=2, |X|=8, |X_m|=4$,
\tm{3} $\alpha =2, \lambda=4, |Y|=6, |X|=4, |X_m|=2$,
\tm{4} $\alpha =4, \lambda=4, |Y|=6, |X|=8, |X_m|=4$.
\end{enumerate}
In all cases, $|X|=2|X_m|$. Therefore, by Lemma~\ref{310714b} and \eqref{300814a}, we conclude that
$X_m=\{x,ys\}$ in cases (1) and~(3), and $X_m = \{x,x^{-1}\varepsilon,sy,sy^{-1}\varepsilon\}$
in cases (2) and~(4). Moreover, since the number $|Y|$ is even, $U\setminus Y$ is a group of order two. Without loss
of generality, we assume that it is $\grp{s}$.\medskip
Let $\pi$ be the quotient epimorphism from $D$ to $D'=D/U$. Then the group $D'$ is cyclic, the S-ring ${\cal A}'={\cal A}_{D'}$ is
circulant\footnote{Any S-ring over a cyclic group is called a circulant one.} and $X'=\pi(X)$ is a non-regular basic set of it (the elements in $X'_m=\pi(X_m)$ and in $X_{}'\setminus X'_m$ have
different orders). However, any non-regular basic set of a circulant S-ring over a $2$-group is a set difference of two its subgroups
(see Subsection~\ref{141114b}). Therefore,
$$
X'=\grp{X'}\setminus\rad(X').
$$
Since $X_{}'\ne X'_m$, this implies that $|X'|\ge 3$. On the other hand, $|X'|=|X|/\lambda$ by the definition of $\lambda$. Thus,
we can exclude cases (1), (3) and (4). In case (2) let $|\rad(X')|=2^i$ for some $i\ge 0$. Then $4=|X'|=2^{i+2}-2^i=3\,2^i$.
Contradiction.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip\medskip
Any basic set $X$ of an S-ring ${\cal A}$ over $D$ that intersects the
group~$E$, must contain an involution. Therefore, such $X$ is rational.
By Theorem~\ref{290714a}, this proves the following statement.
\crllrl{011213d}
Let $X$ be a basic set of an S-ring over $D$. Suppose that $\rad(X)=e$. Then $X$
is either regular or rational.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\end{corollary}
{\bf Proof of Theorem~\ref{170414a}.}\ Suppose that ${\cal A}$ is not regular. Then there exists a highest set $X\in{\cal S}({\cal A})$
that is not regular.
Since $\rad(X)=e$, we conclude by Theorem~\ref{290714a} that the set $X\cap E$ is not empty. Therefore,
$X$ is contained in the ${\cal A}$-group~$H\ge E$ defined in Theorem~\ref{250814a}. But then $H=D$, because
the set $X$ is highest. Now, the first statement follows, because the
S-ring ${\cal A}_H={\cal A}$ is, obviously, rational. Moreover, statements (2) and (3) of Theorem~\ref{250814a} do not
hold, because $\rad({\cal A}_H)=\rad({\cal A})=e$. Thus, the second statement of our theorem is true for $L=e$
(resp. $L=\grp{s}$) if statement~(1) (resp. statement~(4)) of Theorem~\ref{250814a} holds.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip\medskip
From the proof of Theorem~\ref{170414a}, it follows that if one of the highest basic sets of ${\cal A}$ is not regular,
then all highest basic sets are rational. This implies the following statement.
\crllrl{180514a}
Let ${\cal A}$ be an S-ring over the group $D$. Suppose that $\rad({\cal A})=e$. Then either every highest basic set of
${\cal A}$ is regular, or every highest basic set of ${\cal A}$ is rational.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\end{corollary}
\section{S-rings over $D={\mathbb Z}_2\times{\mathbb Z}_{2^n}$: regular case}\label{100414a}
Throughout this section, $C=C_n$ is a cyclic subgroup
of~$D=D_n$ that is isomorphic to ${\mathbb Z}_{2^n}$.
We denote by $c_1$, $c_2$ and $s$, respectively, the unique involution in $C$, one of the two elements of $C$ of order~$4$, one of the two
involutions in $D\setminus C$. The main result is given by the following theorem.
\thrml{230214a}
Let ${\cal A}$ be a regular S-ring over the group $D$. Suppose that $\rad({\cal A})=e$. Then ${\cal A}$ is a cyclotomic S-ring.
More precisely, ${\cal A}=\cyc(K,D)$, where $K\le\aut(D)$ is one of the groups listed in Table~\ref{280414b}.
\end{theorem}
\begin{table}
\centering
\begin{tabular}[tc]{|c|l|c|c|c|}
\hline
$K$ & generators & $|K|$ & $n$ & comment \\
\hline
$K_1$ & $(x,s)\mapsto (x,s)$ & 1 & $n\ge 2$ & $X_1=\emptyset$\\
\hline
$K_2$ & $(x,s)\mapsto (x^{-1},s)$ & 2& $n\ge 3$ & $X_1=\emptyset$ \\
\hline
$K_3$ & $(x,s)\mapsto (c_1x^{-1},s)$ & 2& $n\ge 3$ & $X_1=\emptyset$ \\
\hline
$K_4$ & $(x,s)\mapsto (x^{-1},sc_1)$ & 2 & $n\ge 3$ & $X_1=\emptyset$ \\
\hline
$K_5$ & $(x,s)\mapsto (c_1x^{-1},sc_1)$ & 2 & $n\ge 3$ & $X_1=\emptyset$ \\
\hline
$K_6$ & $(x,s)\mapsto (sc_2x,sc_1)$,\ $(x,s)\mapsto (x^{-1},s)$ & 4 & $n\ge 4$ & $X_a\ne\emptyset$\\
\hline
$K_7$ & $(x,s)\mapsto (sc_2x,sc_1)$,\ $(x,s)\mapsto (c_1x^{-1},s)$ & 4 & $n\ge 4$ & $X_a\ne\emptyset$\\
\hline
$K_8$ & $(x,s)\mapsto (sx^{-1},s)$ & 2 &$n\ge 4$ & $X_a\ne\emptyset$\\
\hline
$K_9$ & $(x,s)\mapsto (sc_1x^{-1},s)$ & 2& $n\ge 4$ & $X_a\ne\emptyset$\\
\hline
$K_{10}$ & $(x,s)\mapsto (sc_2x,sc_1)$ & 2& $n\ge 3$ & $X_a\ne\emptyset$\\
\hline
$K_{11}$ & $(x,s)\mapsto (sc_2x^{-1},sc_1)$ & 2& $n\ge 4$ & $X_a\ne\emptyset$\\
\hline
\end{tabular}\\[2mm]
\caption{The groups of cyclotomic rings with trivial radical}
\label{280414b}
\end{table}
\crllrl{100514d}
Under the assumptions of Theorem~\ref{230214a}, let $K=K_i$, where $i=1,\ldots,11$. Then the following
statements hold:
\begin{enumerate}
\tm{1} $\grp{c_1}$ is an ${\cal A}$-group,
\tm{2} $C$ is an ${\cal A}$-group if and only if $i\le 5$,
\tm{3} $\grp{\varepsilon s}$ with $\varepsilon\in\{e,c_1\}$, is an ${\cal A}$-group if and only if
$i\in\{1,2,3,8,9\}$.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\end{enumerate}
\end{corollary}
In what follows, given a basic set $X\in{\cal S}({\cal A})$, we denote by $X_0$ and $X_1$ the uniquely determined subsets of $C$ for which
$X=X_0\cup sX_1$.\medskip
{\bf Proof of Theorem~\ref{230214a}.} Let $X$ be a highest basic set
of the S-ring~${\cal A}$. Then it is regular by the theorem hypothesis.
By the Schur theorem on multipliers, this implies that
if the set $X_a$ is not empty for some $a\in\{0,1\}$, then $X_a$ is an orbit of
the group $\aut(C)_{\{X_a\}}$. Therefore, $X_a$ is of the form given in
statement~(1) of Lemma~\ref{211113a}. The rest of the proof consists of Lemmas~\ref{040914a},
\ref{040914b} and \ref{040914c} below: in the first one, $X_0$ or $X_1$ is empty, and in the
other two, both $X_0$ and $X_1$ are not empty and $|X_0|=2$ or $1$,
respectively.
\lmml{040914a}
Let $X_0=\emptyset$ or $X_1=\emptyset$. Then ${\cal A}=\cyc(K_i,D)$ with $i\in\{1,2,3,4,5\}$.
\end{lemma}
\noindent{\bf Proof}.\
Without loss of generality, we can assume that $n\ge 3$ and $X_1=\emptyset$. Then $X=X_0$ generates $C$. Therefore, $C$ is
an ${\cal A}$-group and $X$ is a
highest basic set of a circulant S-ring ${\cal A}_C$. Since $\rad(X)=e$, this implies that $\rad({\cal A}_C)=e$. If, in addition,
$\grp{s}$ is an ${\cal A}$-group, then ${\cal A}={\cal A}_C\otimes{\cal A}_{\grp{s}}$
by Lemma~\ref{050813b}, and hence ${\cal A}=\cyc(K_i,D)$ with $i=1,2,3$. Thus, we can assume that
\qtnl{290414a}
s\not\in{\cal A}.
\end{equation}
Let us prove by induction on $n$ that ${\cal A}=\cyc(K_i,D)$ with $i=4$ or $5$. For $n=3$ this statement can be
verified by a computer computation. Let $n>3$. Denote by $X'$ the basic set of ${\cal A}$ that contains $x'=xs$,
where $x\in X$ is a generator of~$C$. Then by the theorem hypothesis,
$X'$ is a regular set with trivial radical. It follows that $C'=\grp{X'}$ is the order $2^n$ cyclic subgroup of $D$
other than $C$. In particular,
$$
\rad({\cal A}_{C'})=\rad(X')=e.
$$
Besides, since $C^2=(C')^2$, the S-rings ${\cal A}_C$ and ${\cal A}_{C'}$ have the same basic sets inside the group $C^2$; in particular,
$|X|=|X'|$. Moreover, these S-rings are not Cayley isomorphic. Indeed, otherwise $X'=sY$, where $Y=X^{(m)}$ for some
odd~$m$. Then $s$ is the only element that appears in the product $\und{Y}^{-1}\und{X'}$ with multiplicity~$|X|$. However,
in this case $s\in{\cal A}$, contrary to~\eqref{290414a}. Thus,
\qtnl{040914f}
X=\{x,\varepsilon x^{-1}\}\quad\text{and}\quad X'=\{sx,sc_1\varepsilon x^{-1}\}.
\end{equation}
Set $i=4$ or $5$ depending on $\varepsilon=e$ or $c_1$, respectively. Then
from~\eqref{040914f} and the Schur theorem on multipliers, it follows that the S-rings ${\cal A}$ and $\cyc(K_i,D)$ have the same
highest basic sets. We also observe that $D_{n-1}$ is an ${\cal A}$-group.\medskip
Since the S-ring ${\cal A}_C$ is cyclotomic, it follows from~\eqref{040914f} that $Y=\{x^2,x^{-2}\}$
is a basic set of ${\cal A}$. Denote by $Y'$ the basic set containing $sx^2$. Then, obviously,
$$
Y'\subseteq X\,X'=\{sx^2,sc_1x^{-2},s,sc_1\}.
$$
However, $|sx^2|\ge 8$, because $n\ge 4$. Thus, $Y'\subseteq\{sx^2,sc_1x^{-2}\} $: otherwise $Y\cap E$ is
not empty and from Theorem~\ref{250814a} it follows that $|Y'|>4$. Therefore,
$Y'$ is regular and $\rad(Y')=e$. Since also $\rad(Y)=e$
and $Y,Y'$ are highest basic sets of the S-ring ${\cal A}_{D_{n-1}}$, the latter
satisfies the hypothesis of Lemma~\ref{040914a}. By the induction, we conclude that
${\cal A}_{D_{n-1}}=\cyc(K_4,D_{n-1})$. Thus, ${\cal A}=\cyc(K_i,D)$ as required.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\lmml{040914b}
Let $|X_0|=2$ and $X_1\ne\emptyset$. Then ${\cal A}=\cyc(K_i,D)$ with $i\in\{6,7\}$.
\end{lemma}
\noindent{\bf Proof}.\ By statement~(1) of Lemma~\ref{211113a}, we have $X_0=\{x,\varepsilon x^{-1}\}$. Since $X$ is regular, this implies that
\qtnl{010514a}
X=\{x,\varepsilon x^{-1},sy,s\varepsilon y^{-1}\}
\end{equation}
for some generator $y$ of the group $C$. For $n=3$, we tested in the computer that no S-ring over $D$
has a highest basic set $X$ such that $X_0=\{x,\varepsilon x^{-1}\}$ and $X_1\ne\emptyset$.
Suppose that $n\ge 4$. Let us prove that the lemma statement holds for $i=6$ or $i=7$ depending on
$\varepsilon=e$ or $\varepsilon=c_1$, respectively. For $n=4$, we tested this statement in the computer. Thus,
in what follows, we can assume that $n\ge 5$.
\lmml{010514b}
In the above notations, the following statements hold:
\begin{enumerate}
\tm{1} $C_{n-1}$ is an ${\cal A}$-group whereas $\grp{s}$ and $\grp{sc_1}$ are not,
\tm{2} $Y_x=\{x^{\pm 2},c_1x^{\pm 2}\}$ and $Z_x=\{s x^{\pm 2}\}$ are ${\cal A}$-sets,
\tm{3} $y=xc_2$ for a suitable choice of $y$ and $c_2$.\footnote{One can interchange $y$ and $y^{-1}$,
and $c_2$ and $c_2^{-1}$.}
\end{enumerate}
\end{lemma}
\noindent{\bf Proof}.\ Since $n\ge 5$, we have $x^2\ne x^{-2}$ and $y^2\ne y^{-2}$. Besides, since neither $s$
nor $sc_1$ belongs to $\rad(X)$, we have $x^2\ne y^{\pm 2}\ne x^{-2}$. Thus,
\qtnl{030514a}
|\{x^2, x^{-2}, y^2, y^{-2}\}|=4.
\end{equation}
However, $X^{[2]}=\{x^2, x^{-2}, y^2, y^{-2}\}$ and $X^{[2]}$ is an ${\cal A}$-set by Theorem~\ref{261009w}.
Thus, the first part of statement~(1) holds, because $C_{n-1}=\grp{X^{[2]}}$. To prove the second part of
statement~(1), suppose on the contrary that $L:=\grp{s\varepsilon'}$ is an ${\cal A}$-group
for some $\varepsilon'\in\{e,c_1\}$. Then the circulant S-ring ${\cal A}_{D/L}$ has a basic set
$$
\pi(X)=\{\pi(x),\pi(\varepsilon x^{-1}),\pi(y),\pi(\varepsilon y^{-1})\},
$$
where $\pi:D\to D/L$ is the quotient epimorphism. However, from~\eqref{030514a} it easily follows that
$|\rad(\pi(X))|\ge 2$. Therefore, one of the quotients $\pi(x)/\pi(\varepsilon x^{-1})$, $\pi(x)/\pi(y)$ or
$\pi(x)/\pi(\varepsilon y^{-1})$ has order~$2$. This implies, respectively, that the order of $\pi(x)$ is $8$,
$\pi(x)=\pi(c_1)\pi(y)$, and $\pi(x)=\pi(c_1)\pi(\varepsilon y^{-1})$. The former case is impossible, because $n\ge 5$,
whereas in the other two, we have $x\in c_1yL$ and $x\in c_1\varepsilon y^{-1}L$ which is impossible due
to~\eqref{030514a}.\medskip
To prove the second part of statement~(2), we observe that $C_{n-1}\cup\tr(X)$ is an ${\cal A}$-set. But the complement to it
in~$D$ coincides with $sC_{n-1}$. Thus, it is also an ${\cal A}$-set. Besides,
\qtnl{110214a}
\und{X}^2\,\circ\,\und{sC_{n-1}}=2\,\und{sX'},
\end{equation}
where $X'=\{(xy)^{\pm 1},\varepsilon(xy^{-1})^{\pm 1}\}$. Therefore, $sX'$ is an ${\cal A}$-set.
However, it is easily seen that $|xy|\ne|\varepsilon xy^{-1}|$. Moreover, $|xy|=2^{n-1}$ or
$|\varepsilon xy^{-1}|=2^{n-1}$, because
$x$ and $y$ are generators of the group $C$ and $n\ge 3$. Thus, the
elements $sxy$ and $s\varepsilon xy^{-1}$ can not belong to the same basic set of ${\cal A}$.
Indeed, otherwise assuming $|xy|=2^{n-1}$, we conclude by Lemma~\ref{290714c} that this basic set contains $sxyc_1$.
Then $xyc_1\in X'$, and hence $xyc_1\in\{(xy)^{-1},\varepsilon x^{-1}y\}$. Consequently, $(xy)^2=c_1$ or $x^2=c_1\varepsilon$. In any
case, $n-1\le 2$. Contradiction. A similar argument leads to a contradiction when $|\varepsilon xy^{-1}|=2^{n-1}$.
In the same way, one can verify that no two elements, one in $\{s(xy)^{\pm 1}\}$ and the other one in $\{s\varepsilon (xy^{-1})^{\pm 1}\}$,
cannot belong to the same basic set of ${\cal A}$. Thus, $sX'$ is a disjoint union
of two ${\cal A}$-sets of the form $\{sz^{\pm 1}\}$ with $z\in C$, and one of them consists of elements
of order $2^{n-1}$. This implies that $Z_x$ is an ${\cal A}$-set, as required.\medskip
To prove statement~(3), suppose on the contrary that $x^4\ne y^{\pm 4}$. Then since $Y:=X^{[2]}$, $Z_x$, and $Z_y$
are ${\cal A}$-sets, the S-ring ${\cal A}$ contains the element
$$
(\und{Y_{}}\,\und{Z_x})\,\circ\, (\und{Y_{}}\,\und{Z_y})
=2s(2e+x^{\pm 2}y^{\pm 2}).
$$
Since $n\ge 3$, this implies that only $s$ appears in the right-hand
side with multiplicity~$4$. By the Schur-Wielandt principle, this implies that $s\in{\cal A}$. However, this contradicts the second part of statement~(1).\medskip
To complete the proof, we note that by statement~(3), we have $Y_x=X^{[2]}$. Since $X^{[2]}$ is an ${\cal A}$-set,
the first part of statement~(2) follows.\hfill\vrule height .9ex width .8ex depth -.1ex
Let us continue the proof of Lemma~\ref{040914b}.
Denote by ${\cal A}_i$ the minimal S-ring over $D$ that contains $X$ as a basic set; we recall that
$i=6$ or $i=7$ depending on $\varepsilon=e$ or $\varepsilon=c_1$. We claim that
\qtnl{040514c}
{\cal A}_i=\cyc(K_i,D).
\end{equation}
Indeed, from statements~(2) and~(3) of Lemma~\ref{010514b}, it follows that the sets
$X$, $Y_x$, and $Z_x$ are orbits of the group $K_i$ (see Table~\ref{tblsg1}, where the generic orbits of the group $K_i$
contained inside $C_{n-1}$ and $sC_{n-1}$ are given).
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|}
\hline
$C_{n-1}$ & $sC_{n-1}$ \\
\hline
$x^{\pm 2},c_1x^{\pm 2}$ & $s\varepsilon x^{\pm 2}$ \\
\hline
$x^4,x^{-4}$ & $sx^{\pm 4},sc_1x^{\pm 4}$ \\
\hline
$\cdots$ & $\cdots$ \\
\hline
$c_2^{},c_2^{-1}$ & $sc_2^{},sc_2^{-1}$ \\
\hline
$c_1$ & $s,sc_1$ \\
\hline
$e $ & \\
\hline
\end{tabular}\\[2mm]
\caption{The orbits of $K_6$ and $K_7$}\label{tblsg1}
\end{center}
\end{table}
Therefore, by the Schur theorem on multipliers, we have
\qtnl{040514a}
({\cal A}_i)_{D\setminus D_{n-2}}=\cyc(K_i,D)_{D\setminus D_{n-2}}.
\end{equation}
Next, $C':=\grp{Z_x}$ is a cyclic ${\cal A}$-group of order $2^{n-1}$ other than $C_{n-1}$. Moreover, $Z_x$ is a highest
basic set of the circulant S-ring $({\cal A}_i)_{C'}$. Therefore, this S-ring has trivial radical. From the results discussed
just after Lemma~\ref{211113a}, it follows that it is the cyclotomic S-ring $\cyc(K',C')$, where
$K'$ is the subgroup of $\aut(C')$ that has $Z_x$ as an orbit. Since $\orb(K_i,C')=\orb(K',C')$,
we conclude that
\qtnl{040514b}
({\cal A}_i)_{C_{n-2}}=\cyc(K_i,D)_{C_{n-2}}.
\end{equation}
Let us complete the proof of~\eqref{040514c}. To do this, taking into account that $\varepsilon(e+c_1)=e+c_1$, we find that
\qtnl{040514u}
\und{Y_x}\,\und{Z_x}=sx^{\pm 4}(e+c_1)+2s(e+c_1).
\end{equation}
Moreover, since $n\ge 5$, the elements $x^{\pm 4},sx^{\pm 4}c_1$
appear in the right-hand side with coefficient~$1$. By the Schur-Wielandt principle, this implies that $\{s,sc_1\}x^{\pm 4}$
and $\{s,sc_1\}$ are ${\cal A}$-sets. Thus,
\qtnl{040514e}
({\cal A}_i)_{sC_m\setminus sC_{m-1}}=\cyc(K_i,D)_{sC_m\setminus sC_{m-1}}\quad\text{and}\quad
({\cal A}_i)_{sC_1}=\cyc(K_i,D)_{sC_1},
\end{equation}
where $m=n-2$. For all $m=n-3,\ldots,2$ the first equality is proved in a similar way by the induction on~$m$; the sets $Y_x$ and
$Z_x$ in equation~\eqref{040514u} are replaced by the ${\cal A}$-sets $\{x^{\pm 2^{m+1}}\}$ and $(e+c_1)\{x^{\pm 2^{m+1}}\}$, respectively.
Thus, the claim follows from~\eqref{040514a}, \eqref{040514b} and~\eqref{040514e}.\medskip
Let us continue the proof of Lemma~\ref{040914b}.
Now, since, obviously, ${\cal A}\ge{\cal A}_i$, we conclude by \eqref{040514c} that ${\cal A}\ge\cyc(K_i,D)$. To verify
the converse inclusion, we have to prove that every $K_i$-orbit $Z'$ belongs to~${\cal S}({\cal A})$. Suppose
on the contrary that some $Z'$ properly contains a set $Z\in{\cal S}({\cal A})$. Then
\qtnl{010314a}
Z\subseteq D_{n-1}\setminus D_1.
\end{equation}
Indeed, from~\eqref{010514a} and statement~(3) of Lemma~\ref{010514b}, it follows that the $K_i$-orbits
outside $D_{n-1}$ are the basic sets of~${\cal A}$. Besides, the orbits $\{e\}$ and $\{c_1\}$ are also basic sets,
because ${\cal A}\ge\cyc(K_i,D)$. Finally, the orbit $\{s,c_1s\}$ belongs to ${\cal S}({\cal A})$ by the second
part of statement~(1) of Lemma~\ref{010514b}.\medskip
From \eqref{010314a} and the first part of statement~(1) of Lemma~\ref{010514b}, it follows that the set $Z$ is regular.
So it is an orbit of an automorphism group of $C$. This implies that $Z$ has cardinality $1$, $2$, or $4$.
The latter case is impossible, because otherwise $Z=Z'$. We claim that the first case is also impossible. Indeed,
otherwise $Z=\{z\}$ for some $z\in D_{n-1}\setminus D_1$. Then
$zX$ is a highest basic set of~${\cal A}$. However, $(zX)_0=\{zx,z\varepsilon x^{-1}\}$.
Therefore, $\varepsilon (zx)^{-1}=z\varepsilon x^{-1}$, and hence $z=z^{-1}$. Thus, $z\in D_1$. Contradiction.\medskip
To complete the proof of Lemma~\ref{040914b}, let $|Z|=2$. Without loss of generality, we
can assume that the order of an element in $Z$
is minimal possible. Clearly, $|Z'|=4$, and hence
$$
Z'=\{z^{\pm 1},c_1z^{\pm 1}\},
$$
where either $z=x^2$, or $z=sx^{2^m}$ with $m\in\{3,\ldots,n-3\}$ (see Table~\ref{tblsg1}). Choose $z\in Z$ so that
$$
Z=\{z,c_1z\}\quad\text{or}\quad Z=\{z,\varepsilon' z^{-1}\},
$$
where $\varepsilon'\in\{e,c_1\}$. In the first case, the singleton $Z^2=\{z^2\}$ is a basic set of ${\cal A}$. By the above
argument, this implies that $z^2\in D_1$. Thus, $z\in D_2$. Contradiction.
In the second case, ${\cal A}$ contains the element
$$
(sz+sz^{-1})(z+\varepsilon' z^{-1})=sz^2+s\varepsilon' z^{-2}+s+s\varepsilon'.
$$
Since $s\not\in{\cal A}$, this implies that $\varepsilon'=c_1$. Therefore, $\{sz^2,s\varepsilon' z^{-2}\}$ cannot be a basic
set of ${\cal A}$, and hence $m\ne 3$. But then, the minimality of $|z|$ implies that
$\{sz^2,s\varepsilon' z^{-2},s,s\varepsilon'\}\in{\cal S}({\cal A})$ that is impossible, because $s+sc_1\in{\cal A}$.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\lmml{040914c}
Let $|X_0|=1$ and $X_1\ne\emptyset$. Then ${\cal A}=\cyc(K_i,D)$
with $i\in\{8,9,10,11\}$.
\end{lemma}
\noindent{\bf Proof}.\ In this case $X=\{x,ys\}$, where $\grp{y}=C$. It follows that $X$ generates $D$. Moreover, $n\ge 3$, because
$c_1\not\in\rad(X)$. Therefore, there are two orthogonals in the S-ring~${\cal A}$: one of them is in
$(X\,X^{-1})\cap Cs$, and another one is in $(X\,X^{-1})\cap C$. Thus, ${\cal A}$ is quasi-thin
by Lemma~\ref{100914a}, and hence schurian by Lemma~\ref{100914y}. The latter implies also that
the stabilizer $K$ of the point $e$ in the group $\aut({\cal A})$, has a faithful regular orbit. Therefore, the index of $D$ in $\aut({\cal A})$
is equal to~$2$. But then $D\trianglelefteq \aut({\cal A})$, and hence $K\le\aut(D)$.
Consequently, ${\cal A}=\cyc(K,D)$ and the group $K$ is generated by an involution
$\sigma\in\aut(D)$. This involution interchanges $x$ and $ys$. Therefore, the automorphism $\sigma$
is uniquely determined. We leave the reader to verify that $\sigma$ is one of automorphisms that are listed
in the rows 8, 9, 10, 11 of Table~\ref {280414b}.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\section{S-rings over $D={\mathbb Z}_2\times{\mathbb Z}_{2^n}$: automorphism groups in regular case}\label{231014a2}
In this section we find the automorphism group of a regular S-ring over $D$ with trivial radical. For this purpose, we
need the following concept introduced in~\cite{KK}: a permutation group is called {\it $2$-isolated} if no other
group is $2$-equivalent to it. The following statement is the main result of this section;
it shows, in particular, that a regular S-ring over $D$ with trivial radical is normal.
\thrml{090514a}
Let ${\cal A}$ be a regular S-ring over the group $D$. Suppose that $\rad({\cal A})=e$. Then for any
${\cal A}$-group $L$ of order at most $2$, the group $\aut({\cal A}_{D/L})$ is $2$-isolated. In
particular, if ${\cal A}=\cyc(K,D)$ for some $K\le\aut(D)$, then
$\aut({\cal A})=DK$.
\end{theorem}
The proof will be given in the end of the section. The following statement provides a sufficient condition
for a permutation group to be $2$-isolated.
\lmml{160514a}
Let ${\cal A}$ be an S-ring and $G=\aut({\cal A})$. Suppose that the point stabilizer of $G$ has a faithful regular
orbit. Then the group $G$ is $2$-isolated.
\end{lemma}
\noindent{\bf Proof}.\ It was proved in \cite[Theorem~$3.5'$]{KK} that $G$ is
$2$-isolated whenever it is $2$-closed and a two-point stabilizer
of $G$ is trivial. However, the latter exactly means that a
point stabilizer of~$G$ has a faithful regular orbit.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip\medskip
To apply Lemma~\ref{160514a}, we need the following auxiliary statement giving a sufficient condition
providing the existence of a faithful regular orbit of a point stabilizer in the automorphism group
of an S-ring.
\lmml{150514d}
Let ${\cal A}$ be an S-ring over an abelian group $H$. Suppose that a set $X\in{\cal S}({\cal A})$ satisfies the
following conditions:
\begin{enumerate}
\tm{1} $\grp{\tr(X)}=H$,
\tm{2} $c_{XY}^Z=1$ for each $Z\in{\cal S}({\cal A})_{\tr(X)}$ and some $Y\in{\cal S}({\cal A})$,
\tm{3} $c_{XY}^X=1$ for each $Y\in{\cal S}({\cal A})_{XX^{-1}}$.
\end{enumerate}
Then $X$ contains a faithful regular orbit of the group $\aut({\cal A})_e$.
\end{lemma}
\noindent{\bf Proof}.\ Denote by ${\cal X}$ the Cayley scheme over $H$ associated with the S-ring~${\cal A}$. Then the
relation $r=R_H(\tr(X))$ is a union of basic relations of ${\cal X}$. Clearly, $r$ is symmetric, and
connected (condition~(1)). Moreover, conditions~(2)
and~(3) imply that the coherent configuration $({\cal X}_e)_{er}$ is semiregular. Thus, by \cite[Theorem~3.3]{P13}
given $x\in X$, the two-element set $\{e,x\}$ is a base of the scheme ${\cal X}$, and hence of the group $\aut({\cal X})$.
This implies that $\{x\}$ is a base of the group $K=\aut({\cal X})_e$. Thus, $x^K\subseteq X$ is a faithful regular
orbit of~$K$.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
{\bf Proof of Theorem~\ref{090514a}.}\ From Theorem~\ref{230214a}, it follows that ${\cal A}=\cyc(K,D)$, where $K=K_i$ is one of the groups
listed in Table~\ref{280414b}, $1\le i \le 11$. Therefore, taking into account that the groups $DK$ and $\aut({\cal A})$ are $2$-equivalent,
we conclude that the second part of the theorem statement immediately follows from the first one. To prove the latter, without loss of generality, we assume that $i\ge 2$ and $n\ge 4$. \medskip
Let $L\le D$ be an ${\cal A}$-group. In what follows, we set $H=D/L$ and $\pi=\pi_L$. To prove that the group $\aut({\cal A}_H)$ is $2$-isolated,
it suffices to verify that its point stabilizer has a faithful regular orbit (Lemma~\ref{160514a}). The remaining part of the proof
is divided into three cases.\medskip
{\bf Case 1:} $L=\grp{\varepsilon s}$, where $\varepsilon\in\{e,c_1\}$. Here, $H$ is a cyclic group and the S-ring ${\cal A}_H=\cyc(\pi(K),H)$
is cyclotomic. Moreover, $i\in\{2,3,8,9\}$ by statement~(3) of Corollary~\ref{100514d}. Therefore, the order of the group $\pi(K)\le\aut(H)$
is at most~$2$. By the implication $(3)\Rightarrow(2)$ of~\cite[Theorem~6.1]{EP02}, this implies
that the group $\aut({\cal A}_H)_e$ has a faithful regular orbit. Thus, the group $\aut({\cal A}_H)$ is $2$-isolated by Lemma~\ref{160514a}.\medskip
{\bf Case 2:} $e\le L \le\grp{c_1}$ and $|K|=2$. Here, $i\not\in\{1,6,7\}$ and each basic set of ${\cal A}$ is of cardinality at most~$2$.
Since ${\cal A}$ is commutative, the latter is also true for the basic sets of~${\cal A}_H$. Therefore, this S-ring is quasi-thin. So by the
second part of
Lemma~\ref{100914y} it suffices to prove that ${\cal A}_H$ has at least two orthogonals.
To do this let $X$ be a basic set of ${\cal A}$ that contains a generator of~$C$. Since
the S-ring ${\cal A}$ is cyclotomic, $\pi(X^{(2)})$ and $\pi(X^{(4)})$ are basic sets of ${\cal A}_H$.
Moreover, they are distinct, because $n\ge 4$. Finally, they are orthogonals,
because $\pi(X^{(2)})\subseteq \pi(X)\,\pi(X^{-1})$ and
$\pi(X)^{(4)}\subseteq \pi(X^{(2)})\,\pi(X^{(-2)})$.\medskip
{\bf Case 3:} $e\le L \le\grp{c_1}$ and $|K|=4$. Here $i=6$ or $7$. It suffices to verify that the hypothesis of
Lemma~\ref{150514d} is satisfied for a highest basic set $X$ of the S-ring~${\cal A}_H$. To do this we first observe
that the sets $X_0$ and $X_1$ are not empty. Therefore,
$\tr(X)=D\setminus D_{n-1}$, and condition~(1) is, obviously, satisfied.\medskip
To verify conditions (2) and (3), suppose first that $L=e$. Then for $x\in X_0$, we have
$$
X=\{x,\varepsilon x^{-1},sc_2x,sc_2^{-1}\varepsilon x^{-1}\},
$$
where $\varepsilon\in\{e,c_1\}$. Since $n\ge 4$, the elements $xy^{-1}$ with $y\in X$
belong to distinct $K$-orbits of cardinalities $1,2,2$ and $4$. A straightforward check
shows that if $Y$ is one of these orbits, then
$$
|Y|c_{XX^{-1}}^Y=4.
$$
Therefore, $4=|Y|c_{XX^{-1}}^Y=|X|c_{X^{}Y^{-1}}^X$, and condition~(3) is satisfied,
because $|X|=4$. Let now $Z\in{\cal S}({\cal A})_{\tr(X)}$. Then
$$
Z=\{xy,\varepsilon (xy)^{-1},sc_2xy,sc_2^{-1}\varepsilon (xy)^{-1}\}
$$
for some $y\in C_{n-1}$. Let $Y$ be the set $\{y^{\pm 1},c_1y^{\pm 1}\}$ ,
$\{y^{\pm 1}\}$, or $\{y\}$ depending on whether $y$ belongs to $C_{n-1}\setminus C_{n-2}$,
$C_{n-2}\setminus C_1$, or $C_1$, respectively. Then $Y$ is a basic set of ${\cal A}$. Moreover,
a straightforward computation shows that in any case, $c_{YZ}^X=1$.
Since $c_{X^{}Y^{}}^{Z^{}}=c_{Y^{-1}Z^{-1}}^{X^{-1}}=c_{Y^{}Z^{}}^{X^{}}$,
condition~(2) is also satisfied.\medskip
Let now $L=\grp{c_1}$. To simplify notations, we identify the group $H=D/L$ with
$D_{n-1}$, write ${\cal A}$ instead of ${\cal A}_H$,\footnote{In our case, the S-ring ${\cal A}_H$ does
not depend on the choice of $i\in\{5,6\}$.} and use the notation $x$ and $s$ for the
$\pi$-images of $x$ and $sc_1$, respectively. Thus, ${\cal A}$ is a cyclotomic
S-ring over $D_{n-1}$ and
$$
X=\{x,x^{-1},sx,sx^{-1}\}
$$
is a highest basic set of ${\cal A}$. It follows that $C_{n-2}$ is an ${\cal A}$-group and any basic set inside $C_{n-2}$ is of
the form $\{z^{\pm 1}\}$ for a suitable $z\in C_{n-2}$. Since $sC_{n-2}$ is an ${\cal A}$-set, the elements $xy^{-1}$
with $y\in X$ belong to distinct basic sets of~${\cal A}$. Therefore, the set $X^{}\,X^{-1}$
consists of basic sets $Y$ for which $c_{XY}^X=1$. Thus, condition~(3) is satisfied.
Let now $Z\in{\cal S}({\cal A})_{\tr(X)}$. Then
$$
Z=\{xy,(xy)^{-1},sxy,s (xy)^{-1}\}
$$
for some $y\in C_{n-2}$. Taking $Y$ to be the basic set $\{y^{\pm 1}\}$, we find that
$c_{YZ}^X=1$. Thus, condition~(2) is also satisfied, and we are done.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\section{S-rings over $D={\mathbb Z}_2\times{\mathbb Z}_{2^n}$: non-trivial radical case}\label{140914b}
In Theorems~\ref{170414a} and~\ref{230214a}, we completely described the structure of an S-ring over~$D$ that has
trivial radical. In this section, we study the remaining S-rings.
\thrml{060414a}
Let ${\cal A}$ be an S-ring over a group $D$. Suppose that $\rad({\cal A})\ne e$. Then ${\cal A}$ is a proper generalized
$S$-wreath product, where the section $S=U/L$ is such that
\qtnl{171114a}
{\cal A}_S={\mathbb Z} S\quad\text{or}\quad |S|=4\quad\text{or}\quad \rad({\cal A}_U)=e\ \text{and}\ |L|=2.
\end{equation}
\end{theorem}
\noindent{\bf Proof}.\ Denote by $U$ the subgroup of~$D$ that is generated by all $X\in{\cal S}({\cal A})$ such that $\rad(X)=e$.
\lmml{170414b}
$U$ is an ${\cal A}$-group and $\rad({\cal A}_U)=e$.
\end{lemma}
\noindent{\bf Proof}.\ The first statement is clear. To prove the second one, without loss of generality, we
can assume that $U=D$. Then there exists a highest set $X\in{\cal S}({\cal A})$ such that
$\rad(X)=e$. Suppose first that $X$ is not regular. Then $X\cap E\ne\emptyset$ by
Theorem~\ref{290714a}. Therefore, $X$ is one of basic sets $X_{c_1}$, $X_s$, or $X_{sc_1}$
from Theorem~\ref{250814a}. Since $\rad(X)=e$, we have $X=X_{c_1}$ and $X=X_s$
in statements~(2) and~(3) of this theorem, respectively. Moreover, since $X$ is highest, $D=\grp{X}$
in statements~(1), (2), (3). Therefore, in these three cases $\rk({\cal A})=2$, and hence $\rad({\cal A})=e$. In the
remaining case (statement~(4) of Theorem~\ref{250814a}), we have $D=H$, and hence
${\cal A}={\cal A}_U\otimes{\cal A}_{\grp{s}}$. Since each of the factors is of rank~$2$, this
implies that again $\rad({\cal A})=e$.\medskip
From now on, we can assume that any highest basic set of~${\cal A}$ with trivial radical,
is regular (Corollary~\ref{011213d}). Moreover, if $X$ is one of them and both $X_0$ and
$X_1$ are not empty, then all highest basic sets are pairwise rationally conjugate and, hence
$\rad({\cal A})=\rad(X)=e$. Thus, we can also assume that $\grp{X}$ is a cyclic group $C$
of order at least~$4$ that has index $2$ in $D$.\medskip
Since $\rad({\cal A}_C)=\rad(X)=e$, the circulant S-ring ${\cal A}_C$ is cyclotomic (see
Subsection~\ref{141114b}). Together with $|C|\ge 4$, this shows that $c_1\in{\cal A}$.
We claim that any $Y\in{\cal S}({\cal A})$ such that $\rad(Y)=e$ and $D=\grp{X,Y}$,
is regular. Indeed, otherwise by Theorem~\ref{290714a}, we have $Y=X_h$,
where $h$ is a non-identity element of the group~$E$. However, $h\ne c_1$: otherwise $Y=\{c_1\}$ by above,
and $\grp{X,Y}=C$, in contrast to the assumption. Since $X_{c_1}=\{c_1\}$ and $\rad(Y)=e$,
only statement~(4) of Theorem~\ref{250814a} can hold. But then, $Y$ is a singleton
in $E$, and hence it is regular. Contradiction.\medskip
To complete the proof, let $Y$ be a regular basic set with trivial radical. We can assume
that $Y\subseteq D\setminus C$, for otherwise $D=C$ and $\rad({\cal A})=\rad(X)=e$.
If $Y$ is not highest, then any basic set $Z\subseteq XY$ is highest, and $\grp{X,Z}=D$.
Moreover, $\rad(Z)=e$ by Lemma~\ref{150414a}. Thus, we can also assume that $Y$ is highest.
Then any highest basic set of ${\cal A}$ is rationally conjugate to either $X$ or $Y$. Thus,
$\rad({\cal A})=e$, as required.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
By the theorem hypothesis and Lemma~\ref{170414b}, we have $U\ne D$. We observe also that
by Theorem~\ref{030414a}, the group $U$ contains every minimal ${\cal A}$-group.
\lmml{130914a}
Suppose that there is a unique minimal ${\cal A}$-group, or $c_1\in\rad(X)$ for all $X\in{\cal S}({\cal A})_{D\setminus U}$. Then the statement of Theorem~\ref{060414a} holds.
\end{lemma}
\noindent{\bf Proof}.\ Let $L$ be a unique minimal ${\cal A}$-group. Then the definition of $U$ implies that
${\cal A}$ is a proper generalized $S$-wreath
product, where $S=U/L$. If, in addition, $|L|\le 2$, then the third statement in~\eqref{171114a} follows
from Lemma~\ref{170414b} and we are done. Let now $|L|>2$. Then $\grp{c_1}$ is not
an ${\cal A}$-group. By statement~(1) of Corollary~\ref{100514d}, this implies that the S-ring
${\cal A}_U$ is not regular. So, by the first part of Theorem~\ref{170414a}, it is rational.
Now, by the second part of this theorem, the uniqueness of~$L$ implies that $U=L$. Thus,
$|S|=1$ and the first statement in~\eqref{171114a} holds.\medskip
To complete the proof, suppose that there are at least two minimal ${\cal A}$-groups. Then, obviously,
one of them, say $H$, contains $c_1$. Therefore, $c_1\in H\le U$. On the other hand, by the
lemma hypothesis, $c_1\in\rad(X)$ for all $X\in{\cal S}({\cal A})_{D\setminus U}$. Thus, ${\cal A}$ is a proper generalized
$S$-wreath product, where $S=U/H$. Without loss of generality, we can assume that $|H|>2$.
If the S-ring ${\cal A}_U$ is rational, then from the second part of
Theorem~\ref{170414a}, it follows that there is another minimal ${\cal A}$-group $L$ of order $2$
and such that
$$
{\cal A}_U={\cal A}_H\otimes{\cal A}_L.
$$
Thus, $|S|=|L|=2$, and the first statement in~\eqref{171114a} holds. In the remaining
case, ${\cal A}_U$ is a regular S-ring by the first part of Theorem~\ref{170414a}. By statement~(1)
of Corollary~\ref{100514d}, this implies that $H=\grp{c_1}$. Thus, $|H|=2$, and
the third statement in~\eqref{171114a} holds.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
Denote by $V$ the union of all sets $X\in{\cal S}({\cal A})$ such that $\rad(X)=e$ or $c_1\in\rad(X)$.
Then, obviously, $U\subseteq V$ and $V$ is an ${\cal A}$-set. By Lemma~\ref{130914a}, we can
assume that $V\ne D$, and that $U$ contains two distinct minimal ${\cal A}$-groups. It is easily seen that in this case
$E\subseteq V$.
\lmml{130914c}
In the above assumptions let $X\in{\cal S}({\cal A})_{D\setminus V}$. Then
\begin{enumerate}
\tm{1} $\rad(X)=\grp{s}$ or $\grp{sc_1}$,
\tm{2} $X$ is a regular set such that both $X_0$ and $X_1$ are not empty.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof}.\ Since $X\not\subseteq U$, we have $\rad(X)\ne e$. Besides, $c_1\not\in\rad(X)$ by the
definition of $V$. Thus, statement (1) holds, because $\grp{s}$ and $\grp{sc_1}$ are the
only subgroups of $D$ that do not contain $c_1$. To prove statement (2), set
$$
L=\rad(X)\quad\text{and}\quad\pi=\pi_{D/L}.
$$
Then $\rad(\pi(X))=e$. However, $D/L$ is a cyclic $2$-group by statement~(1). Therefore,
$\pi(X)$ is the basic set of a circulant S-ring ${\cal A}_{D/L}$. From the description
of basic sets of such an S-ring given in Subsection~\ref{141114b}, it follows
that $\pi(X)$ is regular or is of the form
$$
\pi(X)=\pi(H)^\#
$$
for some ${\cal A}$-group $H\ge D_1$ such that $|H/L|\ge 4$. In the latter case,
$X=H\setminus L$, and hence $L$ is a unique minimal ${\cal A}$-group in contrast to
our assumption on $U$. Thus, the set $\pi(X)$ is regular. This implies that the
set $X$ is also regular. Finally, the fact that $X_0$ and $X_1$ are not empty,
immediately follows from statement~(1).\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
By statement~(2) of Lemma~\ref{130914c}, the union of $\tr(X)$, where $X$ runs over the set
${\cal S}({\cal A})_{D\setminus V}$, is of the form $D\setminus D_k$ for some $k\ge 1$. However,
the set $V$ coincides with the complement to this union. Thus, $V=D_k$ is an ${\cal A}$-group.
A similar argument shows that $D_m$ is an ${\cal A}$-group for all $m\ge k$.
\lmml{130914u}
Let $m=\max\{2,k\}$. Then the group $L:=\rad(X)$ does not depend on the choice of
$X\in{\cal S}({\cal A})_{D\setminus D_m}$.
\end{lemma}
\noindent{\bf Proof}.\ Suppose on the contrary that there exist basic sets $X$ and $Y$ outside $D_m$ such that
$\grp{Y}\subsetneq\grp{X}$ and $\rad(X)\ne\rad(Y)$. Then by statement (1) of Lemma~\ref{130914c},
without loss of generality, we can assume that
\qtnl{181114a}
\rad(X)=\grp{s}\quad\text{and}\quad \rad(Y)=\grp{sc_1}.
\end{equation}
By statement~(2) of that lemma, $X_0$ is a regular non-empty set. Therefore, it
is an orbit of a subgroup of $\aut(C)$. Moreover, from the first equality in \eqref{181114a},
it follows that $\rad(X_0)=e$. Let now $\pi:D\to D/\grp{s}$ be the quotient epimorphism. Then
$\pi(X)=X_0$ is a basic set of a circulant S-ring ${\cal A}'=\pi({\cal A})$. It follows that
${\cal A}'_{\grp{X_0}}$ is a cyclotomic S-ring with trivial radical. Moreover, $Y'=\pi(Y)$ is a
basic set of this S-ring and $|\grp{Y'}|\ge 2^{m+1}\ge 8$. Therefore, by Lemma~\ref{021109a} applied
for ${\cal A}={\cal A}'_{\grp{X_0}}$ and $S=\grp{Y'}/e$, we obtain that
$$
\rad({\cal A}'_{\grp{Y'}})=e.
$$
This implies that $\rad(Y')=e$. On the other hand, by the second equality in~\eqref{181114a},
we have $\rad(Y')=\grp{c_1}\ne e$. Contradiction.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
By Lemma~\ref{130914u}, the S-ring ${\cal A}$ is the generalized $S$-wreath product,
where $S=D_2/L$ if $k=1$, and $S=V/L$ if $k\ge 2$. The only case when this generalized
product is not proper, is $D=D_2$ and $k=1$. However, in this case ${\cal A}$ is, obviously, a proper
$E/L$-wreath product and $|E/L|=2$. Thus, if $k\le 2$, then the first or the second statement
in~\eqref{171114a} holds, and we are done. To complete the proof of Theorem~\ref{060414a},
it suffices to verify that the third statement in~\eqref{171114a} holds whenever $k\ge 3$.
But this immediately follows from the lemma below.
\lmml{220414u}
If $k\ge 3$, then $\rad({\cal A}_V)=e$. In particular, $V=U$.
\end{lemma}
\noindent{\bf Proof}.\ The second statement follows from the first one and statement~(1) of Lemma~\ref{130914c}.
To prove that $\rad({\cal A}_V)=e$, let $X$ be a highest basic set of the S-ring ${\cal A}_V$. Since $V\ne D$, there exists
a set $Y\in{\cal S}({\cal A})_{D\setminus V}$ such that $X\subseteq Y^2$. By Lemma~\ref{130914c},
we have $Y=LY_0$, where $Y_0$ is an orbit of a subgroup of $\aut(C)$ such that $\rad(Y_0)=e$.
However, $Y_0=\{y\}$ or $Y_0=\{y,\varepsilon y^{-1}\}$, where $\varepsilon\in\{e,c_1\}$
(statement~(1) of Lemma~\ref{211113a}). Therefore,
\qtnl{240414b}
Y^2=LY_0^2=L\times\begin{cases}
\{y^2\}, &\text{if $Y_0=\{y\}$},\\
\{\varepsilon,y^{\pm 2}\}, &\text{if $Y_0=\{y,\varepsilon y^{-1}\}$}.\\
\end{cases}
\end{equation}
On the other hand, since $X\subset V$, the definition of $V$ implies that $\rad(X)=e$ or $c_1\in\rad(X)$. In
the former case, $\rad({\cal A}_V)=e$, and we are done. Suppose that $c_1X=X$. Then the set $Y^2$ contains
a $\grp{c_1}$-coset $\{x,c_1x\}$ for all $x\in X$. By~\eqref{240414b}, this implies that
$$
\{x,c_1x\}\subset \{\varepsilon,y^{\pm 2}\}
$$
which is impossible, because $|x|=2^k\ge 8$.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\section{S-rings over $D={\mathbb Z}_2\times{\mathbb Z}_{2^n}$: schurity}\label{231014a3}
In this section, based on the results obtained in Sections~\ref{140914a}--\ref{140914b}, we prove the following main
theorem.
\thrml{090514b}
For any integer $n\ge 1$, every S-ring over the group $D={\mathbb Z}_2\times{\mathbb Z}_{2^n}$ is schurian. In particular,
$D$ is a Schur group.
\end{theorem}
\noindent{\bf Proof}.\
The induction on $n$. An exhaustive computer search of all S-rings over small groups shows that
$D$ is a Schur group for $n\le 4$. Let $n\ge 5$. We have to verify that any S-ring ${\cal A}$
over~$D$ is schurian. However, if $\rad({\cal A})=e$, then this is true by Theorems~\ref{170414a}
and~\ref{230214a}. For the rest of the proof, we need the following result from~\cite{EP12}
giving a sufficient condition for a generalized wreath product of S-rings to be schurian.
\thrml{140914f}{\rm \cite[Corollary~5.7]{EP12}}
Let ${\cal A}$ be an S-ring over an abelian group~$D$. Suppose that ${\cal A}$ is the generalized $S$-wreath
product of schurian S-rings ${\cal A}_{D/L}$ and ${\cal A}_U$, where~$S=U/L$. Then ${\cal A}$ is schurian if and only if there exist two groups
$\Delta_0\ge(D/L)_{right}$ and $\Delta_1\ge U_{right}$, such that
\qtnl{140914r}
\Delta_0\approx_{\scriptscriptstyle 2}} %{\underset{\scriptscriptstyle 2}{\approx}\aut({\cal A}_{D/L})\quad\text{and}\quad
\Delta_1\approx_{\scriptscriptstyle 2}} %{\underset{\scriptscriptstyle 2}{\approx}\aut({\cal A}_U)\quad\text{and}\quad
(\Delta_0)^{U/L}=(\Delta_1)^{U/L}.
\end{equation}
\end{theorem}
\crllrl{110514a}
Under the hypothesis of Theorem~\ref{140914f}, the S-ring ${\cal A}$ is schurian whenever the
group $\aut({\cal A}_S)$ is $2$-isolated.
\end{corollary}
\noindent{\bf Proof}.\ Set $\Delta_0=\aut({\cal A}_{D/L})$ and $\Delta_1=\aut({\cal A}_U)$. Then the first two
equalities in~\eqref{140914r} hold, because the S-rings ${\cal A}_{D/L}$
and ${\cal A}_U$ are schurian. Since the group $\aut({\cal A}_S)$ is $2$-isolated, we have
$
(\Delta_0)^S=\aut({\cal A}_S)=(\Delta_1)^S,
$
which proves the third equality in~\eqref{140914r}. Thus, ${\cal A}$ is schurian
by Theorem~\ref{140914f}.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
Let us turn to the proof of Theorem~\ref{090514b}. Now, we can assume that
$\rad({\cal A})\ne e$. Then by Theorem~\ref{060414a}, the S-ring~${\cal A}$ is a proper generalized $S$-wreath
product, where the section $S=U/L$ is such that formula~\eqref{171114a} holds. Besides, by induction,
the S-rings ${\cal A}_{D/L}$ and ${\cal A}_U$ are schurian. Suppose that ${\cal A}_S={\mathbb Z} S$, or $|S|=4$,
or $|L|=2$ and ${\cal A}_U$ is a regular S-ring with trivial radical. Then the group $\aut({\cal A}_S)$
is $2$-isolated: this is obvious in the first two cases and follows from Theorem~\ref{090514a}
(applied for ${\cal A}={\cal A}_U$) in the third one. Thus, ${\cal A}$ is schurian
by Corollary~\ref{110514a}.\medskip
To complete the proof, we can assume that $|S|=|2^m|$, where $m\ge 3$, and that
${\cal A}_U$ is a non-regular S-ring with trivial radical. Then
${\cal A}_U={\cal A}_H\otimes{\cal A}_L$, where $|H|\ge 4$ and $\rk({\cal A}_H)=2$ (Theorem~\ref{170414a}).
Therefore,
\qtnl{140914w}
\aut({\cal A}_U)^S=(\sym(H)\times\sym(L))^{U/L}=\sym(S).
\end{equation}
On the other hand, $L=\grp{s}$ or $L=\grp{sc_1}$, because $c_1\in H$. Therefore,
the S-ring ${\cal A}_{D/L}$ is circulant. Besides, $S$ is an ${\cal A}_{D/L}$-section of
composite order. By \cite[Theorem~4.6]{EP12}, this implies that
\qtnl{150914a}
\aut({\cal A}_{D/L})^S=\sym(S).
\end{equation}
By \eqref{140914w} and \eqref{150914a}, relations \eqref{140914r} are true
for the groups $\Delta_0:=\aut({\cal A}_{D/L})$ and $\Delta_1:=\aut({\cal A}_U)$. Thus, the
S-ring~${\cal A}$ is schurian by Theorem~\ref{140914f}.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\section{A non-schurian S-ring over $M_{2^n}$}\label{231014a4}
The main result of this section is the following theorem in the proof of which
we construct a non-schurian S-ring over the group $M_{2^n}$
defined in~\eqref{170914a}.
\thrml{071113a}
For any $n\ge 4$, the group $M_{2^n}$ is not Schur.
\end{theorem}
\noindent{\bf Proof}.\ The group $M_{16}$ is not Schur \cite[Lemma~3.1]{PV}. Suppose that $n\ge 5$.
Denote by $e$ the identity of the group~$G=M_{2^n}$, and
by $H$ the
normal subgroup of $G$ that is generated by the elements $c=a^{2^{n-3}}$ and $b$. Then
$H\simeq {\mathbb Z}_4\times {\mathbb Z}_2$ and
\qtnl{110413b}
H=Z_0\,\cup\, Z_1\,\cup\, Z_2,
\end{equation}
where the sets $Z_0=\{e\}$, $Z_1=\{c^2\}$, and $Z_2=H\setminus\grp{c^2}$ are mutually disjoint. Next, let us
fix two other decompositions of $H$ into a disjoint union of subsets:
$$
H=\underbrace{B\,\cup\,Bc^3}_{X_1}\ \cup\ \underbrace{Bc^2\,\cup\,Bc}_{Y_1}
\quad=\quad
\underbrace{B'\,\cup\,B'c}_{X_2}\quad\cup\quad\underbrace{B'c^2\,\cup\,B'c^3}_{Y_2},
$$
where $B$ and $B'$ are the groups of order~$2$ generated by the involutions~$b$ and $b':=c^2b$.
Then a straightforward computation shows that
\qtnl{110413a}
Ha\cup Ha^{-1}=\underbrace{X_1a\cup X_2a^{-1}}_{Z_3}\ \cup\ \underbrace{Y_1a\cup Y_2a^{-1}}_{Z_4},
\end{equation}
in particular, the sets $Z_3$ and $Z_4$ are disjoint. Moreover, $Z_3c^2=Z_4$, because
$Y_1=X_1c^2$ and $Y_2=X_2c^2$. Finally, there are exactly $m'=2^{n-3}-3$ cosets of $H$ in $G$,
other than $H$, $Ha$, and $Ha^{-1}$. Let us combine them in pairs as follows
\qtnl{110413ae}
Z_{i+3}:=Ha^i\cup Ha^{-i},\quad i=2,3,\ldots m,
\end{equation}
where $m=(m'-1)/2+1$ and $Z_{m+1}=Ha^{2^{n-2}}$.
Then the sets $Z_0,Z_1,\ldots,Z_{r-1}$ with $r=m+5$ form
a partition of the group~$G$; denote it by~${\cal S}$. The submodule of ${\mathbb Z} G$ spanned
by the elements $\und{Z_i}$, $i=0,\ldots,r-1$, is denoted by ${\cal A}$.
\lmml{170914s}
The module ${\cal A}$ is an S-ring over $G$. Moreover, ${\cal S}({\cal A})={\cal S}$.
\end{lemma}
\noindent{\bf Proof}.\
From the above definitions, it follows that $Z_i^{-1}=Z_i^{}$ for all $i$. Thus, it suffices
to verify that given $i$ and $j$, the product $\und{Z_i}\,\und{Z_j}$ is a linear combination
of $\und{Z_k}$, $k=0,\ldots,r-1$. However, it is easily seen that
$$
\und{Ha^i}\,\und{Ha^j}=\und{Ha^j}\,\und{Ha^i}=\und{Ha^{i+j}}=\und{a^{i+j}H}
$$
for all $i,j,k$. Therefore, the required statement holds whenever $i,j\not\in\{3,4\}$.
To complete the proof, assume that $i=3$ (the case $i=4$ is considered analogously). Then
a straightforward check shows that
\begin{enumerate}
\item[$\bullet$] $\und{Z_3}\,\und{Z_1}=\und{Z_4}$,\quad $\und{Z_3}\,\und{Z_2}=\und{Z_4}$,\quad $\und{Z_3}\,\und{Z_{r-1}}=4\und{Z_{r-2}}$,
\item[$\bullet$] $\und{Z_3}\,\und{Z_3}=8\,\und{Z_0}+2\,\und{Z_5}+4\,\und{Z_2}$,
\item[$\bullet$] $\und{Z_3}\,\und{Z_4}=8\,\und{Z_1}+2\,\und{Z_5}+4\,\und{Z_2}$,
\item[$\bullet$] $\und{Z_3}\,\und{Z_5}=4\und{Z_3}+4\und{Z_4}+4\und{Z_6}$,
\item[$\bullet$] $\und{Z_3}\,\und{Z_i}=4\und{Z_{i-1}}+4\und{Z_{i+1}}$, $i=6,\ldots r-2$.
\end{enumerate}
Since $\und{Z_3}$ commutes with $\und{Z_j}$ for all $j$, we are done.
\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
By Lemma~\ref{170914s}, the statement of Theorem~\ref{071113a} immediately
follows from the lemma below.
\lmml{231112a}
The S-ring ${\cal A}$ is not schurian.
\end{lemma}
\noindent{\bf Proof}.\ Suppose on the contrary that ${\cal A}$ is schurian. Then it
is the S-ring associated with the group
$\Gamma=\aut({\cal A})$. It follows that the basic set $Z_2$ is an orbit of the one-point
stabilizer of~$\Gamma$. Since $|Z_2|=6$, there exists an element
$\gamma\in\Gamma$ such that
\qtnl{231112b}
|\gamma^{Z_2}|=3.
\end{equation}
On the other hand, due to~\eqref{110413a}, the quotient S-ring ${\cal A}_{G/H}$ is isomorphic
to the S-ring associated with the dihedral group of order $2^{n-2}$ in its natural permutation representation of degree $2^{n-3}$.
Therefore, $\aut({\cal A}_{G/H})$ is a $2$-group. It contains a subgroup $\Gamma^{G/H}$,
and hence the element $\gamma^{G/H}$. So by~\eqref{231112b}, the
permutation~$\gamma$ leaves each $H$-coset fixed (as a set). Therefore,
\qtnl{281112a}
\gamma^{H\cup Ha}\in\aut(C),
\end{equation}
where $C$ is the bipartite graph with vertex set $H\cup Ha$ and the edges $(h,hax)$ with $x\in X_1$. However,
the graph $C$ is isomorphic to the lexicographic product of the empty graph with~$2$ vertices and the
undirected cycle of length~$8$. Therefore, $\aut(C)$ is a $2$-group. By~\eqref{281112a}, this implies
that $|\gamma^H|$ is a power of~$2$. But this contradicts to~\eqref{231112b}, because $Z_2\subset H$.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\section{S-rings over $D=D_{2n}$: divisible difference sets}\label{231014a6}
\sbsnt{Preliminaries.} In the rest of the paper, we deal with S-rings over a
dihedral $2$-group $D=D_{2n}$ of order $2n$. Interesting examples of such rings arise from difference sets. To
construct them, let us recall some definitions from~\cite{P95}.\medskip
Let $T$ be a $k$-subset of a
group $G$ of order $mn$ such that every element outside a subgroup $N$ of order
$n$ has exactly $\lambda_2$ representations as a quotient $gh^{-1}$ with elements
$g,h\in G$, and elements in $N$ different from the identity have exactly
$\lambda_1$ such representations,
\qtnl{161014a}
\underline{T}\cdot\underline{T}^{-1}=
k\cdot e + \lambda_1\underline{N\setminus e}+\lambda_2\underline{G\setminus N}.
\end{equation}
Then $T$ is called an {\it $(m,n,k,\lambda_1,\lambda_2)$-divisible difference set} in $G$
relative to $N$. If $\lambda_1 =0$ (resp. $n = 1$), then we say that $T$
is a {\it relative difference set} or {\it relative $(m,n,k,\lambda_2)$-difference set}
(resp. difference set). A difference set $T$
is {\it trivial} if it equals $G$, $\{x\}$ or $G\setminus\{x\}$, where $x\in G$.
\thrml{231014a}
Let $C$ be a cyclic $2$-group. Then
\begin{enumerate}
\tm{1} any difference set in $C$ is trivial,
\tm{2} there is no relative $(2^a,2,2^a,2^{a-1})$-difference set in $C$.
\end{enumerate}
\end{theorem}
\noindent{\bf Proof}.\ Statement (1) follows from \cite[Theorem~II.3.17]{BJL} and \cite[Theorem~1.2]{DS94}.
Statement~(2) follows from \cite[Theorems~4.1.4,4.1.5]{P95}.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\sbsnt{Constructions.}\label{231014w} Let $D$ be a dihedral group of order $2n$, $C$ the cyclic subgroup
of $D$ of order $n$ and $H$ a subgroup of $C$. Let $T$ be a non-empty subset of~$C$
such that $|T\cap xH|$ does not depend on $x\in T$ ({\it the intersection condition},
cf. Lemma~\ref{090608a}). Set
$$
{\cal S}:=\{e,\ H\setminus e,\ C\setminus H,\ Ts,\ T's\},
$$
where $T'=C\setminus T$. Clearly, ${\cal S}$ is a partition of $D$ such that condition (S1)
is satisfied. Since all the elements of ${\cal S}$ are symmetric, condition (S2) is also
satisfied. Set
\qtnl{161014c}
{\cal A}:={\cal A}(T,C)=\Span\{\und{X}:\ X\in{\cal S}\}.
\end{equation}
\thrml{p0}
In the above notation, ${\cal A}$ is an S-ring over $D$ with ${\cal S}({\cal A})={\cal S}$ if and only if $T$
is a divisible difference set in $C$ relative to $H$.
\end{theorem}
\noindent{\bf Proof}.\ To prove the ``only if'' part, suppose that ${\cal A}$ is an S-ring with ${\cal S}({\cal A})={\cal S}$.
Then $Ts$ is a basic set of ${\cal A}$ and $Ts\,Ts=T\,T^{-1}$ is a subset of~$C$.
Therefore,
\qtnl{191014a}
\underline{T}\cdot\underline{T}^{-1}=
\underline{Ts}^2=
|T|e+\lambda_1\underline{H\setminus e} + \lambda_2\,\underline{C\setminus H},
\end{equation}
where $\lambda_1=c_{Ts\,Ts}^{H\setminus e}$ and
$\lambda_2=c_{Ts\,Ts}^{C\setminus H}$. Thus, $T$ is a divisible difference
set in $C$ relative to~$H$.\medskip
To prove the ``if'' part, suppose that $T$ is a divisible difference set in $C$ relative
to~$H$. It suffices to verify that ${\cal A}\cdot{\cal A}\subseteq {\cal A}$. To do this, denote by ${\cal A}'$
the module spanned by $e$, $\underline{H}$, $\underline{C}$, and $\underline{D}$.
Then, obviously, ${\cal A}'\cdot{\cal A}'\subseteq{\cal A}'$ and ${\cal A} =\Span\{{\cal A}',\underline{sT}\}$.
Thus, we have to check that
$$
\underline{sT}^2\in{\cal A}\quad\text{and}\quad {\cal A}'\cdot\underline{sT}\,\subseteq\, {\cal A}.
$$
The first inclusion follows from~\eqref{161014a}, because $T$ is a divisible difference set.
Routine calculations show that the second inclusion is equivalent to the inclusion
$\und{T}\cdot\und{H}\in{\cal A}$. However, this easily follows from the intersection condition.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
We do not know any divisible difference set over a cyclic $2$-group that satisfies the intersection condition.
However, we can slightly modify the construction by taking the set $T$ to satisfy the intersection condition, but
this time inside $C\setminus H$. Then using the same argument, one can
construct an S-ring of rank $6$ that coincides with ${\cal A}$ on $C$ and has three
basic sets outside $C$: $Ts$, $T's$, and $Hs$, where $T'=C\setminus (T\cup H)$.\medskip
The S-rings of this form do exist. It suffices to take a classical relative
$(q+1,2,q,(q-1)/2)$-difference set $T$ defined as follows (see~\cite[Theorem~2.2.13]{P95}).
Take an affine line $L$ in a $2$-dimensional linear space over finite field ${\mathbb F}_q$ that does not contain
the origin. Then $L$ is a relative $(q+1,q-1,q,1)$-difference set in the
multiplicative group of ${\mathbb F}_{q^2}$. Let $\pi$ be a quotient epimorphism from this group
onto $C$ such that $|\ker(\pi)|=m$ for some divisor $m$ of $q-1$. Then $\pi(L)$ is a
cyclic relative $(q+1,(q-1)/m,q,m)$-difference set in~$C$. When $C$ is a $2$-group, the
number $q+1$ is a $2$-power, and so, $q$ is a Mersenne number. If, in addition,
$m=(q-1)/2$, then we come to the required set $T$.
\crllrl{120614a}
Let ${\cal A}$ be an S-ring over a dihedral $2$-group $D$. Suppose that $C$ is an ${\cal A}$-group. Then
\begin{enumerate}
\tm{1} if $\rk({\cal A}_C)=2$, then ${\cal A}\cong{\cal A}_C\ast{\cal B}$, where $\ast\in\{\wr,\cdot\}$
and ${\cal B}={\mathbb Z}\mZ_2$,
\tm{2} if $\rk({\cal A}_C)=3$ and $\rk({\cal A})=5$, then ${\cal A}={\cal A}(T,C)$, where $T$ is
a divisible difference set in $C$.
\end{enumerate}
\end{corollary}
\noindent{\bf Proof}.\ To prove statement~(1), suppose that $\rk({\cal A}_C)=2$. Let $X$ be a basic set outside $C$.
Then $X=Ts$ for some set $T\subseteq C$. From~\eqref{191014a} with $H=e$, it follows that $T$ is
a difference set in~$C$. By statement~(1) of Theorem~\ref{231014a}, this implies that either
$T=C$, or $T$ or $C\setminus T$ is a singleton. It
is easily seen, that ${\cal A}$ is isomorphic to ${\cal A}_C\wr{\cal B}$ in the former case, and to ${\cal A}_C\otimes {\cal B}$
in the other two.\medskip
To prove statement (2), suppose that $\rk({\cal A}_C)=3$ and $\rk({\cal A})=5$. Then
$$
{\cal A}_C=\Span\{e,\und{H},\und{C}\}
$$
for some ${\cal A}$-group $H<C$. Besides, any $X\in{\cal S}({\cal A})_{D\setminus C}$ is of the form
$X=Ts$ for some set $T\subseteq C$ satisfying the intersection condition (Lemma~~\ref{090608a}).
Thus, ${\cal A}={\cal A}(T,C)$. So, $T$ is a divisible difference set in $C$ by Theorem~\ref{p0}.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\sbsnt{Schurity.}
The main goal of this subsection is to prove the following theorem showing that the first
construction given in the previous subsection, produces mainly non-schurian S-rings.
\thrml{081014w}
Let $T$ be a divisible difference set in a cyclic $2$-group~$C$ relative to
a group $H\le C$. Suppose that the intersection condition holds and
$HT\ne T$. Then the S-ring ${\cal A}(T,C)$ defined in~\eqref{161014c} is not schurian.
\end{theorem}
We will deduce this theorem in the end of this subsection from a general statement on
schurian S-rings over a dihedral $2$-group. This statement shows that if $T$ is a
divisible difference set in a cyclic $2$-group relative to a subgroup $H$, and
the S-ring of rank $6$ associated with $T$, is schurian but not a proper generalized
wreath product, then $H$ is of order~$2$.
\thrml{231014s}
Let ${\cal A}$ be a schurian S-ring over a dihedral $2$-group $D$ and $H<C$ a minimal
${\cal A}$-group. Suppose that ${\cal A}$ is not a proper generalized wreath product. Then $|H|=2$.
\end{theorem}
\noindent{\bf Proof}.\ By the hypothesis, ${\cal S}({\cal A})=\orb(G_e,D)$ for some group $G\leq\sym(D)$ containing $D_{right}$.
Since $H$ is an ${\cal A}$-group, the partition
$D/H$ of the group $D$ into the right $H$-cosets, forms an imprimitivity system for~$G$. Denote by $N$
the stabilizer of this partition in $G$,
$$
N=\{g\in G:\ (Hx)^g=Hx\ \,\text{for all}\ \,x\in D\}.
$$
Since $H_{right}\leq N$, the group $N^X$ is transitive for each block $X\in D/H$.
The following statement can also be deduced from~\cite[Lemma 2.1]{KMM11}.
\lmml{p1}
For each block $X\in D/H$, the group $N^X$ is $2$-transitive.
\end{lemma}
\noindent{\bf Proof}.\ Let $X\in D/H$. Then the group $G^X$ is $2$-transitive, because $\rk({\cal A}_H)=2$. Moreover, it contains a regular cyclic
subgroup isomorphic to $H_{right}$. Since $H$ is a $2$-group,
the classification of primitive groups having a regular cyclic subgroup~\cite{Jo02},
implies that $G^X$ contains
a unique minimal normal subgroup $K$ that is also $2$-transitive. However, $N^X$ is a non-trivial normal subgroup
of $G^X$. Therefore, $N^X$ contains $K$, and hence is $2$-transitive.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip\medskip
Let us define an equivalence relation $\sim$ on the $H$-cosets by setting $X\sim Y$ if and
only if the actions of $N$ on~$X$ and $Y$ have the same permutation character. Then by
Lemma~\ref{p1} and a remark in~\cite[p.2]{C72}, the group $N_e$ acts transitively on each
$X$ not equivalent to $H$. Denote by $U$ the class of $\sim$ that
contains $H$. Then each orbit of $G_e$ outside $U$ is a union of $N$-orbits. Thus,
${\cal A}$ is the generalized $U/H$-wreath product. By the theorem hypothesis, this implies that $U=D$.
Therefore, all classes of the equivalence relation $\sim$ are singletons. So by Lemma~\ref{p1},
we have
$$
|\orb(N,X\times Y)|=2\quad\text{for all}\ X,Y\in D/H.
$$
This yields
us two symmetric block-designs between $X$ and $Y$ which are complementary to each other.
Since $N$ contains a cyclic subgroup $H$ which acts regularly on $X$ and $Y$, these
block-designs are circulant, and so correspond to cyclic difference sets.
By statement~(1) of Lemma~\ref{231014a}, they are trivial. Thus, the group
$N_x$ with $x\in X$, has two orbits on $Y$ of cardinalities $1$ and $|Y|-1$.\medskip
To complete the proof, suppose on the contrary that $|H|>2$. Then, obviously, $|Y| > 2$. Therefore,
the group $N_e$ fixes exactly one point in each $H$-coset~$Y$. This implies that the set $F$ of
all fixed points of $N_e$, is of cardinality $[D:H]$. On the other hand, by
\cite[Proposition~5.2]{KMM11}, the set $F$ is a block of $G$. Therefore, $F$ is a subgroup of~$D$.
Moreover, since $F\cap H=e$, it is a complement for $H$ in $D$. Thus, $H=C$. Contradiction.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
{\bf Proof of Theorem~\ref{081014w}.} Suppose on the contrary that the S-ring ${\cal A}={\cal A}(T,C)$
is schurian. Then the hypothesis of Theorem~\ref{231014s} is satisfied, because
$H$ is a minimal ${\cal A}$-group and $HT\ne T$.
Thus, $|H|=2$. Denote by $x$ the element of order $2$ in~$H$. Then $x\in{\cal A}$, and hence
$x(Ts)=T's$. This implies that $|Ts|=|T's|=m$, where $m=|C|/2$, and that $x$ appears
neither in $\und{Ts}^2$, nor in $\und{T's}^2$. Therefore, $T$ has parameters
$(m,2,m,0,m/2)$. It follows that $T$ is a relative $(m,2,m,m/2)$-difference set in $C$.
However, this contradicts part~(2) of Theorem~\ref{231014a}.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\section{S-rings over $D=D_{2^{n+1}}$: a unique minimal ${\cal A}$-group not in $C$}
In this section we deal with S-rings over a dihedral group $D=D_{2^{n+1}}$ of
order $2^{n+1}$
and keep the notation of Subsection~\ref{011014b}. The main result here is given
by the following statement.
\thrml{011014a}
Let ${\cal A}$ be an S-ring over the dihedral group $D$. Suppose that there
is a unique minimal ${\cal A}$-group $H$, and that $H\not\le C$.
Then ${\cal A}$ is isomorphic to an S-ring over ${\mathbb Z}_{2^{}}\times{\mathbb Z}_{2^n}$.
\end{theorem}
\noindent{\bf Proof}.\ The hypothesis on $H$ implies that every basic set $X$ outside
$H$ is mixed, for otherwise $\grp{X}$ contains a non-identity
${\cal A}$-subgroup of $C$. Moreover, either $H=\grp{s}$ for some $s\in D\setminus C$,
or $H$ is a dihedral group. Let us consider these two cases
separately.\medskip
{\bf Case 1: $H=\grp{s}$} for some $s$. In this case, all basic sets except
for $\{e\}$ and $\{s\}$ are mixed. By statement~(1) of Lemma~\ref{240913a}, this
implies that $X_0^{-1}=X_0^{}$ for all $X\in{\cal S}({\cal A})$. Besides,
$Xs\in{\cal S}({\cal A})$, because $s\in{\cal A}$, and
$(Xs)_0=X_1$ and $(Xs)_1=X_0$.
Thus, $X_1^{-1}=X_1^{}$ also for all $X$.\medskip
Denote by $\sigma$ the automorphism of $D$ that takes $(c,s)$ to $(c^{-1},s)$,
where $c$ is a generator of $C$. Then by the above paragraph, we have
$$
X^\sigma=(X_0\,\cup\,X_1s)^\sigma=X_0^{-1}\,\cup\,X_1^{-1}=
X_0\,\cup\,X_1s=X
$$
for all $X\in{\cal S}({\cal A})$. Therefore, the semidirect product
$D\rtimes\grp{\sigma}\le\sym(D)$ is an automorphism group of~${\cal A}$.
The element $s\sigma$ of this group has order two and commutes with~$c$. Therefore, the group $D'=\grp{s\sigma,c}$
is isomorphic to ${\mathbb Z}_2\times{\mathbb Z}_{2^n}$. On the other hand, $D'$ is a regular subgroup
in $\sym(D)$. Thus, the Cayley scheme over $D$ associated with ${\cal A}$ is
isomorphic to a Cayley scheme over $D'$. Consequently, ${\cal A}$ is isomorphic to an
S-ring over ${\mathbb Z}_{2^{}}\times{\mathbb Z}_{2^n}$.\medskip
{\bf Case 2: $H$ is dihedral}. In this case all basic sets of ${\cal A}$
other than $\{e\}$ are mixed. Moreover, the S-ring ${\cal A}_H$ is primitive
by the minimality of $H$. Therefore, $\rk({\cal A}_H)=2$ by
Theorem~\ref{030414a}. In particular, $\aut({\cal A}_H)=\sym(H)$. Below,
we will prove that
\qtnl{011014c}
H\le\rad(X)\quad\text{for all}\ X\in{\cal S}({\cal A})_{D\setminus H}.
\end{equation}
Then the Cayley scheme associated with ${\cal A}$ is isomorphic to
the wreath product of the scheme associated with ${\cal A}_H$ and a
circulant scheme on the right $H$-cosets. Therefore, the group $\aut({\cal A})$
contains a subgroup isomorphic to $\sym(H)\wr{\mathbb Z}_m$, where $m=[D:H]$.
Since the latter group contains a regular subgroup isomorphic to
${\mathbb Z}_{2^{}}\times{\mathbb Z}_{2^n}$, we are done.\medskip
To complete the proof, we will check statement~\eqref{011014c} in two
steps: first for rational S-rings, and then in general.
\lmml{280813a}
Statement~\eqref{011014c} holds whenever the S-ring ${\cal A}$ is rational.
\end{lemma}
\noindent{\bf Proof}.\ The rationality of ${\cal A}$ implies that it is symmetric, and
hence commutative. Toward to a contradiction,
suppose that $HX\ne X$ for some basic set $X$ contained in $D\setminus H$. Then
the product $HX$ is a union of $m>1$ basic sets $X,Y,\ldots$.
Without loss of generality, we may assume that
$|X|\leq |Y|\leq\cdots$.\medskip
Since $H$ is an ${\cal A}$-group, it follows from Lemma~\ref{090608a} that
the number $\lambda=|X\cap xH|$ does not depend on the choice of $x\in X$. Therefore,
each $x$ appears $\lambda$ times in the product $\und{H_{}}\,\und{X_{}}$,
i.e.
\qtnl{300514a}
\und{H_{}}\,\und{X_{}} = \lambda(\und{X}+\und{Y}+\cdots).
\end{equation}
By the minimality of $X$, this implies that
$|H|\,|X|\geq\lambda m |X|$, and hence $|H|\geq \lambda m$. On the
other hand, $(H\cap C) X_0 = X_0$ by the rationality of $X$. Therefore,
the element $\und{X}$ appears in the product $\und{H}\,\und{X}$
at least $|H\cap C|=|H|/2$ times. Thus, $\lambda\geq |H|/2$,
and
$$
|H|\geq \lambda m\ge m|H|/2.
$$
Due to $m>1$, we have $m=2$ and $\lambda=|H|/2$. Consequently, $H_0=H_1=H\cap C$, because
the group $H$ is dihedral. Therefore,
$$
\und{H_{}}\,\und{X_{}} = \und{H_0}(e+s)(\und{X_0}+s\und{X_1}) =
$$
$$
\und{H_0}\,\und{X_0} +\und{H_0}\,\und{sX_0} +\und{H_0}\,\und{X_1} +\und{H_0}\,\und{sX_1} =
|H_0|\und{X_0} +\und{H_0}\,\und{X_1}+\cdots
$$
By~\eqref{300514a}, all coefficients in the last expression are equal
to $\lambda=|H_0|$. Therefore, the set $H_0X_1\cap X_0$ must be empty.
It follows that $\und{H_0}\,\und{X_1} = |H_0|\und{H_0X_1}$. However,
this means that $H_0\le\rad(X_1)$. Since also $H_0\le\rad(X_0)$,
we conclude that $H_0\leq\rad(X)$. But $H_0\ne e$, because $H$ is
dihedral. Thus, $\rad(X)$ is non-trivial, and so contains the minimal
${\cal A}$-group~$H$. But then, $HX = X$. Contradiction.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
To complete the proof of~\eqref{011014c}, take a basic set $X$ outside $H$. Then $Y:=\tr(X)$ is also
outside $H$. So by Lemma~\ref{280813a}, we have
$$
\und{Y_0}+\und{sY_1}=\frac{1}{|H|}\und{H}\,\und{Y}=
\frac{1}{|H|}(e+s)\,(\und{H_0Y_0}+\und{H_0Y_1}).
$$
Therefore, $|Y_0|=|Y_1|$. On the other hand, for every integer $m$ coprime to $|D|$, we have $|(X^{(m)})_0|=|X_0|$. By
Lemma~\ref{290514b}, this implies that $|(X^{(m)})_1|=|X_1|$. Thus,
$$
k|X_0|=|Y_0|=|Y_1|=k|X_1|,
$$
where $k$ is the number of all distinct sets $X^{(m)}$'s. It follows that $|X_0|=|X_1|$ for each
basic set $X$ outside $H$.\medskip
Denote by $\rho$ the restriction to ${\cal A}$ of the one-dimensional representation of $D$ that takes $s$ and $c$ to $-1$ and
$1$, respectively. Then $\rho$ is an irreducible representation of ${\cal A}$ such that $\rho(e)=1$ and $\rho(\und{H^\#})=-1$. Moreover,
for any basic set $X$ outside $H$, we obtain by above that $\rho(\und{X}) =-|X_1|+|X_0| = 0$. In particular,
$\rho(\und{X^{-1}})=0$. Therefore,
$$
0=\rho(\und{X^{}}\,\und{X}^{-1})=\sum_{Y\in{\cal S}({\cal A})} c_{X^{}X^{-1}}^Y\rho(\und{Y})=
$$
$$
c_{X^{}X^{-1}}^e\rho(e)+c_{X^{}X^{-1}}^{H^\#}\rho(H^\#)=
c_{X^{}X^{-1}}^e-c_{X^{}X^{-1}}^{H^\#}.
$$
It follows that $|X|=c_{X^{}X^{-1}}^e=c_{X^{}X^{-1}}^{H^\#}$. Therefore, $HX = X$ for all basis sets
outside $H$, and we are done.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
\section{Proof of Theorem~\ref{170814c}}\label{231014a7}
Let $D$ be a dihedral $2$-group and $C$ its cyclic subgroup of index $2$.
Let ${\cal A}$ be an S-ring over the group $D$. Suppose that $r:=\rk({\cal A})$ is
at most $5$. For $r=2$, part (1) of Theorem~\ref{170814c} holds trivially. Let $r\ge 3$.
Then from Theorem~\ref{030414a}, it follows that the S-ring ${\cal A}$ is imprimitive; denote
by $H$ a minimal non-trivial ${\cal A}$-group. Then $\rk({\cal A}_H)=2$. Now,
if $r=3$, then ${\cal A}$ is a proper wreath product by Corollary~\ref{021014a}.
Thus, we can assume that $r=4$ or $r=5$.
\lmml{300514e}
If there is a minimal ${\cal A}$-group $L\ne H$, then statement~(2) holds.
\end{lemma}
\noindent{\bf Proof}.\ By the minimality of the groups $H$ and $L$, we have $H\cap L=e$. Therefore, at least one of them intersects~$C$ trivially. Moreover,
if $H\cap C=L\cap C=e$, then $\grp{HL}$ is an ${\cal A}$-group contained in~$C$, and we replace $H$ by a minimal ${\cal A}$-subgroup in
$\grp{HL}$. Thus, without loss of generality, we can assume that
$$
H\cap C\ne e\quad\text{and}\quad L\cap C=e.
$$
Then $L=\grp{s}$ for some involution
$s\in D\setminus C$. Moreover, $sHs=H$ by the minimality of $H$.
Thus, $HL$ is an ${\cal A}$-group and the set ${\cal S}({\cal A}_{HL})$ contains $4$ elements: $\{e\}$, $H^\#$, $\{s\}$,
and $sH^\#$. Therefore,
$$
{\cal A}_{HL}={\cal A}_H\cdot{\cal A}_L.
$$
This implies the required statement for $r=4$, and by Corollary~\ref{021014a}
also for $r=5$.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
By Lemma~\ref{300514e}, we can assume that $H$ is a unique minimal
${\cal A}$-group. If it is not contained in $C$, then statement (1) holds
by Theorem~\ref{011014a}. Thus, from now, on we also assume that $H\le C$.
Denote by $F$ the union of all basic sets of ${\cal A}$ that are not $C$-mixed.
Clearly, $H\subseteq F$.
\lmml{240913c}
$F$ is an ${\cal A}$-group. Moreover, if $r=r({\cal A}_F)+1$, then
${\cal A}$ is a proper wreath product.
\end{lemma}
\noindent{\bf Proof}.\ The second part of our statement follows from the first one and
Corollary~\ref{021014a}. To prove the first statement, denote by $U$
and $V$ the unions of all basic sets of ${\cal A}$ contained
in $C$ and $D\setminus C$, respectively. We have to prove
that $U\cup V$ is a group. Since $U$ is, obviously, an ${\cal A}$-group,
without loss of generality, we may assume that $V$ is not empty. Then
$V=U's$, where $s\in D\setminus C$ is such that $U\cap Us$ is a
subgroup of $D$. It follows that
$$
UU'\subseteq U'\quad\text{and}\quad U'U'\subseteq U.
$$
Since also $U'\subseteq UU'$, the first inclusion implies that $U'=UU'$.
Therefore, $U'$ is a union of some $U$-cosets contained in $C$. Since
the group $C$ is a cyclic $2$-group, this together with the
second inclusion implies $U'=U$. Thus, the set
$U\cup V=U\cup Us$ is a group.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
Suppose first that $F_0=C$. Then from the definition of $F$, it follows that $C$ is an
${\cal A}$-group. By Corollary~\ref{021014a}, we can assume that $r_C=\rk({\cal A}_C)$
is not equal to $r-1$. Since, obviously, $r_C\ge 2$,
we have
$$
(r,r_C)=(4,2),\ (5,2)\ \text{or}\ (5,3).
$$
In the former two cases, we are done by statement~(1) of Corollary~\ref{120614a},
whereas in the third one by statement~(2). Thus, in
what follows, we can assume that
$$
F_0<C\quad\text{and}\quad H=F\ \text{or}\ r_F=3,
$$
where $r_F=\rk({\cal A}_F)$. In particular, there are two
or three basic sets outside $F$ (notice, that they are $C$-mixed).
\lmml{041014a}
Let $X\in{\cal S}({\cal A})_{D\setminus F}$. Suppose that $X$ is rational or
$[V:H]\ge 4$, where $V=\grp{X_0}$. Then $H\le\rad(X)$.
\end{lemma}
\noindent{\bf Proof}.\ It suffices to verify that $H\le\rad(X_0)$. Indeed, then
the coefficient at $\und{X}$ in $\und{H}\,\und{X}$ is at least $|H|$.
Since it can not be larger than $|H|$, we are done.\medskip
Suppose first that $X$ is rational. Then by statement~(2) of Lemma~\ref{240913a} the
set $X_0$ is rational. Since $X_0\subseteq C\setminus H$, we conclude that
$H\le\rad(X_0)$, as required.\medskip
Let now $[V:H]\ge 4$. Then $V\cong{\mathbb Z}_{2^k}$ for some $k\ge 2$. Since $r\le 5$, there are at most two basic
rationally conjugate to $X$. Therefore, the stabilizer of $X_0$ in the group $({\mathbb Z}_{2^k})^*$,
has index at most $2$ in it. It follows that this stabilizer
contains the subgroup of all elements $x\mapsto x^{1+4m}$, $x\in {\mathbb Z}_{2^k}$,
with $m\in {\mathbb Z}_{2^k}$. By statement~(2) of Lemma~\ref{211113a}, this implies that
$\rad(X_0)\ge V^4\ge H$.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
From Lemma~\ref{041014a}, it follows that if all basic sets outside $F$ are
rational, then ${\cal A}$ is a proper wreath product. Indeed, this is obvious when $H=F$.
If $H\ne F$, this is also true, because then $F\setminus H$ is a basic set, the radical
of which equals $H$. Thus, we can assume that two of basic sets outside $F$, say $X$ and $Y$,
are rationally conjugate, and the third one (if exists) is rational. The rest of the proof is divided
into four cases below.\medskip
{\bf Case 1: $F=H$ and $r=4$.}
Using the computer package COCO, \cite{PRZ} we found exactly five and three
S-rings of rank~$4$ over the groups $D_8$ and $D_{16}$, respectively.
In both cases, only two of them have a unique minimal ${\cal A}$-group
contained in $C$ and they are proper wreath products. Thus, in what follows, we assume
that $|D|\ge 32$.\medskip
In our case, the non-trivial basic sets of ${\cal A}$ are $X$, $Y$ and $Z=H^\#$. It is easily seen that
the hypothesis of Lemma~\ref{290514b} is satisfied. Since $X$ and $Y$
are rationally conjugate, there is an algebraic isomorphism of ${\cal A}$ that takes $X$
to $Y$, and $Y$ to $X$. Therefore,
\qtnl{081014a}
\und{H}\,\und{X}=a(\und{X}+\und{Y}),
\end{equation}
where $a=|H|/2$. Consequently, $c_{ZX}^X=a-1$. On the other hand, since $X$ and $Z$ are
symmetric, we have $C_{XX}^Z=\frac{|X|}{|Z|}C_{ZX}^X$. It follows that $|Z|=2a-1$ divides
$$
|X|c_{ZX}^X=\frac{(d-2a)(a-1)}{2},
$$
where $d=|D|$. However, by Lemma~\ref{041014a}, without loss of generality,
we can assume that $|C:H|=2$. Therefore, $2a=|H|=d/4$. It follows that
$2a-1$ divides $3a(a-1)$. But $a$ being a power of $2$, must be coprime to $2a-1$.
Consequently, $2a-1$ divides $3a-3$. Since this is possible only for $a\le 2$, i.e.
when $d\le 16$, we are done.\medskip
{\bf Case 2: $F=H$ and $r=5$.} Denote by $Z$ the basic set in ${\cal S}({\cal A})_{D\setminus H}$
other than $X$ and $Y$. It is easily seen that the hypothesis of Lemma~\ref{290514b}
is satisfied. Since $X$ and $Y$ are rationally conjugate, there is an algebraic isomorphism of ${\cal A}$
that takes $X$ to $Y$, $Y$ to $X$ and leaves $Z$ fixed. Therefore, the rational closure of
${\cal A}$ is of rank~$4$. So by Lemma~\ref{041014a}, it is the
wreath product ${\cal A}_H\wr{\cal B}$, where ${\cal B}$ is isomorphic to the rational closure of ${\cal A}_{D/H}$.
Therefore, $\rk({\cal B})=3$, and hence there exists a non-trivial ${\cal B}$-group. Denote by $U$
its preimage in ${\cal A}$. Then, obviously, $H<U<D$.\medskip
Since $X$ and $Y$ are rationally conjugate, we have $X\cup Y\subseteq U$ or
$X\cup Y\subseteq D\setminus U$. However, $\rk({\cal A})=r=5$. Therefore, $Z=D\setminus U$ in the former
case, and $Z=U\setminus H$ in the latter one. In any case, $H\le\rad Z$. By Lemma~\ref{041014a},
this implies that if $Z=U\setminus H$, then
\qtnl{140215a}
\rad(X)=\rad(Y)\ge H,
\end{equation}
and ${\cal A}$ is a proper wreath product. Let now, $Z=D\setminus U$. Then $\rk({\cal A}_U)=4$, and ${\cal A}_U$ is
the wreath product ${\cal A}_H\wr{\cal A}_{U/H}$. Therefore, again \eqref{140215a} holds, and ${\cal A}$ is a proper wreath product.\medskip
{\bf Case 3: $C\not\ge F>H$.} In this case, $F=H\cup Hs$. So by Lemma~\ref{041014a},
the sets $X_0$ and $Y_0$ are orbits of an index $2$ subgroup of $\aut(C)$, unless
${\cal A}$ is a proper wreath product.
This implies that the group $\rad(X_0)=\rad(Y_0)$ has index $2$ in $H$. Therefore,
equality~\eqref{081014a} holds with $a=|H|/2$. Exactly as in Case~2, we conclude
that $2a-1$ divides
$$
|X|c_{ZX}^X=\frac{(d-4a)(a-1)}{2},
$$
where $Z=H^\#$ and $d=|D|=4|H|=8a$. Thus, $2a-1$ divides $2a(a-1)$. Contradiction.\medskip
{\bf Case 4: $C\ge F>H$.} In this case, $F=F_0<C$ by the above assumption. Therefore,
$X_0$ (and also $Y_0$) contains a generator of $C$. It follows that
$$
[\grp{X_0}:H]=[C:H]\ge [C:F][F:H]\ge 4.
$$
By Lemma~\ref{041014a}, this implies that \eqref{140215a} holds. Since also
$F\setminus H$ is the basic set and $H=\rad(F\setminus H)$, the group $H$ is contained
in the radical of every basic set outside~$H$. Thus, ${\cal A}$ is a proper wreath product.\hfill\vrule height .9ex width .8ex depth -.1ex\medskip
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,595
|
Q: Gradle build not working: Execution failed for task ':MyApp:compileDebug' I'm trying to migrate a project to Gradle, but I get an exception raised everytime I run the command
gradle build --stacktrace
I'm using Gradle 1.6. Here's the stack trace for the error:
:MyApp:compileDebug FAILED
FAILURE: Build failed with an exception.
* What went wrong:
Execution failed for task ':MyApp:compileDebug'.
> Compilation failed; see the compiler error output for details.
* Try:
Run with --info or --debug option to get more log output.
* Exception is:
org.gradle.api.tasks.TaskExecutionException: Execution failed for task ':MyApp:compileDebug'.
at org.gradle.api.internal.tasks.execution.ExecuteActionsTaskExecuter.executeActions(ExecuteActionsTaskExecuter.java:69)
at org.gradle.api.internal.tasks.execution.ExecuteActionsTaskExecuter.execute(ExecuteActionsTaskExecuter.java:46)
at org.gradle.api.internal.tasks.execution.PostExecutionAnalysisTaskExecuter.execute(PostExecutionAnalysisTaskExecuter.java:35)
at org.gradle.api.internal.changedetection.state.CacheLockReleasingTaskExecuter$1.run(CacheLockReleasingTaskExecuter.java:35)
at org.gradle.internal.Factories$1.create(Factories.java:22)
at org.gradle.cache.internal.DefaultCacheAccess.longRunningOperation(DefaultCacheAccess.java:179)
at org.gradle.cache.internal.DefaultCacheAccess.longRunningOperation(DefaultCacheAccess.java:232)
at org.gradle.cache.internal.DefaultPersistentDirectoryStore.longRunningOperation(DefaultPersistentDirectoryStore.java:142)
at org.gradle.api.internal.changedetection.state.DefaultTaskArtifactStateCacheAccess.longRunningOperation(DefaultTaskArtifactStateCacheAccess.java:83)
at org.gradle.api.internal.changedetection.state.CacheLockReleasingTaskExecuter.execute(CacheLockReleasingTaskExecuter.java:33)
at org.gradle.api.internal.tasks.execution.SkipUpToDateTaskExecuter.execute(SkipUpToDateTaskExecuter.java:58)
at org.gradle.api.internal.tasks.execution.ContextualisingTaskExecuter.execute(ContextualisingTaskExecuter.java:34)
at org.gradle.api.internal.changedetection.state.CacheLockAcquiringTaskExecuter$1.run(CacheLockAcquiringTaskExecuter.java:39)
at org.gradle.internal.Factories$1.create(Factories.java:22)
at org.gradle.cache.internal.DefaultCacheAccess.useCache(DefaultCacheAccess.java:124)
at org.gradle.cache.internal.DefaultCacheAccess.useCache(DefaultCacheAccess.java:112)
at org.gradle.cache.internal.DefaultPersistentDirectoryStore.useCache(DefaultPersistentDirectoryStore.java:134)
at org.gradle.api.internal.changedetection.state.DefaultTaskArtifactStateCacheAccess.useCache(DefaultTaskArtifactStateCacheAccess.java:79)
at org.gradle.api.internal.changedetection.state.CacheLockAcquiringTaskExecuter.execute(CacheLockAcquiringTaskExecuter.java:37)
at org.gradle.api.internal.tasks.execution.ValidatingTaskExecuter.execute(ValidatingTaskExecuter.java:57)
at org.gradle.api.internal.tasks.execution.SkipEmptySourceFilesTaskExecuter.execute(SkipEmptySourceFilesTaskExecuter.java:41)
at org.gradle.api.internal.tasks.execution.SkipTaskWithNoActionsExecuter.execute(SkipTaskWithNoActionsExecuter.java:51)
at org.gradle.api.internal.tasks.execution.SkipOnlyIfTaskExecuter.execute(SkipOnlyIfTaskExecuter.java:52)
at org.gradle.api.internal.tasks.execution.ExecuteAtMostOnceTaskExecuter.execute(ExecuteAtMostOnceTaskExecuter.java:42)
at org.gradle.api.internal.AbstractTask.executeWithoutThrowingTaskFailure(AbstractTask.java:282)
at org.gradle.execution.taskgraph.DefaultTaskPlanExecutor.executeTask(DefaultTaskPlanExecutor.java:48)
at org.gradle.execution.taskgraph.DefaultTaskPlanExecutor.processTask(DefaultTaskPlanExecutor.java:34)
at org.gradle.execution.taskgraph.DefaultTaskPlanExecutor.process(DefaultTaskPlanExecutor.java:27)
at org.gradle.execution.taskgraph.DefaultTaskGraphExecuter.execute(DefaultTaskGraphExecuter.java:89)
at org.gradle.execution.SelectedTaskExecutionAction.execute(SelectedTaskExecutionAction.java:29)
at org.gradle.execution.DefaultBuildExecuter.execute(DefaultBuildExecuter.java:61)
at org.gradle.execution.DefaultBuildExecuter.access$200(DefaultBuildExecuter.java:23)
at org.gradle.execution.DefaultBuildExecuter$2.proceed(DefaultBuildExecuter.java:67)
at org.gradle.api.internal.changedetection.state.TaskCacheLockHandlingBuildExecuter$1.run(TaskCacheLockHandlingBuildExecuter.java:31)
at org.gradle.internal.Factories$1.create(Factories.java:22)
at org.gradle.cache.internal.DefaultCacheAccess.useCache(DefaultCacheAccess.java:124)
at org.gradle.cache.internal.DefaultCacheAccess.useCache(DefaultCacheAccess.java:112)
at org.gradle.cache.internal.DefaultPersistentDirectoryStore.useCache(DefaultPersistentDirectoryStore.java:134)
at org.gradle.api.internal.changedetection.state.DefaultTaskArtifactStateCacheAccess.useCache(DefaultTaskArtifactStateCacheAccess.java:79)
at org.gradle.api.internal.changedetection.state.TaskCacheLockHandlingBuildExecuter.execute(TaskCacheLockHandlingBuildExecuter.java:29)
at org.gradle.execution.DefaultBuildExecuter.execute(DefaultBuildExecuter.java:61)
at org.gradle.execution.DefaultBuildExecuter.access$200(DefaultBuildExecuter.java:23)
at org.gradle.execution.DefaultBuildExecuter$2.proceed(DefaultBuildExecuter.java:67)
at org.gradle.execution.DryRunBuildExecutionAction.execute(DryRunBuildExecutionAction.java:32)
at org.gradle.execution.DefaultBuildExecuter.execute(DefaultBuildExecuter.java:61)
at org.gradle.execution.DefaultBuildExecuter.execute(DefaultBuildExecuter.java:54)
at org.gradle.initialization.DefaultGradleLauncher.doBuildStages(DefaultGradleLauncher.java:166)
at org.gradle.initialization.DefaultGradleLauncher.doBuild(DefaultGradleLauncher.java:113)
at org.gradle.initialization.DefaultGradleLauncher.run(DefaultGradleLauncher.java:81)
at org.gradle.launcher.exec.InProcessBuildActionExecuter$DefaultBuildController.run(InProcessBuildActionExecuter.java:64)
at org.gradle.launcher.cli.ExecuteBuildAction.run(ExecuteBuildAction.java:33)
at org.gradle.launcher.cli.ExecuteBuildAction.run(ExecuteBuildAction.java:24)
at org.gradle.launcher.exec.InProcessBuildActionExecuter.execute(InProcessBuildActionExecuter.java:35)
at org.gradle.launcher.exec.InProcessBuildActionExecuter.execute(InProcessBuildActionExecuter.java:26)
at org.gradle.launcher.cli.RunBuildAction.run(RunBuildAction.java:50)
at org.gradle.api.internal.Actions$RunnableActionAdapter.execute(Actions.java:171)
at org.gradle.launcher.cli.CommandLineActionFactory$ParseAndBuildAction.execute(CommandLineActionFactory.java:201)
at org.gradle.launcher.cli.CommandLineActionFactory$ParseAndBuildAction.execute(CommandLineActionFactory.java:174)
at org.gradle.launcher.cli.CommandLineActionFactory$WithLogging.execute(CommandLineActionFactory.java:170)
at org.gradle.launcher.cli.CommandLineActionFactory$WithLogging.execute(CommandLineActionFactory.java:139)
at org.gradle.launcher.cli.ExceptionReportingAction.execute(ExceptionReportingAction.java:33)
at org.gradle.launcher.cli.ExceptionReportingAction.execute(ExceptionReportingAction.java:22)
at org.gradle.launcher.Main.doAction(Main.java:48)
at org.gradle.launcher.bootstrap.EntryPoint.run(EntryPoint.java:45)
at org.gradle.launcher.Main.main(Main.java:39)
at org.gradle.launcher.bootstrap.ProcessBootstrap.runNoExit(ProcessBootstrap.java:50)
at org.gradle.launcher.bootstrap.ProcessBootstrap.run(ProcessBootstrap.java:32)
at org.gradle.launcher.GradleMain.main(GradleMain.java:26)
Caused by: org.gradle.api.internal.tasks.compile.CompilationFailedException: Compilation failed; see the compiler error output for details.
at org.gradle.api.internal.tasks.compile.jdk6.Jdk6JavaCompiler.execute(Jdk6JavaCompiler.java:42)
at org.gradle.api.internal.tasks.compile.jdk6.Jdk6JavaCompiler.execute(Jdk6JavaCompiler.java:33)
at org.gradle.api.internal.tasks.compile.NormalizingJavaCompiler.delegateAndHandleErrors(NormalizingJavaCompiler.java:95)
at org.gradle.api.internal.tasks.compile.NormalizingJavaCompiler.execute(NormalizingJavaCompiler.java:48)
at org.gradle.api.internal.tasks.compile.NormalizingJavaCompiler.execute(NormalizingJavaCompiler.java:34)
at org.gradle.api.internal.tasks.compile.DelegatingJavaCompiler.execute(DelegatingJavaCompiler.java:29)
at org.gradle.api.internal.tasks.compile.DelegatingJavaCompiler.execute(DelegatingJavaCompiler.java:20)
at org.gradle.api.internal.tasks.compile.IncrementalJavaCompilerSupport.execute(IncrementalJavaCompilerSupport.java:33)
at org.gradle.api.internal.tasks.compile.IncrementalJavaCompilerSupport.execute(IncrementalJavaCompilerSupport.java:24)
at org.gradle.api.tasks.compile.Compile.compile(Compile.java:68)
at org.gradle.api.internal.BeanDynamicObject$MetaClassAdapter.invokeMethod(BeanDynamicObject.java:216)
at org.gradle.api.internal.BeanDynamicObject.invokeMethod(BeanDynamicObject.java:122)
at org.gradle.api.internal.CompositeDynamicObject.invokeMethod(CompositeDynamicObject.java:147)
at org.gradle.api.tasks.compile.JavaCompile_Decorated.invokeMethod(Unknown Source)
at org.gradle.util.ReflectionUtil.invoke(ReflectionUtil.groovy:23)
at org.gradle.api.internal.project.taskfactory.AnnotationProcessingTaskFactory$StandardTaskAction.doExecute(AnnotationProcessingTaskFactory.java:217)
at org.gradle.api.internal.project.taskfactory.AnnotationProcessingTaskFactory$StandardTaskAction.execute(AnnotationProcessingTaskFactory.java:210)
at org.gradle.api.internal.project.taskfactory.AnnotationProcessingTaskFactory$StandardTaskAction.execute(AnnotationProcessingTaskFactory.java:199)
at org.gradle.api.internal.AbstractTask$TaskActionWrapper.execute(AbstractTask.java:526)
at org.gradle.api.internal.AbstractTask$TaskActionWrapper.execute(AbstractTask.java:509)
at org.gradle.api.internal.tasks.execution.ExecuteActionsTaskExecuter.executeAction(ExecuteActionsTaskExecuter.java:80)
at org.gradle.api.internal.tasks.execution.ExecuteActionsTaskExecuter.executeActions(ExecuteActionsTaskExecuter.java:61)
... 67 more
I have the following structure defined in settings.gradle:
include ':MyApp'
include ':Libraries:LibProject1'
include ':Libraries:LibProject2'
include ':Libraries:LibProject3'
include ':Libraries:LibProject4'
And my project's directories are like this:
- My Repository
-- settings.gradle
- My App
-- build.gradle
- Libraries
- LibProject1
-- build.gradle
- LibProject2
-- build.gradle
- LibProject3
-- build.gradle
- LibProject4
-- build.gradle
My App > build.gradle file is like this:
buildscript {
repositories {
mavenCentral()
}
dependencies {
classpath 'com.android.tools.build:gradle:0.5.+'
}
}
apply plugin: 'android'
dependencies {
compile fileTree(dir: 'libs/main', include: '*.jar')
compile project(':Libraries:LibProject1')
compile project(':Libraries:LibProject2')
compile project(':Libraries:LibProject3')
compile project(':Libraries:LibProject4')
}
android {
compileSdkVersion "Google Inc.:Google APIs:19"
buildToolsVersion "19.0.1"
sourceSets {
main {
manifest.srcFile 'AndroidManifest.xml'
java.srcDirs = ['src/main/java', 'src-gen/main/java', 'generated']
resources.srcDirs = ['src']
aidl.srcDirs = ['src']
renderscript.srcDirs = ['src']
res.srcDirs = ['res']
assets.srcDirs = ['assets']
}
}
}
And all LibProject# > build.gradle files are like this:
buildscript {
repositories {
mavenCentral()
}
dependencies {
classpath 'com.android.tools.build:gradle:0.5.+'
}
}
apply plugin: 'android-library'
android {
compileSdkVersion 19
buildToolsVersion "19.0.1"
sourceSets {
main {
manifest.srcFile 'AndroidManifest.xml'
java.srcDirs = ['src']
resources.srcDirs = ['src']
aidl.srcDirs = ['src']
renderscript.srcDirs = ['src']
res.srcDirs = ['res']
assets.srcDirs = ['assets']
}
}
}
I've already tried several possible solutions found in other threads, but nothing seems to work.
I've also compared my project's structure and all its .gradle files to another project that's successfully building, and I couldn't find anything missing.
Unlike other problems I've found in related threads, the stack trace here doesn't seem to provide anything specific on why the exception is raised. Any ideas?
A: If you don't use Android Annotations but still cannot see what the reason is, you can actually do what the message says (check the compiler output) by:
*
*going to the build tab
*
*Switch output to compiler output
A: I finally found out what the problem was. The project uses Android Annotations, and I had included its generated sources folder in my build.gradle file.
When Gradle tried to generate the sources again, it would fail because of the duplicated classes.
Here's the correct build.gradle:
buildscript {
repositories {
mavenCentral()
}
dependencies {
classpath 'com.android.tools.build:gradle:0.5.+'
}
}
apply plugin: 'android'
dependencies {
compile fileTree(dir: 'libs/main', include: '*.jar')
compile project(':Libraries:LibProject1')
compile project(':Libraries:LibProject2')
compile project(':Libraries:LibProject3')
compile project(':Libraries:LibProject4')
}
android {
compileSdkVersion "Google Inc.:Google APIs:19"
buildToolsVersion "19.0.1"
sourceSets {
main {
manifest.srcFile 'AndroidManifest.xml'
java.srcDirs = ['src/main/java', 'src-gen/main/java']
resources.srcDirs = ['src']
aidl.srcDirs = ['src']
renderscript.srcDirs = ['src']
res.srcDirs = ['res']
assets.srcDirs = ['assets']
}
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,793
|
{"url":"https:\/\/tex.stackexchange.com\/questions\/376641\/intersection-point-between-line-and-arc","text":"# Intersection point between line and arc\n\nI am using tikzpicture for a long time now and making intersections between lines nicely. Even if I make an intersection between an circle and line works nicely and I also find some threads about that topic online. However, I did not manage to make the intersection between an arc and line. I checked this here (How to Intersect an arc with a line without calculating the angle) but they always work with circles and not arcs. The intersection I try to make is as follows:\n\n\\usepackage[utf8]{inputenc}\n\\usepackage{tikz}\n\\usetikzlibrary{intersections, calc}\n\\begin{document}\n\\begin{figure}[h]\n\\begin{tikzpicture}\n\n% Points\n\\coordinate (a) at (0,0);\n\\coordinate (b) at (0,4);\n\\coordinate (c) at (0,6);\n\\coordinate (d) at (4,0);\n\\coordinate (e) at (6,0);\n\n% Point for analyse line\n\\coordinate (f) at (7,7);\n\n% Lines\n\\draw [-] (b) -- node [rotate=90, above] {Symmetry plane} (c);\n\\draw [-] (d) -- node [below] {Symmetry plane} (e);\n\n% Arcs\n\\draw [-, name=R1] (d) arc [radius=4, start angle=0, end angle=90];\n\\draw [-] (e) arc [radius=6, start angle=0, end angle=90];\n\n\\path[name=L1] (a) -- (f);\n\n% Intersection point (NOT WORKING)\n\\path[name intersections={of=L1 and R1, by=B}];\n\n\\end{tikzpicture}\n\\end{figure}\n\\end{document}\n\nThe problem I have here is, that the intersection can not be calculated. Maybe it is not working with arcs but I guess it should work. Using a full circle as a path is working but then the figure is getting bigger based on the circle (hope you get the point). Any idea suggestion is welcomed.\n\n\u2022 Sorry for opening that question. I don't know why I was missing the correct syntax. Thanks Heiko for the fast answer. \u2013\u00a0Tobi Jun 24 '17 at 20:44\n\nThe option is called name path instead of name for naming a path to use it for the intersection:\n\n\\draw [-, name path=R1]\n\\path[name path=L1]\n\u2022 Oh my good. What a stupid question I asked. In my other examples I have always [name path=FOO]. Thank you very much. \u2013\u00a0Tobi Jun 24 '17 at 20:43","date":"2020-02-28 01:08:54","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8291153907775879, \"perplexity\": 1107.2475656236218}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2020-10\/segments\/1581875146907.86\/warc\/CC-MAIN-20200227221724-20200228011724-00293.warc.gz\"}"}
| null | null |
\section{Introduction}
As the techniques for analyzing large data sets continue to grow,
diverse quantitative sciences -- including computational biology,
observation astronomy, and high energy physics -- are becoming
increasingly data driven. Moreover, modern business decision making
critically depends on quantitative analyses such as community
detection and consumer behavior prediction. Consequently, statistical
learning has become an indispensable tool for modern data
analysis. Data acquired from various experiments are usually organized
into an $n\times m$ matrix, where the number $n$ of features typically
far exceeds the number $m$ of samples. In this view, the $m$ samples,
corresponding to the columns of the data matrix, are naturally
interpreted as points in a high-dimensional feature space
$\mathbb{R}^{n}$. Traditional statistical modeling approaches often
lose their power when the feature dimension is high.
To ameliorate this problem, Lafferty and Lebanon
proposed a multinomial interpretation of non-negative feature vectors
and an accompanying transformation of the multinomial simplex to a
hypersphere, demonstrating that using the heat kernel on this
hypersphere may improve the performance of kernel support vector
machine (SVM)
\cite{Lafferty:2005uy,Hastie:2013fd,Evgeniou:2001hc,Boser:1992fz,Cortes:1995fs,
Freund:1999dh,Guyon:1993ub}. Despite the interest that this idea has
attracted, only approximate heat kernel is known to date. We here
present an exact form of the heat kernel on a hypersphere of arbitrary
dimension and study its performance in kernel SVM classifications of
text mining, genomic, and stock price data sets.
To date, sparse data clouds have been extensively analyzed in the flat
Euclidean space endowed with the $L^{2}$-norm using traditional
statistical learning algorithms, including KMeans, hierarchical
clustering, SVM, and neural network \cite{Hastie:2013fd,
Kaufman:2009dk,
Evgeniou:2001hc,Boser:1992fz,Cortes:1995fs,Freund:1999dh,Guyon:1993ub};
however, the flat geometry of the Euclidean space often poses severe
challenges in clustering and classification problems when the data
clouds take non-trivial geometric shapes or class labels are spatially
mixed. Manifold learning and kernel-based embedding methods attempt to
address these challenges by estimating the intrinsic geometry of a
putative submanifold from which the data points were sampled and by
embedding the data into an abstract Hilbert space using a nonlinear
map implicitly induced by the chosen kernel, respectively
\cite{Anonymous:U64WGXbS,Aronszajn:1950bl,Paulsen:2016wz}. The
geometry of these curved spaces may then provide novel information
about the structure and organization of original data points.
Heat equation on the data submanifold or transformed feature space
offers an especially attractive idea of measuring similarity between
data points by using the physical model of diffusion of relatedness
(``heat'') on curved space, where the diffusion process is driven
by the intrinsic geometry of the underlying space. Even though such
diffusion process has been successfully approximated as a discrete-time,
discrete-space random walk on complex networks, its continuous formulation
is rarely analytically solvable and usually requires complicated asymptotic
expansion techniques from differential geometry \cite{Berger:wXGLCovN}.
An analytic solution, if available, would thus provide a valuable
opportunity for comparing its performance with approximate asymptotic
solutions and rigorously testing the power of heat diffusion for geometric
data analysis.
Given that a Riemannian manifold of dimension $d$ is locally
homeomorphic to $\mathbb{R}^{d}$, and that the heat kernel is a
solution to the heat equation with a point source initial condition,
one may assume in the short diffusion time limit ($t\downarrow 0$) that most
of the heat is localized within the vicinity of the initial point and
that the heat kernel on a Riemannian manifold locally resembles the
Euclidean heat kernel. This idea forms the motivation behind the
parametrix expansion, where the heat kernel in curved space is
approximated as a product of the Euclidean heat kernel in normal
coordinates and an asymptotic series involving the diffusion time and
normal coordinates. In particular, for a unit hypersphere, the
parametrix expansion in the limit $t\downarrow 0$ involves a modified
Euclidean heat kernel with the Euclidean distance $\left\Vert
\mathbf{x}\right\Vert $ replaced by the geodesic arc length
$\theta$. Computing this parametrix expansion is, however,
technically challenging; even when the computation is tractable,
applying the approximation directly to high-dimensional clustering and
classification problems may have limitations. For example, in order to
be able to group samples robustly, one needs the diffusion time $t$ to
be not too small; otherwise, the sample relatedness may be highly
localized and decay too fast away from each sample. Moreover, the
leading order term in the asymptotic series is an increasing function
of $\theta$ and diverges as $\theta$ approaches $\pi$, yielding an
incorrect conclusion that two antipodal points are highly similar. For
these reasons, the machine learning community has been using only the
Euclidean diffusion term without the asymptotic series correction; how
this resulting kernel, called the parametrix kernel
\cite{Lafferty:2005uy}, compares with the exact heat kernel on a
hypersphere remains an outstanding question, which is addressed in
this paper.
Analytically solving the diffusion equation on a Riemannian manifold
is challenging \cite{Hsu:2002vq,Varopoulos:1987vx,Berger:wXGLCovN}.
Unlike the discrete analogues \textendash{} such as spectral clustering
\cite{Ng:2002tj} and diffusion map \cite{Coifman:2006cy}, where
eigenvectors of a finite dimensional matrix can be easily obtained
\textendash{} the eigenfunctions of the Laplace operator on a Riemannian
manifold are usually intractable. Fortunately, the high degree of
symmetry of a hypersphere allows the explicit construction of eigenfunctions,
called hyperspherical harmonics, via the projection of homogeneous
polynomials \cite{Atkinson:2012tz,Wen:1985fm}. The exact heat kernel
is then obtained as a convergent power series in these eigenfunctions.
In this paper, we compare the analytic behavior of this exact heat
kernel with that of the parametrix kernel and analyze their performance
in classification.
\section{Results}
The heat kernel is the fundamental solution to the heat equation
$(\partial_{t}-\Delta_{x})u(x,t)=0$ with an initial point source
\cite{GOLDBART:2016uz}, where $\Delta_{x}$ is the Laplace operator;
the amount of heat emanating from the source
that has diffused to a neighborhood during time $t>0$ is
used to measure the similarity between the source and proximal
points. The heat conduction depends on the geometry of feature space,
and the main idea behind the application of hyperspherical geometry to
data analysis relies on the following map from a
non-negative feature space to a unit hypersphere:
\begin{defn}
A hyperspherical map $\varphi:\mathbb{R}_{\ge0}^{n}\setminus\{0\}\rightarrow S^{n-1}$
maps a vector $\mathbf{x}$, with $x_{i}\ge0$ and $\sum_{i=1}^{n}x_{i}>0$,
to a unit vector $\hat{x}\in S^{n-1}$ where $(\hat{x})_{i}\equiv\sqrt{x_{i}/\sum_{j=1}^{n}x_{j}}$.
\end{defn}
We will henceforth denote the image of a feature vector $\mathbf{x}$
under the hyperspherical map as $\hat{x}$. The notion of
neighborhood requires a well-defined measurement of distance on the
hypersphere, which is naturally the great arc length \textendash{} the
geodesic on a hypersphere. Both parametrix approximation and exact
solution employ the great arc length, which is related to the
following definition of cosine similarity:
\begin{defn}
The generic cosine similarity between two feature vectors
$\mathbf{x},\mathbf{y}\in\mathbb{R}^{n}\setminus\{0\}$ is
\[
\cos\theta\equiv\frac{\mathbf{x}\cdot\mathbf{y}}{\left\Vert
\mathbf{x}\right\Vert \left\Vert \mathbf{y}\right\Vert },
\]
where $\left\Vert \cdot\right\Vert $ is the Euclidean $L^{2}$-norm,
and $\theta\in[0,\pi] $ is the great arc length on $S^{n-1}$. For
unit vectors $\hat{x}=\varphi(\mathbf{x})$ and
$\hat{y}=\varphi(\mathbf{y})$ obtained from non-negative feature
vectors $\mathbf{x},\mathbf{y}\in\mathbb{R}_{\ge0}^{n}\setminus\{0\}$
via the hyperspherical map, the cosine similarity reduces
to the dot product $\cos\theta=\hat{x}\cdot\hat{y}$; the
non-negativity of $\bf x$ and $\bf y$ guarantees that
$\theta\in[0,\pi/2]$ in this case.
\end{defn}
\subsection{Parametrix expansion}
The parametrix kernel $K^{{\rm prx}}$ previously used in the literature
is just a Gaussian RBF function with $\theta=\arccos\hat{x}\cdot\hat{y}$
as the radial distance \cite{Lafferty:2005uy}:
\begin{defn}
The parametrix kernel is a non-negative function
\[
K^{{\rm prx}}(\hat{x},\hat{y};t)={\rm e}^{-\frac{\arccos^{2}\hat{x}\cdot\hat{y}}{4t}}={\rm e}^{-\frac{\theta^{2}}{4t}},
\]
defined for $t>0$ and attaining global maximum $1$ at $\theta=0$.
\end{defn}
\noindent Note that this kernel is assumed to be restricted to the positive orthant.
The normalization factor $(4\pi t)^{-\frac{n-1}{2}}$ is
numerically unstable as $t\downarrow0$ and complicates hyperparameter
tuning; as a global scaling factor of the kernel can be absorbed into
the misclassification $C$-parameter in SVM, this overall normalization
term is ignored in this paper. Importantly, the parametrix kernel
$K^{{\rm prx}}$ is merely the Gaussian multiplicative factor without
any asymptotic expansion terms in the full parametrix expansion
$G^{{\rm prx}}$ of the heat kernel on a hypersphere
\cite{Lafferty:2005uy,Berger:wXGLCovN}, as described below.
The Laplace operator on manifold $\mathcal M$ equiped with a
Riemannian metric
$g_{\mu\nu}$ acts on a function $f$ that depends only on the geodesic
distance $r$ from a fixed point as
\begin{equation}\label{eq:radialLaplacian}
\Delta f(r) =f''(r)+\left(\log\sqrt{g}\right)'f'(r),
\end{equation}
where $g\equiv \det(g_{\mu\nu})$ and $'$ denotes the radial
derivative. Due to the nonvanishing metric derivative in
Equation~\ref{eq:radialLaplacian}, the canonical diffusion function
\begin{equation}
G(r,t)=\left(\frac{1}{4\pi t}\right)^{\frac{d}{2}}\exp\left(-\frac{r^{2}}{4t}\right) \label{eq:gaussianrbf}
\end{equation}
does not satisfy the heat equation; that is,
$(\Delta-\partial_t)G(r,t) \neq 0$ (Supplementary Material, Section
S2). For sufficiently small time $t$ and geodesic distance $r$, the
parametrix expansion of the heat kernel on a full hypersphere proposes an approximate
solution
\[
K_p(r,t)=G(r,t)\left(u_{0}(r)+u_{1}(r)t+u_{2}(r)t^{2}+\cdots + u_p(r) t^p\right),
\]
where the functions $u_i$ should be found such that $K_p$ satisfies
the heat equation to order $t^{p-d/2}$, which is small for $t\ll1$ and
$p>d/2$; more precisely, we seek $u_i$ such that
\begin{equation}
(\Delta - \partial_t) K_p = G \,t^p\, \Delta u_p. \label{eq:parametrix}
\end{equation}
Taking the time derivative of $K_p$ yields
\[
\partial_{t}K_p=G\cdot\left[\left(-\frac{d}{2t}+\frac{r^{2}}{4t^{2}}\right)\left(u_{0}+u_{1}t+u_{2}t^{2}+\cdots+u_pt^p\right)+\left(u_{1}+2u_{2}t+\cdots+pu_pt^{p-1}\right)\right],
\]
while the Laplacian of $K_p$ is
\[
\Delta K_p=\left(u_{0}+u_{1}t+\cdots+u_pt^p\right)\Delta G+G\Delta\left(u_{0}+u_{1}t+\cdots+u_pt^p\right)+2G'\left(u_{0}+u_{1}t+\cdots+u_pt^p\right)'.
\]
One can easily compute
\[
\Delta G=\left[\left(-\frac{1}{2t}+\frac{r^{2}}{4t^{2}}\right)-\frac{r}{2t}(\log\sqrt{g})'\right]G
\]
and
\[
G'\left(u_{0}+u_{1}t+\cdots\right)'=-\frac{r}{2t}\left(u_{0}'+u_{1}'t+\cdots\right)G.
\]
The left-hand side of Equation~\ref{eq:parametrix} is thus equal to
$G$ multiplied by
\begin{align*}
\left(u_{0}+\cdots+u_pt^p\right)\left[-\frac{r}{2t}(\log\sqrt{g})'+\frac{d-1}{2t}\right]+
\Delta\left(u_{0}+\cdots+u_pt^p\right)+&\\
-\frac{r}{t}\left(u_{0}'+\cdots+u_p't^p\right)-\left(u_{1}+2u_{2}t+\cdots+pu_pt^{p-1}\right)&,
\end{align*}
and we need to solve for $u_i$ such that all the coefficients of
$t^q$ in this expression, for $q<p$, vanish.
For $q=-1$, we need to solve
\[
u_{0}\frac{r}{2}\left[-(\log\sqrt{g})'+\frac{d-1}{r}\right]=ru'_{0} \ ,
\]
or equivalently,
\[
\left(\log u_{0}\right)'=-\frac{1}{2}(\log\sqrt{g})'+\frac{d-1}{2r}.
\]
Integrating with respect to $r$ yields
\[
\log u_{0}=-\frac{1}{2}\left[\log\sqrt{g}-(d-1)\log r\right]+{\rm const.},
\]
where we implicitly take only the radial part of $\log \sqrt{g}$.
Thus, we get
\[
u_{0}={\rm const.}\times\left(\frac{\sqrt{g}}{r^{d-1}}\right)^{-\frac{1}{2}} \propto \left(\frac{\sin r}{r}\right)^{-\frac{d-1}{2}}
\]
as the zeroth-order term in the parametrix expansion.
Using this expression of $u_{0}$, the remaining terms become
\[
r\left[\left(u_{1}+u_{2}t+\cdots\right)(\log u_{0})'-\left(u_{1}'+u_{2}'t+\cdots\right)\right]+
\]
\[
+\left(\Delta u_{0}+t\Delta u_{1}+\cdots\right)-\left(u_{1}+2u_{2}t+\cdots\right),
\]
and we obtain the recursion relation
\[
u_{k+1}(\log u_{0})'-u_{k+1}'=-\frac{\Delta u_{k}-(k+1)u_{k+1}}{r}.
\]
Algebraic manipulations show that
\[
(\log r^{k+1}-\log u_{0}+\log u_{k+1})'u_{k+1}=r^{-1}\Delta u_{k}\, ,
\]
from which we get
\[
\left(\frac{u_{k+1}r^{k+1}}{u_{0}}\right)'=r^{(k+1)-1}u_{0}^{-1}\Delta
u_{k}.
\]
Integrating this equation and rearranging terms, we finally get
\begin{equation}
u_{k+1}=r^{-(k+1)}u_{0}\int_{0}^{r}d\tilde{r}\:\tilde{r}^{k}u_{0}^{-1}\Delta u_{k}. \label{eq:u_k+1}
\end{equation}
Setting $k=0$ in this recursion equation, we find the second correction term to be
\begin{eqnarray*}
u_{1} & = & \frac{u_{0}}{r}\int_{0}^{r}d\tilde{r}\:u_{0}^{-1}\Delta u_{0}\\
& = & \frac{u_{0}}{r}\int_{0}^{r}d\tilde{r}\:u_{0}^{-1}\left(u_{0}''+u_{0}'(\log\sqrt{g})'\right).
\end{eqnarray*}
From our previously obtained solution for $u_0$, we find
\[
u_{0}'=\frac{1}{2}\left(\frac{d-1}{r}-\frac{g'}{2g}\right)u_{0}.
\]
and
$$
u_{0}'' = \frac{1}{4}\left[\frac{(d-1)(d-3)}{r^{2}}-\frac{g'(d-1)}{gr}-\frac{g''}{g}+\frac{5}{4}\left(\frac{g'}{g}\right)^{2}\right]u_{0}.
$$
Substituting these expressions into the recursion relation for $u_1$ yields
$$
u_{1} = \frac{u_{0}}{4r}\int_{0}^{r}dr\left[\frac{(d-1)(d-3)}{r^{2}}-\frac{g''}{g}+\frac{3}{4}\left(\frac{g'}{g}\right)^{2}\right].
$$
For the hypersphere $S^{d}$, where $d \equiv n-1$ and $g={\rm const.}\times\sin^{2(d-1)}r$,
we have
\[
\frac{g'}{g}=\frac{2(d-1)}{\tan r}
\]
and
\[
\frac{g''}{g}=2(d-1)\left(\frac{2d-3}{\tan^{2}r}-1\right).
\]
Thus,
\begin{eqnarray}
u_{1}&=&\frac{u_{0}}{4r}\int_{0}^{r}d\tilde{r}\left[\frac{(d-1)(d-3)}{\tilde{r}^{2}}-
(d-1) \left( \frac{d-3}{\tan^2 \tilde{r}} -2\right)
\right]\nonumber\\
&=& \frac{u_0 (d-1)}{4r^2} \left[ 3-d + (d-1) r^2 + (d-3) r \cot r\right].\label{eq: para_u1}
\end{eqnarray}
Notice that $u_1(r)=0$ when $d=1$ and
$u_1(r)=u_0(r)$ when $d=3$. For $d=2$, $u_1/u_0$ is an increasing function in
$r$ and diverges to $\infty$ at $r=\pi$. By contrast, for $d>3$, $u_1/u_0$ is
a decreasing function in $r$ and diverges to $-\infty$ at
$r=\pi$; $u_1/u_0$ is relatively
constant for $r<\pi$ and starts to decrease rapidly only near
$\pi$.
Therefore, the first order correction is not able to remove the
unphysical behavior near $r=0$ in high dimensions where, according to
the first order parametrix kernel, the surrounding area is hotter than the heat
source.
Next, we apply Equation \ref{eq:u_k+1} again to obtain $u_2$ as
\begin{eqnarray*}
u_{2} & = & \frac{u_{0}}{r^{2}}\int_{0}^{r}d\tilde{r}\:\tilde{r}u_{0}^{-1}\Delta u_{1}\\
& = & \frac{u_{0}}{r^{2}}\int_{0}^{r}d\tilde{r}\:\tilde{r}u_{0}^{-1}\left(u_{1}''+u_{1}'(\log\sqrt{g})'\right).
\end{eqnarray*}
After some cumbersome algebraic manipulations, we find
\begin{eqnarray}
\frac{u_{2}}{u_0}&=&\frac{d-1}{32}\left[(d-3)^{3}+\frac{(d-3)(d-5)(d-7)}{r^{4}}-\frac{(d-3)^{2}(d-5)}{r^{3}\tan r}\right.\nonumber\\
&&\left.+\frac{2(d-1)^{2}(d-3)}{r\tan r}+\frac{(d+1)(d-3)(d-5)}{r^{2}\sin r}\right].\label{eq: para_u2}
\end{eqnarray}
\noindent Again, $d=1$ and $d=3$ are special dimensions, where $u_{2}(r)=0$ for $d=1$,
and $u_{2}(r)=u_{0}/2$ for $d=3$; for other dimensions, $u_{2}(r)$ is
singular at both $r=0$ and $\pi$.
Note that on $S^1$, the metric in geodesic polar coordinate is $g_{11} = 1$, so all parametrix expansion
coefficients $u_k(r)$ must vanish identically, as we have explicitly shown above.
Thus, the full $G^{{\rm prx}}$ defined on a hypersphere, where the
geodesic distance $r$ is just the arc length $\theta$, suffers from
numerous problems. The zeroth order correction term $u_0 =
(\sin\theta/\theta)^{-\frac{n-2}{2}}$ diverges at $\theta=\pi$; this
behavior is not a major problem if $\theta$ is restricted to the range
$[0,\frac{\pi}{2}]$. Moreover, $G^{{\rm prx}}$ is also unphysical as
$\theta\downarrow0$ when $(n-2)t>3$; this condition on dimension and
time is obtained by expanding ${\rm
e}^{-\theta^{2}/4t}=1-\frac{\theta^{2}}{4t}+{\mathcal
O}(\theta^{4})$ and
$(\sin\theta/\theta)^{-\frac{n-2}{2}}=1+\frac{\theta^{2}}{12}(n-2)+{\mathcal
O}(\theta^{3})$, and noting that the leading order $\theta^{2}$ term
in the product of the two factors is a non-decreasing function of
distance $\theta$ when $\frac{n-2}{12}\ge\frac{1}{4t}$, corresponding
to the unphysical situation of nearby points being hotter than the
heat source itself. As the feature dimension $n$ is typically very
large, the restriction $(n-2)t<3$ implies that we need to take the
diffusion time to be very small, thus making the similarity measure
captured by $G^{{\rm prx}}$ decay too fast away from each data point
for use in clustering applications.
In this work, we further computed the first and second order correction terms,
denoted $u_1$ and $u_2$ in Equation~\ref{eq: para_u1} and Equation~\ref{eq:
para_u2}, respectively
In high dimensions, the divergence of
$u_1/u_0$ and $u_2/u_0$ at $\theta=\pi$ is not a major problem, as we expect the
expansion to be valid only in the vicinity $\theta\downarrow0$; however, the divergence
of $u_2/u_0$ at $\theta=0$ (to $-\infty$ in high dimensions) is pathological, and thus, we truncate
our approximation to ${\mathcal O}(t^2)$. Since $u_1(\theta)$ is not able to
correct the unphysical behavior of the parametrix kernel near $\theta=0$ in
high dimensions,
we conclude that the parametrix approximation fails in high dimensions.
Hence, the
only remaining part of $G^{{\rm prx}}$ still applicable to SVM
classification is the Gaussian factor, which is clearly not a heat
kernel on the hypersphere. The failure of this perturbative expansion
using the Euclidean heat kernel as a starting point suggests that
diffusion in $\mathbb{R}^{d}$ and $S^{d}$ are fundamentally different
and that the exact hyperspherical heat kernel derived from a
non-perturbative approach will likely yield better insights into the
diffusion process.
\subsection{Exact hyperspherical heat kernel}
By definition, the exact heat kernel $G^{{\rm ext}}(\hat{x},\hat{y};t)$
is the fundamental solution to heat equation $\partial_{t}u+\hat{L}^{2}u=0$
where $-\hat{L}^{2}$ is the hyperspherical Laplacian \cite{GOLDBART:2016uz,Grigoryan:1999du,Hsu:2002vq,Varopoulos:1987vx}.
In the language of operator theory, $G^{{\rm ext}}(\hat{x},\hat{y};t)$
is an integral kernel, or Green's function, for the operator $\exp\{-\hat{L}^{2}t\}$
and has an associated eigenfunction expansion. Because $\hat{L}^{2}$
and $\exp\{-\hat{L}^{2}t\}$ share the same eigenfunctions, obtaining
the eigenfunction expansion of $G^{{\rm ext}}(\hat{x},\hat{y};t)$
amounts to solving for the complete basis of eigenfunctions of $\hat{L}^{2}$.
The spectral decomposition of the Laplacian is in turn facilitated
by embedding $S^{n-1}$ in $\mathbb{R}^{n}$ and utilizing the global
rotational symmetry of $S^{n-1}$ in $\mathbb{R}^{n}$. The Euclidean
space harmonic functions, which are the solutions to the Laplace
equation $\nabla^{2}u=0$ in $\mathbb{R}^{n}$, can be projected to
the unit hypersphere $S^{n-1}$ through the usual separation of radial
and angular variables \cite{Atkinson:2012tz,Wen:1985fm}. In this
formalism, the hyperspherical Laplacian $-\hat{L}^{2}$ on $S^{n-1}$
naturally arises as the angular part of the Euclidean Laplacian on
$\mathbb{R}^{n}$, and $\hat{L}^{2}$ can be interpreted as the squared
angular momentum operator in $\mathbb{R}^{n}$ \cite{Wen:1985fm}.
The resulting eigenfunctions of $\hat{L}^{2}$ are known as the hyperspherical
harmonics and generalize the usual spherical harmonics in $\mathbb{R}^{3}$
to higher dimensions. Each hyperspherical harmonic is equipped with
a triplet of parameters or ``quantum numbers'' $(\ell,\{m_{i}\},\alpha)$:
the degree $\ell$, magnetic quantum numbers $\{m_{i}\}$ and $\alpha=\frac{n}{2}-1$.
In the eigenfunction expansion of $\exp\{-\hat{L}^{2}t\}$, we use
the addition theorem of hyperspherical harmonics to sum over the magnetic
quantum number $\{m_{i}\}$ and obtain the following main result:
\begin{thm}
\label{thm:The-exact-hyperspherical-kernel-expansion}The exact hyperspherical
heat kernel $G^{{\rm ext}}(\hat{x},\hat{y};t)$ can be expanded as
a uniformly and absolutely convergent power series
\[
G^{{\rm ext}}(\hat{x},\hat{y};t)=\sum_{\ell=0}^{\infty}{\rm
e}^{-\ell(\ell+n-2)t}\frac{2\ell+n-2}{n-2}\frac{1}{A_{S^{n-1}}}C_{\ell}^{\frac{n}{2}-1}(\hat{x}\cdot\hat{y})
\]
in the interval $\hat{x}\cdot\hat{y}\in[-1,1]$ and for $t>0$, where
$C_{\ell}^{\alpha}(w)$ are the Gegenbauer polynomials and $A_{S^{n-1}}=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}$
is the surface area of $S^{n-1}$. Since the kernel depends on $\hat{x}$
and $\hat{y}$ only through $\hat{x}\cdot\hat{y}$, we will write
$G^{{\rm ext}}(\hat{x},\hat{y};t)=G^{{\rm ext}}(\hat{x}\cdot\hat{y};t)$.
\end{thm}
\noindent\emph{Proof.}
We will obtain an eigenfunction expansion of the exact heat
kernel by using the lemmas proved in Supplementary Material Section S2.5.3.
The completeness of hyperspherical harmonics (Lemma 1) states that
\begin{equation}\label{eq:completeness}
\delta(\hat{x},\hat{y})=\sum_{\ell=0}^{\infty}\sum_{\{m\}}Y_{\ell\{m\}}(\hat{x})Y_{\ell\{m\}}^{*}(\hat{y}).
\end{equation}
Applying the addition theorem (Lemma 2) to Equation~\ref{eq:completeness},
we get
$$
\delta(\hat{x},\hat{y})=\frac{1}{A_{S^{n-1}}}\sum_{\ell=0}^{\infty}\frac{2\ell+n-2}{n-2}C_{\ell}^{\frac{n}{2}-1}(\hat{x}\cdot\hat{y}).
$$
Next, we apply time evolution operator $e^{-t\hat{L}^2}$ on this
initial state to generate the
heat kernel
\begin{align}
G(\hat{x}\cdot\hat{y};t) & ={\rm e}^{-\hat{L}^{2}t}\delta(\hat{x},\hat{y})\\
& =\sum_{\ell=0}^{\infty}{\rm e}^{-\ell(\ell+n-2)t}\frac{2\ell+n-2}{n-2}\frac{1}{A_{S^{n-1}}}C_{\ell}^{\frac{n}{2}-1}(\hat{x}\cdot\hat{y}).
\end{align}
To show that it is a uniformly and absolutely convergent
series for $t>0$, note that
$$
|G(w;t)|\le\frac{1}{(n-2)A_{S^{n-1}}}\sum_{\ell=0}^{\infty}{\rm e}^{-\ell(\ell+n-2)t}(2\ell+n-2)\left|C_{\ell}^{\frac{n-2}{2}}(w)\right|,
$$
where $w=\hat{x}\cdot\hat{y}$.
The terms involving Gegenbauer
polynomials can be bounded by using Lemma 3
as
\begin{align*}
\left|C_{\ell}^{\frac{n-2}{2}}(w)\right| & \le\left[w^{2}\frac{\Gamma(\ell+n-2)}{\Gamma(n-2)\Gamma(\ell+1)}+(1-w^{2})\frac{\Gamma(\frac{\ell+n-2}{2})}{\Gamma(\frac{n-2}{2})\Gamma(\frac{\ell}{2}+1)}\right]\\
& =\left[\frac{\Gamma(\frac{\ell+n-2}{2})}{\Gamma(\frac{n-2}{2})\Gamma(\frac{\ell}{2}+1)}+\left(\frac{\Gamma(\ell+n-2)}{\Gamma(n-2)\Gamma(\ell+1)}-\frac{\Gamma(\frac{\ell+n-2}{2})}{\Gamma(\frac{n-2}{2})\Gamma(\frac{\ell}{2}+1)}\right)w^{2}\right]\\
& \le\frac{\Gamma(\frac{\ell+n-2}{2})}{\Gamma(\frac{n-2}{2})\Gamma(\frac{\ell}{2}+1)}+\left|\frac{\Gamma(\ell+n-2)}{\Gamma(n-2)\Gamma(\ell+1)}-\frac{\Gamma(\frac{\ell+n-2}{2})}{\Gamma(\frac{n-2}{2})\Gamma(\frac{\ell}{2}+1)}\right|w^{2}\\
& \le\frac{\Gamma(\frac{\ell+n-2}{2})}{\Gamma(\frac{n-2}{2})\Gamma(\frac{\ell}{2}+1)}+\left|\frac{\Gamma(\ell+n-2)}{\Gamma(n-2)\Gamma(\ell+1)}-\frac{\Gamma(\frac{\ell+n-2}{2})}{\Gamma(\frac{n-2}{2})\Gamma(\frac{\ell}{2}+1)}\right|\\
& \equiv M_{\ell}.
\end{align*}
We thus have
\begin{align*}
|G(w;t)| & \le\frac{1}{(n-2)A_{S^{n-1}}}\sum_{\ell=0}^{\infty}{\rm e}^{-\ell(\ell+n-2)t}(2\ell+n-2)\left|C_{\ell}^{\frac{n-2}{2}}(w)\right|\\
& \le\frac{1}{(n-2)A_{S^{n-1}}}\sum_{\ell=0}^{\infty}{\rm e}^{-\ell(\ell+n-2)t}(2\ell+n-2)M_{\ell}\\
& \equiv\frac{1}{(n-2)A_{S^{n-1}}}\sum_{\ell=0}^{\infty}Q_{\ell}.
\end{align*}
But, in the large $\ell$ limit, the asymptotic expansion
$$
M_{\ell}\sim \frac{\ell^{n-3}}{(n-3)!}\,
$$
implies that
$$
\lim_{\ell\rightarrow\infty}\frac{Q_{\ell+1}}{Q_{\ell}}=\lim_{\ell\rightarrow\infty}\frac{{\rm
e}^{-(2\ell+n-1)t}(2\ell+n)M_{\ell+1}}{(2\ell+n-2) M_{\ell}}= 0<1,
$$
for any $t>0$.
The sequence $\{Q_{\ell}\}$ is thus convergent, and hence, the
Weiestrass M-test implies that the eigenfunction expansion of the heat
kernel is uniformly and absolutely convergent in the indicated
intervals. {\hfill Q.E.D.}
Note that the exact kernel $G^{{\rm ext}}$ is a Mercer kernel
re-expressed by summing over the degenerate eigenstates indexed by $\{ m\}$.
As before, we will rescale the kernel by self-similarity and define:
\begin{defn}
The exact kernel $K^{{\rm ext}}(\hat{x},\hat{y};t)$ is the exact
heat kernel normalized by self-similarity:
\[
K^{{\rm ext}}(\hat{x},\hat{y};t)=\frac{G^{{\rm ext}}(\hat{x}\cdot\hat{y};t)}{G^{{\rm ext}}(1;t)},
\]
which is defined for $t>0$, is non-negative, and attains global maximum
$1$ at $\hat{x}\cdot\hat{y}=1$.
\end{defn}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=6in]{Fig1}
\par\end{centering}
\caption{\label{fig:HK-randomwalks}(A) Color maps of the exact kernel $K^{{\rm ext}}$
on $S^{2}$ at rescaled time $t^{*}=0.5,1.0,2.0$; the white paths are simulated random walks on $S^{2}$
with the Monte Carlo time approximately equal to $t=t^{*}\log3/3$. (B)
Plots of the parametrix kernel $K^{{\rm prx}}$ and exact kernel $K^{{\rm ext}}$
on $S^{n-1}$, for $n=3,100,200$,
as functions of the geodesic distance.}
\end{figure}
Note that unlike $K^{{\rm prx}}(\hat{x},\hat{y};t)$, $K^{{\rm ext}}(\hat{x},\hat{y};t)$ explicitly depends on the feature dimension $n$.
In general, SVM kernel hyperparameter tuning can be computationally costly
for a data set with both high feature dimension and large sample size.
In particular, choosing an appropriate diffusion time scale is an
important challenge. On the one hand, choosing a very large value
of $t$ will make the series converge rapidly; but, then, all points
will become uniformly similar, and the kernel will not be very useful.
On the other hand, a too small value of $t$ will make most data pairs
too dissimilar, again limiting the applicability of the kernel. In
practice, we thus need a guideline for a finite time scale at which
the degree of ``self-relatedness'' is not singular, but still larger
than the ``relatedness'' averaged over the whole hypersphere. Examining
the asymptotic behavior of the exact heat kernel in high feature dimension
$n$ shows that an appropriate time scale is $t\sim{\mathcal O}(\log n/n)$;
in this regime the numerical sum in Theorem \ref{thm:The-exact-hyperspherical-kernel-expansion}
satisfies a stopping condition at low orders in $\ell$ and the sample points are in
moderate diffusion proximity to each other so that they can be accurately
classified (Supplementary Material, Section S2.5.4)
Figure~\ref{fig:HK-randomwalks}A illustrates the diffusion process
captured by our exact kernel $K^{{\rm ext}}(\hat{x},\hat{y};t)$ in
three feature dimensions at time $t=t^{*}\log 3/3$, for
$t^{*}=0.5,1.0, 2.0$. In Figure~\ref{fig:HK-randomwalks}B, we systematically compared
the behavior
of (1) dimension-independent parametrix kernel $K^{{\rm prx}}$ at
time $t=0.5, 1.0,2.0$ and (2) exact kernel $K^{{\rm ext}}$ on $S^{n-1}$
at $t=t^{*}\log n/n$ for $t^{*}=0.5,1.0,2.0$ and $n=3,100,200$.
By symmetry, the slope of $K^{{\rm ext}}$
vanished at the south pole $\theta=\pi$ for any time $t$ and dimension
$n$. In sharp contrast, $K^{{\rm prx}}$ had a negative slope at $\theta=\pi$, again highlighting a singular behavior of the parametrix kernel. The ``relatedness'' measured by $K^{{\rm ext}}$ at the sweet
spot $t=\log n/n$ was finite over the whole hypersphere with
sufficient contrast between nearby and far away points. Moreover, the characteristic behavior of $K^{{\rm ext}}$ at $t=\log n/n$
did not change significantly for different values of the feature dimension $n$, confirming that the optimal
$t$ for many classification applications will likely reside near the ``sweet spot'' $t=\log n/n$.
\subsection{SVM classifications}
Linear SVM seeks a separating hyperplane that maximizes the margin,
i.e. the distance to the nearest data point. The primal formulation of
SVM attempts to minimize the norm of the weight vector $\mathbf{w}$
that is normal to the separating hyperplane, subject to either hard or
soft margin constraints. In the so-called Lagrange dual formulation of
SVM, one applies the Representer Theorem to rewrite the weight as a
linear combination of data points; in this set-up, the dot products of
data points naturally appear, and kernel SVM replaces the dot product
operation with a chosen kernel evaluation. The ultimate hope is that
the data points will become linearly separable in the new feature
space implicitly defined by the kernel.
We evaluated the performance of kernel SVM using the
\begin{enumerate}
\item linear kernel $K^{{\rm lin}}(\mathbf{x},\mathbf{y})=\mathbf{x}\cdot\mathbf{y}$,
\item Gaussian RBF $K^{{\rm rbf}}(\mathbf{x},\mathbf{y};\gamma)=\exp\{-\gamma|\mathbf{x}-\mathbf{y}|^{2}\}$,
\item cosine kernel $K^{{\rm cos}}(\hat{x},\hat{y})=\hat{x}\cdot\hat{y}$,
\item parametrix kernel $K^{{\rm prx}}(\hat{x},\hat{y};t)$, and
\item exact kernel $K^{{\rm ext}}(\hat{x},\hat{y};t)$,
\end{enumerate}
on two independent data sets: (1) WebKB data of websites from four
universities (WebKB-4-University) \cite{Craven:1998tq}, and (2)
glioblastoma multiforme (GBM) mutation data from The Cancer Genome
Atlas (TCGA) with 5-fold cross-validations (CV)
(Supplementary Material, Section S1).
The WebKB-4-University data contained 4199 documents in
total comprising four classes: student (1641), faculty (1124), course
(930), and project (504); in our analysis, however, we selected an
equal number of representative samples from each class, so that the
training and testing sets had balanced classes. Table~\ref{tab:WebKB} shows the
average optimal prediction accuracy scores of the five kernels for a
varying number of representative samples, using 393 most frequent word
features (Supplementary Material, Section S1).
The exact kernel outperformed the
Gaussian RBF and parametrix kernel, reducing the error by
$41\%\sim45\%$ and by $1\%\sim7\%$, respectively.
Changing the feature dimension did not affect the performance much
(Table \ref{tab:SI-WebKB-4-University}).
\begin{table}
\centering
\begin{tabular}{lccccc}
\hline\hline
$m_{{\rm r}}$ & lin & rbf & cos & prx & ext\\
\hline
100 & 74.2\% & 75.1\% & 84.4\% & 85.4\% & \textbf{85.6\%}\\
200 & 80.9\% & 82.0\% & 89.2\% & 89.6\% & \textbf{89.9\%}\\
300 & 83.2\% & 84.1\% & 89.9\% & 90.5\% & \textbf{91.1\%}\\
400 & 86.7\% & 86.1\% & 91.3\% & 91.7\% & \textbf{92.3\%}\\
\hline\hline
\end{tabular}
\caption{
\label{tab:WebKB}
WebKB-4-University Document Classification.
Performance test on four-class (\emph{student}, \emph{faculty}, \emph{course}, and \emph{project})
classification of WebKB-4-University word count data with different
number $m_{{\rm r}}$ of representatives for each class, for $m_{{\rm r}}=100,200,300,400$.
The entries show the average of optimal 5-fold cross-validation mean
accuracy scores of five runs. The exact kernel (ext) reduced the error of
parametrix kernel (prx) by $1\%\sim7\%$ and the Gaussian RBF (rbf)
by $41\%\sim 45\%$; the cosine kernel (cos) also reduced the error
of linear kernel (lin) by $34\%\sim 43\%$.
}
\end{table}
\begin{table}[htb]
\centering
\begin{tabular}{llccccc}
\hline\hline
$n$ & $m_{{\rm r}}$ & lin & rbf & cos & prx & ext\\\hline
393 & 400 & 86.73\% & 86.27\% & 91.57\% & 91.99\% & \textbf{92.44\%}\\
726 & 400 & 86.78\% & 86.95\% & 92.62\% & 92.91\% & \textbf{93.00\%}\\
1023 & 400 & 85.56\% & 86.11\% & 92.62\% & 92.74\% & \textbf{92.91\%}\\
1312 & 400 & 85.78\% & 86.75\% & 92.56\% & 92.81\% & \textbf{93.03\%}\\\hline\hline
\end{tabular}
\caption{\label{tab:SI-WebKB-4-University}WebKB-4-University Document Classification. Comparison of kernel SVMs on the WebKB-4-University data
with a fixed sample size $m_{{\rm r}}$, but varying feature
dimension $n$. To account for the randomness in
selecting the representative samples using KMeans
(Supplementary Material, Section S1),
we performed fives runs of representative selection, and
then performed CV using the training and test sets obtained from
each run.
Finally, we averaged the five mean CV scores to assess the
performance of each classifier on the imbalanced
WebKB-4-University data set. The exact (ext) and cosine (cos)
kernels outperformed the Gaussian RBF (rbf) and linear (lin) kernels
in various feature dimensions $n=393,726,1023,$ and $1312$, with
fixed and balanced class size $m_{{\rm r}}=400$.
A word was selected as a feature if its total count was greater than
1/10, 1/20, 1/30 or 1/40 times the total number of web pages in the
WebKB-4-University data set, with the different thresholds
corresponding to the different rows in the table.
The exact kernel reduced
the errors of Gaussian RBF and parametrix kernels by $45\sim48\%$ and
$1\sim6\%$, respectively; the cosine kernel reduced the errors of
linear
kernel by $36\sim49\%$.}
\end{table}
\begin{table}
\centering
\begin{tabular}{rccccc}
\hline \hline
& lin & rbf & cos & prx & ext \\
\hline
\textit{ZMYM4} & 82.9\% & 84.0\% & 83.6\% & 84.1\% & \textbf{85.1\%}\\
\textit{ADGRB3} & 75.7\% & \textbf{81.0\%} & 78.0\% & 79.5\% & 79.3\%\\
\textit{NFX1} & 73.0\% & 81.2\% & 80.9\% & \textbf{82.7\%} & 82.5\%\\
\textit{P2RX7} & 79.2\% & 84.1\% & \textbf{85.0\%} & 84.0\% & \textbf{85.0\%}\\
\textit{COL1A2} & 68.4\% & 70.5\% & 72.9\% & 73.9\% & \textbf{74.2\%}\\
\hline \hline
\end{tabular}
\caption{\label{tab:TCGA}TCGA-GBM Genotype Imputation. Performance test on binary classification of \emph{mutant} vs.~\emph{wild-type} in TCGA-GBM
mutation count data. The rows are different genes, the mutation statuses
of which were imputed using $m_{{\rm r}}$ samples in each mutant and
wild-type class. The entries show the average of optimal 5-fold cross-validation mean accuracy
scores of five runs.
}
\end{table}
In the TCGA-GBM data, there were 497 samples, and we aimed to impute
the mutation status of one gene -- i.e., mutant or wild-type -- from the mutation
counts of other genes. For each imputation target, we first counted
the number $m_{{\rm r}}$ of mutant samples and then selected an equal number of wild-type samples
for 5-fold CV. Imputation tests were performed for top 102 imputable
genes (Supplementary Material, Section S1).
Table~\ref{tab:TCGA} shows the
average prediction accuracy scores for 5 biologically interesting
genes known to be important for cancer \cite{Hanahan:2011gu}:
\begin{enumerate}
\item \emph{ZMYM4} ($m_{{\rm r}}=33$) is implicated in an antiapoptotic activity;
\cite{Smedley:1999uq,Shchors:2002hu};
\item \emph{ADGRB3} ($m_{{\rm r}}=37$) is a brain-specific angiogenesis
inhibitor \cite{Zohrabian:2007wv,Kaur:2010jf,Hamann:2015hv};
\item \emph{NFX1} ($m_{{\rm r}}=42$) is a repressor of \emph{hTERT}
transcription \cite{Yamashita:2016dm} and is thought to regulate inflammatory
response \cite{Song:1994bh};
\item \emph{P2RX7} ($m_{{\rm r}}=48$) encodes an ATP receptor which plays
a key role in restricting tumor growth and metastases \cite{Adinolfi:2015kk,GomezVillafuertes:2015fd,LinanRico:2016gu};
\item \emph{COL1A2} ($m_{{\rm r}}=61$) is overexpressed in the medulloblastoma
microenvironment and is a potential therapeutic target \cite{Anderton:2008gh,Liang:2007kw,Schwalbe:2011ic}.
\end{enumerate}
\begin{figure}[ht]
\begin{centering}
\includegraphics[width=6in]{Fig2}
\par\end{centering}
\caption{\label{fig:HK-SI-scores}Comparison of the classification
accuracy of SVM using linear (lin), cosine (cos), Gaussian RBF
(rbf), parametrix (prx), and exact (ext) kernels on TCGA mutation
count data. The plots show the ratio of accuracy scores for two
different kernels. For visualization purpose, we excluded one gene
with $m_{{\rm r}}=250$. The ratios rbf/lin, prx/cos, and ext/cos
were essentially constant in class size $m_{{\rm r}}$ and greater than 1; in other words, the Gaussian RBF (rbf)
kernel outperformed the
linear (lin) kernel, while the exact (ext) and parametrix (prx) kernels
outperformed the cosine (cos) kernel uniformly over all values of class size
$m_{{\rm r}}$. However, the more negative slope in the linear fit
of cos/lin hints that the accuracy scores of
cosine and linear kernels may depend on the class
size $m_{{\rm r}}$; the exact kernel also tended to outperform Gaussian RBF kernel when $m_{\rm r}$ was small. }
\end{figure}
For the remaining genes, the exact kernel generally outperformed the
linear, cosine and parametrix kernels (Figure~\ref{fig:HK-SI-scores}). However, even though
the exact kernel dramatically outperformed the Gaussian RBF in the
WebKB-4-University classification problem, the advantage of the exact
kernel in this mutation analysis was not evident (Figure~\ref{fig:HK-SI-scores}). It is
possible that the radial degree of freedom $\sum_{i=1}^{n}x_{i}$ in
this case, corresponding to the genome-wide mutation load in each
sample, contained important covariate information not captured by the
hyperspherical heat kernel. The difference in accuracy between the
hyperspherical kernels (cos, prx, and ext) and the Euclidean kernels
(lin and rbf) also hinted some weak dependence on class size $m_{\rm
r}$ (Figure~\ref{fig:HK-SI-scores}), or equivalently the sample size $m=2m_{\rm r}$. In
fact, the level of accuracy showed much stronger correlation with the
``effective sample size'' $\tilde{m}$ related to the empirical
Vapnik-Chervonenkis (VC) dimension
\cite{Vladimir:l-V4Juw7,Boser:1992fz,Guyon:1993ub,Vapnik:1994dk,Anonymous:2001vo}
of a kernel SVM classifier (Figure~\ref{fig:HK-SI-VC}A-E); moreover,
the advantage of the exact kernel over the Guassian RBF kernel
grew with the effective sample size ratio $\tilde{m}_{\rm
cos}/\tilde{m}_{\rm lin}$ (Figure~\ref{fig:HK-SI-VC}F, Supplementary Material, Section S2.5.5).
\begin{figure}[t]
\begin{centering}
\includegraphics[width=6in]{Fig3}
\par\end{centering}
\caption{\label{fig:HK-SI-VC}(A) A strong linear relation is seen
between the VC-bound for cosine kernel $\mu_{{\rm VC}}^{*\cos}$ and class
size $m_{{\rm r}}$. The dashed line marks $y=x$; the VC-bound
for linear kernel, however, was a constant $\mu_{{\rm VC}}^{*{\rm
lin}}=439$. (B-E) The scatter plots of accuracy scores for
cosine (cos), linear (lin), exact (ext), and Gaussian RBF (rbf)
kernels vs. the effective sample size $\tilde{m} = 2m_{{\rm
r}}/\mu_{{\rm VC}}^{*}$; the accuracy scores of exact and cosine
kernels increased with the effective sample size, whereas those of
Gaussian RBF and linear kernels tended to decrease with the effective
sample size. (F) The ratio of ext vs. rbf accuracy scores is
positively correlated with the ratio $\tilde{m}_{\rm cos}/\tilde{m}_{\rm lin}$
of effective sample sizes.}
\end{figure}
By construction, our definition of the hyperspherical map exploits
only the positive portion of the whole hypersphere, where the
parametrix and exact heat kernels seem to have similar
performances. However, if we allow the data set to assume negative
values, i.e.~the feature space is the usual
$\mathbb{R}^{n}\backslash\{0\}$ instead of
$\mathbb{R}_{\ge0}^{n}\backslash\{0\}$, then we may apply the usual
projective map, where each vector in the Euclidean space is normalized
by its $L^{2}$-norm. As shown in
Figure~\ref{fig:HK-randomwalks}B, the parametrix kernel is
singular at $\theta=\pi$ and qualitatively deviates from the exact
kernel for large values of $\theta$. Thus, when data points
populate the whole hypersphere, we expect to find more significant
differences in performance between the exact and parametrix
kernels. For example, Table \ref{tab:SI-SP500} shows the kernel SVM classifications
of 91 S\&P500 \emph{Financials} stocks against 64 \emph{Information
Technology} stocks ($m=155$) using their log-return instances
between January 5, 2015 and November 18, 2016 as features. As long as
the number of features was greater than sample size, $n>m$, the exact
kernel outperformed all other kernels and reduced the error of
Gaussian RBF by $29\sim51\%$ and that of parametrix kernel by
$17\sim51\%$.
\begin{table}[htb]
\centering
\begin{tabular}{rrccccc}
\hline\hline
$n$ & $m$ & lin & rbf & cos & prx & ext\\\hline
475 & 155 & 98.06\% & 98.69\% & 98.69\% & 98.69\% & \textbf{99.35\%}\\
238 & 155 & 95.50\% & 96.77\% & 94.82\% & 96.13\% & \textbf{98.06\%}\\
159 & 155 & 94.86\% & 95.48\% & 95.48\% & 96.13\% & \textbf{96.79\%}\\
119 & 155 & 92.86\% & 93.53\% & 91.57\% & \textbf{94.15\%} & \textbf{94.15\%}\\
95 & 155 & 91.55\% & \textbf{95.50\%} & 94.19\% & 94.15\% & 94.79\%\\\hline\hline
\end{tabular}
\caption{\label{tab:SI-SP500}S\&P500 Stock Classification. Classifications were performed on $m=155$ stocks from S\&P500 companies:
91 \emph{Financial} vs.~64 \emph{Information Technology}. The 475
log-return instances between January 5, 2015 and November
18, 2016 were used as features. We uniformly subsampled the
instances to generate variations in the feature dimension $n$. Here, we
report the mean 5-fold CV accuracy score for each kernel.
Although the two classes were slightly imbalanced, all scores were much
larger than the ``random score'' $91/155\approx58.7\%$, calculated
from the
majority class size and sample size. For $n>m$, the exact (ext) kernel outperformed all
other kernels and reduced the errors of Gaussian RBF (rbf) and parametrix
(prx) kernels by $29\sim51\%$ and $17\sim51\%$, respectively. When
$n<m$, the exact kernel started to lose its advantage over the Gaussian RBF kernel.}
\end{table}
\section{Discussion}
This paper has constructed the exact hyperspherical heat kernel using
the complete basis of high-dimensional angular momentum eigenfunctions
and tested its performance in kernel SVM. We have shown that the exact
kernel and cosine kernel, both of which employ the hyperspherical
maps, often outperform the Gaussian RBF and linear kernels. The
advantage of using hyperspherical kernels likely arises from the
hyperspherical maps of feature space, and the exact kernel may
further improve the decision boundary flexibility of the raw cosine
kernel. To be specific, the hyperspherical maps remove the less
informative radial degree of freedom in a nonlinear fashion and
compactify the Euclidean feature space into a unit hypersphere where
all data points may then be enclosed within a finite radius. By
contrast, our numerical estimations using TCGA-GBM data show that for
linear kernel SVM, the margin $M$ tends to be much smaller than the
data range $R$ in order to accommodate the separation of strongly
mixed data points of different class labels; as a result, the ratio
$R/M$ was much larger than that for cosine kernel SVM. This insight
may be summarized by the fact that the upper bound on the empirical
VC-dimension of linear kernel SVM tends to be much larger than that
for cosine kernel SVM, especially in high dimensions, suggesting that
the cosine kernel SVM is less sensitive to noise and more
generalizable to unseen data. The exact kernel is equipped with an
additional tunable hyperparameter, namely the diffusion time $t$,
which adjusts the curvature of nonlinear decision boundary and thus
adds to the advantage of hyperspherical maps. Moreover, the
hyperspherical kernels often have larger effective sample sizes than
their Euclidean counterparts and, thus, may be especially useful for
analyzing data with a small sample size in high feature dimensions.
The failure of the parametrix expansion of heat kernel, especially in
dimensions $n\gg 3$, signals a dramatic difference between diffusion
in a non-compact space and that on a compact manifold. It remains to
be examined how these differences in diffusion process, random walk
and topology between non-compact Euclidean spaces and compact
manifolds like a hypersphere help improve clustering performance as supported by the results of this paper.
\section*{Funding}
This research was supported by a Distinguished
Scientist Award from Sontag Foundation and the Grainger Engineering
Breakthroughs Initiative.
\section*{Acknowledgments}
We thank Alex Finnegan and Hu Jin for critical reading of the manuscript
and helpful comments. We also thank Mohith Manjunath for his help with the TCGA data.
\newpage
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\centerline{\LARGE\bf Supplementary Material}
\section{Data preparation and SVM classification}\label{appendix_data}
The WebKB-4-University raw webpage data were downloaded from \url{http://www.cs.cmu.edu/afs/cs/project/theo-20/www/data/} and
processed with the python packages Beautiful Soup and Natural Language
Toolkit (NLTK). Our feature extraction excluded punctuation marks and
included only letters and numerals where capital letters were all
converted to lower case and each individual digit 0-9 was represented
by a ``\#.'' Very infrequent words, such as misspelled words,
non-English words, and words mixed with special characters, were
filtered out. We selected top $393$ most frequent words as features
in our classification tests; the cutoff was chosen to select frequent
words whose counts across all webpage documents are greater than
$10\%$ of the total number of documents. There were 4199 documents in
total: student (1641), faculty (1124), course (930), and project
(504).
The TCGA-GBM data were downloaded from the GDC Data Portal under the
name TCGA-GBM Aggregated Somatic Mutation. The mutation count data set
was extracted from the MAF file, while ignoring the detailed types of
mutations and counting only the total number of mutations in each
gene. Very infrequently, mutated genes were filtered out if the total
number of mutations in one gene across all samples is less than $10\%$
of the total number of samples ($m=497$ samples and $n=439$ genes). We
imputed the mutation status of one gene, mutant or wild-type, from the
mutation counts of the remaining genes. The most imputable genes were
selected using 5-fold cross-validation linear kernel SVM. Most of the
mutant and wild-type samples were highly unbalanced, the ratio being
typically around $1:9$; therefore, unthresholded area-under-the-curve
(AUC) of the receiver operating characteristic (ROC) curve was used to
quantify the classification performance of the linear kernel
SVM. Mutated genes with AUC greater than $60\%$ were selected for the
subsequent imputation tests.
To balance the sample size between classes, we performed K-means
clustering of samples within each class, with a specified number
$m_{{\rm r}}$ of centroids and took the samples closest to each
centroid as representatives. For the WebKB document classifications,
we used $m_{{\rm r}}\le\min\{m_{{\rm student }},m_{{\rm
faculty}},m_{{\rm course }},$ $m_{\rm project}\}$, and
K-means clustering was performed in each of the four classes
separately; for the TCGA-GBM data, $m_{{\rm r}}$ was chosen to be the
number of samples in each mutant (minority) class, and K-means
clustering was performed in the wild-type (majority) class. Since
K-means might depend on the random initialization, we performed the
clustering 50 times and selected the top $m_{{\rm r}}$ most frequent
representatives. Five-fold stratified cross-validations (CV) were
performed on the resulting balanced data sets, where training and test
samples were drawn without replacement from each class. The mean CV
accuracy scores across the five folds were recorded.
\section{Hyperspherical Heat Kernel}
\subsection{Laplacian on a Riemannian manifold}
The Laplacian on a Riemannian manifold ${\mathcal M}$ with metric $g_{\mu\nu}$
is the operator
$$
\Delta: C^{\infty}({\mathcal M}) \rightarrow C^{\infty}({\mathcal M})
$$
defined as
\begin{equation}\label{eq:Laplacian}
\Delta\equiv\frac{1}{\sqrt{g}}\partial_{\mu}\left(\sqrt{g}g^{\mu\nu}\partial_{\nu}\right),
\end{equation}
where $g=|\det g|$, and the Einstein summation convention is used.
It can be also written in terms of the covariant derivative $\nabla_{\mu}$ as
\begin{equation}\label{eq:Laplacian2}
\Delta=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}.
\end{equation}
The covariant derivative satisfies the following
properties
$$
\nabla_{\mu}f=\partial_{\mu}f,\quad f\in C^{\infty}({\mathcal M})
$$
$$
\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma_{\:\lambda\mu}^{\nu}V^{\lambda},\quad V\in T_{p}{\mathcal M}
$$
$$
\nabla_{\mu}\omega_{\nu}=\partial_{\mu}\omega_{\nu}-\Gamma_{\:\nu\mu}^{\lambda}\omega_{\lambda},\quad\omega\in T_{p}^{*}{\mathcal M},
$$
where $\Gamma_{\:\alpha\beta}^{\lambda}$ is the Levi-Civita connection
satisfying
$\Gamma_{\:\alpha\beta}^{\lambda}=\Gamma_{\:\beta\alpha}^{\lambda}$
and $\nabla_{\lambda}g_{\mu\nu}=0$.
To show Equation~\ref{eq:Laplacian2},
recall that the Levi-Civita connection is uniquely
determined by the geometry, or the metric tensor, as
$$
\Gamma_{\:\alpha\beta}^{\lambda}=\frac{1}{2}g^{\lambda\rho}\left(\partial_{\alpha}g_{\beta\rho}+\partial_{\beta}g_{\alpha\rho}-\partial_{\rho}g_{\alpha\beta}\right).
$$
Using the formula for determinant
differentiation
$$
\left[\log\left(\det\mathbf{A}\right)\right]'={\rm tr}\left(\mathbf{A}'\mathbf{A}^{-1}\right),
$$
we can thus write
$$
\Gamma_{\:\lambda\mu}^{\lambda} =\partial_{\mu}\log\sqrt{g}.
$$
Hence, for any $f\in C^{\infty}({\mathcal M})$,
\begin{align*}
g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}f & = \nabla_{\mu}(g^{\mu\nu}\partial_{\nu}f)\\
&=\partial_{\mu}(g^{\mu\nu}\partial_{\nu}f) + \Gamma_{\:\lambda\mu}^{\lambda} (g^{\mu\nu}\partial_{\nu}f)\\
&=\partial_{\mu}(g^{\mu\nu}\partial_{\nu}f) + (\partial_{\mu}\log\sqrt{g} )(g^{\mu\nu}\partial_{\nu}f)\\
& =\frac{1}{\sqrt{g}}\partial_{\mu}\left(\sqrt{g}g^{\mu\nu}\partial_{\nu}\right),
\end{align*}
proving the equivalence of Equation~\ref{eq:Laplacian} and Equation~\ref{eq:Laplacian2}.
\subsection{The induced metric on $S^{n-1}$}
The $(n-1)$-sphere embedded in $\mathbb{R}^{n}$ can be parameterized as
\begin{eqnarray*}
x_{1} & = & \cos\theta_{1}\\
x_{2} & = & \sin\theta_{1}\cos\theta_{2}\\
x_{3} & = & \sin\theta_{1}\sin\theta_{2}\cos\theta_{3}\\
& \vdots & \\
x_{n-1} & = & \sin\theta_{1}\cdots\sin\theta_{n-2}\cos\theta_{n-1}\\
x_{n} & = & \sin\theta_{1}\cdots\sin\theta_{n-2}\sin\theta_{n-1},
\end{eqnarray*}
where $0\leq\theta_i\leq\pi$, for $i=1,\ldots,n-2$, and $0\leq\theta_{n-1}\leq2\pi$.
Let $\lambda:=(\partial x_{i}/\partial\theta_{j})$ denote
the $n\times(n-1)$ Jacobian matrix for the above
coordinate transformation.
The square of the line element in $\mathbb{R}^{n}$ is given by
$$
ds_{n}^{2}=\sum_{i=1}^{n}dx_idx_i.
$$
Restricted to $S^{n-1}$,
$$
dx_i=\sum_{j=1}^{n-1}\frac{\partial x_{i}}{\partial\theta_{j}}d\theta_{j}=\sum_{j=1}^{n-1}\lambda_{ij}d\theta_{j}.
$$
Therefore, on $S^{n-1}$, we have
\begin{align*}
ds_{n-1}^{2} & =\sum_{i=1}^{n}\sum_{j,j'=1}^{n-1}\lambda_{ij}\lambda_{ij'}d\theta_{j}d\theta_{j'}\\
&
=\sum_{j,j'=1}^{n-1}\left(\sum_{i=1}^{n}\lambda_{ij}\lambda_{ij'}\right)d\theta_{j}
d\theta_{j'}\,.\\
\end{align*}
Hence, the induced metric on $S^{n-1}$ embedded in $\mathbb{R}^{n}$ is
$$
g_{\mu\nu}= \left(\lambda^T\lambda\right)_{\mu\nu}.
$$
After some algebraic manipulations, it can be shown that the metric is in fact diagonal
and its determinant takes the form
\begin{equation}\label{eq:detg}
g=\sin^{2(n-2)}\theta_{1}\sin^{2(n-3)}\theta_{2}\cdots\sin^{4}\theta_{n-3}\sin^{2}\theta_{n-2}.
\end{equation}
The geodesic arc length $\theta$ between $\hat{x}$ and
$\hat{x}'$ on $S^{n-1}$ is the angle given by
$$
\theta\equiv\arccos\hat{x}\cdot\hat{x}'=\arccos\sum_{i=1}^{n}\hat{x}_{i}\hat{x}_{i}'.
$$
\subsection{Laplacian in geodesic polar coordinates}
In geodesic polar coordinates $(r,\xi)$ around a point, one can show
using Equation~\ref{eq:Laplacian2} that the Laplacian on a
$d$-dimensional Riemannian manifold ${\mathcal M}$ takes the form
$$
\Delta = \partial_r^2 + (\partial_r \log \sqrt{g}) \partial_r + \Delta_{S_r^{d-1}},
$$
where $\Delta_{S_r^{d-1}}$ is the Laplacian induced on the geodesic
sphere $S_r^{d-1}$ of radius $r$. If function $f$ depends only on the geodesic
distance $r$ from the fixed point, then
\begin{equation}
\Delta f(r) =f''(r)+\left(\log\sqrt{g}\right)'f'(r),
\end{equation}
where $'$ denotes the radial derivative.
For the special case when ${\mathcal M}$ is $S^{n-1}$, the coordinates
$\theta_1,\ldots,\theta_{n-1}$ described above correspond to the
geodesic polar coordinates around the north pole, with $r=\theta_1$. From
Equation~\ref{eq:detg}, we get
\begin{align*}
\log\sqrt{g(x)} & =(n-2)\log\sin r+(n-3)\log\sin\theta_{2}+\cdots\\
& +\log\sin\theta_{n-2}.
\end{align*}
Note that only the first terms contributes to the radial derivative.
\subsection{Euclidean heat kernel}
Heat kernels in general are solutions to the heat equation
$$
\left(\partial_{t}-\Delta\right)\phi=0
$$
with a point-source (Dirac delta) initial condition. The heat kernel
in $\mathbb{R}^{d}$ is easily found to be
\begin{equation}\label{eq:euclideanHeatKernel}
G(\mathbf{x},\mathbf{y};t)=\left(\frac{1}{4\pi t}\right)^{\frac{d}{2}}K(\mathbf{x},\mathbf{y};t)
\end{equation}
where
$$
K(\mathbf{x},\mathbf{y};t)=\exp\left(-\frac{\|\mathbf{x}-\mathbf{y}\|^{2}}{4t}\right).
$$
$K$ is known as the Gaussian RBF kernel with parameter
$\gamma=1/4t$. $G(\mathbf{x},\mathbf{y};t)$ is the solution to the
heat equation satisfying the initial condition
$G(\mathbf{x},\mathbf{y};0)=\delta(\mathbf{x}-\mathbf{y})$. Note
that formally,
$$
G(\mathbf{x},\mathbf{y};t) = {\rm e}^{t \Delta} \delta(\mathbf{x}-\mathbf{y});
$$
using the Fourier transform representation of the right-hand side then
yields the expression in Equation~\ref{eq:euclideanHeatKernel}.
\subsection{Exact hyperspherical heat kernel}\label{appendix_proof}
We treat the hypersphere $S^{n-1}$ as being embedded in
$\mathbb{R}^{n}$ and use the induced metric on $S^{n-1}$ to define the
Laplacian. The Laplacian in $\mathbb{R}^{n}$ takes the usual form
\begin{equation}\label{eq:LSeparation}
\Delta=\frac{1}{r^{n-1}}\partial_{r}\left(r^{n-1}\partial_{r}\right)-\frac{\hat{L}^{2}}{r^{2}}
\end{equation}
where the differential operator $\hat{L}^{2}$ depends only on the angular
coordinates. $-\hat{L}^{2}$ is the spherical Laplacian operator \cite{Wen:1985fm}.
\subsubsection{Spherical Laplacian and its eigenfunctions}
For $n=3$, the Laplacian on $\mathbb{R}^{3} $ is
$$
\Delta=\frac{1}{r^{2}}\partial_{r}\left(r^{2}\partial_{r}\right)-\frac{\hat{L}^{2}}{r^{2}}
$$
where $\hat{L}^2$ is the squared orbital angular momentum operator in quantum
mechanics. Restricted to
$r=1$, the Laplacian reduces to the spherical Laplacian on $S^{2}$,
which is exactly the operator $-\hat{L}^{2}$ whose eigenfunctions
are the spherical harmonics $Y_{lm}(\theta,\phi)$ with eigenvalue
$-\ell(\ell+1)$. In this setting, $Y_{lm}(\theta,\phi)$ can be viewed
as the angular component of homogeneous harmonic polynomials in $\mathbb{R}^3$, and
this perspective will be used in the subsequent discussion of
hyperspherical Laplacian. By convention, our spherical
harmonics satisfy the normalization condition
$$
\sum_{m=-\ell}^{\ell}|Y_{\ell m}(\theta,\phi)|^{2}=\frac{2\ell+1}{4\pi}
$$
and the completeness condition
$$
\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{\ell m}(\theta,\phi)Y_{\ell m}^{*}(\theta',\phi') = \delta(\cos\theta-\cos\theta')\delta(\phi-\phi').
$$
Analogous to the Euclidean case, applying the evolution operator
$\exp(-\hat{L}^{2}t)$ on the initial delta distribution yields the
following eigenfunction expansion of the heat kernel on $S^{2}$:
$$
G(\hat{x},\hat{y};t)=\sum_{l=0}^{\infty}{\rm e}^{-\ell(\ell+1)t}\sum_{m=-\ell}^{\ell}Y_{\ell m}(\hat{x})Y_{\ell m}(\hat{y})^{*}.
$$
Applying the addition theorem of spherical harmonics,
$$
\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}Y_{\ell
m}(\hat{x})Y_{\ell m}(\hat{y})^{*} = P_{\ell}(\hat{x}\cdot\hat{y}),
$$
we finally get
$$
G(\hat{x}\cdot\hat{y};t)=\sum_{\ell=0}^{\infty}\left(\frac{2\ell+1}{4\pi}\right){\rm e}^{-\ell(\ell+1)t}P_{\ell}(\hat{x}\cdot\hat{y}).
$$
\subsubsection{Generalization to $S^{n-1}$}
Similar to the spherical harmonics, the hyperspherical harmonics arise as the
angular part of degree-$\ell$ homogeneous harmonic polynomials
$h_{\ell}$ that satisfy $\Delta h_{\ell} =0$. In spherical coordinates
$(r,\xi)$, we can decompose $h_\ell (\mathbf{x})=r^{\ell}\tilde{Y}_{\ell}(\xi)$
\cite{Atkinson:2012tz,Wen:1985fm}, where $\tilde{Y}_{\ell}(\xi)$ is
the desired hyperspherical harmonic. Using the spherical coordinate
Laplacian in $\mathbb{R}^n$ shown in
Equation~\ref{eq:LSeparation}, we get
$$
0=\Delta h_{\ell}(\mathbf{x})=\tilde{Y}_{\ell}(\hat{x})\frac{1}{r^{n-1}}\partial_{r}\left(r^{n-1}\partial_{r}r^{\ell}\right)
- r^{\ell-2}\hat{L}^{2}\tilde{Y}_{\ell}(\xi),
$$
which can be simplified to yield the following eigenvalue equation for the
hyperspherical Laplacian:
$$
\hat{L}^{2}Y_{\ell\{m\}}=\ell(\ell+n-2)Y_{\ell\{m\}},
$$
where the set $\{m\}$ indexes the degenerate eigenstates.
\subsubsection{Lemmas for the proof of convergence}
To construct the eigenfunction expansion of the exact heat kernel and prove its convergence, we need
the following lemmas \cite{Atkinson:2012tz,Wen:1985fm,Lorch:1984fw}:
\begin{lem}
The hyperspherical harmonics are complete on $S^{n-1}$ and resolve
the $\delta$-function
\begin{equation}
\delta(\hat{x},\hat{y})=\sum_{\ell=0}^{\infty}\sum_{\{m\}}Y_{\ell\{m\}}(\hat{x})Y_{\ell\{m\}}^{*}(\hat{y}).
\end{equation}
\end{lem}
\begin{lem}The hyperspherical harmonics satisfy the
generalized addition theorem
$$
\sum_{\{m\}}Y_{\ell\{m\}}(\hat{x})Y_{\ell\{m\}}(\hat{y})^{*}=\frac{1}{A_{S^{n-1}}}\frac{2\ell+n-2}{n-2}C_{\ell}^{\frac{n}{2}-1}(\hat{x}\cdot\hat{y}),
$$
where $C_{\ell}^{\nu}(w)$ are the Gegenbauer polynomials and $A_{S^{n-1}}=2\pi^{n/2}/\Gamma\left(\frac{n}{2}\right)$
is the surface area of $S^{n-1}$.
\end{lem}
\begin{lem}
\label{lem:The-Gegenbauer-polynomials}The Gegenbauer polynomials
$C_{\ell}^{\alpha}(w)$ with $\alpha>0$ and $\ell\ge0$ are bounded
in the interval $w\in[-1,1]$: in particular, $C_{0}^{\alpha}(w)=1$,
$C_{1}^{\alpha}(w)=\alpha w$, and thus, $|C_{1}^{\alpha}(w)|\le\alpha$
for $w\in[-1,1]$. Finally, for $\ell\ge2$,
$$
|C_{\ell}^{\alpha}(w)|\le\left[w^{2}c_{2\ell,2\alpha}+(1-w^{2})c_{\ell,\alpha}\right],
$$
where
$$
c_{\ell,\alpha}=\frac{\Gamma(\frac{\ell}{2}+\alpha)}{\Gamma(\alpha)\Gamma(\frac{\ell}{2}+1)}.
$$
\end{lem}
\subsubsection{The sweet spot of $t$}
Choosing an appropriate diffusion time $t$ for the heat kernel is important for machine
learning applications. Here, we use the degree of self-similarity
measured by the heat kernel as a function of $t$, and propose a choice
for which the self-similarity is neither too large nor too small.
If $t$ is too large, then the self-similarity is roughly the
uniform similarity $1/A_{S^{n-1}}$, thereby losing contrast between neighbors
and outliers. By contrast, as $t$ approaches 0, the
self-similarity becomes infinite, and the sense of neighborhood
becomes too localized. We thus need an intermediate value of $t$, for
which the self-similarity interpolates between the two limits.
The self-similarity is a special value of the heat kernel
\begin{eqnarray*}
G(1;t) & = & \sum_{\ell=0}^{\infty}{\rm e}^{-\ell(\ell+n-2)t}\frac{2\ell+n-2}{n-2}\frac{1}{A_{S^{n-1}}}C_{\ell}^{\frac{n}{2}-1}(1)\\
& = & \frac{1}{A_{S^{n-1}}}\sum_{\ell=0}^{\infty}{\rm e}^{-\ell(\ell+n-2)t}\frac{2\ell+n-2}{n-2}\frac{\Gamma(\ell+n-2)}{\Gamma(\ell+1)\Gamma(n-2)}.
\end{eqnarray*}
Because the series converges rapidly for sufficiently large $t$, we can truncate the series at
$\ell=\ell_{\max}$; i.e.
$$
G(1;t)
\approx \frac{1}{A_{S^{n-1}}}\sum_{\ell=0}^{\ell_{\max}}{\rm e}^{-\ell(\ell+n-2)t}\frac{2\ell+n-2}{n-2}\frac{\Gamma(\ell+n-2)}{\Gamma(\ell+1)\Gamma(n-2)}.
$$
\noindent In the large $n$ limit, we can bound the sum as
\begin{align*}
G(1;t) \leq \frac{1}{A_{S^{n-1}}}\sum_{\ell=0}^{\ell_{\max}}\left({\rm
e}^{-nt}\right)^{\ell}\frac{n^{\ell}}{\ell!} \leq \frac{\exp\left(n{\rm e}^{-nt}\right)}{A_{S^{n-1}}}.
\end{align*}
To keep the self-similarity finite, but larger than the uniform
similarity, suggests the choice for $t$ of order $\log n/n$, at which the self-similarity
is roughly ${\rm e}/A_{S^{n-1}}$. We thus search for an optimal value of
$t$ around $\log n/n$.
\subsubsection{SVM Classification\label{appendix_svm}}
In the main text, we denoted the parametrix and exact heat kernels
normalized by self-similarity as the ``parametrix kernel'' and ``exact
kernel,'' respectively. We then used the linear (lin), Gaussian RBF (rbf),
cosine (cos), parametrix (prx), and exact (ext) kernels in SVM to (1)
classify WebKB-4-University web pages into four classes:
\emph{student}, \emph{faculty}, \emph{course}, and \emph{project}; and
(2) impute the binary mutation status of genes in TCGA-GBM data. The kernel
SVM classification results shown in the main text indicated that the
cosine kernel usually outperformed the linear kernel, most likely
as a pure consequence of the hyperspherical geometry, as we argue below.
The exact kernel outperformed the Gaussian RBF kernel for the WebKB
document data, but the advantage of exact kernel diminished in the
TCGA mutation count data.
Figure 2 compares the
accuracy of SVM using different kernels on the TCGA-GBM data, where
the accuracy ratios rbf/lin, cos/lin, ext/lin, prx/cos, and ext/cos were
greater than 1 for most class sizes $m_{{\rm r}}$. Interestingly, the ratio
cos/lin showed some dependence on the sample size $m_{{\rm r}}$, and the exact kernel
also tended to outperform the Gaussian RBF kernel when $m_{{\rm r}}$
was small; in general, we noted that the hyperspherical kernels tended
to outperform the Euclidean kernels in small-sample-size
classification problems. This pattern may be understood by examining
the generalization error of kernel SVM as follows.
Intuitively, if a generic classifier were closely acquainted with the population
distribution of data through a large sample size, then its predictions
would be more generalizable to unseen samples. The ``largeness'' of
sample size $m$, however, is not explicitly quantifiable unless we
have a natural unit for it. Statistical learning theory
\cite{Vladimir:l-V4Juw7,Vapnik:1994dk,Guyon:1993ub} provides such a unit
associated with a probabilistic upper bound on generalization
errors. That is, with probability at least $1-\eta$, the
generalization error of a binary SVM
classification is bounded from above by
$$ F(\tilde{m};\mu_{{\rm VC}},\eta)=\sqrt{\frac{1}{\tilde{m}}\left[\left(\log2\tilde{m}+1\right)-\frac{\log\frac{\eta}{4}}{\mu_{{\rm VC}}}\right]} $$
where $\mu_{{\rm VC}}$ is the VC-dimension of the classifier, and
$\tilde{m}=m/\mu_{{\rm VC}}$ is the effective sample size. The
derivative of $F(\tilde{m};\mu_{{\rm VC}},\eta)$ with respect to
$\tilde{m}$ is proportional to a positive factor times
$-\log\left[(2\tilde{m})^{\mu_{{\rm VC}}}4/\eta\right]$. Thus, the
upper bound decreases with $\tilde{m}$ when $(2\tilde{m})^{\mu_{{\rm
VC}}}>\eta/4$, and increases otherwise; the critical effective
sample size $\tilde{m}_{{\rm
crt}}=\frac{1}{2}\cdot(\eta/4)^{1/\mu_{{\rm
VC}}}\approx\frac{1}{2}$ for typical values of $\mu_{{\rm VC}}>100$ and
$\eta\in[10^{-3},0.1]$. The VC dimension of a linear kernel SVM can be
estimated using an empirical upper bound
\cite{Vapnik:1994dk,Anonymous:2001vo}
$$
\mu_{VC}\le\mu_{VC}^{*}=\min\left\{ n,\frac{R^{2}}{M^{2}}\right\} +1,
$$
where $n$ is the feature space dimension, $R$ is the radius of the
smallest ball in feature space that encloses all data points, and $M$
is the SVM margin. We evaluated the bound $\mu_{{\rm VC}}^{*}$ for the
TCGA-GBM mutation count data with $C=1$, and found that the linear
kernel had $R^{2}/M^{2}\approx6\times10^{3}$ and thus that $\mu_{{\rm
VC}}^{*{\rm lin}}=n+1\approx4\times10^{2}$. By contrast, the
cosine kernel, which is a linear kernel in the hyperspherically
transformed space with $R\le1$, had $\mu_{{\rm VC}}^{*{\rm cos}}$
approximately in the range $20\sim100\ll\mu_{{\rm VC}}^{*{\rm lin}}$,
as shown in Figure~3A.
This reduction
in the VC-dimension is likely responsible for the classification
improvement of the cosine kernel over the linear kernel. We thus found
that $\tilde{m}_{\cos}=2m_{{\rm r}}/\mu_{{\rm VC}}^{*\cos}>\tilde{m}_{\rm crt}$, while
$\tilde{m}_{{\rm lin}}=2m_{{\rm r}}/\mu_{{\rm VC}}^{*{\rm lin}}< \tilde{m}_{\rm crt}$
for the TCGA-GBM data, and that the cosine kernel accuracy increased
with effective sample size, whereas the linear kernel accuracy tended
to decrease (Figure~3B,C,
consistent with the analysis of
the upper bound on generalization error $F(\tilde{m};\mu_{{\rm
VC}},\eta)$. In addition, the Gaussian RBF and exact kernels
followed similar trends as the linear and cosine kernels, respectively
(Figure~3D,
). Similar to the cosine kernel, the exact
kernel likely inherited the reduction in VC-dimension from the
hyperspherical map; as a result, the accuracy of the exact kernel also
increased with $\tilde{m}_{\cos}$, but with slightly higher accuracy
due to the additional tunable parameter $t$ that can adjust the
curvature of nonlinear decision boundaries. Moreover, the cases of
small sample size where the exact kernel outperformed the Gaussian RBF
kernel corresponded to the cases of larger effective sample size ratio
$\tilde{m}_{\rm cos}/ \tilde{m}_{\rm lin}$ (Figure~
F).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8
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Q: Export a plist in a readable fashion I have a class following NSCoding protocols.
It is designed to store an array, containing objects of a Custom-class type.
I currently can retrieve and view this information in a table view, retrieving and passing the relevant objects.
However, a more accessible, or readable fashion is desirable, As to manually copy this information would be tedious
Is there some way i can retrieve either the entire plist, or take the data from a specific object in it, then send it via email?
or some other export method, perhaps to create a spreadsheet?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,510
|
Anton Eger
Jasper Høiby
Piano Trio Album
Release Date: 8th April 2016
Phronesis | Parallax
LP, CD & Download (MP3 & FLAC).
Bandcamp: http://bit.ly/parallax-EditionStore
IVO NEAME piano
ANTON EGER drums
1. 67000 mph – Anton Eger
2.Ok Chorale – Ivo Neame
3. Stillness – Jasper Høiby LISTEN | WATCH
4. Kite For Seamus – Ivo Neame
5. Just 4 Now – Jasper Høiby
6. Ayu – Anton Eger
7. A Silver Moon – Jasper Høiby
8. Manioc Maniac – Ivo Neame
9. Rabat – Anton Eger
Extraordinary communication between the players is often the foundation of extraordinary music making – as Phronesis prove every time they play. Years of performing, touring and recording have given the three members of the trio a matchless rapport. That inspires an ever-flowing fountain of new music, captured to perfection on this, the Anglo-Scandinavian trio's sixth CD and their fourth for Edition Records.
Recorded in a single day at London's fabled Abbey Road studios, Parallax is a brilliantly realised piece of music-making. It has all the standout features of the trio's work. Rhythmic drive. Constant shifts in mood and texture. Drama heightened by dazzlingly fast reactions. All leavened by a melodic sense all three draw on as much when improvising as composing.
There are new elements to enjoy, too. Pianist Ivo Neame's Manioc Maniac displays a stronger spicing of humour. Jasper Høiby's Stillness, featuring the composer's bowed bass, explores a contemplative mood less prominent in their earlier work. Fans of their more up tempo excursions also have plenty to enjoy, though – witness percussionist Anton Eger's opener, 67,000mph, named for the earth´s speed around the sun.
All are enhanced by the endless rewarding details the three together offer as each piece unfolds. It's an indispensible feature of their live shows which is captured here to perfection in a studio recording. This is truly collective music-making, in composition – with each of the trio contributing three of the nine titles – and in supremely interactive performances that display a brand of musical wit that captivates audiences.
All three players have remarkable facility, giving the unadorned, purely acoustic trio some of the strongest bass playing, fleetest piano, and most orchestral percussion you have ever heard. The music they make is lithe and supple, with the added spark that takes such fluid mobility beyond mere gymnastics on to balletic heights.
Each player writes for the trio knowing that their work will be taken somewhere different each time it is performed. They negotiate sometimes complex arrangements while retaining a spontaneity of suggestion and response that keeps the listener on the edge of their seat every time. The nine versions of new pieces captured here achieve a rare thing that only jazz at its very best can offer – music, created in the moment, that lasts.
Reviews of Parallax:
"The chemistry joining pianist Ivo Neame – with his Django Batesian rhythm-swaps hitched to classic jazz roots in Bill Evans and Chick Corea – to Jasper Hoiby's double-bass muscle and the maniacally personal sound of drummer Anton Eger make Phronesis a wonderful live band. But though live recordings have represented them best, this single-day Abbey Road session gets very close." THE GUARDIAN (UK)
"Brilliant synergy between the players, striking themes and dramatic and intricate ideas…a really exciting experience…" BBC RADIO 3, JAZZ LINE UP, CLAIRE MARTIN (UK)
"Their sixth album, recorded in the hallowed acoustics of Abbey Road, reveals a group that, 10 years on, is still growing and maturing, achieving the sort of deep rapport that other, more temporary groupings, can never replicate." THE IRISH TIMES (EIRE)
"The album possesses enormous bite and an exhilarating in-the-room immediacy." THE ARTS DESK (UK)
"The tension between abandon and control is pitch-perfect, reflecting a band in its prime…" THE FINANCIAL TIMES (UK)
"This is a classy affair, but it's far removed from the insipid post-EST piano minimalism of Go Go Penguin and the like. While the band's themes have that bright and breezy quality associated with contemporary Scandinavian piano music, there's a refreshing vigour to the playing that always keeps things exciting." THE QUIETUS (UK)
"Another exceptional Phronesis album, then? Oh yes." THE JAZZ BREAKFAST (UK)
""Parallax" is an excellent album, with the trio continuing to astound at times, but it is perhaps becoming a little familiar. Too much of a good thing maybe? Who knows. Are they still making brilliant music? Are they still intergalactically interconnected in their interplay? Are they still full of energy and entertainment? Yes to all of the above." UK VIBE (UK)
"They've been real favourites on the show right from the beginning…looking forward to hearing the whole album. I imagine it's going to be as fiery as that!" JAMIE CULLUM BBC RADIO 2 (UK)
"…Parallax opens a new chapter full of musical riches from this most complete and compelling of bands." JAZZWISE **** (UK)
"A genuinely scintillating display of intuitive ensemble interaction where composition, technique and emotional expression achieve a near-perfect, symbiotic balance." RECORD COLLECTOR (UK)
"…[the] music is absolutely gripping…In truth, Phronesis are one of the most exciting jazz trios around…the sheer energy that's generated from this album is simply phenomenal." ALL ABOUT JAZZ (UK)
"Parallax is easily one of the stand-out jazz releases of the spring a filip for the UK jazz scene…" MARLBANK (UK)
"Phronesis have excellent rapport, continuously feeding off each other, whether live or on disc. Parallax, recorded in a single day at London's Abbey Road Studios, characterises the trio's many features such as their rhythmic drive and a common melodic sense exemplified by the constant shifts of mood and texture." BEBOP SPOKEN HERE (UK)
"HOW'S THIS for supreme confidence? One of the world's foremost jazz trios recording an hour of explosive, original material in a single day at London's renowned Abbey Road Studios… and ab-so-lute-ly nailing it! …having honed and established such a distinctive identity and direction over more than a decade of democratic collaboration, it's remarkable how they raise the bar still higher with each new release." AP REVIEWS (UK)
"There's something soothingly addictive about this album. It gets under the skin without you noticing." JAZZ STANDARD (UK)
"…a work of three powerful and talented musicians spewing forth their art. There is no room for star soloists here. This is a light, bright, sparkling work of jazz that will lift the day of anyone who is lucky enough to hear it." THE AUDIOPHILE MAN (UK)
"The fascination with this group lies not so much in the ballads, but in the budding melody lines and eruptive rhythms. It is no coincidence that the drummer Anton Eger successful compositions frames the album. The lively end track, 'Rabat', combines oriental rhythms with the special kind of advanced power-jazz, which Phronesis have mastered to perfection. However it's in the incorporation of lyrical moments that the music really gets interesting. This represents a path that it might be interesting to hear the group pursue further." POLITIKEN (DK)
"…..once again, in the best finest traditions of improvisational music – maintained in the correct conventional form." WESTZEIT (DE)
"The fact is: the small scale, bass-heavy composition and the further nine improvised tracks are masterpieces of patchwork jazz forming larger units from the fragments, reflecting a new world view, far away from any attempt at an overarching synthesis. So the name 'Phronesis' fits the band concept because they bring the atomised parts of the trio together according to the rules of contemporary art." RONDO (DE
"Another extremely interesting release from Edition Records. Phronesis is a very experienced and harmonious team, and Parallax -in my opinion – is their best disk so far. These musicians could, retaining their existing strengths, climb to even higher levels." JAZZPRESS (POL)
"We can confidently call Phronesis the most interesting jazz trio of our time. And in their new album Parallax there is the clear evidence of this…Phronesis is a trio that stands out every time with their super-passionate game and especially exciting compositions that really last. Parallax is therefore a new highlight in their already super rich oeuvre." WRITTEN IN MUSIC (NED)
"This is an energetically charged, electrifying music…" JAZZ'HALO (BE)
"Phronesis is a jazz trio that qualifies for the Champions League. The Danish-Anglo-Norwegian trio, which in many ways has taken over where Esbjörn Svensson Trio slack, certainly has the qualities – both technically and artistically to conquer the world on stage and on record…Parallax is jazz music with speed – not a forced pace, covering over a lack of ability, but a controlled ferocity with an underlying network of terrific details." GAFFA (DK)
"The trio has existed for 10 years. They are now one of the best examples in Europe that trio jazz can be both sophisticated and charming without taking a dallying and boring position. It is a very commendable record." JAZZNYT (DK)
"A key reason for the success of this disc in being a fine record is the coherent and purposeful band sound which at the same time clearly emphasises each instrument." VALON KUVIA (FI)
"The British trio Phronesis has in recent years been one of the hottest on the British jazz scene…Therefore, one should also expect to get served a top-class recording when these three musicians put their mind to recording a new disc…" SALT PEANUTS (NO)
"Although the music has its own intricate and distinctive character – I think Phronesis is the most exciting trio since the Esbjörn Svensson Trio of around a decade ago…There is a drive and an intensity in their music that's rare and… startlingly good." TOR DE JAZZ BLOG (NO)
"Here, the trio truly excels in its uniquely diverse play through unceasing changes of rhythm, expression and dynamics, which alternate nu-jazz groovy, minimalist pieces, odd rhythms, loosing almost free-jazz or even ecstatic attacks or vice versa a coolly lyrical mood …A just wonderful album!" HIS VOICE (CZE)
"When Phronesis gave the album name as a term of geometry, hence astronomy and optics, certainly it was a clear intention. An imaginary line – views of three observers of the music world – again unmistakably intersect." ROZHLAS (CZE)
"Once again, Edition Records lives up to its reputation for great European jazz with Parallax of Phronesis. From cover to cover, the musical texture is transparent and touches the listener's heart." REDCURTAIN MAGAZINE (TR)
'Phronesis is at the world forefront of accomplished jazz piano trios, with high-calibre individuals combining perfectly on their originals in a decidedly Eurojazz style'. THE AUSTRALIAN (AUS)
"This is one of the best piano trios of our day releasing a new album. It's some of the the best, most explosive jazz around and they still don't stop turning heads." NEXTBOP (ONLINE)
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Le Končistá est un sommet de la Slovaquie et de la chaîne des Hautes Tatras. Il culmine à d'altitude.
Première ascension
La première ascension connue date de 1874 et fut réalisée par A. Münich.
Notes et références
Sommet dans le parc national des Tatras (Slovaquie)
Sommet des Hautes Tatras
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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{"url":"https:\/\/ocw.mit.edu\/courses\/18-013a-calculus-with-applications-spring-2005\/pages\/assignments\/","text":"# Assignments\n\n## Review Exercises 1 - 5\n\n### Exercise 1\n\nGiven the following five vectors: A = (1, 2, 3); B = (2, -3, 5); C = (x, y, z); D = (cos t, sin t, t 2); E = (-2, 1, 0).\nDo each of the following:\n\n\u2022 Form the sum: A + B + C.\n\u2022 Compute AB.\n\u2022 Compute A\u00d7(B+C).\n\u2022 Find values for x, y and z for which CA = 0 and CB = 0.\n\u2022 Find the cosine of the angle between A and B. Between B and D (the answer will be a function of t).\n\u2022 Find the projection of E on B.\n\u2022 Find the determinant whose columns are A, B and E; also find the determinant whose columns are A, B and C.\n\u2022 Suppose the point P has coordinates x = 1, y = 2, z = 3. What are its spherical coordinates \u03c1, \u03b8 and \u03a6?\n\u2022 What is the volume of the parallelepiped with edges A, B and E?\n\u2022 Find the projection of D into the xy plane. What is its length?\n\n### Exercise 2\n\nConsider the line containing the points A and B above.\n\n\u2022 Give a parametric representation of the points on that line.\n\n\u2022 Find a unit length \u201ctangent vector\u201d that points in the direction of the line.\n\n\u2022 Find two directions normal to that vector.\n\n\u2022 Consider the plane containing the points A, B and E:\n\n\u2022 Find a (two parameter) parametric representation of the plane.\n\u2022 Find a normal to the plane.\n\u2022 Find an equation that points on the plane all obey.\n\u2022 Suppose we have a new and different product of vectors V@W that has the property V@V = 0 for all V and @ is linear in each argument so that you can apply the distributive law.\n\n\u2022 Deduce something about V@W + W@V by applying same to (V + W)@(V + W).\n\n### Exercise 3\n\nDifferentiate the following functions with respect to the indicated variables:\n\n\u2022 sin (2x).\n\u2022 (sin xy)ex+y with respect to x for fixed y.\n\u2022 x2 + y2 - 3xy with respect to y for fixed x.\n\u2022 (sin (y + s sin t))e-(x+s cos t) with respect to s everything else fixed.\n\u2022 Find the gradient of (sin y)e-x.\n\u2022 Find the directional derivative of this function in the direction whose unit vector is (cos t, sin t).\n\u2022 Find the linear approximation to sin (ex) at x = 0.\n\u2022 Evaluate the derivative with respect to t of (r\u00d7v) where v is dr\/dt; suppose that dv\/dt is in the direction of r. What then is the answer?\n\u2022 Where is 1\/x not differentiable? Where is tan x not differentiable? Where is |x| not differentiable?\n\u2022 Find the derivative of an inverse function to sin (ex) (to define an inverse function completely you have to specify a range; ignore that here).\n\n### Exercise 4\n\n\u2022 Find the gradient of the function r = (x2 + y2)1\/2 and \u03c1 = (x2 + y2 + z2)1\/2.\n\u2022 Find the gradient of 1\/\u03c1.\n\u2022 Find the gradients of cos \u03b8 and of \u03b8.\n\u2022 Find the curl of (y, z, x).\n\u2022 Find the divergence of \u03c13 (remember that \u03c1 = (x, y, z)).\n\u2022 Find the curl of same.\n\n### Exercise 5\n\n\u2022 Find the quadratic approximation to sin xy at x = 1, y = 2 (radians).\n\u2022 Where does this function have critical points (both partial derivatives are 0).\n\u2022 Find at least one saddle point.\n\u2022 Evaluate (a\u00d7b)\u2022(a\u00d7b) by switching a dot and cross product and expressing the triple cross product according the rule for doing same, to get an alternate expression for the same thing entirely in terms of dot products.\n\u2022 Which of the following functions can be defined at x = 0? (1-cos x)\/x2, x2\/sinx, (sin x cos x)\/x2?\n\n#### Learning Resource Types\n\nlaptop_windows Simulations","date":"2022-05-23 17:07:40","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8206660151481628, \"perplexity\": 926.1395188724621}, \"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-2022-21\/segments\/1652662560022.71\/warc\/CC-MAIN-20220523163515-20220523193515-00237.warc.gz\"}"}
| null | null |
\section{Introduction}
In recent years, progressively more people are interacting with virtual voice assistants, such as Siri, Alexa, Cortana and Google Assistant. Interfaces that ignore a user's emotional state or fail to manifest the appropriate emotion can dramatically impede performance and risks being perceived as cold, socially inept, untrustworthy, and incompetent \cite{brave2009emotion}. Because of this, speech emotion recognition is becoming an increasingly relevant task.
Several datasets have been created over the years for training and evaluating emotion recognition models, including SAVEE \cite{savee}, RAVDESS \cite{ravdess}, EMODB \cite{emodb}, IEMOCAP \cite{iemocap}, and MSP-Podcast \cite{lotfian-msppodcast-2019}. With the exception of MSP-Podcast, these datasets are relatively small in size, usually including only a few dozen speakers.
In terms of modeling techniques, many different traditional approaches have been proposed, using hidden Markov models (HMMs) \cite {schuller-hmm-2003, sato-hmm-2007}, support vector machines (SVMs) \cite {shen-svm-2011,rao-svm-2013}, decision trees \cite {borchert-dt-2005, lee-dt-2011}, and, more recently, deep neural networks (DNNs) \cite {mirsamadi-rnn-2017, satt-cnnlstm-2017, sarma-raw-2018}. Yet, while DNNs have shown large gains over traditional approaches on tasks like automatic speech recognition (ASR) \cite{gulati-conformer-2020} and speaker identification \cite{matvejka-speakerid-2016}, the gains observed on emotion recognition are limited, likely due to the small size of the datasets.
A common strategy when dealing with small training datasets is to apply transfer learning techniques.
One approach for transfer learning is to use a model learned for a certain auxiliary task for which large datasets are available for training to improve robustness for the task of interest for which data is scarce. The model learned on the auxiliary task can be used as feature extractor or fine-tuned, after replacing some of its final layers, to the task of interest.
Recently, transfer learning approaches have been explored in the field of speech emotion recognition. In \cite{zhao-tl-2021}, a deep neural network based on transformers \cite{vaswani-attention-2017} is pretrained on LibriSpeech \cite{librispeech} using multiple self-supervised objectives at different time scales. Then, the model is used as a feature extractor or fine-tuned for speech emotion recognition, among other downstream tasks. Similarly, in \cite{li-tl-2021} and \cite{zhang-tl-2021}, a deep encoder is pretrained in a contrastive predictive coding task, and the resulting embeddings are tested in speech emotion datasets.
Recently, several models for automatic speech recognition (ASR) which use self-supervised pretraining have been released, including wav2vec \cite{wav2vec} and VQ-wav2vec \cite{vq-wav2vec}. A few recent studies \cite{siriwardhana-vqw2v-2020, macary-w2v-2020, boigne-w2v-2020} have successfully applied representations from these models as features for emotion recognition.
In line with those works, in this paper, we explore the use of the wav2vec 2.0 model \cite{wav2vec2}, an improved version of the original wav2vec model, as a feature extractor for speech emotion recognition.
The main contributions of our paper are (1) the use of wav2vec 2.0 representations for speech emotion recognition which, to our knowledge, had never been done for this task, (2) a novel approach for the downstream model which leverages information from multiple layers of the wav2vec 2.0 model and leads to significant improvements over previous approaches, and (3) an analysis of the importance of the different layers in wav2vec 2.0 for the emotion recognition task. Our results are superior to others in the literature for models based only on acoustic information for IEMOCAP and RAVDESS. The code to replicate the results of this paper will soon be released at https://github.com/habla-liaa/ser-with-w2v2.
\section{Methods}
In our study, we extracted features from two released wav2vec 2.0 models and used them for speech emotion recognition. In this section, we describe the wav2vec 2.0 model, the datasets used for training and evaluation, and the downstream models.
\vspace{-0.1cm}\subsection{Wav2vec 2.0 model architecture}
\vspace{-0.1cm}
Wav2vec 2.0 \cite{wav2vec2} is a framework for self-supervised learning of representations from raw audio. The model consists of three stages. The first stage is a {\bf local encoder}, which contains several convolutional blocks and encodes the raw audio into a sequence of embeddings with a stride of 20 ms and a receptive field of 25 ms. Two models have been released for public use, a large one and a base one where the embeddings are 1024- and 768-dimensional, respectively. The second stage is a {\bf contextualized encoder}, which takes the local encoder representations as input. Its architecture consists of several transformer encoder blocks \cite{vaswani-attention-2017}. The base model uses 12 transformer blocks with 8 attention heads each, while the large model uses 24 transformer blocks with 16 attention heads each. Finally, a {\bf quantization module}, takes the local encoder representations as input and consists of 2 codebooks with 320 entries each. A linear map is used to turn the local encoder representations into logits. Given the logits, Gumbel-Softmax \cite{jang-gumbelsoftmax-2016} is applied to sample from each codebook. The selected codes are concatenated and a linear transformation is applied to the resulting vector leading to a quantized representation of the local encoder output, which is used in the objective function, as explained below.
\subsection{Wav2Vec 2.0 pretraining and finetuning}
\vspace{-0.1cm}
The wav2vec 2.0 model is pretrained in a self-supervised setting, similar to the masked language modelling used in BERT \cite{devlin-bert-2018} for NLP. Contiguous time steps from the local encoder representations are randomly masked and the model is trained to reproduce the quantized local encoder representations for those masked frames at the output of the contextualized encoder.
The training objective is composed by terms of the form
\begin{equation}
L_m = -\log \frac{\exp(\similarity(c_t,q_t)/\kappa)}{\sum_{\tilde{q} \in \tilde{Q}}{\exp(\similarity(c_t,\tilde{q})/\kappa)}}
\label{eq:wav2vec2_loss}
\end{equation}
where $\similarity(c_t,q_t)$ is the cosine distance between the contextualized encoder outputs $c_t$ and the quantized local encoder representations $q_t$. $t$ is the time step, $\kappa$ is the temperature and \~{Q} is the union of a set of K \emph{distractors} and $q_t$. The distractors are outputs of the local encoder sampled from masked frames belonging to the same utterance as $q_t$. The contrastive loss is then given by $L_m$ summed over all masked frames. Finally, terms to encourage diversity of the codebooks and L2 regularization are added to the contrastive loss.
The main goal of the wav2vec 2.0 paper was to use the learned representations to improve ASR performance, requiring less data for training and enabling its use for low resource languages.
To this end, the model trained as described above is finetuned for ASR using a labelled speech corpus like LibriSpeech. A randomly initialized linear projection is added at the output of the contextual encoder and the connectionist temporal classification (CTC) loss \cite{graves-ctc-2006} is minimized. The models finetuned in 960 hours of LibriSpeech reach state of the art results in automatic speech recognition when evaluated in different subsets of LibriSpeech. Even when finetuning using considerably less hours, wav2vec 2.0 models reach a performance comparable to the state of the art.
In this paper we compared the performance in speech emotion recognition when using both the wav2vec 2.0 base model pretrained in Librispeech without finetuning\footnote{https://dl.fbaipublicfiles.com/fairseq/wav2vec/wav2vec\_small.pt} (we will call this model Wav2vec2-PT), and a model finetuned for ASR using a 960-hour subset of Librispeech\footnote{https://dl.fbaipublicfiles.com/fairseq/wav2vec/wav2vec\_small\_960h.pt} (Wav2vec2-FT). In both cases, we used the base model as we did not see significant performance improvements when using the large one, and it allowed us to reduce computational requirements.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{figs/is2021-downstreams-3.pdf}
\vspace{-0.1cm}
\caption{Downstream models used for this work. The green blocks represent trainable layers. We call the model in figure a) Dense, when using a pointwise convolutional layer, and LSTM, when using an LSTM layer. We call the model in b) the Fusion model. Blocks for each time-step have a height dimension for the features, and an optional depth dimension for the layers in the wav2vec 2.0 model. }
\label{fig:downstream-models}
\end{figure}
\vspace{-0.1cm}
\subsection{Features}
\vspace{-0.1cm}
We compare results when using the output of the local and the contextualized encoders as input to the downstream models. We also propose to use a weighted average of the outputs of the 12 transformer blocks, along with those of the local encoder. The weights of this average are trained along with the rest of the downstream model, as explained in the next subsection.
As baseline features, we also calculated magnitude spectrograms with a hanning window of 25 ms and a hop size of 10 ms, and eGeMAPS \cite{eyben-geneva-2015} low level descriptors (LLD), which are commonly used in the emotion recognition literature. The eGeMAPS features were extracted using opensmile-python~\cite{eyben-opensmile-2010}. In order to match the sequence lengths of the eGeMAPS features, which use a stride of 10 ms, with the lengths of the wav2vec 2.0 features, which have a stride of 20 ms, we downsampled the eGeMAPS LLDs by averaging every 2 consecutive frames.
All features were normalized by subtracting the mean and dividing by the standard deviation of that feature over all the data for the corresponding speaker. When disabling speaker normalization for comparison of results, we replaced it with global normalization per feature. In this case, the statistics are computed over the training data only, excluding the test data.
\subsection{Downstream models}
\vspace{-0.1cm}
Our downstream model is inspired by \cite{boigne-w2v-2020} due to its simplicity, which should reduce the chance of overfitting.
The model we will refer to as \emph{Dense}, consists of 2 dense layers with 128 neurons, ReLU activation and dropout with a probability of 0.2, applied independently to each frame of the input features sequence~$f$. This is equivalent to using 1D pointwise convolutional layers (with a kernel size of 1) and 128 filters. The outputs are averaged over time resulting in a vector of size 128. Finally, another dense layer with softmax activation returns probabilities for each of the emotion classes. During the training of the downstream model, the weights of the wav2vec 2.0 model remain unaltered, so it serves as a feature extractor.
For computational reasons, we used a maximum sequence length of 400 for IEMOCAP and 250 for RAVDESS, as input to the network. This is equivalent to 8 seconds and 5 seconds, respectively. Note that the output of the contextualized encoder can contain information from the full input waveform which might be longer than the sequence seen by the network. This is because each output of the contextual encoder has a receptive field given by the whole input waveform, since this encoder is a transformer. On the other hand, the local encoder has a limited receptive field, so its outputs can only capture local information.
For the case in which we take as features the activations from both the wav2vec 2.0 local encoder and the transformer blocks, we incorporate a trainable weighted average as the first layer. This layer learns the weights $\alpha_i$ for each of the wav2vec 2.0 layer activations, $f_i$, where $f_0$ corresponds to the local encoder output, $f_1$ through $f_{11}$ are the outputs of the internal blocks in the transformer and $f_{12}$ is the output of the contextualized encoder. Then, the activations are combined as follows
\begin{equation}
f = \frac{\sum_{i=1}^{N}{\alpha_i f_i}}{\sum_{i=1}^{N}{\alpha_i}}
\end{equation}
The weights $\alpha_i$ are initialized with 1.0.
Note that the layer activations can be combined in this way because they are all the same size.
This way of extracting features from a pretrained model is similar to the approach used in ELMO \cite{peters-elmo-2018} for NLP.
In the results section, we compare performance of the Dense model with the LSTM model, in which the second dense layer is replaced by an LSTM layer. Finally, we also evaluate a third model, called Fusion model, which incorporates a branch taking as input eGeMAPS features. In this last model, the outputs of the first dense layer are concatenated before applying the second dense layer. The described downstream models can be seen in Figure \ref{fig:downstream-models}.
The downstream models were trained using batches of 32 utterances, Adam optimizer with a learning rate of 0.001, and early stopping with a patience of 4 epochs monitoring the validation loss.
\subsection{Datasets}
\vspace{-0.1cm}
The Ryerson Audio-Visual Database of Emotional Speech and Song
(RAVDESS) \cite{ravdess} is a multi-modal database of
emotional speech and song. It features 24 different actors (12 males and 12 females) enacting 2 statements: "Kids are
talking by the door" and "Dogs are sitting by the door." with 8 different emotions: happy, sad, angry, fearful, surprise,
disgust, calm, and neutral. These emotions are expressed in two different intensities: normal and strong, except for neutral (normal only). Each of the combinations was spoken and sung, and repeated 2 times, leading to 104 unique vocalizations per actor.
Following \cite{venkataramanan2019emotion}, we merged the neutral and calm emotions, resulting in 7 emotions, and used the first 20 actors for training, actors 20-22 for validation to do early stopping, and actors 22-24 for test.
The Interactive Emotional Dyadic Motion Capture
(IEMOCAP) dataset \cite{iemocap} has a length of approximately 12 hours and consists of scripted and improvised dialogues by 10 speakers. It is composed of 5 sessions, each including speech from an actor and an actress. Annotators were asked to label each sample choosing one or more labels from a pool of emotions. In this work, we used 4 emotional classes: anger, happiness, sadness and neutral, and following the work in \cite{Fayek2017}, we relabeled excitement samples as happiness. Instances from other classes and with no majority label across the annotations were discarded.\footnote{Note that discarding no-agreement samples and samples from non-target emotions is not an ideal practice \cite{riera2019}. Here, we decided to do this since it is standard practice in emotion recognition literature, facilitating comparisons across papers.} We also trimmed the waveforms to 15 seconds to reduce the computational requirements when extracting wav2vec 2.0 features. Only 2\% of the waveforms were longer than 15 seconds.
To evaluate the models in IEMOCAP, we performed 5-fold cross-validation, leaving one session out for each of the folds, as it is a standard practice with this dataset. We used one of the 4 training sessions to perform early stopping for each cross-validation model.
\begin{table}[t]
\centering
\caption{Average recall (\%) obtained for the different features and models explored in IEMOCAP and RAVDESS. All results are obtained with the Dense downstream model.}
\begin{tabular}{llcc}
\toprule
Pretrained & Features & \multicolumn{2}{c}{Dataset} \\
model & & IEMOCAP & RAVDESS\\
\midrule
None & eGeMAPS& 52.4 $\pm$ 0.1 & 57.0 $\pm$ 2.4\\
& Spectrogram& 49.8 $\pm$ 1.0 & 44.5 $\pm$ 0.8\\
\midrule
Wav2vec2-PT & Local enc. & 60.3 $\pm$ 0.7 & 65.4 $\pm$ 1.7\\
& Cont. enc. & 58.5 $\pm$ 0.6& 69.0 $\pm$ 0.2\\
& All layers & \textbf{67.2} \boldsymbol{$\pm$} \textbf{0.7} & \textbf{84.3} \boldsymbol{$\pm$} \textbf{1.7}\\
\midrule
Wav2vec2-FT & Local enc. & 57.3 $\pm$ 1.0& 58.8 $\pm$ 2.7\\
& Cont. enc. & 44.6 $\pm$ 1.0& 37.5 $\pm$ 3.0\\
& All layers & 63.8 $\pm$ 0.3& 68.7 $\pm$ 0.9\\
\bottomrule
\end{tabular}
\label{tab:results-features}
\end{table}
\section{Results and discussion}
We calculated the performance of our models by training them 5 times with different seeds. Table \ref{tab:results-features} shows the average recall over all emotion classes obtained using the different features extracted from wav2vec 2.0 models. We also show two systems based on eGeMAPS and spectrogram features, which can be considered as baselines. In all cases, features are normalized by speaker and the Dense model architecture in Figure \ref{fig:downstream-models} is used as downstream model. We can see that features for both wav2vec 2.0 models, the one finetuned in 960 hours of Librispeech (wav2vec2-FT) and the one that is not finetuned (wav2vec2-PT), the local encoder representations lead to better results than both of the baseline features. It is worth noting that eGeMAPS, spectrograms and the wav2vec 2.0 local encoder representations contain information restricted only to a local window around each frame, of 60 ms for eGeMAPS, and 25 ms for the others. Also, the downstream model combines information from consecutive frames using just global average, which is a very simple approach that might be suboptimal because it cannot take into account temporal patterns in the features. In spite of that, the local encoder representations, particularly the ones obtained from the PT model, reach a performance comparable to much more complex models like the one proposed in \cite{satt-cnnlstm-2017}, which is a CNN with Bi-LSTM layers trained in a fully supervised setting with spectrograms as input.
We can also see that the wav2vec2-PT features perform better than wav2vec2-FT features in all cases. In particular, the model using only the contextualized encoder outputs for the wav2vec2-FT model has the worst results in the table. We hypothesize that this is because when the model is finetuned for an ASR task, information that is not relevant for that task but might be relevant for speech emotion recognition is lost from the embeddings. For example, information about the pitch might not be important for speech recognition, while it is essential for speech emotion recognition.
Further, Table \ref{tab:results-features} shows that using a weighted average of the transformer blocks outputs along with the local encoder outputs (rows labelled ''All layers'') results in better performance than using only the local or the contextual encoder outputs. Using only the local encoder representations might not give information about events occurring at time scales larger than the receptive field (25 ms). Further, the output of the contextual encoder might be similar to the quantized local encoder representations, since the contrastive loss objective is designed to achieve this goal. Hence, using both of these layers, along with all intermediate layers in the transformer provides additional information that is valuable to the model.
Figure \ref{fig:layerweights} shows the weights $\alpha_i$ from the weighted average layer once the downstream models are trained.
We can see that the middle layers are given larger weights. This could be because these layers have already contextualized enough information from the local encoder, but they are not yet too specific to the pretraining or finetuning task as the last layers. Also, in the case of the feature extracted from the wav2vec2-PT model, the weights tend to be larger in the layers closer to the output when compared with the weights for the features from the wav2vec2-FT model. This again suggests that the layers of the wav2vec2-FT model closer to the output are less useful for emotion recognition than those of the wav2vec2-PT model.
Finally, note that in both datasets, the different training seeds lead to very similar weights, as observed from the error bars in Figure \ref{fig:layerweights}, from which we can conclude that these weights are not too sensitive to the neural network initialization.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/fig-layerweights.pdf}
\vspace{-0.5cm}
\caption{Weights of the trainable weighted average layer of the dense model. Index 0 corresponds to the local encoder output, and indices 1 through 12 correspond to the outputs of the transformer blocks in the contextual encoder, with layer 12 being the output layer. Solid lines correspond to the wav2vec2-PT features, while dashed lines correspond to the wav2vec2-FT features
}
\label{fig:layerweights}
\end{figure}
Finally, we experimented with several variations of the best performing model in Table~\ref{tab:results-features}. The first line in Table~\ref{tab:variant_results} shows the results for that system, corresponding to the fifth line in Table~\ref{tab:results-features}.
The rest of the lines in Table~\ref{tab:variant_results} show results for that model and the others in Figure \ref{fig:downstream-models} applying global normalization instead of speaker-dependent normalization.
This is the scenario that is most commonly used in papers as is the one with less assumptions about the available data, treating all samples from a speaker as independent from each other, not assuming that additional samples from a speaker are available at test time.
Comparing the first and the second line in Table \ref{tab:variant_results}, we can see that results significantly degrade when doing global normalization. This is expected since normalization by speaker helps the model focus on the emotion characteristics by eliminating part of the speaker information. The degradation is larger in RAVDESS probably because, in this dataset, audios from different emotions do not have much variation in lexical content. Hence, by reducing the effect of speaker in the features all that is left is the variation due to the emotions. On the other hand, in IEMOCAP data, the variation due to lexical content is still in the samples after speaker normalization.
The third and fourth lines in Table \ref{tab:variant_results} show the results obtained using the LSTM model, and the Fusion model. The latter fuses eGeMAPS with the wav2vec 2.0 features.
We can see that using an LSTM layer before the global pooling, does not seem to bring improvements over using a simple dense layer. This might be because wav2vec 2.0 features are already contextualized and have global information about the full utterance. Also, LSTMs might be more prone to overfitting or optimization problems than a simpler dense layer. Finally, the table shows some modest improvements from the addition of eGeMAPS features, suggesting that wav2vec 2.0 features may be lacking some of the information present in eGeMAPS.
\begin{table}[t]
\centering
\caption{Average recall from the model variants in Figure \ref{fig:downstream-models} using the wav2vec2-PT features from all layers, with two types of normalization (top block) compared with results from other works, which all use global normalization (bottom block). In bold are the best results using global normalization.}
\begin{tabular}{lcc}
\toprule
Model - Norm & IEMOCAP & RAVDESS \\
\midrule
Dense - Speaker & 67.2 $\pm$ 0.7 & 84.3 $\pm$ 1.7 \\
Dense - Global & 65.8 $\pm$ 0.3 & 75.7 $\pm$ 2.3 \\
LSTM - Global & 64.8 $\pm$ 1.9 & 74.6 $\pm$ 3.7 \\
Fusion - Global & \textbf{66.3} \boldsymbol{$\pm$} \textbf{0.7}& \textbf{77.5} \boldsymbol{$\pm$} \textbf{1.0}\\
\midrule
BiLSTM w. attention \cite{mirsamadi-rnn-2017} & 58.8 & - \\
CNN-BiLSTM \cite{satt-cnnlstm-2017} & 59.4 & - \\
TDNN-LSTM w. attention \cite{sarma-raw-2018} & 60.7 & - \\
Wav2Vec \cite{boigne-w2v-2020} & 64.3 & - \\
\bottomrule
\end{tabular}
\label{tab:variant_results}
\end{table}
Table \ref{tab:variant_results} also shows some of the results obtained in other works on the IEMOCAP dataset, using the same experimental setup we are using. For a fair comparison with our work, we restrict the comparison to models in the literature that do not use automatic or manual transcriptions.
We can see that all of our models perform better than the state of the art.
Finally, we also compare our results with those in \cite{issa-ser-2020}, which consists of a deep convolutional neural network using acoustic features as input. For this comparison, we used the dense downstream model with global normalization and imitated their experimental setup both for IEMOCAP and RAVDESS. For IEMOCAP, they only use the improvised sessions and full agreement utterances. For RAVDESS, they use 5 fold cross-validation, dividing the data randomly. Moreover, they do not merge calm and neutral emotions, so the total number of emotions to be predicted is 8. We outperformed the models in \cite{issa-ser-2020} for both datasets obtaining an average recall of 84.1 $\pm$ 1.2 \% in RAVDESS, and 72.1 $\pm$ 0.9\% in IEMOCAP, compared to the results in the paper which are 64.3\%
and 71.6\%, respectively.
\section{Conclusions}
In this work, we explored different ways of extracting and modeling features from pretrained wav2vec 2.0 models for speech emotion recognition. We proposed to combine the different layers in the wav2vec 2.0 model using trainable weights and model the resulting features with a simple DNN with a time-wise pooling layer. We evaluated our models on two standard emotion datasets, IEMOCAP and RAVDESS, and showed superior results on both cases, compared to those in recent literature. We found that the combination of information from different layers in the wav2vec 2.0 model led to improved results over using only the encoder outputs, as in previous works.
Further, we found that the combination of the wav2vec 2.0 features with a set of prosodic features gave additional gains, suggesting that the wav2vec 2.0 model does not contain all the prosodic information needed for emotion recognition.
Finally, we showed that a wav2vec 2.0 model finetuned for the task of ASR worked worse than the one trained only with the self-supervised task, indicating that the finetuning eliminates information from the embeddings that is useful for emotion recognition.
\section{Acknowledgements}
This material is based upon work supported by a Google Faculty Research Award, 2019, and an Amazon Research Award, 2019.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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SimonBurchell said:
It will be worse than that, your car will completely shut down for an update just while you are in the middle of switching lanes having overtaken an articulated lorry on the motorway.
I was once driving a mercedes sprinter van that did that on the M1.
Reactions: IbisNibs, Trevp666, Victory and 2 others
SimonBurchell
Somewhere in the labyrinth
You see, who needs Microsoft?
Reactions: IbisNibs, Trevp666, Victory and 1 other person
Planes are as bad. I was due to fly out to Guernsey from Stansted, 5 or 6 years ago, and all us passengers got to the little bit before the steps down to the tarmac, ready to board the plane, and the staff member there told us to sit and wait for a little while.
About 45 minutes later we were allowed to board, whereupon the captain informed us that the aircraft had refused to move until it had completed a software update.
Reactions: Floyd1, IbisNibs, Victory and 3 others
TBH I reckon everyone was relieved it had decided that it needed an update before it took off.
Reactions: Spudrick68, Floyd1, Mythopoeika and 1 other person
Min Bannister
Possessed dog
The Spanish government recently ran an ad campaign aimed at women with the aim of showing that all women's bodies are "beach ready" no matter size or shape. It features a selection of ladies badly photoshopped onto a picture of a beach. It has transpired however that at least two of the women had their bodies used without their consent. To add the icing on the cake, one of them had her prosthetic leg edited out as apparently "all women" are only beach ready if they possess a full complement of limbs.
You couldn't make it up!
A British model says she was left furious after the Spanish government used her image without permission and edited out her prosthetic leg.
The photo was used as part of a body positivity campaign launched by Spain's equality ministry.
But Sian Green-Lord, 32, said she only found out about the campaign when friends messaged her about it.
She is the second woman featured in the campaign to say her photos were used without permission.
On Friday, Nyome Nicholas-Williams, from London, told the BBC that an image from her Instagram page was used in the poster released by the Spanish Institute for Women
More at the link.
"Given the - justified - controversy over the image rights in the illustration, I have decided that the best way to make amends for the damages that may have resulted from my actions is to share out the money I received for the work and give equal parts to the people in the poster," the artist said.
You mean she got paid for that?
Reactions: GerdaWordyer, IbisNibs, Victory and 2 others
Min Bannister said:
Beginners error, surely, by someone with no idea about copyright law... otherwise you wouldn't plaster it all over a national poster campaign.
Reactions: Min Bannister
Twitter: Calum Macdonald @CalumAM
Yes I have just taken delivery of my new Wi-Fi enabled dishwasher (thank you, landlord). And yes it does mean that after I have stood right beside it, loading it with dishes, I can then leave the kitchen and turn it on from elsewhere in the flat. CONVENIENT, I'm sure you'll agree
Also Twitter:
I was discussing this topic with a Scottish colleague a couple of days ago.
He said he'd just bought an Internet-enabled Candy tumble dryer, so you can stop/start and monitor the drying process from your smartphone.
I did ask him if it's been known to sing "Donald, where's yer troosers?".
Reactions: Spudrick68, IbisNibs and Victory
The Spanish government recently ran an ad campaign
All we need now is to find out that the picture of the beach that was used, was in fact a beach in (eg) Cornwall.
Reactions: Victory, Min Bannister and SimonBurchell
From a bishop's church in Utrecht:
Reactions: JamesWhitehead, IbisNibs, pandacracker and 2 others
IbisNibs
Exotic animal, sort of . . .
Outside my comfort zone.
Sooo . . . what exactly do you two do for a living that leaves you with so much time on your hands your colleague has to stay busy by keeping tabs on his dryer? What does this contraption provide him with, humidity readings? All you need for knowing when the drying cycle is over is a timer — but a timer's not good enough, huh? Not enough interruptions for him huh? A timer goes off only once, and he prefers to keep looking at his phone over and over and over throughout the drying process, huh?
Reactions: Trevp666, PeteS, Floyd1 and 1 other person
IbisNibs said:
Far easier to take your laundry down to the canal like I do.
Reactions: Trevp666 and PeteS
What, throw it in and buy new stuff online?
Reactions: Trevp666, PeteS and Floyd1
Dang @SimonBurchell you beat me to it - I was going to say 'throw it in and never see it again', lol.
Reactions: SimonBurchell and Floyd1
Rugby star died 'after suffering heart attack while beating up girlfriend'
A former British Rugby League player 'viciously beat up' his girlfriend of three years just moments before his death at a four-star hotel in Italy.
Prosecutors investigating the attack in the early hours of July 16 had to wait more than a week to talk to Jennie due to the extent of her injuries.
Her lawyer Stefano Goldstein told MailOnline: 'My client Jennie was the victim of terrible assault and needed several operations during a lengthy stay in hospital.
'She has cooperated fully with the authorities who are investigating the matter and we expect the case to be closed within the next few days but for Jennie the trauma of what happened will not end as she will need medical care in England.
'Ricky had suffered mental health issues in recent times and experienced a serious downturn in the last six months,' a family friend added.
Although toxicology results are yet to be confirmed, police say the rugby star had taken cocaine and alcohol which triggered his death.
A post-mortem examination confirmed he died from a heart attack.
Reactions: Mythopoeika
My sympathy levels are low.
Reactions: CALGACUS03 and Spudrick68
I find myself in agreement. By all means partake in alcohol & cocaine if you want, but beating up your girlfriend is stepping over the line.
At the Dokumenta in Kassel. All over the place they have these adverts for oriental chicken shops. This one was very ironic. (This is art BTW.)
Reactions: GerdaWordyer, Tunn11, Trevp666 and 2 others
https://www.goodreads.com/book/show...s?ac=1&from_search=true&qid=VyEfQSrBRp&rank=1
I find this funny:
Just as patients do not come to therapy with a "genuine desire to change," they do not come with a "genuine desire for self-knowledge." Although at the outset many patients express a desire to know what went wrong, what they are doing that keeps backfiring, why their relationships always fall apart, and so on, there is-Lacan suggests-a more deeply rooted wish not to know any of those things. Once patients are on the verge of realizing exactly what it is they have done or are doing to sabotage their lives, they very often resist going any further and flee therapy. When they begin to glimpse their deeper motives and find them hard to stomach, they often drop out. Avoidance is one of the most basic neurotic tendencies.
Freud occasionally talked about a drive to know (Wissentrieb), but Lacan restricts such a drive to children's curiosity about sex ("Where do babies come from?"). In therapy, Lacan says, the analysand's basic position is one of a refusal of knowledge, a will not to know (a ne rien vouloir savoir)."' The analysand wants to know nothing about his or her neurotic mechanisms, nothing about the why and wherefore of his or her symptoms. Lacan even goes so far as to classify ignorance as a passion greater than love or hate: a passion not to know."
https://slate.com/culture/2022/08/c...&utm_source=article&utm_content=twitter_share
There's something adolescent about the belief that trauma automatically makes a person interesting or deep, but Hoover's characterization needs all the help it can get to make her heroes and heroines seem more than generic. Often the only distinguishing traits they possess are their traumas: past abuse, burn scars, the death of a child (or children), a conviction for manslaughter in the accidental death of a former lover, a loveless childhood, and so on.
The blandness of Hoover's characters makes them easy for anyone to identify with, and the smooth, featureless quality of her prose makes her novels easy to breeze through in a day or two. They are built of clichés, which is not necessarily a drawback in romance fiction, where the deployment of familiar devices feels comforting.
EnolaGaia
I knew the job was dangerous when I took it ...
(ACCOUNT RETIRED)
A popular Saint Louis area water park has been forced to close for the season as a result of flooding from recent record rains.
Aquaport closed for season following flooding damage
Aquaport decided to close for the season on Monday following the damage they sustained from flooding on July 26.
The outdoor water park in Maryland Heights said, "While we were initially hopeful the facility would be able to open at some point this summer, it has become apparent that the cleanup and repairs needed to safely operate the facility will not be able to be completed this season." ...
FULL STORY: https://fox2now.com/news/missouri/aquaport-closed-for-season-following-flooding-damage/
Reactions: sherbetbizarre, ramonmercado and Ascalon
Respected snake researcher dies from rattlesnake bite
A respected snake researcher who had been making significant discoveries about the species since childhood has died after being bitten by a timber rattler.
William H. "Marty" Martin died Aug. 3 after being bitten the day before by a captive snake on the property at his home in Harpers Ferry, West Virginia.
https://apnews.com/article/science-mountains-virginia-obituaries-23be2c4051e18c447f1ed2a228e4aa09
Reactions: Spudrick68, ramonmercado and Min Bannister
maximus otter said:
Hazards of the job, I suppose.
Good job they've got 'windows-747, 20' already onboard. . . !
Twitter, Nassim Taleb:
Reactions: Spudrick68, ramonmercado and Mythopoeika
Must be a very worrying time for anyone who has dumped a body in a reservoir.
Reactions: Spudrick68, pandacracker, Ascalon and 2 others
Not sure I agree with this but it shows how much Governments rely on taxes from alcohol and tobacco;
Japan urges its young people to drink more to boost economy.
Japan's young adults are a sober bunch - something authorities are hoping to change with a new campaign.
The younger generation drinks less alcohol than their parents - a move that has hit taxes from beverages like sake (rice wine).
So the national tax agency has stepped in with a national competition to come up with ideas to reverse the trend.
The "Sake Viva!" campaign hopes to come up with a plan to make drinking more attractive - and boost the industry.
https://www.bbc.co.uk/news/world-asia-62585809
Reactions: Ronnie Jersey and ramonmercado
Reactions: Floyd1, Trevp666, sherbetbizarre and 1 other person
Igor Schatz @Copernicus2013
China demolishing unfinished buildings.. beautiful Keynesianism at work
Reactions: pandacracker and Trevp666
China is very wasteful with its building programs. Property speculation will crash their economy.
Reactions: Trevp666
https://www.pcgamer.com/mark-zucker...e-got-was-this-stupid-selfie/?utm_source=digg
I don't want to start this post by personally attacking Mark Zuckerberg's eyes, which have in years past been described as "two weird lil black marbles" and "vacant, black shark eyes." Who wouldn't look utterly bereft of a soul after trying to explain the internet to the United States Senate? I also don't need to suggest Mark Zuckerberg may in fact be a robot, since there's an entire meme community devoted to that joke. I am avoiding these easy, obvious dunks on ol' Zuck because his latest "metaverse" selfie looks so bad, he's already owned himself harder than I possibly could.
Zuckerberg posted a screenshot from Meta's Horizon Worlds (opens in new tab) on Facebook earlier this week, saying that he's "looking forward to seeing people explore and build immersive worlds" in the VR "metaverse." The screenshot is not just bad: it is exquisitely bad. It is ironic clipart bad. It is stock-cratering, anyone-other-than-a-billionaire-CEO-would-immediately-be-fired-for-this bad. Meta spent $10 billion on developing whatever the hell it's doing with the metaverse last year, and all it's got to show for it is a baby doll-faced Zuckerberg hovering in front of a miniature Eiffel Tower.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: Calling fragment from ViewPager fragment I am getting error while calling Fragment from tab fragment.
I have MainActivity which use TabFragment which use ViewPager.
In ViewPager I have 4 tabs. I am calling another fragment from 1st tab fragment.
MainActivity
TabFragment tabFrag = new TabFragment();
mFragmentManager = getSupportFragmentManager();
mFragmentTransaction = mFragmentManager.beginTransaction();
mFragmentTransaction.replace(R.id.containerView,new TabFragment()).commit();
Now from TabFragment which extends Fragment I am calling 4 fragment on different tabs.
** @Override
public Fragment getItem(int position){
Fragment fragment=null;
switch (position){
case 0 : {
fragment= new PostFragment();
Bundle bundle = new Bundle();
bundle.putString("fragmentName","0");
fragment.setArguments(bundle);return fragment;
}
case 1 : {fragment= new EventFragment();
Bundle bundle = new Bundle();
bundle.putString("fragmentName","1");
fragment.setArguments(bundle);return fragment;}
case 2 : {fragment= new ServiceFragment();
Bundle bundle = new Bundle();
bundle.putString("fragmentName","2");
fragment.setArguments(bundle);return fragment;
}
case 3 : {fragment= new AccountFragment();
Bundle bundle = new Bundle();
bundle.putString("fragmentName","3");
fragment.setArguments(bundle);return fragment;
}
}
return fragment;
}
Now In PostFragment I am calling another fragment PostQuestionFragment on button click.
public void onClick(View view) {
int id = view.getId();
switch (id){
case R.id.post:
animateFAB();
break;
case R.id.postQue:
{
FragmentManager fragmentManager = getFragmentManager();
FragmentTransaction fragmentTransaction = fragmentManager.beginTransaction();
PostQuestionFragment postQuestionFragment = new PostQuestionFragment();
fragmentTransaction.replace(R.id.containerView, postQuestionFragment);
fragmentTransaction.addToBackStack(null);
fragmentTransaction.commit();
break;
}
case R.id.fab2:
break;
}
}
I am getting error on fragmentTransaction.replace(R.id.containerView, postQuestionFragment); Below is the log information.
4836/com.lab.bankersbank E/AndroidRuntime: FATAL EXCEPTION: main
Process: com.lab.bankersbank, PID: 14836
java.lang.IllegalArgumentException: No view found for id 0x7f0c007f (com.lab.bankersbank:id/containerView) for fragment PostQuestionFragment{85a5fe #2 id=0x7f0c007f}
at android.support.v4.app.FragmentManagerImpl.moveToState(FragmentManager.java:1059)
at android.support.v4.app.FragmentManagerImpl.moveToState(FragmentManager.java:1252)
at android.support.v4.app.BackStackRecord.run(BackStackRecord.java:742)
at android.support.v4.app.FragmentManagerImpl.execPendingActions(FragmentManager.java:1617)
at android.support.v4.app.FragmentManagerImpl$1.run(FragmentManager.java:517)
at android.os.Handler.handleCallback(Handler.java:746)
at android.os.Handler.dispatchMessage(Handler.java:95)
at android.os.Looper.loop(Looper.java:148)
at android.app.ActivityThread.main(ActivityThread.java:5443)
at java.lang.reflect.Method.invoke(Native Method)
at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:728)
at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:618)
Updated :
MainActivity.java
private static final String VIEW_KEY1 ="key1" ;
private static final String VIEW_KEY2 ="key2" ;
FragmentManager mFragmentManager;
FragmentTransaction mFragmentTransaction;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_main);
Toolbar toolbar = (Toolbar) findViewById(R.id.toolbar);
setSupportActionBar(toolbar);
FloatingActionButton notifBtn = (FloatingActionButton) findViewById(R.id.post);
FloatingActionButton postBtn = (FloatingActionButton) findViewById(R.id.post);
DrawerLayout drawer = (DrawerLayout) findViewById(R.id.drawer_layout);
ActionBarDrawerToggle toggle = new ActionBarDrawerToggle(
this, drawer, toolbar, R.string.navigation_drawer_open, R.string.navigation_drawer_close);
drawer.setDrawerListener(toggle);
toggle.syncState();
NavigationView navigationView = (NavigationView) findViewById(R.id.nav_view);
navigationView.setNavigationItemSelectedListener(this);
TabFragment tabFrag = new TabFragment();
mFragmentManager = getSupportFragmentManager();
mFragmentTransaction = mFragmentManager.beginTransaction();
mFragmentTransaction.replace(R.id.containerView,new TabFragment()).commit();
}
@Override
public void onBackPressed() {
DrawerLayout drawer = (DrawerLayout) findViewById(R.id.drawer_layout);
if (drawer.isDrawerOpen(GravityCompat.START)) {
drawer.closeDrawer(GravityCompat.START);
} else {
super.onBackPressed();
}
}
@Override
public boolean onCreateOptionsMenu(Menu menu) {
getMenuInflater().inflate(R.menu.main, menu);
return true;
}
@Override
public boolean onOptionsItemSelected(MenuItem item) {
int id = item.getItemId();
if (id == R.id.action_settings) {
return true;
}
return super.onOptionsItemSelected(item);
}
@SuppressWarnings("StatementWithEmptyBody")
@Override
public boolean onNavigationItemSelected(MenuItem item) {
DrawerLayout drawer = (DrawerLayout) findViewById(R.id.drawer_layout);
drawer.closeDrawer(GravityCompat.START);
return true;
}
activity_main.xml
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="match_parent"
android:layout_height="match_parent">
<include
layout="@layout/app_bar_main"
android:layout_width="match_parent"
android:layout_height="match_parent" />
<android.support.v4.widget.DrawerLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
android:id="@+id/drawer_layout"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_marginTop="80dp"
tools:openDrawer="start">
<FrameLayout
android:id="@+id/containerView"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical">
</FrameLayout>
<android.support.design.widget.NavigationView
android:id="@+id/nav_view"
android:layout_width="wrap_content"
android:layout_height="match_parent"
android:layout_gravity="start"
android:fitsSystemWindows="true"
app:headerLayout="@layout/nav_header_main"
app:menu="@menu/activity_main_drawer" />
</android.support.v4.widget.DrawerLayout>
</RelativeLayout>
Please help.
Thanks
Deepak
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,199
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Q: JPA Specification with null parameter I have created JPA Specification like
public class CarFilterSpecs {
public static Specification<Car> carStatusIn(Collection<String> carStatus) {
return (root, query, builder) ->
CollectionUtils.isEmpty(carStatus)
? builder.conjunction()
: carStatus.contains(-1)
? builder.isNull(root.get("carStatus").get("id"))
: root.get("carStatus").get("id").in(carStatus);
}
public static Specification<Car> carVisibilityStatusIn(Collection<String>
carVisibilityStatus) {
return (root, query, builder) ->
CollectionUtils.isEmpty(carVisibilityStatus)
? builder.conjunction()
: carVisibilityStatus.contains(-1)
? builder.isNull(root.get("carVisibilityStatus"))
: root.get("carVisibilityStatus").in(carVisibilityStatus);
}
}
and models looks like
public class Status {
String id;
String name;
}
public class Car {
String id;
Status status;
...
}
and I am consuming the spec like
import org.springframework.data.jpa.domain.Specification;
Specification<Car> specs = Specification.where(CarSpecs.carStatusIn(status))
.and(CarSpecs.carVisibilityStatusIn(visibilityStatus));
carRepository.findAll(specs);
till here works perfectly but now I am failing to get cars with some status and as well cars with status null(there is a chance of having status null)
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| 5,703
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Q: Checking for spaces in C++? I'm writing an encryption program for class. I need to be able to track the position of spaces in a string, as well as adding 13 to the ASCII value of each character of the string. I keep getting an error in line 46, and I don't see what I'm doing wrong.
/*
program 4
use an array to store the string
convert decimal values to ASCII, add 13 to the ASCII values,
convert back to decimal
the character for the space shoud be determined by the character
that came before it.
The character before it should have 4 added to it and
placed after itself to act as the space.
*/
#include <iostream>
#include <fstream>
#include <string>
#include <conio.h>
using namespace std;
int valueChange(string array), length(string array);
int spacePosition[0];
string message[0], encrypt[0];
int main()
{
cout << "Please enter your message."; //start with this
cin >> message[0];
int value = length(message[0]);
int count=0;
/*
store message as an array
loop through array, adding 13 to each value, use tolower() on each value
if value is
*/
for (int i=0; i < value; i++)
{
valueChange(message[i]);
if (message[i] == ' ') //checks for spaces in the string
{
spacePosition[count] = i + 1; //records the placement of spaces
//in the string
//have the array cast the i value to a new int value
//store space positions in an array
count++;
}
}
cout << "&";
cout << "Message encrypted and transmitted."; //final message
getch();
return 0;
}
int valueChange(int array[])
{
array[0] += 13;
if (array[0] > 122)
{
array[0] - 122;
}
return (array[0]);
}
int length(string array)
{
return (array.length() - 1);
}
A: If you need to search for spaces you can use string::find_first_of(string). You just need to write
`
std::string yourstring;
yourstring.find_first_of(" ");
`
then you need to iterate through the whole string using iterator.
And why are you declearing array if zero length? You can simply use simple std::string class. If you are not allowed to then simply use find_first_of(....) function available in standerd library. It basically returns the iterator to the first matching element.
A: int spacePosition[0];
string message[0], encrypt[0];
Probably want those to be 1's... or not use an array at all. But you are declaring arrays with 0 length, so yeah, you're going to segfault when you try writing to them.
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| 2,227
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Q: Change h.264 quality when using SinkWriter to encode video I am using Microsoft Media Foundation to encode a H.264 video file.
I am using the SinkWriter to create the video file. The input is a buffer (MFVideoFormat_RGB32) where I draw the frames and the output is a MFVideoFormat_H264.
The encoding works and it creates a video file with my frames in it. But I want to set the quality for that video file. More specifically, I want to set the CODECAPI_AVEncCommonQuality property on the H.264 encoder.
In order to get a handle to the H.264 encoder, I call GetServiceForStream on the SinkWriter. Then I set the CODECAPI_AVEncCommonQuality property.
The problem is that my property change is ignored. As stated in the documentation:
To set this parameter in Windows 7, set the property before calling IMFTransform::SetOutputType. The encoder ignores changes after the output type is set.
The problem is that I don't create the H.264 encoder manually. I set the input and the output type on the SinkWriter, and the SinkWriter creates the H.264 encoder automatically. As soon as it creates the encoder, it calls the IMFTransform::SetOutputType method, and I can't change the CODECAPI_AVEncCommonQuality property anymore. The documentation also says that the property change isn't ignored in Windows 8, but I need this to run on Windows 7.
Do you know how I can change the quality for the encoded file while using SinkWriter on Windows 7?
PS: Someone asked the same question on the msdn forums, and he didn't seem to get an answer.
A: As the documentation says, you just can't change the CODECAPI_AVEncCommonQuality property after the output type is set, and the SinkWriter sets the output type before you can get a hand on the encoder.
In order to bypass this problem I managed to create a class factory and register it in Media Foundation, so that the SinkWriter uses it to create a new encoder. In my class factory, I create a new H264 encoder and set whatever properties I want before passing it on to the SinkWriter.
I have written in more detail the steps I took to create this class factory on the MSDN forums, here: http://social.msdn.microsoft.com/Forums/en-US/mediafoundationdevelopment/thread/6da521e9-7bb3-4b79-a2b6-b31509224638
That was the only way I could get around my problem on Windows 7.
A: CODECAPI_AVEncCommonRateControlMode and CODECAPI_AVEncCommonQuality can be passed to the h.264 encoder using IMFSinkWriter->SetInputMediaType(/* ... */,, IMFAttributes pEncodingParameters). I suspect other CODECAPI_ values would work as well.
CComPtr<IMFAttributes> pEncAttrs;
ATLENSURE_SUCCEEDED(MFCreateAttributes(&pEncAttrs, 1));
ATLENSURE_SUCCEEDED(pEncAttrs->SetUINT32(CODECAPI_AVEncCommonRateControlMode, eAVEncCommonRateControlMode_Quality));
ATLENSURE_SUCCEEDED(pEncAttrs->SetUINT32(CODECAPI_AVEncCommonQuality, 40));
ATLENSURE_SUCCEEDED(writer->SetInputMediaType(sink_stream, mtSource, pEncAttrs));
// ^^^^^^^^^
|
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| 7,163
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Arsenal 1968-70 – Relative calm before the storm
October 12, 2019 October 14, 2019 Neil Fredrik Jensen
WHEN Arsenal won the "double" in 1970-71, they received very little praise for their considerable efforts. Were it not for Charlie George's messianic fall to the ground in the FA Cup final, Arsenal would be remembered for merely grinding-out results and out-slugging Leeds United's relentless machine.
But the seeds of Arsenal's triumph were sown in three seasons leading up to 1970-71. From 1953, when Arsenal won their seventh Football League title, the club had declined from its golden period and had been usurped by North London rivals Tottenham. The Gunners became a mid-table club with crowds in 1965-66 at Highbury dipping below 30,000 for the first time since before the arrival of Herbert Chapman.
The 1965-66 season saw Arsenal finish 14th despite the presence of former England skipper Billy Wright as manager. Wright was not a success as manager and in June 1966, the club made the surprise choice of appointing physiotherapist Bertie Mee as his replacement. Mee wasn't totally convinced, either, and accepted with the caveat that if it didn't work out, he could resume his physio duties.
Mee was a disciplinarian but essentially, a modest man with limited football knowledge. To compensate, he surrounded himself with good coaches, men like Dave Sexton and Don Howe. Initially, Mee took the job on a trial basis, but in March 1967, he made the move permanent.
By this time, Mee had started to mould the Arsenal side more to his liking. In October 1966, Arsenal signed full back Bob McNab from Huddersfield Town and within two years, he had been capped by England. At the same time, Mee acquired George Graham from Chelsea in a deal that saw Tommy Baldwin travel in the opposite direction. Mee also had in mind a number of young players, some of whom – such as Pat Rice and Sammy Nelson – had played for the Gunners' FA Youth Cup final teams in 1965 and 1966. Arsenal ended 1966-67 season well, but they were never in the hunt for major honours.
The first signs of an Arsenal revival came in 1967-68. Mee's team reached the Football League Cup final and faced Leeds United in a bad tempered game at Wembley. They lost 1-0 but it started a feud between the two clubs that reached its zenith in the early 1970s. That same week, Arsenal paid £ 90,000 for Coventry City's Bobby Gould. The hard-working Gould had a lot of hope riding on him, but he never truly settled at Highbury and he didn't stay too long.
Mee and Howe had built a team that was increasingly hard to beat and adept at shutting out the opposition. In 1968-69, Arsenal showed the first signs of their potential. They were unbeaten in their first nine league games, conceding just six goals in the process. Once again, they enjoyed a good run in the Football League Cup, beating neighbours Tottenham in the semi-final over two legs to reach a second successive final. Red-hot favourites they may have been against third division Swindon Town, but Mee's men came unstuck on a dreadful Wembley surface and were victims of a major giant-killing.
A Don Rogers-inspired Swindon won 3-1 against an Arsenal team that was hampered by illness and just a little over confident. Arsenal were lambasted in the press, but this setback served to make the squad tighter and more determined to end the club's trophy drought. And it didn't stop Arsenal from finishing fourth in the table thanks to an excellent away record and a rock solid defence that conceded just 27 goals in 42 games.
Arsenal had qualified for European competition for only the second time in their history. Their previous appearance, in 1963-64, was in the Inter-Cities Fairs' Cup but ended briefly in the second round in Liege. Spurs, Chelsea and West Ham had all enjoyed good European campaigns, but it had largely passed Arsenal by.
Mee made some changes to his squad at the start of 1969-70, letting centre half Ian Ure, who was at the heart of the Swindon defeat, go to Manchester United. His replacement was supposed to be John Roberts, signed from Northampton Town. But Roberts failed to establish himself at Arsenal and moved on in 1972 to Birmingham. As the season progressed, Gunners fans also saw the last of Northern Irishman Terry Neill as a player. In the summer, he would join Hull City as player-manager.
The most significant player movement for Arsenal was the arrival of two youngsters, Charlie George and Ray Kennedy, both of whom were blooded during 1969-70. There was no place for Gould, though, and he made less than a dozen appearances before finding himself on his way out of Highbury.
In the first weeks of 1970, the club attempted to add some flair to an otherwise workmanlike team, paying £100,000 for Peter Marinello, Hibernian's George Best lookalike. With his long hair, trendy clothes, model girlfriend and jinking wing play, Marinello arrived in London on a wave of hype. "We've just signed the nearest thing that football has to the Beatles," said one Arsenal director. Marinello was just 19 years' old and although he scored on his debut, at Old Trafford of all places, the move to Arsenal seemed to overawe the young man from Edinburgh.
Arsenal's league form was far from satisfactory. Marinello's first appearance came during a 10-game run without a win. Arsenal had also gone out of the FA Cup cheaply, losing 1-2 at home to Blackpool and after two years in the Football League Cup final, the 1969-70 run ended in round three. Marinello was supposed to be about the future of Arsenal.
Europe provided a welcome distraction from a very average league campaign. The Fairs' Cup could be an attritional competition and any team with ambitions of lifting the trophy would have to play 12 games. Glentoran of Northern Ireland were beaten 3-1 on aggregate before Sporting Lisbon were overcome 3-0. A tough tie with Rouen of France saw Jon Sammels score the only goal over the two games to squeeze Arsenal through. Dinamo Bacau were then thrashed 9-1 on aggregate before a very daunting semi-final with Ajax Amsterdam and Johan Cruyff, the clear flavour of the month in European football.
Ajax were not yet the triple-crown winners of Europe, though, which was fortunate for Arsenal. This was Charlie George's moment, for he netted twice in a 3-0 win against the Dutch. They lost the second leg 1-0 in Holland, but Arsenal were through to the final. Their opponents would be Belgium's Anderlecht, who had disposed of three British clubs on the way, Coleraine, Newcastle United (the holders) and Dunfermline Athletic. They had also beaten Inter Milan in the semi-final. Furthermore, they had some genuine talent in their line-up, including Belgian international Paul Van Himst and the Dutchman Jan Mulder.
It was Mulder that tormented Arsenal in Brussels in the first leg of the final. He scored twice, following up on Johan Devrint's opener to give Anderlecht a 3-0 lead. Arsenal looked buried, but substitute Ray Kennedy headed a consolation goal eight minutes from time to give the Londoners a glimmer of hope.
The second leg, on April 28, 1970, saw Arsenal go ahead after 25 minutes through Eddie Kelly, another young player who had made his bow in 1969-70. Arsenal remained patient, although Mulder hitting the woodwork jangled Highbury nerves. It wasn't until the final quarter of an hour that Arsenal broke through, John Radford and Sammels scoring within a minute of each other. Arsenal had won, producing a miraculous comeback that had not looked possible a week earlier in the Belgian capital. The crowd streamed onto the Highbury pitch as Don Howe declared that victory was the start of a new dawn at the club.
He was right, to some extent. Arsenal won the "double" in 1971, but that triumph signalled the peak of that team's achievement, not the beginning of a dynasty. The period between 1968 and 1970 had been the build-up, 1971 the climax.
Sadly, the triumph of 1970 has too often been overlooked. It didn't help that 24 hours later, Chelsea and Leeds fought out the most compelling FA Cup final of recent times and Manchester City won the European Cup-Winners' Cup in Vienna. But Arsenal had broken a 17-year famine and had secured a magnificent piece of silverware.
Just for the record, the team that won 3-0 in London that night was: Bob Wilson, Peter Storey, Bob McNab, Eddie Kelly, Frank McClintock, Peter Simpson, George Armstrong, Jon Sammels, John Radford, Charlie George, George Graham.
www.gameofthepeople.com
Categories: TeamsTags: 1970s, Arsenal, Bertie Mee, Fairs Cup, great reputations
Previous PostNeville, Neville, how could they know? Next PostRangers 1963-64 – a final flourish for the 60s
Johnno says:
Most people who were there that night against Anderlecht will tell you that was the best atmosphere ever witnessed at Highbury. The double followed and then Mee decided to break up the team shortly afterwards which led to a shocking few years where the club even flirted with relegation.
Shane Wright says:
Great article. The logical next piece int he series (if this is indeed a series) should be on how Everton's 1970 championship team was frittered away so wastefully over the two years that followed.
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| 4,902
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{"url":"https:\/\/delong.typepad.com\/sdj\/2005\/03\/a_positive_prog.html","text":"## A Positive Program for Social Security\n\nI'm writing a piece for Berkeley's GSPP's magazine, Policy Matters. It's trying to be a positive piece. Basically, I'm advocating Diamond-Orszag plus, where the \"plus\" includes automatic savings from your tax refund and a stronger Social Security Commission so that the Congress doesn't have to be too brave all at once.\n\n#### A Genuinely Good Deal for Social Security\n\nU.C. Berkeley\nMarch, 2005\n\nDraft 1.2\n\nSuppose that we lived in some bizarre parallel universe in which everything was topsy-turvy, and that as a result we were having an informed debate about what to do with our Social Security system that made sense. I know that this is total fantasy, but bear with me a minute: there is method to my madness. Suppose that this were so: what questions would we be trying to answer?\n\n1. Roughly what proportion of pre-retirement income should we guarantee people will have after they retire--no matter how well their investments do, no matter how thrifty or feckless they were during their working years? In short, how big--measured as a share of pre-retirement income--should the Social Security system be?\n\nWe should guarantee a basic Social Security benefit of roughly half of pre-retirement after-tax income--a replacement rate for the median worker, counting state and local as well as federal taxes, of roughly 30 percent of pre-retirement pre-tax income. Anything less runs a substantial risk of producing a lot of elderly poverty: the feckless and the unlucky will, when they are old, live much worse than is common in the surrounding society. Now as society grows richer this elderly poverty will be a relative phenomenon: people will be much poorer than their neighbors, but few of them will be absolutely poor in the sense of living in boxes or eating catfood. Nevertheless, relative elderly poverty is real elderly poverty, and it is something that a good society should protect against.\n\n2. How progressive should the Social Security system be? Those who make less than the average should probably have a higher replacement rate--receive a higher share of pre-retirement income--than those who make more than the average. But how much more?\n\nThe Social Security system should be somewhat but not extremely more progressive than is our current system. I see no case for exempting the top 15% of wage income from the Social Security tax base. I see no case for exempting non-wage income from the Social Security tax base. On the benefits side, we already have substantial progressivity: benefits relative to scaled lifetime taxes paid--the \"Primary Insurance Amount\"--rise at a rate of 0.9 for roughly the first $600 a month in your Social Security check, at a rate of 0.32 for roughly the next$900 a month, and at a rate of 0.15 thereafter. It would be good if the system were more progressive, covered everyone, and offered a minimum benefit to those whose taxes paid in were zero. But otherwise the system seems in good shape as well as progressivity is concerned.\n\n3. How should this basic Social Security system be financed? Should it be pay-as-you-go, in that each generation of taxpayers pays for the last generation's retirement? Or should it be a funded system, that builds up enormous assets and so owns large chunks of the economy, and uses the returns on those assets to finance large chunks of benefits?\n\nThe answer to this question depends on the shape of future economic growth and demographic change. When the economy is growing faster than the interest rate, pay-as-you-go systems are attractive: they offer a low-cost way of moving wealth up the generations from the (richer) future to the (poorer) present, and so raising social welfare. When the economy is growing much more slowly than the interest rate, the burdens placed on workers by a pay-as-you-go system are much harder to justify, and funded systems--those that build up lots of assets to finance part or all of this generation's benefits--become much more attractive. Currently, the Social Security Administration's actuaries have a set of assumptions about economic growth, interest rates, and equity returns that I believe are inappropriate, and make pre-funded systems look artificially good and pay-as-you-go systems look artificially bad. There is definitely a strong case for pre-funding some of the cost of Social Security for the large baby-boom generation. There is not a case for prefunding much else of the basic benefit, and in my view there will not be a strong case for prefunding much of the basic benefit until immigration into the United States begins to decline significantly from its current relatively high levels.\n\n4. What additional steps should the government take to make it easy--or perhaps mandatory--for people to save in their own private accounts, so that they reach retirement with more than their basic Social Security benefit to draw on?\n\nThis question is, I think, the most interesting. Americans do not take nearly as much advantage of tax-preferred and other savings vehicles as we economists think that they should. I am one of those who believes that America's national savings rate is dangerously low. The bottom half of America's income distribution has essentially no wealth invested in the stock market, and that cannot be right. It seems to me that these are problems that the government should address them.\n\nHow should it address them? The government should address them by making add-on savings out of payroll into individuals' private accounts the default--not mandatory, but the default: you have to fill out a form and check out a box in order not to make your contribution to your private account. The government should sweeten the pot: provide a partial match for funds directed into private accounts. The government should also provide a simple and reasonable default option for investing private accounts: half in a low-fee stock index fund, and half in a low-free bond fund. It should offer little else in the way of investment options: the danger that private account holders will be on the least informed side of trades is great, and the danger that private account holders will degrade their account through fees and transaction costs is great as well.\n\n5. What kind of bureaucracy should govern and administer this system? And how much flexibility should it have--what adjustments should it be able to make on its own without going back to Congress for revised authorizing legislation?\n\nIt seems clear that the system needs more flexibility than current law allows. Fertility waxes and wanes, economic growth speeds up and slows down, returns increase and decrease. A pay-as-you-go system thought of as a defined-benefit program will always be sliding into deficit or surging into surplus. Americans' entitlement in retirement to their share of pay-as-you-go Social Security revenues is more equity-like than debt-like. Because there is no residual claimant or debtor (besides the U.S. government), the system should be operated more like a credit union or a mutual association, with payouts that naturally rise and fall with resources. Such a system is, I think, best operated with a Social Security Board of Trustees with a fiduciary duty to maintain the long-run actuarial balance of the system, and the power to alter benefit levels (and, within limits, contribution rates) to achieve that long-run actuarial balance.\n\nAs for the add-on system, we already have the bureaucracies to run it. The IRS is a natural place to receive the add-on contributions: a check box on form 1040 to opt in--or, better yet, opt out--to the savings program. And an expanded Thrift Savings Plan to manage the money. If it's a good enough system for members of congress and senior administration officials, it should be good enough for all Americans.\n\nThus the outlines of a potential deal on Social Security--a potential reform--that would be genuinely good for the country are clear:\n\n1. Shift responsibility for maintaining actuarial balance off of the Congress and onto a Social Security Administration that has added discretion.\n\n2. Uncap FICA--increase the Social Security tax base to all wage income and perhaps further--and apply the extra resources to sweeten private add-on accounts, to add a little more progressivity to benefits for the poor, and to serve other purposes (like boosting benefits for widows).\n\n3. Make enrollment in private accounts automatic (it's done automatically on your 1040) but voluntary (you can fill in an extra form to get the money the IRS earmarks for your account back as part of your refund).\n\n4. Use the government's existing Thrift Savings Plan as a vehicle for managing private add-on accounts--and keep its choices restricted: churning and extra administrative costs caused by asset shuffling are not your friend.\n\nSuch a plan should satisfy everyone. It would satisfy optimists who believe SSA's projections are much too pessimistic and that no benefit cuts ever are required: if they are right, it would impose no benefit cuts. This would satisfy pessimists who worry that there is no mechanism to finance the existing level of benefits: if they are right, the SSA will have the fiduciary duty and the power to cut benefits. This would satisfy Congress: if there are benefit cuts, their fingerprints aren't anywhere nearby. This would satisfy believers in boosting national savings: the revenues from uncapping FICA and the money flowing into private accounts from people's choosing the default option will boost national savings. This would satisfy those scared that private accounts would be churned and looted by unscrupulous brokers: the TSP is a good operation that provides powerful protections.\n\nWhat's not to like?","date":"2020-08-11 21:53:20","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\": 0.1881738156080246, \"perplexity\": 2897.4171183049057}, \"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-2020-34\/segments\/1596439738855.80\/warc\/CC-MAIN-20200811205740-20200811235740-00321.warc.gz\"}"}
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{"url":"https:\/\/www.physicsforums.com\/threads\/vectors-as-a-differential-op-and-covectors-as-differenential.791576\/","text":"# Vectors as a differential op and covectors as differenential\n\n1. Jan 10, 2015\n\n### binbagsss\n\nHey,\n\nI'm new to this , and I understand the derivation of the transition laws for overlapping regions of a manifold for covectors and vectors starting from thinking of them as a differential and a differential operator respectively, but I don't really have a clue where this comes from...\n\nAny assistance or a point toward some good source, greatly appreciated !\n\n2. Jan 10, 2015\n\n### Fredrik\n\nStaff Emeritus\nThere's a more intuitive definition of the tangent space at p that isn't used much, because it's harder to work with. By this definition, a tangent vector at p is an equivalence class of curves through p. The equivalence relation is defined using a coordinate system $x:U\\to\\mathbb R^n$ such that $p\\in U$. We say that two curves $C:\\mathbb R\\to M$ and $B:\\mathbb R\\to M$ such that $C(0)=B(0)=p$ are equivalent if\n$$(x\\circ C)'(0)=(x\\circ B)'(0).$$ It turns out that this definition is independent of the chosen coordinate system. We can then define addition and scalar multiplication on the set of equivalence classes to turn it into a vector space. And then we can prove that the tangent space defined this way is isomorphic to the already familiar tangent space at p. The isomorphism is the map $\\phi$ defined by\n$$\\phi([C])(f)=(f\\circ C)'(0)$$ for all equivalence classes [C] and all smooth functions f.\n\nThere's more information in my posts in this thread and the one I linked to from there.\n\nThe fact that $\\big((\\mathrm dx^1)_p,\\dots,(\\mathrm dx^n)_p\\big)$ is the dual of $\\big(\\frac{\\partial}{\\partial x^1}\\big|_p,\\dots,\\frac{\\partial}{\\partial x^n}\\big|_p\\big)$ is a straighforward consequence of the definition of the $\\mathrm d$ operation. The definition is $(\\mathrm df)_p(v)=v(f)$ for all smooth $f:M\\to\\mathbb R$ and all $v\\in T_pM$, so we have\n$$(\\mathrm dx^i)_p\\left(\\frac{\\partial}{\\partial x^j}\\bigg|_p\\right) = \\frac{\\partial}{\\partial x^j}\\bigg|_p x^i =\\delta^i_j.$$\n\nLast edited: Jan 10, 2015\n3. Jan 11, 2015\n\n### lavinia\n\nAlthough Fredrik has explained it all, I thought this might be helpful as well.\n\nIn ordinary calculus in Euclidean space, vectors operate on functions by directional derivatives. If v is a vector, its action on a function,f, is\n\nlim t ->0 (f( x + tv) - f(x))\/t\n\nConversely one can define vectors at a point,x, as linear operators on functions that obey the Leibniz rule, that is : (v.fg) = (v.f)g(x) + (v.g)f(x)\n\nOne can show that such a \"derivation\" is equal to a directional derivative for some vector.\n\nSimilarly df(v) is is defined by the same limit. The way I learned it was that the Jacobian matrix of a function is a linear map between vector spaces. For instance, the gradient of a function is a linear map from n space to the real numbers.The notation df is the invariant way to describe the Jacobian.\n\nAs Fredrik explained, on a manifold one only has directional derivatives in a coordinate chart so some notion of equivalence under change of charts is needed. One way to define vectors invariantly without reference to coordinate charts is as equivalence classes of smooth curves at a point, another is as derivations.\n\nLast edited: Jan 12, 2015","date":"2017-08-21 23:05:10","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\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9280763864517212, \"perplexity\": 234.92864694322236}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-34\/segments\/1502886109670.98\/warc\/CC-MAIN-20170821211752-20170821231752-00040.warc.gz\"}"}
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{"url":"http:\/\/www.swmath.org\/?term=weighted%20likelihood","text":"\u2022 # AIS-BN\n\n\u2022 Referenced in 25 articles [sw02223]\n\u2022 importance function, and (3) a dynamic weighting function for combining samples from different stages ... general purpose sampling algorithms, likelihood weighting and self-importance sampling. We used in our tests...\n\u2022 # AWS\n\n\u2022 Referenced in 43 articles [sw04034]\n\u2022 package aws: Adaptive Weights Smoothing. The package contains R-functions implementing the Propagation-Separation Approach ... Spokoiny (2006), Propagation-Separation Approach for Local Likelihood Estimation, Prob. Theory and Rel. Fields...\n\u2022 # cmm\n\n\u2022 Referenced in 5 articles [sw24365]\n\u2022 Quite extensive package for maximum likelihood estimation and weighted least squares estimation of categorical marginal...\n\u2022 # ForestFit\n\n\u2022 Referenced in 1 article [sw31035]\n\u2022 moment, percentile, rank correlation, least square, weighted maximum likelihood, U-statistic, weighted least square ... three-parameter case are: maximum likelihood, modified moment type 1, modified moment type 2, modified ... product spacing, T-L moment, and weighted maximum likelihood. III) The Bayesian estimators...\n\u2022 # TagProp\n\n\u2022 Referenced in 13 articles [sw11682]\n\u2022 model to exploit labeled training images. Neighbor weights are based on neighbor rank or distance ... metric learning by directly maximizing the log-likelihood of the tag predictions in the training ... word specific sigmoidal modulation of the weighted neighbor tag predictions to boost the recall...\n\u2022 # MAMSE\n\n\u2022 Referenced in 0 articles [sw18460]\n\u2022 weights can be used in a weighted likelihood or to define a mixture of empirical ... package includes functions for the MAMSE weighted Kaplan-Meier estimate and for MAMSE weighted...\n\u2022 # RAxML-Light\n\n\u2022 Referenced in 2 articles [sw29595]\n\u2022 supercomputers under maximum likelihood. It implements a light-weight checkpointing mechanism, deploys 128-bit (SSE3 ... level message passing interface parallelization of the likelihood function. To demonstrate scalability and robustness...\n\u2022 # N-way Toolbox\n\n\u2022 Referenced in 28 articles [sw12996]\n\u2022 using expectation maximization); Fitting models with a weighted least squares loss function (including MILES); Predicting ... generalized multiplicative ANOVA, MILES for maximum likelihood fitting, conload for congruence and correlation loadings, eemscat...\n\u2022 # selectMeta\n\n\u2022 Referenced in 1 article [sw06784]\n\u2022 discuss how to estimate a decreasing weight function in the above model and illustrate ... examples. Some basic properties of the log-likelihood function and computation of a $p$-value ... weight function are indicated. In addition, we provide an approximate selection bias adjusted profile likelihood...","date":"2020-08-09 07:58:23","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\": 0.274873822927475, \"perplexity\": 7432.645262062744}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439738523.63\/warc\/CC-MAIN-20200809073133-20200809103133-00245.warc.gz\"}"}
| null | null |
Vegan
Chocoholic
Philip Hochuli
# Vegan
# Chocoholic
Cakes, Biscuits,
Desserts and Quick Sweet Snacks
Grub Street • London
Published in 2016 by Grub Street
4 Rainham Close
London
SW11 6SS
Email: food@grubstreet.co.uk
Web: www.grubstreet.co.uk
Twitter: @grub_street
Facebook: Grub Street Publishing
Copyright this English language edition © Grub Street 2016
Copyright © AT Verlag, Aarau and Munich, 2015
Published originally in German
Photos: Alexandra Schubert, www.myshoots.de
Image editing: Vogt-Schild Druck, Derendingen
A CIP record for this title is available from the British Library.
ISBN 978-1-910690-32-1
eISBN 978-1-911621-63-8
Mobi ISBN 978-1-911621-63-8
The moral right of the author has been asserted.
All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the prior written permission of both the copyright owner and the above publisher of this book.
# Contents
## Foreword
## Introduction
### About the recipes
### Some important ingredients
## Chocolate: Essentials, Production, and Products
### The correct way to heat and melt chocolate
## Recipes
### Spreads and Jams
### Biscuits, Muffins, Cupcakes and Brownies
### Cakes, Tarts and Cheesecakes
### Chocoholic Favourites
### No-Bake Chocolate Treats
## ... the savoury versions
## Foreword
'A day without chocolate – unthinkable!'
### Dear chocoholics
Few foods or beverages are as alluring and fascinating as chocolate, and not just for every generation, but also in almost every nation.
This book will take you on a culinary adventure through the world of chocolate. Had the word 'vegan' not appeared in its title, you would hardly suspect that this book contained purely vegan recipes, but this was always my intention. As regards taste, vegan cuisine is in no way inferior to the 'classics'. Rather, my experience even shows that the opposite is true: vegan cuisine represents an enormous culinary enrichment. The fact is that there is no need for sacrifice, particularly as far as sweet creations are concerned, and that chocolate's place in vegan cuisine is at least as important as it is in conventional cuisine, as this book demonstrates in some sixty recipes.
Personally, the most important thing is to have fun with cooking and baking, which I think are some of the nicest things you can do, an area in which I've been involved in professionally for several years. The subject of this book is a food preparation that has an extremely important place in my life.
A day without chocolate for me would be incomplete. I've never been as enthusiastic about any other food as I have about chocolate.
The recipes in this book range from classic cakes and biscuits to exquisite dessert creations and quick and delicious treats such as truffles, pralines and home-made chocolate yoghurt. Refined chocolate can also be used in spicy dishes, as shown in a small selection of savoury main courses. You won't find any fancy ingredients or long ingredient lists. You should be able to find between 95-98 per cent of the ingredients in supermarkets. Most of the recipes have a very simple design and are really simple to make. I consider these two points to be the most important things to ensure the recipes are suitable for daily use and that the joy of cooking isn't spoilt by endless instructions. My personal tip: leaf through the pages, and when the urge takes you, just cook – in that order. Information on the level of difficulty and preparation time is provided to help you decide.
The book is completed with an introductory section offering important information on chocolate and practical tips for cooking. You should always pay attention to this advice before starting on the recipe. I won't go into why going vegan can or should make sense. If you're a vegan, you already know; and if you want to know about it, there's plenty of information available on the Internet and in books. What I find fascinating about vegan cuisine is the incredible variation and diversity of flavours, and the fact that vegan dishes are simply a lot of fun to make. That's what I want to convey in this book.
And finally: This book isn't a guide to healthy living and is definitely not a book for people looking to get fit. Its purpose is to bring you fun and pleasure. And one more thing: Don't forget to wipe off the chocolate from around your mouth before you leave home...
## Introduction
### About the recipes
The recipes in this book are marked according to their difficulty with stars, with 1 star being the lowest level and 3 stars representing the highest level of difficulty. The stars and the preparation time are a good indication of the effort required for each recipe. The fact that very few recipes have 3 stars demonstrates the basic idea behind the book and the suitability of the recipes for everyday use.
Unless otherwise stated, the following information applies to baking: Bake on the middle shelf with top and bottom heat, in conventional (static) oven mode.
The recipes that contain no gluten, soya or sugar are marked with the following symbols:
### Some important ingredients
#### Bourbon vanilla powder (ground vanilla pods)
This product is a great alternative to vanilla pods and is particularly easy to use, since you don't have to split a pod and scrape out the seeds every time. Vanilla powder is found in small jars and available in supermarkets and organic food shops. Tightly closed, it keeps almost indefinitely. In these recipes, vanilla powder can always be substituted by vanilla pods and vice versa. Convert with this rule: The seeds from a vanilla pod are equivalent to ⅓ of a teaspoon of vanilla powder.
#### Cocoa powder
In a book focussing on nothing but chocolate, a good cocoa powder is essential. Bear this in mind when buying a particularly high-quality product. On no account should cocoa powder be confused with chocolate powder. The latter contains cocoa and other ingredients, mainly sugar. It's also important to distinguish between the lightly defatted variety (about 20 g/¾ oz fat for every 100 g/3¾ oz) and the extensively defatted variety (about 10 g/¼ oz fat for every 100 g). The lightly defatted variety has an intense flavour and more rounded mouthfeel, which is why it is used exclusively in my recipes (more on this can be found on p. 13). The labelling doesn't clearly state which variety it contains, so it's useful to check the fat content provided on the package.
#### Coconut oil
Coconut oil is one of the few fats or oils that is solid at room temperature. For this reason, and because it withstands high temperatures, it is very popular for use in cooking. The choice is between the cold-pressed or virgin form of the oil, which has a very strong coconut flavour, and refined oil, which has a relatively neutral taste. The latter can be found in most supermarkets. Unless otherwise stated on the package, you can assume it isn't cold pressed. Virgin coconut oils are available at gourmet and organic food shops. I very rarely use virgin coconut oil because its intense coconut flavour is usually undesirable in dishes, and coconut oil is typically used for its other qualities. Owing to this difference in flavour, virgin coconut oil should on no account be used as a substitute for refined coconut oil, otherwise the dish will most likely be unsuccessful. Unlike cold-pressed coconut oil, coconut oil with the additional labelling of 'mild' can be used.
#### Flour
Choosing the right flour plays an important and decisive role in my recipes. Two particular points need to be taken into consideration:
The type of grain and the extraction rate.
The importance of the type of grain: The recipes basically use wheat flour or the alternative of spelt flour, as these have the best properties for baking and binding of all different types of flour. The use of other flours with these recipes might result in unpleasant surprises. In the case of gluten intolerance, use special flours that are as close as possible to the properties of wheat or spelt flours.
The importance of extraction rate: A distinction is usually made between the extraction rate, or the quantity of the whole grain used, between flours with low extraction rate (light flour) and flours with a high extraction rate (dark flour). Light (white) wheat and spelt flours have excellent baking and binding properties, particularly the latter, so they can easily be used to replace eggs in certain recipes, which is important for vegan cuisine. These are also the flours of choice for the recipes in this book. I get the best results in baking using white wheat flour. This is the equivalent of plain flour, pastry flour or cake flour.
#### Neutral vegetable oil
These include liquid plant oils with little or no taste and therefore will not influence the flavour of dishes. There are many different commercially available oils that meet this requirement. The following oils have been tried and tested in my kitchen: neutral, refined sunflower oil for low and medium temperatures; high oleic sunflower oil and refined peanut (groundnut) oil for high temperatures.
#### Plant-based milk
Plant-based alternatives to cow's milk range from the well-known but – as far as taste is concerned – questionable soya milk and others made from spelt, rice and oats to almond and cashew milk. The choice depends on different factors. When it comes to baking and cooking, soya, almond and cashew milks are suitable. The characteristics of rice, spelt and oat milks mean that they aren't ideal for cooking or baking, and I don't recommend them. Soya milk is the easiest to work with, as its protein and fat content is the closest to cow's milk. Its questionable flavour almost completely evaporates during cooking or baking, making it extremely versatile.
Certain recipes require the use of soya milk because their success depends on its protein content, and the other alternatives contain no protein. For example, this applies to the lemon and 'cream cheese' slice and the saffron cream pie.
#### Vegan margarine
In a book with a large proportion of baked dishes, margarine plays an important part. Margarines can be found in any supermarket. However, many of them unexpectedly contain dairy ingredients, so it's a good idea to check the list of ingredients. Given that cheap margarines contain trans (hydrogenated) fats, and also because of the environmentally questionable use of palm oil, an ingredient in practically all margarines, the question is whether there is an alternative to margarine. In my culinary experience, there's no alternative to margarine; coconut oil isn't a suitable alternative, and so far there are practically no palm oil-free margarines on the market. There is now a butter-flavoured rapeseed oil that can replace margarine in some recipes, but by no means in all of them. The environmentally questionable use of palm oil can be overcome by purchasing organic margarine certified by reliable organic labels, made with environmentally friendly palm oil. As for avoiding unhealthy trans fats, simply don't go for the cheap margarines, because usually only the cheapest margarines contain these discredited fatty acids. So it's worth having a good look on the back of the package.
#### Vegan dark chocolate
Besides cocoa, good vegan dark chocolate is the essential ingredient in a book on chocolate. It is therefore fundamental to ensure the use of a high-quality product. I always recommend Swiss chocolate, not because I'm being patriotic, but because qualitative experience has shown that it's in fact very difficult to beat. Cheap cooking chocolate should not be used; as with other ingredients, care must be taken with quality. There's also vegan milk and white chocolate. Since it's always important that my recipes are designed for everyday use, neither is used in my recipes. These products are sometimes available in organic food shops, but are often only found in gourmet shops and online.
## Chocolate: Essentials, Production and Products
### From bean to chocolate
'In the beginning was the bean' is more or less how it goes. This is how the story of chocolate starts. However, there are many steps in the manufacturing process that takes the harvested cocoa beans to the three main end products of cocoa butter, cocoa powder and chocolate confectionery. The following is a brief outline of the main steps and a description of the individual intermediate and end products, which are also of interest for the recipes.
#### The cocoa bean – where it all begins
Cocoa beans are the starting point for each of the following products. The main producers of cocoa beans are Ivory Coast, Ghana and Indonesia. Cocoa beans themselves can actually be eaten (the broken up beans are known as cocoa nibs), although this use is not widespread and, for many, it takes a lot of getting used to. The beans are a valuable and appreciated resource that further processing will turn into a series of products. The cocoa beans are prepared for further processing by cleaning and roasting. Basically, roasting brings out the aroma of chocolate, making this an essential step. The beans are then shelled and ready for the next step in the process.
#### Cocoa mass
The roasted and shelled beans are milled and turned into a liquid, known as cocoa mass or cocoa liquor. This is only an intermediate product and forms the base of the subsequent production of cocoa butter, cocoa powder and, finally, chocolate confectionery, which will involve more processing with additional cocoa butter and, depending on the type of chocolate, with a wide variety of other ingredients.
#### Cocoa butter
Cocoa butter is the oil extracted from cocoa mass. Cocoa butter is solid at room temperature and ideally should not contain any water, which is why it's also suitable for use where it needs to be melted with chocolate (unlike butter or margarine; for more see 'the correct way to heat and melt chocolate' on p. 14). Cocoa butter is a yellowish-white colour, which is a surprise for many who see it for the first time. It is the same colour as white chocolate, which is no coincidence, since the two products are directly related. Only cocoa butter goes into white chocolate. No cocoa powder is used. This is why white chocolate has the characteristic colour of cocoa butter and not the dark colour of cocoa powder. The following principle basically applies: the lighter the chocolate, the higher the proportion of cocoa butter.
#### Cocoa powder
Aside from chocolate confectionery, cocoa powder is probably the most widely used and popular product made from cocoa beans. It's extremely versatile, easy to use and immediately adds a wonderful chocolatey aroma. After the cocoa butter is extracted from cocoa mass, cocoa powder is left as a dry product. Unlike cocoa butter, cocoa powder has a dark, chocolatey colour. Depending on how thoroughly the cocoa mass is defatted, i.e. depending on how much cocoa butter is extracted from it, the result is lightly defatted or extensively defatted cocoa powder. The difference between both varieties is very significant. Lightly defatted cocoa powder is characterised by an intense and rounded flavour, while the extensively defatted variety is more soluble in water owing to its lower residual oil content, and is therefore more widely used in beverages. My recipes only make use of lightly defatted cocoa powder.
#### Chocolate as an end product
It is a long process to make the internationally prized and popular end product that is high-quality chocolate. After the desired amount of cocoa butter and any other ingredients are added to the cocoa mass, the process involves quite a few more steps. The first step is about achieving a fine texture. Absolutely no rough lumps should be felt on the tongue that will spoil the enjoyment of the chocolate. To achieve this, the chocolate is passed through several rollers and, depending on preference, is rolled down to thinness of a few micrometres. Only this refining process will ensure that the chocolate will feel soft as it melts in the mouth, and not have a rough feel. The final two steps are conching and tempering. Broadly speaking, the first of these steps involves the chocolate being heated to a high temperature and mixed for many hours. The aim of this is to allow any moisture to escape and to achieve a smooth flavour. The last step, tempering, is ultimately responsible for the chocolate having the right crystalline structure. Last but not least, it gives chocolate its characteristic gloss and the popping sound we expect from a good chocolate when broken. Crystals can form in as many as six different ways, but only one is desirable. After conching, tempering involves gentle, precise and controlled cooling of the chocolate and temporarily maintaining the liquid chocolate at a certain temperature. The conched and tempered chocolate is then poured into the desired mould and slowly allowed to cool, before being packaged. Depending on the cocoa mass content, a distinction is made between white, milk and dark chocolate.
### The correct way to heat and melt chocolate
How to melt chocolate the right way and why in certain circumstances there can be undesirable results from the melting process are questions that are often asked. Here is a quick lesson on the basics of melting chocolate correctly.
There are two ways of melting chocolate: the conventional way over a bain-marie, and in a microwave. When melting chocolate over a bain-marie, the bowl should not come into contact with the water; the chocolate is melted slowly by the rising steam. The best thing is to use a chrome-plated steel bowl that is a bit larger than the diameter of the saucepan. Chrome-plated steel distributes the heat the best and ensures even melting. The chocolate should be stirred gently, so that not too many air bubbles will form in the chocolate. For this reason, a rubber scraper or spatula is better for stirring than a whisk.
Melting chocolate in the microwave is very popular and is often the easier option. Caution: Use a low setting and check from time to time that the bottom of the bowl isn't too hot. Otherwise, there is a risk of burning the chocolate. Here's another tip: The more finely you chop the chocolate before melting, the more evenly it will melt, both in the microwave and over a bainmarie.
Regardless of the option you choose, there are two basic principles that you need to bear in mind.
1. Chocolate is extremely sensitive to heat and should be melted slowly on a low heat or at a low setting. Ideally, dark chocolate should never be heated to over 40ºC/104ºF. Milk chocolate requires a lower temperature. The molecular structure of chocolate can be modified at high temperatures, causing it to thicken, for instance. This may not be so important for most of the recipes as the melted chocolate will be incorporated into a mixture anyway. However, it is important in two cases: when the recipe calls for very runny melted chocolate (for example, the chocolate sushi), or when the melted chocolate is going to be cooled again to give it the same properties as chocolate confectionery, for instance, if you ever want to make your own chocolates.
2. Chocolate and water don't get on well. The addition of water or ingredients with a high water content (such as milk or soya milk) causes the chocolate to thicken, or even worse, the cocoa butter to separate from the rest of the chocolate, giving it a gritty consistency.
If this happens, the chocolate is beyond saving, so you'll have to start over again. This is the reason for using cocoa powder when making cakes. The batter is usually made with ingredients that contain a lot of water. And this is also why the bowl should completely cover the saucepan when melting chocolate over a bain-marie. Otherwise, there's a risk that steam will enter the bowl of chocolate and ruin the result.
If the chocolate for a recipe has to be very runny or needs to retain its original qualities, it's essential to ensure that no water, steam or any ingredients that contain water come into contact with the chocolate.
3. This principle also applies, at least in theory, for melting chocolate together with margarine. As margarine contains some water (as does butter), there's also a risk that the chocolate will thicken. Chocolate should only be melted with margarine if any possible thickening doesn't pose a problem for the recipe, such as when making a cake. Chocolate that has thickened, and even chocolate that has passed its use-by date are suitable for this use. It is common to combine chocolate with margarine to 'lighten' the chocolate or to give it the aroma of butter, and you will find a number of recipes in this book where chocolate and margarine are melted together. Of course, this is only possible when thickening isn't a problem for the finished product.
4. A tip for making a simple version of 'lightened' dark chocolate without the risk of thickening and without the use of margarine: cocoa butter makes an excellent alternative. Cocoa butter is available in organic food shops, gourmet food shops and online.
### Spreads and Jams
Pages 16/ clockwise from the left: Dark Chocolate and Orange Cream, Dark Chocolate Spread, Chocolate Coconut Spread, Vegan Nutella
### Vegan Nutella
This vegan nut spread can keep in the refrigerator for several weeks and will gain in intensity.
Makes 1 jar
Difficulty *
Preparation time: 5 minutes plus cooling time
75 g /3 oz vegan dark chocolate (45–55 % cocoa solids)
60 g/5 tbsp vegan margarine
75 g/6 tbsp hazelnut butter (see tip)
30 ml/2 tbsp agave syrup or 40 g/⅓ cup sifted icing sugar
5 g/1 tsp white almond butter (optional)
Slowly melt the chocolate with the margarine over a bain-marie. Add the hazelnut butter and agave syrup or icing sugar, keeping the bowl over the bain-marie.
Use a whisk to stir the mixture until there are no lumps. Optionally, add white almond butter and stir again. Remove from the heat, pour into the jar and leave to cool. (Picture on p. 17)
### Tips
As an alternative to hazelnut butter, lightly toast 65 g/9 tbsp of ground hazelnuts in a non-stick frying pan for 2–3 minutes and add to the melted chocolate. You can also add finely chopped nuts to the finished spread to make a 'crunchy' version.
This spread also makes a wonderful filling for the chocolate yeast rolls (see p. 82).
### Dark Chocolate Spread
You can make this recipe for a delicious vegan dark chocolate spread in a flash. It also has a very long shelf life when stored outside the refrigerator.
Makes 1 jar
Difficulty *
Preparation time: 5 minutes plus cooling time
90 g/7 tbsp refined coconut oil
90 g/¾ cup sifted icing sugar
40 g/⅓ cup lightly defatted cocoa powder (see tip)
1 pinch salt 1 pinch Bourbon vanilla powder (optional)
Slowly heat the coconut oil in a frying pan until completely melted. Add the other ingredients and stir with a whisk to a smooth cream. Remove from the heat, pour into the jar and leave to cool. The spread is quite creamy at room temperature and will firm up in the refrigerator. (Picture on p. 16)
### Tip
I prefer to use 50 g/½ cup of cocoa powder because I like a strong taste of chocolate. But if you prefer a milder flavour, use 40 g/⅓ cup.
### Chocolate Coconut Spread
Makes 1 small jar
Difficulty *
Preparation time: 5 minutes plus cooling time
75 g/⅔ cup sifted icing sugar
4 tbsp coconut milk, or another plant-based milk
50 g/¼ cup cold-pressed coconut oil (see tip)
4 tsp desiccated coconut
1 pinch salt
2 tbsp lightly defatted cocoa powder
Dissolve the icing sugar completely in the coconut milk. Heat the coconut oil until it is completely melted. Add the other ingredients and stir everything well with a whisk. Allow to cool, and then fill a jar.
The spread is quite creamy at room temperature and will firm up in the refrigerator. Although it needs to be stored in the refrigerator, it should be taken out some time before use. The spread will keep in the refrigerator for about 2 weeks. (Picture on p. 17)
### Tip
The use of unrefined coconut oil will give this spread a strong coconut flavour. If you can't find virgin coconut oil, you can use refined coconut oil, but the coconut flavour won't be so intense.
In that case, you'll have to use coconut milk instead of another plant-based milk.
### Dark Chocolate and Orange Cream
This cream is really quick to make and is the perfect topping for bread or pancakes.
Makes 1 small jar
Difficulty *
Preparation time: 5 minutes plus at least 2 hours cooling time
50 g/¼ cup vegan margarine or refined coconut oil
75 g/⅔ cup sifted icing sugar
50 ml/¼ cup freshly squeezed orange juice
25 g/¼ cup lightly defatted cocoa powder
1 pinch salt
⅛ tsp Bourbon vanilla powder
Melt the margarine in a saucepan or in the microwave. Dissolve the icing sugar completely in the orange juice. Add all the ingredients to the melted margarine and stir with a whisk.
Pour into a jar and leave to cool for at least 2 hours. Stir well before use. This cream can keep in the refrigerator for about 1 week. (Picture on p. 16)
### Quick Strawberry Jam with Dark Chocolate
Makes 1 jar
Difficulty *
Preparation time: 10 minutes plus cooling time
250 g/2 cups ripe strawberries
1 tbsp freshly squeezed lemon juice
200 g/⅞ cup jam sugar
40 g/1½ oz vegan dark chocolate (45–55 % cocoa solids)
Wash, pat dry and hull the strawberries, and purée thoroughly with a hand-held blender. Mix with the lemon juice and jam sugar, and then combine with the chocolate in a saucepan. Bring to the boil and leave to boil for 5 minutes over a medium to high heat. Immediately pour into a clean jar and seal tightly. Leave to cool. Store in the refrigerator after opening.
Quick Strawberry Jam with Dark Chocolate
### Caramel Royale Sauce with Chocolate Shavings
Caramel lovers would love to bathe in this sauce. The sauce tastes good on pancakes, cakes, muffins, bread, or just on a spoon.
Makes 1 jar
Difficulty **
Preparation time: 10 minutes plus at least 2 hours cooling time
200 ml/scant 1 cup coconut milk
150 g/⅝ cup sugar
1 dash lemon juice
100 g/3¾ oz vegan dark chocolate (45–55 % cocoa solids)
Combine 100 ml/scant ½ cup of coconut milk with the sugar and lemon juice in a saucepan and bring to the boil over a high heat. Slightly reduce the heat while keeping at a vigorous boil, and stir constantly with a whisk until the colour changes from white to light brown (this should take about 5 minutes). Be very careful because this change is very sudden. When hardly any more bubbles come up, this is a sign that it is almost ready.
Then continue to cook until the caramel turns a deep golden colour. If you remove the caramel from the heat too soon, it will be too light and its flavour won't be very intense, but you don't want it to cook for too long and become too dark.
Now add the rest of the coconut milk, return the saucepan to the heat and stir the sauce constantly for 2 minutes until smooth. Pour the sauce into a jar, leave to cool, and then refrigerate for at least 2 hours.
Finely chop or grate the chocolate, fold into the sauce and return it to the refrigerator. Stir briefly before each use.
### Tip
Making caramel takes a little practice, so don't despair if it isn't perfect the first time.
### Biscuits, Muffins, Cupcakes and Brownies
Best-Ever American Brownies
Raspberry and Chocolate Brownies
### Best-Ever American Brownies
Makes 20–30
Difficulty *
Preparation time: 10 minutes plus 25-35 minutes baking time
115 g/1 cup shelled walnuts
150 g/5 oz vegan dark chocolate (45–55 % cocoa solids)
290 g/2½ cups light wheat or spelt flour
300 g/1¼ cups sugar
1½ tsp baking powder
90 g/¾ cup lightly defatted cocoa powder
1 tsp Bourbon vanilla powder
1 level tsp salt
2 tsp white wine vinegar
200 ml/scant 1 cup refined rapeseed oil
Preheat the oven to 180°C/350°F.
Coarsely chop the walnuts and lightly roast in a dry frying pan. They shouldn't turn too dark, but they should develop the typical roasted aroma.
Coarsely or finely chop the chocolate, according to preference, and mix together with all the dry ingredients in a bowl, including the roasted walnuts. Add 250 ml/1 cup of water, the vinegar and oil and beat together briskly until smooth.
Spread out the mixture over a baking tray lined with baking parchment to form a 30 x 33 cm/12 x 13 in rectangle, and bake for 25-35 minutes. After cooling, cut into squares of the desired size. The brownies should still be a little moist. (Picture on p. 26)
### Raspberry and Chocolate Brownies
Makes 20–30
Difficulty *
Preparation time: 10 minutes plus 25–30 minutes baking time
290 g/2½ cups light wheat or spelt flour
75 g/⅔ cup lightly defatted cocoa powder
2 (10 g/0.35 oz) sachets vanilla sugar
1 level tsp salt
150 g/⅝ cup sugar
2 tsp baking powder
125 g/4½ oz vegan dark chocolate (45–55 % cocoa solids), grated or finely chopped
250 g/scant 1 cup seedless raspberry jam (see tip)
200 ml/scant 1 cup neutral vegetable oil
1 tsp white wine vinegar
215 g/1¾ cups raspberries, fresh or frozen
Preheat the oven to 180°C/350°F.
Mix all the dry ingredients in a bowl. Add the jam, oil, vinegar and 200 ml/scant 1 cup of water, and mix well. Carefully fold the raspberries into the batter.
Spread out the mixture over a baking tray lined with baking parchment to form a 30 x 33 cm/12 x 13 in rectangle, and bake for 25–30 minutes. After cooling, cut into squares of the desired size. The brownies should still be a little moist. (Picture on p. 27)
### Tip
If you can't find seedless raspberry jam, you can use normal raspberry jam and press it through a fine sieve.
### Banana and Dark Chocolate Volcano
Makes 9-12
Difficulty *
Preparation time: 10 minutes plus 22–25 minutes baking time
3 medium bananas
225 g/1 cup sugar
1 pinch salt
50 ml/¼ cup neutral vegetable oil
240 g/2 cups plus 2 tbsp light wheat or spelt flour
1½ tsp baking powder
9–12 tsp dark chocolate spread (see recipe on p. 19 or use another dark chocolate spread)
Vegan chocolate sprinkles or grated vegan dark chocolate for sprinkling (optional, as needed)
Preheat the oven to 180°C/350°F.
Peel the bananas, put into a shallow dish and mash as finely as possible with a fork. Whisk together with the sugar, salt and oil.
Mix the flour and baking powder together, combine with the banana mixture and mix to make a batter. Put a small amount of batter into the bottom of 9–12 greased cavities of a muffin mould, and then add 1 scant teaspoon of dark chocolate spread. Cover each with 1½–2 teaspoons of batter. The muffins can be sprinkled with chocolate sprinkles or finely grated chocolate.
Bake for 22–25 minutes. The chocolate centres should have broken through the surface by the time the muffins have finished baking. If this is not the case, the chocolate would have been placed too far down inside the mould or there would have been too much batter placed over the top. (Picture on p. 30)
### Gingerbread Cake
This gingerbread cake is quick to make.
Difficulty *
Preparation time: 15 minutes plus about 20 minutes baking time
For the gingerbread:
240 g/2 cups plus 2 tbsp wholemeal spelt flour or wholemeal wheat flour
150 g/⅝ cup whole cane sugar
2 tbsp gingerbread spice blend
1½ tsp baking powder
1 heaped tbsp lightly defatted cocoa powder
1 pinch salt
250 ml/1 cup soya milk
2 tbsp neutral vegetable oil
For the icing:
125 g/4¼ oz vegan dark chocolate (45–55 % cocoa solids)
50 g/¼ cup vegan margarine
Preheat the oven to 180°C/350°F.
Mix the dry ingredients in a bowl. Add the soya milk and oil, and mix well. Spread out the mixture over a baking tray lined with baking parchment to form a 20 x 30 cm/8 x 12 in rectangle, and bake for 20 minutes. Prick with a wooden skewer. It should come out clean when you pull it out. Otherwise, bake for a few more minutes.
For the icing, completely melt the chocolate with the margarine over a bain-marie. Spread evenly over the gingerbread and leave to cool.
Cut the gingerbread cake into rectangles. Store in a metal box or plastic container. (Picture on p. 31)
Banana and Dark Chocolate Volcano
Gingerbread Cake
### Lemon and Chocolate Chip Mini Muffins
Refreshing little muffins! Use fresh lemons for this recipe, to give them their special flavour. If you want to make full-sized muffins, double the ingredients to make the same number of them.
Makes about 20
Difficulty *
Preparation time: 15 minutes plus about 20 minutes baking time
For the muffins:
190 g/1⅔ cups light wheat or spelt flour
150 g/⅝ cup sugar
25 g/1 oz vegan dark chocolate chips or
25 g/1 oz vegan dark chocolate (44–55 % cocoa), chopped
1 tsp baking powder
1 pinch salt
1 untreated lemon for grated zest and 3 tbsp juice
3 tbsp neutral vegetable oil
100 ml/scant ½ cup soya milk
For the icing:
90 g/¾ cup icing sugar
2–3 tsp lemon juice
Preheat the oven to 180°C/350°F.
To make the batter, mix all the dry ingredients together with the lemon zest. Add the oil, lemon juice and soya milk, and mix well. Fill the greased cavities of a mini muffin mould with the batter and bake for 20 minutes (a little longer for full-sized muffins).
For the icing, mix the icing sugar with the lemon juice until smooth. The icing should be thick, so be very careful when measuring out the lemon juice. Spread the icing over the still warm muffins, and then leave the muffins to cool.
### Chestnut Muffins with a Chocolate and Cinnamon Glaze
Makes 8–10
Difficulty *
Preparation time: 15 minutes plus 20–25 minutes baking time
For the muffins:
100 g/⅜ cup sugar
150 g/1¼ cups light wheat or spelt flour
1 tbsp baking powder
1 pinch salt
2 tsp white wine vinegar
200 g/⅔ cup frozen chestnut purée, thawed (see tip)
4 tbsp neutral vegetable oil
150ml/⅔ cup almond or soya milk
For the icing:
80 g/3¼ oz vegan dark chocolate (45–55 % cocoa solids)
20 g/1½ tbsp vegan margarine
1 tsp ground cinnamon
Vegan chocolate sprinkles (optional, as needed)
Preheat the oven to 180°C/350°F.
For the batter, combine the sugar, flour, baking powder and salt in a bowl, and mix well. In a separate bowl, combine the vinegar, chestnut purée, oil and almond or soya milk, and mix to a smooth cream.
Combine both mixtures and mix well to make a batter. Divide the batter evenly into 8–10 greased cavities of a muffin mould. Bake for 20–25 minutes.
For the icing, slowly melt the chocolate with the margarine over a bain-marie or in the microwave. Mix in the cinnamon, and then spread the icing over the still warm muffins. The muffins can be decorated with chocolate sprinkles.
Leave to cool.
### Tip
Chestnut purée is best bought frozen. If this isn't available, you can replace it as follows: Make a purée by blending vacuum sealed, cooked sweet chestnuts with a little sugar (1 tablespoon per 115 g/1 cup) and water. You can make a larger amount than you need, and then freeze the rest so that you have some on hand.
### Chocolate Muffins with a Liquid Centre
Makes 4
Difficulty *
Preparation time: 10 minutes plus 10–13 minutes baking time
80 g/3¼ oz vegan dark chocolate (45–55 % cocoa solids)
50 g/¼ cup vegan margarine
50 g/¼ cup sugar
90 g/¾ cup light wheat or spelt flour
2 tsp baking powder
1 pinch salt
1 tsp white wine vinegar
Preheat the oven to 180°C/350°F.
Slowly melt the chocolate with the margarine over a bain-marie or in the microwave.
Mix the dry ingredients in a bowl. Add the melted chocolate mixture, 3 tablespoons of water and the vinegar, and mix slowly but thoroughly.
Divide the batter into four individual ramekins or cups (if baked in a mould with individual cavities, line with muffin cases). Bake for 10–13 minutes. The centres should still be liquid. Serve immediately. (Picture on p. 36)
### Chocolate Amaretti
Makes about 16
Difficulty *
Preparation time: 15 minutes plus 14–18 minutes baking time
100 g/generous ¾ cup ground almonds
100 g/generous ¾ cup sifted icing sugar
40 g/⅓ cup light wheat or spelt flour
1 tsp baking powder
½ tsp cornflour
1 large pinch salt
1½ tbsp neutral vegetable oil
50 ml/¼ cup almond or soya milk
2–3 dashes lemon juice
½ tsp bitter almond flavouring
35 g/1¼ oz vegan dark chocolate (45–55 % cocoa solids), finely grated
20 g/¾ oz vegan dark chocolate, melted (optional)
Preheat the oven to 200°C/400°F.
Mix the ground almonds, icing sugar, flour, baking powder, cornflour and salt in a bowl. Add the oil, soya or almond milk, lemon juice and bitter almond flavouring, and mix briskly until smooth. Finally, fold the finely grated chocolate into the batter.
On a baking tray lined with baking parchment, make about 16 small mounds, spaced out evenly over the tray and with sufficient space around them, as they will expand as they bake.
Bake the amaretti for 14–18 minutes, and then leave to cool. Optionally, fill a clean plastic bag with the melted chocolate. Cut out a small hole and pipe chocolate over the amaretti. Leave to cool again. (Picture on p. 37)
Chocolate Muffin with a Liquid Centre
Chocolate Amaretti
### Chocolate and Peanut Muffins
Makes 10-14
Difficulty **
Preparation time: 25 minutes plus 14–17 minutes baking time
For the muffins:
125 g/generous ½ cup sugar
175 g/1½ cups light wheat or spelt flour
2 tbsp lightly defatted cocoa powder
1 large pinch salt
1½ tsp baking powder
50 g/2 oz vegan dark chocolate, chopped
1 tbsp cornflour
75 ml/⅓ cup cold water
120 ml/½ cup neutral vegetable oil
1 tsp white wine vinegar
For the icing:
110 g/½ cup peanut butter
50 g/4 tbsp soft vegan margarine (leave out of the refrigerator to soften before use)
115 g/1 cup sifted icing sugar
⅔ tsp salt
1–2 tsp plant-based milk
Chopped peanuts and grated chocolate to sprinkle (optional, as needed)
Preheat the oven to 190°C/375°F.
For the muffins, mix the sugar, flour, cocoa powder, salt, baking powder and chopped chocolate together in a bowl. Dissolve the cornflour completely in cold water, and add together with the oil and vinegar to the flour mixture. Mix well to obtain a smooth batter. If the batter is too dry, add a little more water.
Line 10–14 cavities of a muffin mould with muffin cases and fill with the batter. Bake for 14–17 minutes. The muffin centres should still be moist after baking.
For the icing, mix the peanut butter and margarine until smooth. Add the icing sugar, salt and milk, and beat to a fluffy cream. If the icing is too runny, add more icing sugar; and if it's too firm, add more milk.
Use a piping bag to decorate the muffins with the icing. Optionally, decorate with chopped peanuts and grated chocolate.
### Chocolate Cookies
Easy-to-make cookies in an instant. Whole cane sugar (not to be confused with raw sugar) is critical for its aroma. This sugar is made from the juice of sugar cane including the molasses. Unlike almost all other kinds of sugar, because whole cane sugar is unrefined, it contains many nutrients, particularly calcium, magnesium and iron.
Makes about 12
Difficulty *
Preparation time: 15 minutes plus 12–14 minutes baking time
150 g/1¼ cups light wheat or spelt flour
⅓ tsp Bourbon vanilla powder
¾ tsp salt
60 g/generous ¼ cup sugar
60 g/generous ¼ cup whole cane sugar
25 g/¼ cup lightly defatted cocoa powder
90 g/7 tbsp vegan margarine
1 tbsp plant-based milk of your choice
Preheat the oven to 200°C/400°F.
Mix the dry ingredients in a bowl. Add the margarine and milk and knead well to form a compact dough ball. This takes some time, so keep kneading patiently until the dough no longer feels wet.
Shape the dough into 12 mounds and space them out evenly over a baking tray lined with baking parchment with sufficient space around them. Press lightly with a fork to give the cookies their typical texture. Don't press too hard; the cookies should be about 1 cm/½ in thick before baking. Bake for 12–14 minutes.
### Chocolate Chip Cookies
Makes about 12
Difficulty *
Preparation time: 10 minutes plus 8–12 minutes baking time
150 g/1¼ cups light wheat or spelt flour
¾ tsp Bourbon vanilla powder
¾ tsp baking powder
¾ tsp salt
60 g/generous ¼ cup sugar (see tip)
60 g/generous ¼ cup whole cane sugar
80 g/3¼ oz vegan dark chocolate (45–55 % cocoa solids), chopped, or 80 g/3¼ oz vegan chocolate chips
90 g/7 tbsp vegan margarine
1 tbsp plant-based milk of your choice
Preheat the oven to 200°C/400°F.
Combine the dry ingredients in a bowl and mix well. Add the margarine and milk and knead well to form a compact dough ball.
Shape the dough into 12 mounds and space them out evenly over a baking tray lined with baking parchment with sufficient space around them. Press lightly with a fork to give the cookies their typical texture. Don't press too hard; the cookies should be about 1 cm/½ in thick before baking.
Bake for 8-12 minutes. They should turn golden brown. Take care that they don't become too dark.
### Tip
If you prefer more of a caramel flavour, you can replace the white sugar in this recipe with whole cane sugar.
### Cakes, Tarts and Cheesecakes
### My Sachertorte
'If life gives you chocolate, make Sachertorte with it!' The perfect vegan Sachertorte.
I've been polishing this recipe like no other, and for so long.
Surprise yourself!
For one 22 cm/8¾ in diameter springform cake tin
Difficulty **
Preparation time: 35 minutes plus 45–60 minutes baking time and approx. 3½ hours resting time
For the cake:
185 g/6½ oz vegan dark chocolate (45–55 % cocoa solids)
200 g/⅞ cup sugar
1 (9 g/0.32 oz) sachet vanilla sugar
200 g/scant 1 cup soft vegan margarine (leave out of the refrigerator for 1 hour to soften before use)
240 g/2 cups plus 2 tbsp light wheat flour
25 g/¼ cup cornflour
1 tbsp baking powder
¼ tsp salt
150 ml/⅔ cup water
2 tsp white wine vinegar
140 g/½ cup apricot jam
For the icing:
200 g/7 oz vegan dark chocolate (45–55 % cocoa solids)
90 g/7 tbsp vegan margarine
Preheat the oven to 180°C/350°F. Line the springform cake tin with baking parchment.
Slowly melt the chocolate over a bain-marie or in the microwave.
Combine the sugar, vanilla sugar and margarine in a bowl and beat until very fluffy. Fold in the melted chocolate.
Mix the flour, cornflour, baking powder and salt in a bowl. Add the flour mixture, water and vinegar to the chocolate mixture. Slowly and carefully fold together by hand with a scraper or in a food processor to a smooth batter. Do not mix too vigorously.
Pour the batter into the prepared cake tin and place on a shelf positioned on the second runner from the bottom. Bake for 45–60 minutes. Check that the cake is cooked through by pricking with a wooden or metal skewer. It should come out clean when you pull it out. Note: don't do this until the cake has been baking for at least 45 minutes, otherwise it may sink.
Turn off the oven and allow the cake to rest inside with the door slightly ajar for 30–40 minutes. The cake may sink a little. If this should happen, it can be fixed with icing. Unmould the cake, cover it completely with aluminium foil and leave to cool for at least 3 hours.
Use a large, sharp knife or a thin plastic thread or metal wire to carefully cut through the cake at one third of its height. Use two long knives or two rubber scrapers to carefully lift off the top and set it aside. Spread 90 g/⅓ cup of jam over the bottom, and then gently replace the top part. Spread the rest of the jam over the top of the cake.
For the icing, slowly melt the chocolate with the margarine over a bain-marie or in the microwave. Use a brush or palette knife to spread the icing evenly over the top and sides of the cake. Leave to cool completely. (Picture on p. 46)
### Tip
After cooling, cover the cake well with aluminium foil so that it doesn't dry out. If stored in the refrigerator – this is absolutely unnecessary – take it out 1½ hours before serving.
### 5-Minute Microwave Mug Cakes
Nothing could be simpler or quicker.
In 5 minutes you'll have a delicious chocolate cake.
These cakes are best enjoyed while they're still warm.
For a medium mug
Difficulty *
Preparation time: 5 minutes
65 g/9 tbsp light wheat or spelt flour
1½ tbsp sugar
2 tsp lightly defatted cocoa powder
1 large pinch Bourbon vanilla powder
¾ tsp baking powder
1 pinch salt
1½ tsp desiccated coconut (optional)
3 tbsp neutral sunflower or rapeseed oil
½ tsp white wine vinegar
Mix all the dry ingredients in a small bowl. Add 75 ml/⅓ cup of water, the vinegar and oil, and mix to a smooth batter. Transfer the batter to a mug of sufficient size. Bear in mind that the cake will roughly double in size in the microwave.
Put the mug in the microwave and cook for 2–3 minutes on a medium–high setting. The cooking time can vary depending on the microwave, so you'll need to pay attention the first time. The bottom of the cake should be pleasantly moist, not hard. (Picture on p. 47)
My Sachertorte
5-Minute Microwave Mug Cake
### Saffron Cream Pie with a Biscuit Base
A biscuit base is a great alternative to the conventional pastry crust.
It stays fresher longer and goes wonderfully well with creamy fillings.
The little extra effort – you can even make the biscuits yourself – is definitely worthwhile.
The wonderful saffron cream ranks among my greatest achievements.
For one 24 cm/9½ in diameter springform cake tin
Difficulty **
Preparation time: 15 minutes plus 8–10 minutes baking time
For the pie crust:
1 portion chocolate chip cookies (see recipe p. 40, or 350 g/12 oz chocolate cookies)
40 g/3 tbsp vegan margarine, room temperature or melted
For the cream:
175 ml/¾ cup soya milk (the high protein content is important, so no other plant-based milk can be used)
115 g/½ cup refined coconut oil, melted
70 g/¼ cup plus 2 tbsp sugar
1 medium lemon, for juice (about 50 ml/¼ cup)
1 large pinch saffron threads
4–5 (10 g/0.35 oz) sachets whipped cream stabiliser
½ tsp salt
Preheat the oven to 200°C/400°F. Line the springform cake tin with baking parchment.
Finely crush the cookies with a rolling pin or in a food chopper. Add the margarine and 1–2 tablespoons of water and knead well until it forms a cohesive dough-like ball. Spread the biscuit base evenly over the prepared cake tin and bake for 8-10 minutes. Take the pie crust out of the oven and leave to cool for a short time.
Combine all the ingredients for the cream and mix well for 2–3 minutes with an electric hand mixer. Spread the cream out evenly over the slightly cooled pie crust. Refrigerate for at least 2 hours for the flavour to develop. Serve cold.
### Tip
For a particularly decorative effect, scatter saffron threads over the pie.
### Death by Chocolate
The ultimate chocolate cake! Along with the chocolate sushi, this is probably the most elaborate recipe in the book, but for all chocolate junkies – I don't hesitate to count myself as one – this will make your chocolate dreams come true.
For one 24 cm/9½ in sided square springform cake tin at least 5 cm/2 in deep (alternatively, use a 26 cm/10¼ in diameter round tin)
Difficulty ***
Preparation time: 1 hour plus 35–45 minutes baking time and at least 4½ hours resting time
For the cake batter:
300 g/1¼ cups soft vegan margarine (leave out of the refrigerator for 1 hour to soften before use)
300 g/1¼ cups sugar
40 g/⅓ cup cornflour
200 ml/scant 1 cup cold soya milk
240 g/2 cups plus 2 tbsp light wheat or spelt flour
90 g/¾ cup lightly defatted cocoa powder
1 tbsp baking powder
1 large pinch salt
The seeds from 1 vanilla pod
2 tsp white wine vinegar
For the icing:
200 g/7 oz vegan dark chocolate (45–55 % cocoa solids)
90 g/7 tbsp vegan margarine
For the cream filling:
200 g/7 oz vegan dark chocolate (45–55 % cocoa solids)
200 g/1 cup natural soya yoghurt, either sweetened or unsweetened
50 g/2 oz vegan white chocolate, grated (optional)
Preheat the oven to 180°C/350°F. Line the springform cake tin with baking parchment.
For the batter, beat the margarine with the sugar in a large bowl until fluffy.
Combine the flour, cocoa powder, baking powder, salt and vanilla seeds in a bowl, and mix well. Dissolve the cornflour completely in the cold soya milk. Add the flour mixture, soya milk and vinegar to the sugar and margarine mixture and carefully mix to a smooth batter. If the batter is too dry, add a 1–3 more tablespoons of soya milk.
Pour the batter into the cake tin and smooth the surface. Bake for 35–45 minutes. Check that the cake is cooked by pricking with a wooden skewer. If it doesn't come out clean, bake for a little longer. After baking, turn off the oven and allow the cake to rest inside with the door slightly ajar for 30–40 minutes.
In the meantime, make the icing. Slowly melt the chocolate with the margarine over a bain-marie or in the microwave. Take the cake out of the oven, but leave it in the tin. Pour the icing over the cake and spread over the top evenly. Leave to cool for a short while, and then cover the cake in the tin with aluminium foil and refrigerate for at least 2 hours (better longer).
For the cream filling, slowly melt the chocolate over a bain-marie or in the microwave. Immediately stir the chocolate into the yoghurt to obtain a smooth cream. Leave to cool.
After cooling the cake, remove the springform tin. Use a sharp knife to cut the cake into about 10 rectangular pieces, and then cut the pieces in half through the middle. Carefully take off the tops and set aside. Spread the cream over the bottoms and carefully replace the tops. Cover well with foil and refrigerate again for at least 2 hours.
Take out of the refrigetor 1 hour before serving. Decorate with grated vegan white chocolate if you like. (Picture on p. 52)
### Chocolate Cake – The Classic
The classic: super easy and quick to make, and always popular.
For one 900 g/2 lb loaf tin
Difficulty *
Preparation time: 15 minutes plus about 50 minutes baking time
225 g/1 cup sugar
210 g/1¾ cups light wheat flour
1 tbsp baking powder
2 pinches salt
75 g/⅔ cup lightly defatted cocoa powder
1½ tbsp cornflour
175 ml/¾ cup cold water
150 ml/⅔ cup neutral sunflower or rape seed oil
1 tsp white wine vinegar
Preheat the oven to 180°C/350°F. Line the loaf tin with baking parchment.
Mix the sugar, flour, baking powder, salt and cocoa powder in a bowl.
Dissolve the cornflour completely in cold water. Add the cornflour, oil and vinegar to the flour mixture and stir briskly with a tablespoon to a smooth batter.
Pour the batter into the tin and smooth the surface. Bake for about 50 minutes. Check that the cake is cooked through by pricking with a wooden or metal skewer. It should come out clean when you pull it out. (Picture on p. 53)
Death by Chocolate
Chocolate Cake – The Classic
### Chocolate Orange Marble Cake
For one 900 g/2 lb loaf tin
Difficulty *
Preparation time: 20 minutes plus 55–65 minutes baking time
1–2 large, untreated orange(s), for grated zest and juice
250 g/1⅛ cup soft vegan margarine
125 g/generous ½ cup sugar
1 large pinch salt
290 g/2½ cups light wheat or spelt flour
1 tbsp baking powder
25 g/¼ cup lightly defatted cocoa powder
50 g/2 oz vegan dark chocolate (45–55 % cocoa solids), finely grated (optional)
100 g/generous ¾ cup icing sugar
Preheat the oven to 180°C/350°F. Line the loaf tin with baking parchment.
Rinse 1 orange under hot water and dry. Completely grate the zest and set aside. Squeeze the juice and set aside. The second orange will be used only if more juice is needed.
Beat the margarine with the sugar and salt in a bowl until fluffy. Mix the flour and baking powder, then add with the orange juice to the sugar and margarine mixture. Mix well to obtain a smooth batter.
Halve the batter. Stir the grated orange zest to one half of the batter. Add the cocoa powder, 4–5 tablespoons of water and the grated chocolate, if required, to the other half and mix until smooth.
Pour the light batter into the bottom of the loaf tin and smooth the surface. Spread the dark batter evenly over the top. Drag a fork through the batter to create the typical marble patterns.
Bake for 55–65 minutes. Check that the cake is cooked through by pricking with a wooden or metal skewer. It should come out clean when you pull it out.
For the icing, mix the icing sugar with 2 tablespoons of orange juice. The icing should be quite thick and definitely not too thin, so be careful when adding the orange juice. Spread the icing over the still warm cake, and then leave to cool completely.
### Swedish Coffee Cake
Coconut and coffee meet chocolate in this Swedish cake.
Super easy to make and a real treat to have with a coffee or cappuccino.
For one 23 cm/9 in diameter springform cake tin
Difficulty *
Preparation time: 20 minutes plus 35–40 minutes baking time and 40 minutes resting time
For the cake batter:
210 g/1¾ cups light wheat or spelt flour
200 g/⅞ cup sugar
1 tbsp baking powder
25 g/¼ cup lightly defatted cocoa powder
1 (9 g/0.32 oz) sachet vanilla sugar
1 pinch salt
100 ml/scant ½ cup neutral vegetable oil
200 ml/scant 1 cup soya milk or almond milk
2 tsp white wine vinegar
For the icing:
75 g/6 tbsp vegan margarine, melted
160 g/generous 1¼ cups sifted icing sugar
¾ tsp vanilla sugar
1 tbsp lightly defatted cocoa powder
1 tsp instant coffee dissolved in 4 tbsp hot water (or 4 tbsp very strong coffee, eg espresso)
50–100 g/½–1 cup desiccated coconut for sprinkling
Preheat the oven to 190°C/375°F. Line the springform cake tin with baking parchment.
For the batter, combine the dry ingredients in a bowl and mix well. Add the oil, soya milk and vinegar, and mix until smooth. Pour the batter into the tin and smooth the surface. Bake for about 35–40 minutes (check that it is cooked through by pricking with a skewer). Leave the cake to cool for 20 minutes in the tin, then remove the tin.
In the meantime, combine the ingredients for the icing and mix well, preferably with a whisk. Shortly before use, heat the icing in a saucepan to leave it nice and runny. Cover the cake completely with the runny icing, allowing it to run down the sides of the cake to form a nice crust. Then put the springform tin back on the cake. If you don't want to cover the sides with icing, you can ice the cake once it is back in the tin.
Leave the icing for 20 minutes to harden, then sprinkle generously with desiccated coconut. Remove the springform pan.
### Tip
The cake tastes even better the next day when some of the icing has been absorbed by the cake and a nice crust has formed.
### Healthy Lifestyle Chocolate and Orange Tart
This tart is a real treat, and especially refreshing in summer. It needs to be kept in the freezer and should be taken out shortly before serving to allow it to thaw a little.
The chocolate literally melts in the mouth, a dream for everybody who – like myself – loves the combination of chocolate and orange.
For one 28 cm/11 in diameter springform cake tin
Difficulty **
Preparation time: 30 minutes plus 3 hours freezing time
For the tart base:
175 g/1 cup pitted dates, as soft as possible (pitted weight, see tip on p. 60)
150 g ground almonds
50 g ground hazelnuts
75 g/6 tbsp refined coconut oil, melted
For the caramel filling:
220 g /1¼ cups pitted soft Medjool dates (pitted weight, see tip on p. 60) or other soft dates
⅓ tsp sea salt
1 heaped tbsp concentrated pear juice or maple syrup
For the chocolate cream:
40 g/⅓ cup lightly defatted cocoa powder
75 g/6 tbsp refined coconut oil, melted
1 tbsp concentrated pear juice or maple syrup
30 ml/2 tbsp agave syrup or 25 g/¼ cup sifted icing sugar
2 untreated oranges for grated zest and juice (total 200 ml/scant 1 cup)
25 g/2 tbsp white almond butter
Line the springform cake tin with baking parchment.
Finely chop the dates for the base. Mix with the ground almonds, hazelnuts, 1½ tablespoons of water and the melted coconut oil. Knead vigorously to obtain a dough. Spread the dough evenly over the bottom of the prepared cake tin and press firmly to leave a 1 cm/½ in border around the sides.
For the caramel filling, combine the dates with 250 ml/1 cup of water, the salt and concentrated pear syrup in a deep container, and blend with a hand-held blender until you have a smooth cream without any lumps. Spread the cream over the tart base. Place in the freezer for at least 1 hour.
In the meantime, make the chocolate cream. Combine all the ingredients in a bowl and blend well. After the caramel layer has frozen, take the tart base out of the freezer and spread the chocolate cream over it. Return the tart to the freezer for at least another 2 hours.
### Healthy Lifestyle Chocolate Cheesecake
So delicious and so healthy! This cheesecake can be made in a 26 cm/10¼ in diameter springform cake tin with double the given measures.
For one 16 cm/6¼ in square cake tin or an 18 cm/7 in diameter springform cake tin
Difficulty **
Preparation time: 40 minutes plus 6 hours freezing time and a few hours refrigeration time
For the base:
60 g/⅓ cup pitted dates, as soft as possible (pitted weight, see tip)
25 g/¼ cup cocoa powder
60 g/½ cup ground hazelnuts
25 g/2 tbsp refined coconut oil, melted
For the filling:
1 medium banana
200 g/1¾ cups cashew nuts (see tip)
75 g/scant ½ cup pitted dates, as soft as possible (pitted weight, see tip)
1 tbsp lemon juice
40 g/⅓ cup cocoa powder
¾ tsp Bourbon-vanilla powder or the seeds from ½ vanilla pod
1 pinch salt
25-40 g/2–3 tbsp refined coconut oil, melted
The banana for the filling should be ripe enough to smell through the skin, but not overripe. Peel and slice the banana, and freeze for at least 6 hours.
Line the cake tin with baking parchment.
For the base, slice the dates finely with a knife. Mix with the cocoa powder and ground hazelnuts in a bowl. Add 2 teaspoons of water and the coconut oil, and knead vigorously to a dough. Spread the dough over the bottom of the cake tin.
For the filling, take the banana out of the freezer and leave to thaw slightly. Combine with the cashews, dates, lemon juice and 5 tablespoons of water in a blender and blend to a smooth cream. If necessary, you can also let the banana thaw out a little more and purée it with a hand-held blender.
Mix in the cocoa powder, vanilla, salt and melted coconut oil, and spread the mixture over the base. Refrigerate for a few hours so that it becomes firm.
### Tips
You can buy dates that are already soft. As a suitable alternative, Medjool dates are naturally soft.
If the cashews are soaked overnight in water, the amount of water required for the filling can be reduced by 1–2 tablespoons.
### Chocolate Cheesecake
For a not so healthy lifestyle.
For one 18 cm/7 in diameter springform cake tin
Difficulty **
Preparation time: 30 minutes plus 12 hours draining time, 1 hour baking time and 1½ hours refrigeration time
For the pastry:
125 g/generous 1 cup light wheat or spelt flour
1 pinch salt
1 tbsp sugar
60 g/5 tbsp vegan margarine
For the filling:
500 g/2½ cups natural soya yoghurt, unsweetened (see tip)
100 ml/scant ½ cup soya milk
1½ tbsp lemon juice
40 g/⅓ cup cornflour
130 g/9 tbsp vegan margarine, melted
100 g/⅜ cup sugar
2 tbsp concentrated pear juice or maple syrup
25 g/¼ cup lightly defatted cocoa powder
½ tsp Bourbon vanilla powder
1 pinch salt
2 tbsp rum (optional)
Leave the yoghurt to drain overnight in a sieve lined with 2 layers of kitchen paper.
The next day, make the pastry by mixing the flour with the salt and sugar. Knead in the margarine with your fingers until no lumps can be felt.
Add 2 tablespoons of water and knead a little longer. Refrigerate for at least 30 minutes.
Preheat the oven to 180°C/350°F. Line the springform cake tin with baking parchment.
Spread the dough evenly over the bottom of the prepared cake tin and bake for 10 minutes.
In the meantime, make the filling by mixing the soya milk with the lemon juice and cornflour. Transfer the drained yoghurt to a bowl.
Add the soya milk mixture and the other ingredients for the filling. Mix well with a whisk to obtain a nice, creamy, lump-free mixture. Spread the filling over the pre-baked pastry base.
Raise the oven temperature to 190°C/375°F and bake the cheesecake for 50 minutes. Leave to cool, and then refrigerate for at least 1 hour.
### Tip
Sweetened soya yoghurt can naturally be used for the filling, but the sugar in the recipe will have to be reduced to 70 g/¼ cup plus 2 tbsp.
### Berry Pie with a Chocolate Crust
For one 28 cm/11 in diameter round baking tin
Difficulty *
Preparation time: 20 minutes plus about 30 minutes refrigeration and the same for baking
For the pastry:
240 g/2 cups plus 2 tbsp light wheat or spelt flour
½ tsp salt
3 tbsp sugar
25 g/¼ cup lightly defatted cocoa powder
50 g/2 oz vegan dark chocolate (45–55 % cocoa solids), grated
90 g/7 tbsp vegan margarine
1 tsp white wine vinegar
For the glaze:
300 ml/1¼ cups almond or soya milk
40 g/3 tbsp white almond butter
The seeds from 1 vanilla pod or ⅓ tsp Bourbon vanilla powder
2½ tbsp freshly squeezed lemon juice
70 g/¼ cup plus 2 tbsp sugar
2 tbsp cornflour
1 pinch salt
For the filling:
5 tbsp ground almonds
500 g/4 cups assorted frozen berries (raspberries, blackberries, redcurrants, strawberries)
5 tbsp raspberry or blackberry jam
Preheat the oven to 220°C/425°F.
For the pastry, combine the flour, salt, sugar, cocoa powder and grated chocolate in a bowl, and mix well. Add the margarine and work it in with your fingers, without kneading, until no lumps can be felt.
Mix the vinegar and 100 ml/scant ½ cup of cold water and add it to the dough. Briefly knead, roll the dough into a compact ball, cover with aluminium foil and refrigerate for at least 30 minutes.
Combine all the ingredients for the glaze and blend well with an electric hand mixer.
Grease the baking tin and spread the pastry over evenly. Spread the ground almonds out over the pastry base. Prick the pastry repeatedly with a fork. Cover the pastry with the berries, then spread the jam over the top. Finally, cover with an even layer of glaze. Place the pie on a shelf positioned on the second runner from the bottom. Bake for 30–35 minutes.
### Chocoholic Favourites
### Chocoholic Custard Slices
Traditional millefeuille pastries filled with vanilla custard, are a Swiss classic.
But honestly, who needs vanilla? This chocolate version will knock your socks off!
Makes 6
Difficulty **
Preparation time: 40 minutes plus 12–15 minutes baking time
1 rectangular sheet vegan puff pastry (320 g; 36 x 24 cm/14¼ x 9½ in)
For the filling:
75 g/3 oz vegan dark chocolate (45–55 % cocoa solids)
300 ml/1¼ cups soya whipping cream
1½–2 (10 g/0.35 oz) sachets whipped cream stabiliser
3 tbsp sugar
¼ tsp Bourbon vanilla powder
1 pinch salt
25 g/1 oz vegan dark chocolate (45–55 % cocoa solids), grated, or 25 g/1 oz vegan chocolate sprinkles
For the topping:
60 g/¼ cup apricot jam
100 g/3¾ oz vegan dark chocolate (45–55 % cocoa solids)
50 g/¼ cup vegan margarine
Preheat the oven to 200°C/400°F.
Roll out the puff pastry over a baking tray lined with baking parchment, and trim to the required size. Prick the pastry repeatedly with a fork so that it rises as little as possible. Cut the pastry across its width to make 3 uniform 12 x 24 cm/4½ x 9½ in strips. Bake for 12–15 minutes until light golden.
For the chocolate filling, slowly melt the chocolate over a bain-marie or in the microwave. Whip the soya cream for 1–2 minutes with an electric hand mixer. Sprinkle in the stabiliser and whip for another 2 minutes until the cream is firm. Depending on the cream you use, you may need more or less stabiliser.
Fold the melted chocolate, sugar, vanilla and salt into the whipped cream to make a nice chocolate mousse. Finally, gently fold the grated chocolate or chocolate sprinkles into the mousse.
The mousse must be firm, otherwise refrigerate for 1 or 2 hours.
Spread half of the mousse over one of the three pastry strips, right to the edge. Take the second strip, place it over the first and spread the other half of the mousse over it. Finally cover with the last strip, with the smooth side facing upwards.
For the topping, lightly warm the apricot jam and carefully brush over the top of the pastry. Slowly melt the chocolate with the margarine over a bain-marie or in the microwave, and spread this icing over the top of the pastry. Don't spread this icing right to the edges as none of it should cover the sides. Wait until the chocolate has firmed a little, but still remaining soft, then use a brush to spread the icing to the edges.
Use a sharp serrated knife to cut the pastry into six 12 x 4 cm/4½ x 1½ in slices. Allow to cool fully. Best served cold.
### Super Easy Chocolate Croissants
Makes 8–10
Difficulty *
Preparation time: 10 minutes plus 14–20 minutes baking time
150 g/5 oz vegan dark chocolate (45–55 % cocoa solids)
1 round sheet vegan puff pastry (275 g/8 oz, 32 cm/12½ in in diameter), very cold
1 tsp plant-based cream with ½ tsp ground turmeric (optional)
Use a sharp knife to chop 100 g/3¾ oz of the chocolate into very small pieces.
Cut the pastry into 8–10 uniform segments (like slices of a pie), depending on the size of the croissants. Note: the pastry should be refrigerated for at least 2 hours before use, and can be used directly from the refrigerator.
Preheat the oven to 220°C/425°F.
Place 1–1½ teaspoons of chopped chocolate in the middle of each piece of puff pastry, about ½ cm/¼ in from the bottom edge. Carefully roll them up from the bottom edge towards the tip. Push the sides in and press lightly. Bend slightly into a crescent – the typical croissant shape. Lay the croissants over a baking tray lined with parchment paper.
Optionally, you can brush a little cream with turmeric lightly over the croissants. Place the baking tray in the oven and position an oven shelf in the runner immediately above it to stop the pastry rising too much. Bake the croissants for 14-20 minutes until golden brown.
Melt the rest of the chocolate over a bain-marie or in the microwave, Brush the chocolate over both ends of the croissants or decorate the croissants with it as you please.
### Tip
The croissants should be allowed to cool a little and are best served warm.
### Crunchy Chocolate and Cinnamon Granola
Nothing beats home-made granola. This version has a particularly aromatic flavour.
The combination of cinnamon and chocolate is brought to full effect during baking.
Granola, accompanied by spelt or rice milk, is one of my favourite snacks.
Dry, it also makes a good snack to have on the go.
Makes 1 kg
Difficulty **
Preparation time: 15 minutes plus 35 minutes baking time and 3–4 hours resting time
2 x 125 g/generous 1 cup coarse oatmeal
2 x 125 g/generous 1 cup pinhead oatmeal
2 x 40 g/½ cup puffed oats
2 x 25 g/¼ cup desiccated coconut
2 x ⅔ tsp ground cinnamon
2 x ⅓ tsp Bourbon vanilla powder
65 g/9 tbsp lightly defatted cocoa powder
75–120 ml/⅓–½ cup concentrated pear juice or maple syrup, depending on the desired sweetness
100 ml/scant ½ cup neutral vegetable oil
Sultanas and corn flakes (as needed)
Combine each portion of the coarse and pinhead oatmeal, puffed oats, desiccated coconut, cinnamon and vanilla, respectively in two separate bowls and mix well. Add the cocoa powder to one of the bowls and mix well.
Heat the concentrated pear juice, oil and 120 ml/scant ½ cup of water in a saucepan. Bring to the boil, then pour equal amounts in to both bowls. Make sure that the oil floating on the surface doesn't get poured into only one of the bowls. Now mix until the liquid is as evenly distributed as possible.
Combine both mixtures and spread over a baking tray. Place in a cold oven. Set the temperature to 180ºC/350ºF and bake for 35 minutes. Stir the mixture after 25 minutes. After baking, turn off the oven. Leave the granola in the still warm oven with the door closed for 3–4 hours. This step will produce a particularly even crispiness.
Take the granola out of the oven and mix with sultanas and corn flakes.
### Tip
You can also make up double the measures of this recipe. Simply spread it out over two baking trays and bake one at the top and the other at the bottom of the oven.
### NB
Because they contain very little gluten compared to other cereals, oats are relatively well tolerated by people affected by coeliac disease, when consumed in moderate amounts. When shopping, it is absolutely essential to look out for special gluten-free oat products that haven't been contaminated with gluten from other products. However, people who are allergic to avenin, a protein present in oats, should totally avoid oats and oat products.
### Gingerbread Mousse Tartlets
Makes 7-9
Difficulty *
Preparation time: 25 minutes plus at least 1½ hours refrigeration time and 15–17 minutes baking time
For the mousse:
200 ml/scant 1 cup soya whipping cream, unsweetened (see tip)
1 (9 g/0.32 oz) sachet whipped cream stabiliser
40 g/1½ oz vegan dark chocolate (45–55 % cocoa solids)
1 tsp gingerbread spice blend
1–1½ tbsp maple syrup (grade A)
1 pinch salt
1 tbsp rum or 2 drops rum flavouring (optional)
Cocoa powder for sprinkling
For the tartlet cases:
150 g/1¼ cups light wheat flour
1½ tbsp sugar
1 pinch salt
75 g/6 tbsp cold vegan margarine
Preheat the oven to 200°C/400°F.
For the mousse, whip the cream with an electric hand mixer on high speed. Sprinkle in the stabiliser and whip for another 2 minutes.
Slowly melt the chocolate over a bain-marie or in the microwave. Add the gingerbread spice, maple syrup, salt, 2 tablespoons of the whipped cream and, optionally, the rum or rum flavouring, and mix to a smooth cream. Carefully fold the cream into the remaining whipped cream to obtain a smooth mousse. Refrigerate the mousse for at least 1 hour, although preferably for 2–3 hours.
For the tartlet cases, mix the flour with the sugar and salt. Add the margarine and work it in with your fingers, without kneading, until there are no more lumps. Add 50 ml/¼ cup of water, knead briefly and roll into a compact ball. Wrap in aluminium foil and refrigerate for at least 30 minutes.
Grease 7–9 tartlet moulds (or the cavities of a muffin mould) and line each one with the pastry, leaving a border of 1–1½ cm/½ –⅝ in around the sides. Prick the pastry repeatedly with a fork, and then bake for 15–17 minutes. Leave to cool. Fill a piping bag with the mousse and pipe a uniform amount into the cooled tartlet cases. Before serving, dust with cocoa powder. Store the mousse-filled tartlets in the refrigerator until it is time to serve.
### Tip
If you prefer to use a sweetened plant-based whipping cream, simply reduce the amount of maple syrup to 2–3 teaspoons.
### Pear and Chocolate Gratin with Hazelnut Streusel
Serves 4
Difficulty *
Preparation time: 20 minutes plus 35–40 minutes baking time
For the gratin:
450 g/1 lb ripe pears
375 ml/generous 1½ cups soya milk
2 tbsp cornflour
25 g/¼ cup lightly defatted cocoa powder
40 g/3 tbsp hazelnut butter or brown almond butter
60 g/generous ¼ cup sugar
1 pinch salt
1½ tsp vanilla sugar
1 tbsp freshly squeezed lemon juice
For the streusel:
60 g/½ cup ground hazelnuts
1½ tsp sugar
2–3 tsp lemon juice
1 tsp grated lemon zest (untreated lemon)
Preheat the oven to 200°C/400°F.
Wash, core and slice the pears. Arrange the pear slices in a large, greased gratin dish or divide evenly between several smaller moulds.
Combine the remaining ingredients for the gratin and mix well with an electric hand mixer to a smooth sauce.
For the streusel, mix the ground hazelnuts with the sugar. Add the lemon juice and grated zest and knead vigorously with your fingers. The streusel should not be moist, nor should it be too dry. Adjust the amount of lemon juice accordingly.
Pour the sauce over the pears, put the dish in the oven and bake for 35–40 minutes.
After baking for 20 minutes, take the gratin out of the oven and crumble the streusel over the top. Return the dish to the oven and finish baking. The streusel should turn golden brown.
### Marzipan and Chocolate Bar
Makes 8–10
Difficulty **
Preparation time: 20 minutes plus at least 30 minutes cooling time and 20 minutes baking time
For the biscuit:
100 g/generous ¾ cup light wheat or spelt flour
¼ tsp baking powder
45 g/scant ¼ cup sugar
2 pinches salt
75 g/6 tbsp vegan margarine
For the filling and coating:
375 g/13 oz soft marzipan
100 g/3¾ oz vegan dark chocolate (45–55 % cocoa solids)
20 g/1½ tbsp vegan margarine
Preheat the oven to 200°C/400°F.
For the biscuit, combine the flour, baking powder, sugar and salt, and mix well. Add the margarine and work in with your fingers. Add 4 teaspoons of water, knead vigorously for a short time, and then roll the dough into a ball. If the dough is very dry, add another teaspoon of water. It should be moist but not wet. Wrap the dough in aluminium foil and refrigerate for at least 30 minutes.
Halve the dough and roll each half out on a sheet of baking parchment under a sheet of cling film to make two rectangles measuring 26 x 10 cm/10¼ x 4 in. Trim with a knife if necessary. Bake for 20 minutes, and then leave to cool.
Use a sharp knife to cut the rectangles across their width to make 8–10 bars.
Roll out the marzipan and cut into pieces the same size as the bars and place them over the biscuit base. Take care that the marzipan doesn't break.
For the coating, slowly melt the chocolate with the margarine over a bain-marie and brush it over the bars.
### Chocolate Biscuit Slices with Lemon 'Cream Cheese' Filling
Makes 6
Difficulty **
Preparation time: 25 minutes plus 10 minutes baking time and 4 hours cooling time
For the biscuit:
100 g/generous ¾ cup light wheat or spelt flour
1 (9 g/0.32 oz) sachet vanilla sugar
1 (9 g/0.32 oz) sachet baking powder
2 tbsp lightly defatted cocoa powder
2 tbsp cornflour
70 g/¼ cup plus 2 tbsp sugar
1 pinch salt
1 tsp white wine vinegar
120 ml/½ cup soya milk
2 tbsp neutral vegetable oil
Fat/oil for greasing
For the cream:
120 ml/½ cup soya milk (the high protein content is important, so no other plant-based milk can be used)
75 g/6 tbsp refined coconut oil, melted
50 g/¼ cup sugar
½ untreated lemon, for grated zest
60 ml/4 tbsp freshly squeezed lemon juice
4 (10 g/0.35 oz) sachets whipped cream stabiliser 1 pinch salt
Preheat the oven to 200°C/400°F.
For the biscuit, combine the dry ingredients in a bowl and mix well. Add the vinegar, soya milk and oil, and mix well. Spread the wet dough out over a greased baking tray into a 24 cm/9½ in square. Bake for 10 minutes, and then leave to cool. Cut the biscuit lengthways into two 12 cm/4½ in wide strips. Cut each of the halves into six 4 x 12 cm/1½ x 4½ in strips.
For the cream, combine all the ingredients and mix well for 2–3 minutes with an electric hand mixer. Refrigerate for at least 4 hours.
Spread about 1 tablespoon of the cream over the porous side of one strip and cover with a second strip. These slices will keep for about 2–3 days in the refrigerator covered in aluminium foil.
### Chocolate Yeast Rolls
There's nothing like a fresh chocolate roll! With this recipe you can get everybody out of bed as soon as the smell of baking bread comes wafting from the oven.
Makes 8
Difficulty **
Preparation time: 30 minutes plus 45–60 minutes proving time and 15–18 minutes baking time
20 g/¾ oz fresh yeast
1 tsp plus 1 tbsp sugar
100 ml/scant ½ cup lukewarm almond milk or soya milk
210 g/1¾ cups light wheat or spelt flour
¼ tsp salt
2 tbsp desiccated coconut (optional)
50 g/¼ cup soft vegan margarine
Version 1: Vegan Nutella (see recipe on p. 18) or another chocolate spread
Version 2: 75 g/3 oz vegan chocolate chips or 75 g/3 oz vegan dark chocolate, finely chopped
Pearl sugar (optional, as needed)
Fully dissolve the yeast and 1 teaspoon of sugar in the lukewarm almond milk.
Combine the flour with 1 tablespoon of sugar, the salt and, optionally, the desiccated coconut, and mix well. Add the yeast mixture and the soft margarine, and knead vigorously for 10 minutes to a smooth dough. Transfer the dough to a bowl, cover with a cloth and leave to prove in a warm place (at least 22ºC/72ºF, maximum 50ºC/122ºF) for 45–60 minutes until it doubles in size.
Preheat the oven to 200°C/400°F.
Divide the dough into 8 uniform balls and place them on a baking tray lined with baking parchment.
For version 1, press the balls flat and place 1 generous teaspoon of vegan Nutella in the middle. Fold up the sides of the dough to the middle and seal the well, so that the filling doesn't leak out when baked. Turn the rolls over so that the join is resting on the tray.
For version 2, knead the chocolate chips or finely chopped chocolate into the dough.
You can prove the dough balls again if you like, so that they become especially fluffy. But if you'd rather not wait, you can put them straight into the oven and bake for 15–18 minutes, until they turn golden. You can also decorate them with pearl sugar. Best served a little warm.
### Chocolate Focaccia with Rosemary and Sea Salt
This focaccia has a light but not overpowering chocolate flavour that makes it a culinary treat when combined with salty and savoury food.
Makes 1 focaccia
Difficulty **
Preparation time: 10 minutes plus 1–1½ hours proving time and 22–25 minutes baking time
20 g/¾ oz fresh yeast
1 tsp sugar
75 ml/⅓ cup lukewarm plant-based milk (preferably soya or almond milk)
100 g/3¾ oz vegan dark chocolate (45–55 % cocoa solids)
240 g/2 cups plus 2 tbsp light wheat or spelt flour
Sea salt
1 tbsp fresh rosemary, finely chopped
1 tbsp mild olive oil
Flour for the work surface
Fully dissolve the yeast and the sugar in the lukewarm milk.
Slowly melt the chocolate over a bain-marie or in the microwave.
Mix the flour with a pinch of salt and the chopped rosemary. Add the dissolved yeast, 50 ml/¼ cup of water, the melted chocolate and oil to the flour mixture and knead together vigorously until the dough no longer sticks.
Transfer the dough to a bowl, cover with a cloth and leave to prove – either at room temperature (ideally 22–35ºC/72–95ºF) for 1–1½ hours, or in a preheated oven at 50ºC/112ºF for 45 minutes – until it doubles in size.
Preheat the oven to 200°C/400°F.
Knead the dough again briefly, and then roll it out over a floured work surface to a 20 x 30 cm/8 x 12 in rectangle. Transfer to a baking tray lined with parchment paper. Use a knife to score the surface with a diamond pattern, and sprinkle with sea salt. Bake for 22–25 minutes.
### No-Bake Chocolate Treats
### My Bounties
These bars are one of my favourite sweet surprises.
I always have a portion stored away in my freezer to enjoy whenever I want, or to give away.
Makes 15–30, depending on size
Difficulty **
Preparation time: 30 minutes plus at least 2 hours cooling time
For the coconut mixture:
150 g/1½ cups desiccated coconut
¾ (9 g/0.32 oz) sachet vanilla sugar
70 g/¼ cup plus 2 tbsp sugar
1 pinch salt
75 g/6 tbsp refined coconut oil
75 ml/⅓ cup almond or soya milk
150 ml/⅔ cup coconut milk
1½ tbsp cornflour
For the coating:
185 g/6½ oz vegan dark chocolate (45–55 % cocoa solids)
50 g/¼ cup vegan margarine
Combine all the ingredients for the coconut mixture, except the cornflour, in a saucepan and heat slowly until the coconut oil is completely melted. Then stir in the cornflour. Leave the mixture to cool completely, then refrigerate for at least 1 hour to harden.
After refrigerating, stir the coconut mixture for a short time, and then use your hands to shape into very compact bars of the desired size. It's a good idea to refrigerate them again (if you're in a hurry, you can also use them as they are).
For the chocolate coating, slowly melt the chocolate and margarine completely over a bain-marie over a low heat. Use two teaspoons to dip the coconut bars in the chocolate and lay them on a sheet of baking parchment. Leave to cool completely. They should be allowed to harden again in the refrigerator.
### Spiced Chocolate Mousse
Nutmeg, cloves and cinnamon – three spices which, together with cocoa and chocolate, call to mind the unforgettable flavour of Magenbrot, Swiss alpine gingerbread.
If you like Magenbrot, you'll love this mousse!
Serves 3–4
Difficulty *
Preparation time: 10 minutes plus at least 1 hour cooling time
400 ml/1⅔ cups soya whipping cream
2½-3 (10 g/0.35 oz) sachets whipped cream stabiliser
60 g/2¼ oz vegan dark chocolate (45–55 % cocoa solids)
2 tsp lightly defatted cocoa powder
⅓ tsp ground nutmeg
⅔ tsp ground cloves
1 tsp ground cinnamon
1 large pinch salt
50 ml/¼ cup concentrated pear juice, or concentrated apple juice or maple syrup
100 g/3¾ oz vegan Magenbrot (optional, see tips)
Whip the cream with an electric hand mixer for 2 minutes. Sprinkle in the stabiliser and whip for another 2 minutes.
Slowly melt the chocolate over a bain-marie. Add the cocoa powder, spices, salt and concentrated pear juice, and mix to a smooth cream. Add 2 tablespoons of the whipped cream and mix until smooth. Carefully fold this cream into the remaining whipped cream to obtain a smooth, fluffy mousse. Refrigerate the mousse for at least 1 hour, although preferably for 2–3 hours.
Optionally, you can crumble some Magenbrot with your fingers and sprinkle it over the mousse.
### Tips
It's very important when following this recipe to be extremely precise when measuring out the spices. Even small deviations can mean that the spices will either be overpowering or unnoticeable.
You may find vegan Magenbrot in some supermarkets and organic food shops, or online.
### Frozen Mocha and Chocolate Cream Tart with a Hazelnut and Date Crust
I've never really been into coffee as a drink, but coffee ice cream has always fascinated me.
This frozen mocha tart is a real summer delight.
For one 18 cm/7 in diameter springform cake tin or another mould of a similar size but with high sides
Difficulty **
Preparation time: 20 minutes plus at least 3 hours cooling time
For the tart base:
100 g/generous ½ cup pitted soft dates (pitted weight, see tip)
90 g/¾ cup ground hazelnuts
½ tsp ground cinnamon
For the chocolate cream:
3 tbsp lightly defatted cocoa powder
3 tbsp neutral vegetable oil
3 tbsp maple syrup or concentrated pear juice
For the mocha filling:
2 tsp instant coffee powder
70 g/¼ cup plus 2 tbsp sugar
40 g/3 tbsp cashew butter or white almond butter
1 pinch salt
The seeds from ½ vanilla pod
4 tsp coffee beans
Coffee beans, cranberries and pistachios to decorate (optional, as needed)
For the tart base, chop the dates and knead together with the ground hazelnuts and cinnamon. If necessary, add teaspoons of water to make the dough easier to knead, but it shouldn't be too moist. Line the bottom of the tin with baking parchment and cover with the pastry.
For the chocolate cream, combine all the ingredients and mix to a smooth cream. Spread the cream over the tart base.
For the mocha filling, combine all the ingredients, except the coffee beans, in a saucepan with 300 ml/1¼ cups of water and bring to the boil to completely dissolve the sugar and coffee. Leave to cool for a short time. Grind the coffee beans in a food chopper and fold into the mocha cream. Pour this cream over the chocolate cream and spread smoothly.
Place the tart in the freezer for at least 3 hours. Keep it in the freezer until it is time to serve. Take it out of the freezer some 10–15 minutes before serving and allow it to thaw slightly. You can decorate it with coffee beans, cranberries and/or pistachios if you like.
### Tip
Soft dates can be found in supermarkets. They're perfect for this recipe.
Alternatively, you can buy Medjool dates, which are naturally very soft and easy to work with.
### Peanut Tartlets
If you love peanut and chocolate bars, you'll adore these tartlets.
Makes 12-14
Difficulty *
Preparation time: 15 minutes plus 3 hours refrigeration time
For the peanut base:
75 g/6 tbsp refined coconut oil
150 g/10 tbsp peanut butter or 130 g/9 tbsp peanut butter plus 20 g/1½ tbsp coconut oil
100 g/generous ¾ cup sifted icing sugar
The seeds from 1 vanilla pod
60 g/½ cup salted and roasted peanuts
¾ tsp salt
For the filling:
150 g/5 oz vegan dark chocolate (45–55 % cocoa solids)
50 g/¼ cup peanut butter
For the peanut base, combine the coconut oil with the peanut butter in a saucepan and place over a low heat until both ingredients are fully melted. Add the icing sugar, vanilla seeds, peanuts and salt, and stir until the sugar is completely dissolved.
Spread the peanut mixture evenly to a height of 1 cm/½ in in 12–14 paper or silicone cupcake cases. Place the cases on a tray and refrigerate for 1 hour. If you're in a hurry, put them in the freezer for 30 minutes. Ideally, the peanut base shouldn't be too firm for the next step, but slightly runny.
For the filling, slowly melt the chocolate and peanut butter over a bain-marie and stir to a smooth cream. Pour a layer of chocolate cream over the peanut base in each cupcake case.
If the base is still runny, drag a fork once or twice across the base through the layers to create a pretty pattern. This is purely decorative and isn't a necessary step. Refrigerate for at least another 2 hours. The tart-lets can be served directly from the refrigerator.
### Tip
These tartlets will freeze very well, and they can be thawed and enjoyed at will.
### Almond and Chocolate Panna Cotta with a Berry Sauce
Makes 4
Difficulty *
Preparation time: 20 minutes plus at least 1 hour cooling time
For the panna cotta:
400 ml/1⅔ cups water
25 g/2 tbsp white almond butter
40 g/1½ oz vegan dark chocolate (45–55 % cocoa solids)
¾ tsp Bourbon vanilla powder or the seeds from ½ vanilla pod
45 g/scant ¼ cup sugar
2 tsp lemon juice
10 g/2 tsp brown almond butter (optional)
1 tsp agar-agar
For the berry sauce:
215 g/1¾ cups assorted frozen berries (blackberries, blueberries, raspberries, strawberries), thawed (with collected juice)
100 ml/scant ½ cup grape juice
2 tbsp sugar
For the panna cotta, combine the water, white almond butter, chocolate, vanilla, sugar, lemon juice and, optionally, the brown almond butter in a saucepan and bring to the boil. Simmer over a medium heat and stir with a whisk until the chocolate is completely melted. Sprinkle in the agar-agar and simmer for another 1–2 minutes while stirring constantly.
Fill four small glasses, cups or small plastic bowls and leave to cool, and then refrigerate for at least 1 hour.
For the sauce, combine the thawed berries, their juice, the grape juice and sugar in a saucepan and bring to the boil. Simmer over a medium heat for 2–3 minutes uncovered.
Unmould the panna cotta on plates and serve with the sauce.
### Tip
If a panna cotta doesn't want to come out of its mould, it helps to dip the bowl halfway in hot water or, failing that, run a thin, sharp knife around the edge to loosen it, then it will slide out easily.
### Grandma's Chocolate-Covered Dates
Chocolate-covered dates were my maternal grandmother's speciality, and they still remind me of Christmas as a child.
I took her recipe and developed it. Of course, the version I present here is vegan.
Dates are particularly used at Christmas to decorate biscuits and confectionery.
Makes 30
Difficulty **
Preparation time: 30 minutes
30 blanched almonds
1 tbsp concentrated pear juice or maple syrup
1 tsp gingerbread spice blend
30 dates (Deglet Nour variety, don't use very moist dates)
90 g/3½ oz vegan dark chocolate (45–55 % cocoa solids)
40 g/3 tbsp vegan margarine
grated vegan dark chocolate and/or vegan white chocolate (optional, as needed)
Roast the almonds in a dry non-stick frying pan on a high heat for 3–5 minutes, turning regularly, until they turn a light golden brown. On no account should they turn black. Remove the pan from the heat, and immediately add the concentrated pear juice and gingerbread spices. Mix well. Leave the almonds to stand in the pan off the heat for 1–2 minutes, and then transfer to a sheet of baking parchment. Spread the almonds out so that they don't stick to each other.
Make a slit in the dates and remove the stone, but do not cut completely in half. Insert an almond into each date.
Slowly melt the chocolate with the margarine over a bain-marie. Dip each date individually in the melted chocolate and transfer to a plate or a sheet of baking parchment. You can also dredge the chocolate-coated dates in grated chocolate if you want, and leave them to cool.
### Chocolate and Hazelnut Pralines
Makes about 10
Difficulty *
Preparation time: 15 minutes plus cooling time
100 g/3¾ oz vegan dark chocolate (45–55 % cocoa solids)
75 g/6 tbsp vegan margarine
50 g/½ cup sifted icing sugar
100 g/scant 1 cup ground roasted hazelnuts (see tip)
Slowly melt the chocolate with the margarine over a bain-marie or in the microwave. Add the icing sugar and ground hazelnuts, stir briskly while over the bain-marie.
Allow the mixture to cool completely before shaping into pralines. Keep refrigerated until use.
### Tip
If you can't find roasted ground hazelnuts, you can use the unroasted variety and roast them yourself in a dry non-stick frying pan for a few minutes.
### Oat and Chocolate Crunchies with Sunflower Seeds
These crunchies make a great snack for when you're on the go or between meals, and they're really quick to make.
Makes about 30 small crunchies
Difficulty *
Preparation time: 10 minutes
25 g/2 tbsp refined coconut oil
3 tbsp sunflower seeds
6 tbsp coarse oatmeal (see note on p. 72)
35 g/1¼ oz vegan dark chocolate (45–55 % cocoa solids)
1 tbsp concentrated pear juice or maple syrup
Heat the coconut oil in a non-stick frying pan. Add the sunflower seeds and oatmeal, turn the heat to medium, and toast until light golden.
Remove from the heat, add the chocolate and concentrated pear juice and stir until the chocolate is fully melted and is evenly distributed. Leave to stand for 1 more minute. Spread the mixture out over a sheet of baking parchment and make sure there are no large clumps. Leave to cool for at least 10 minutes. (Picture on p. 102)
Chocolate and Hazelnut Pralines
Oat and Chocolate Crunchies with Sunflower Seeds
Chocolate Sushi
### Chocolate Sushi
A sweet reinterpretation of the classic Japanese dish.
Makes 4 rolls of each variety
Difficulty ***
Preparation time: 45 minutes plus at least 2 hours cooling time
For the chocoholic makizushi:
About 215 g/1¾ cups ripe kiwifruit, strawberries, mango, pears, grapes or any combination of these fruits
100 g/3¾ oz vegan dark chocolate (45–55 % cocoa solids)
150 g/¾ cup sushi rice or rice pudding
1 tsp lemon juice
2 tbsp sugar
For the chocoholic California rolls:
White sesame seeds
About 215 g/1¾ cups ripe kiwifruit, strawberries, mango, pears, grapes or any combination of these fruits
100 g/3¾ oz vegan dark chocolate (45–55 % cocoa solids)
150 g/¾ cup sushi rice or rice pudding
1 tsp lemon juice
2 tbsp sugar
For the chocoholic makizushi, prepare a clean and smooth work surface.
Draw four rectangles measuring 17 x 11 cm/6½ x 4¼ in and four more of 17 x 10 cm/6½ x 4 in with a pencil on baking parchment and cut them out.
Wash or peel and clean the fruit, as required, and slice thinly. Slowly melt the chocolate over a bain-marie (not in the microwave because the chocolate will be too thick for this recipe).
In the meantime, bring 175 ml/¾ cup of water and the sushi rice to the boil in a small saucepan. Lower the temperature, cover with a lid and simmer for 10 minutes. Remove from the heat, remove the lid, and leave the rice to stand for 15 minutes. Add the lemon juice and sugar and mix well.
Now, use a quarter of the melted chocolate to coat the four 17 x 10 cm/6½ x 4 in rectangles completely, including the edges. Spread the still warm sushi rice over one of the 17 x 11 cm/6½ x 4¼ in rectangles, leaving a 1 cm/½ in border at the top and bottom along the longer edges without covering. Make a thin line of fruit of your choice (don't use too much fruit) along the middle of the rice, parallel to the longer edge. Roll the rice with the parchment, starting from one of the longer sides. It should come loose from the paper. Lay the rice roll over the chocolate-covered baking parchment with the edges flush, and then roll the chocolate over the top of the rice. Roll this parchment up slowly and carefully to the edge, and leave it on. Ideally, the ends of the paper should be quite flush.
With the help of a large knife, lay the rolls (baking parchment still attached), as they are made, on a plate with the join underneath, and refrigerate for 2 hours.
After cooling, carefully remove the baking parchment from the rolls. Use a sharp knife to cut the rolls into slices of about 2 cm/¾ in wide, dipping the knife in hot water between cuts. The chocoholic makizushi is ready.
For the chocoholic California roll, prepare a clean and smooth work surface.
Draw four 17 x 13 cm/6½ x 5 in rectangles on a sheet of baking parchment and cut them out. Sprinkle the work surface with sesame seeds. The sushi rolls will be rolled in it later.
Wash or peel and clean the fruit, as required, and slice thinly. Slowly melt the chocolate over a bain-marie (see opposite).
Prepare the sushi rice as described opposite.
Spread the still warm sushi rice over one of the four 17 x 11 cm/6½ x 4¼ in rectangles, leaving a 1 cm/½ in border at the top and bottom along the longer edges without covering. Spread the runny chocolate over the rice, also leaving a 1 cm/½ in border without covering, otherwise the chocolate will run out. Make a thin line of fruit of your choice (don't use too much fruit) along the middle of the rice, parallel to the longer edge.
Roll the rice with the parchment, starting from one of the longer sides. It should come loose from the paper. As they are made, carefully lay the rolls over the sprinkled sesame seeds, and use both hands to carefully turn them in the sesame seeds. With the help of a large flat knife, transfer the rolls to a plate. Refrigerate the sushi rolls for at least 2 hours.
After cooling, use a sharp knife to cut the rolls into slices of about 2 cm/¾ in wide, dipping the knife in hot water between cuts. (Picture on p. 103)
### Tips
Precision is crucial for the success of this recipe. It is important that the rice is still hot and the chocolate is still runny when starting to make the rolls. It's a good idea to keep the saucepan/bain-marie with the chocolate close at hand and the chocolate over the bain-marie so it stays runny as you work.
To prevent the chocolate and/or the rice from cooling too soon, always make one roll at a time. Don't spread all the rice or chocolate over the baking parchment rectangles at the beginning.
A tip about cutting: The rolls are easier to cut if you take them out of the refrigerator about half an hour beforehand, as the chocolate will be less brittle. Also, don't cut the rolls slowly. Instead cut quickly by pressing the knife down from top to bottom.
### Frozen Caramel and Truffle Tart
Warning, addictive! This recipe requires you to have sufficient space in your freezer.
For a tart with a diameter of about 26 cm/10¼ in
Difficulty **
Preparation time: 25 minutes plus at least 3 hours freezing time
For the cream:
150 g/5 oz vegan dark chocolate (45–55 % cocoa solids)
60 g/5 tbsp cashew butter or white almond butter
⅓ tsp salt
For the caramel:
300 ml/1¼ cups coconut milk
225 g/1 cup sugar
1 tsp lemon juice
For the cream, slowly melt the chocolate over a bain-marie. Add the cashew butter and mix well, keeping the bowl over the bain-marie. Add 50 ml/¼ cup of water and the salt, and slowly but thoroughly mix until smooth. Take the bowl off the bain-marie and transfer the cream to a shallow mould lined with baking parchment. Leave to cool for a short time, and then put the mould in the freezer for at least 1 hour.
For the caramel, combine 200 ml/scant 1 cup of coconut milk with the sugar and lemon juice in a saucepan and bring to the boil over a high heat. Slightly reduce the heat while keeping at a vigorous boil, and stir constantly with a whisk until the colour changes from white to light brown (this should take about 5 minutes). Be very careful because this change is very sudden. When hardly any more bubbles come up, this is a sign that it is almost ready.
Then continue to cook until the caramel turns a deep golden colour. If you remove the caramel from the heat too soon, it will be too light and its flavour won't be very intense, but you don't want it to cook for too long and become too dark. Now add the rest of the coconut milk, return the saucepan to the heat and stir the cream constantly for 2 minutes until smooth.
Immediately spread the liquid caramel (this is important) over the chocolate cream layer and immediately return the mould to the freezer for at least 2 hours.
### Tip
Making caramel takes a little practice, so don't despair if it isn't perfect the first time.
### Chocolate and Corn Flake Crunchies with Orange and Sea Salt
These crunchies are extremely fast to make. Don't store them in a plastic container or with cling film, otherwise they will lose all their crunch.
It's better to use a metal container, or leave them uncovered.
Makes 12
Difficulty *
Preparation time: 15 minutes
115 g/4 oz vegan dark chocolate (45–55 % cocoa solids)
20 g/1½ tbsp vegan margarine
2 tbsp orange juice
35 g/scant ½ cup corn flakes
40 g/½ cup candied orange peel
about ½ tsp coarse sea salt
Slowly melt the chocolate with the margarine over a bain-marie. Add the orange juice and mix to a smooth cream, keeping the bowl over the bain-marie.
Mix the cornflakes, candied orange peel and sea salt in a bowl. Pour the melted chocolate over the mixture and slowly mix. Use a tablespoon to form 12 compact mounds, space them out over a sheet of baking parchment and leave to cool completely.
### Raspberry and Cocoa Ice Shake
Totally refreshing and with a light cocoa taste.
Using frozen fruit is essential to ensure the shake is at the ideal temperature.
Best served ice cold with a straw.
Makes 330 ml (1 glass)
Difficulty *
Preparation time: 5 minutes
90 g/¾ cup frozen raspberries
25 g/2 tbsp white almond butter
150 ml/⅔ cup water
2 ice cubes
¼ tsp Bourbon vanilla powder
¾ tsp lightly defatted cocoa powder
2–3 tsp concentrated pear juice or maple syrup
Combine all the ingredients in a deep and narrow container, blend with a hand-held blender, and serve immediately.
Chocolate and Corn Flake Crunchies with Orange and Sea Salt
Coconut Hot Chocolate
Cold Blueberry Breakfast Shake
### Coconut Hot Chocolate
It doesn't have to be winter for this chocolate drink to lift your spirits.
The combination of coconut and chocolate is definitely one of my favourites.
Makes 250 ml/1 cup (serves 1)
Difficulty *
Preparation time: 5 minutes
200 ml/scant 1 cup coconut milk
25 ml/1½ tbsp agave syrup or 2 tbsp sugar
1–2 tsp lemon juice
4 tsp desiccated coconut
25 g/1 oz vegan dark chocolate (45–55 % cocoa), grated
Combine all the ingredients in a small saucepan, place over the heat, and stir slowly and constantly until the chocolate is completely melted. Serve immediately. (Picture on p. 110)
### Cold Blueberry Breakfast Shake
Fancy a bit of variety at breakfast? Here you have it.
This shake is definitely what you're after.
Combine it with oat and chocolate crunchies for a light and enjoyable start to your day, especially in summer.
This shake also makes a fantastic dessert.
Variety guaranteed.
Makes 330 ml/1⅓ cups (serves 1)
Difficulty *
Preparation time: 20 minutes
100–125 g/scant 1 cup – generous cup blueberries
25 g/2 tbsp white almond butter
2 tbsp concentrated pear juice or maple syrup
1 tsp vanilla sugar or ⅓ tsp Bourbon vanilla powder
1 tbsp lemon juice
120 ml/½ cup water, as cold as possible
Oat and chocolate crunchies with sunflower seeds (see recipe on p.100)
Wash, clean and dry the berries. Combine all the ingredients, except the oat and chocolate crunchies, in a deep and narrow container, and blend with a hand-held blender. Serve with the oat and chocolate crunchies. (Picture on p. 111)
### Chocolate and Pineapple Cream
Serves 3–4
Difficulty *
Preparation time: 5 minutes plus at least 4 hours cooling time
250 ml/1 cup soya milk
1½ tbsp lightly defatted cocoa powder
125 g/generous 1 cup tinned pineapple
4 tbsp pineapple juice (from the tin)
60 g/5 tbsp refined coconut oil, melted
45 g/scant ¼ cup sugar
1 pinch salt
3 (10 g/0.35 oz) sachets whipped cream stabiliser
Combine all the ingredients in a large, deep mixing jug and blend well for 3–4 minutes. Refrigerate for at least 4 hours.
(Picture on p. 114)
### Avocado and Chocolate Buttercream with Orange and Black Pepper
This is probably the healthiest chocolate cream you can imagine.
In addition to iron and calcium, avocados are high in vitamins A and B, unsaturated fats and other nutrients, and leave you feeling sated longer.
Serves 2–3
Difficulty *
Preparation time: 5 minutes
80 g/3¼ oz vegan dark chocolate (45–55 % cocoa solids)
1 ripe avocado
1 untreated orange, for juice and grated zest
⅓ tsp sea salt
2 tbsp maple syrup or 2½ tbsp icing sugar
Freshly ground pepper
Slowly and completely melt the chocolate in the microwave or over a bain-marie.
Halve the avocado, remove the stone and scoop out the flesh.
Combine the avocado with the other ingredients in a deep and narrow container. Add the melted chocolate and blend well with a hand-held blender.
The cream can be served immediately or kept in the refrigerator until later. (Picture on p. 115)
Chocolate and Pineapple Cream
Avocado and Chocolate Buttercream with Orange and Black Pepper
### Home-Made Chocolate Yoghurt
My favourite yoghurt has always been the chocolate-flavoured variety. But unfortunately, there are no satisfactory vegan alternatives available to buy.
This recipe allows you to make your own chocolate yoghurt, and it leaves all other shop-bought versions in the shade.
You can also use sweetened yoghurt as the base for this recipe, as some supermarkets only offer this.
In this case, simply leave out the additional sugar in the recipe.
Makes 600 g
Difficulty *
Preparation time: 10 minutes plus at least 1 hour cooling time
For the quick and easy version:
80 g/3¼ oz vegan dark chocolate (45–55 % cocoa solids)
100 ml/scant ½ cup soya milk
400 g/2 cups natural soya yoghurt, unsweetened
1 tsp sugar
⅔ tsp vanilla sugar
For the deluxe version:
125 g/4 oz vegan dark chocolate (45–55 % cocoa solids)
120 ml/½ cup soya milk
400 g/2 cups natural soya yoghurt, unsweetened
1 tsp sugar
⅔ tsp vanilla sugar
1 tsp lightly defatted cocoa powder
For the quick and easy version, combine the chocolate and soya milk in a saucepan and heat while stirring constantly, until the chocolate is completely melted. Combine with the other ingredients in a bowl and mix well (preferably with a whisk) and leave to cool completely.
Store in the refrigerator. This yoghurt can keep in the refrigerator for 1–2 weeks.
For the deluxe version, combine the chocolate and soya milk in a saucepan and heat while stirring constantly, until the chocolate is completely melted. Put 3 tablespoons of the chocolate mixture in a bowl and add the yoghurt, sugar and vanilla sugar. Mix briskly and thoroughly (preferably with a whisk) until smooth.
Combine 2 tablespoons of the yoghurt mixture and cocoa powder with the remaining 3 tablespoons of melted chocolate in a saucepan and mix to obtain a dark cream.
Pour the dark cream into the bottom of two or three glasses, then use a spoon to cover it with the lighter yoghurt cream. Refrigerate for at least 1 hour.
### Chocolate Risotto with Raspberry Sauce
Serves 2
Difficulty *
Preparation time: 1 hour
For the risotto:
750 ml/3 cups almond or soya milk
The seeds from ½ vanilla pod
1½ tbsp sugar
1 pinch salt
125 g/⅝ cup risotto rice
40–50 g/1½–2 oz vegan dark chocolate (45–55 % cocoa solids, depending on how strong you like it)
1 tbsp cognac (optional)
Cocoa powder and a few raspberries (optional) to decorate
For the raspberry sauce:
225 g/generous 1¾ cups raspberries (see tip)
1½–2 tbsp sugar
1 tbsp lemon juice
For the risotto, combine the almond or soya milk with the vanilla seeds, sugar and salt, and bring to the boil. Sprinkle in the rice, cover and cook, stirring from time to time, until the rice has absorbed almost all the liquid (about 40 minutes). About 10 minutes before the end of the cooking time, add the chocolate and if you like stir in the cognac.
For the raspberry sauce, wash, clean and dry the raspberries, and purée with a hand-held blender. Heat the sugar in a frying pan and make a golden caramel. Add the puréed raspberries and reduce for 2–3 minutes until the caramelised sugar has completely dissolved from the bottom of the pan. Stir in the lemon juice.
Serve the risotto in a small bowl and decorate with a few fresh raspberries if you like. Pour the sauce over the rice and dust with cocoa powder.
### Tip
If you don't like the idea of having raspberry seeds in the sauce, simply press the puréed raspberries through a sieve.
##... the savoury versions
### Chocolate and Chilli Meatballs with a Quick Tomato Salsa
Serves 2
Difficulty *
Preparation time: 20 minutes
For the meatballs:
250 g/9 oz tofu
50 g/1 cup dry breadcrumbs
2 tbsp wheat or spelt flour
1 tsp soy sauce
1 level tsp salt
2 tsp lemon juice
½ tbsp smooth mustard
1 tsp sweet paprika
25 g/1 oz vegan dark chocolate (45–55 % cocoa solids), finely grated
½–1 small chilli pepper, finely chopped
Peanut (groundnut) oil or high oleic sunflower oil
For the salsa:
2 large ripe tomatoes, finely diced
1 small onion, finely chopped
1½ tbsp chopped fresh basil
2 tsp freshly squeezed lemon juice
About ¾ tsp salt
Freshly ground pepper
Mix all the ingredients for the salsa in a bowl and leave to stand for 10 minutes.
For the meatballs, combine all the ingredients, except the oil, in a large bowl. Knead vigorously with both hands for 3–5 minutes. Thorough kneading is crucial for the success of this dish. Shape the mixture into small balls.
Heat a generous amount of oil in a non-stick frying pan and fry the meatballs over a medium–high heat for about 5 minutes. There should always be sufficient oil in the pan. Top up the oil if necessary. Serve the meatballs with the salsa.
### Tip
For a decorative effect, garnish the meatballs with chilli threads.
### Chestnut and Tofu Schnitzel with a Red Wine and Chocolate Sauce
Serves 2, makes 5–6 small patties
Difficulty *
Preparation time: 35 minutes
For the schnitzel:
200 g/1¾ cups frozen, peeled chestnuts or vacuum-packed cooked chestnuts
200 g/7 oz firm tofu
1 tbsp lemon juice
1½ tsp untreated lemon zest
Freshly ground pepper
1 level tsp salt
2 tsp cornflour
3 tbsp light wheat or spelt flour
2 tsp quality balsamic vinegar
1 tbsp chopped fresh parsley
Refined olive oil for frying
Salt
For the red wine and chocolate sauce:
1 small onion, finely chopped
1 tbsp olive oil
½ tsp cornflour
50 ml/¼ cup cold water
1 tsp vegetable stock powder
150 ml/⅔ cup red wine
1 tsp quality balsamic vinegar
10 g/2 tsp white almond butter
2 squares vegan dark chocolate (45–55 % cocoa solids)
1–2 tsp fresh thyme leaves (optional)
Salt, freshly ground pepper
For the schnitzel, if you're using frozen chestnuts, put them in a saucepan and cover with water. Cover with the lid, place over medium heat and cook for about 15 minutes, until they are soft and falling apart. If you're using vacuum-packed chestnuts, they can be used as they are for the next step, where they are kneaded with the other ingredients.
Drain the cooked chestnuts well and combine with the rest of the ingredients, except the olive oil, in a bowl. Leave to cool if necessary. Vigorously knead the ingredients together for at least 3 minutes. Shape the mixture into 5–6 large patties.
Heat the olive oil in a frying pan and fry the schnitzel over medium–high heat for 4 minutes on each side until golden brown and crispy. Don't touch the schnitzels when frying for the first 2 minutes. Even after turning, leave them alone for the first 1–2 minutes. Note: There must always be enough oil in the pan, otherwise the schnitzels will stick to the bottom. Top up the oil if necessary.
For the sauce, sweat the chopped onions in the oil until translucent. Dissolve the cornflour completely in cold water. Add the cornflour and other ingredients to the onions in the pan, and simmer uncovered over a medium heat. Season with salt and freshly ground pepper.
### Seitan Ragout with Chocolate and Dried Apricots
Serves 2
Difficulty *
Preparation time: 45 minutes
2–3 tbsp peanut (groundnut) oil or high oleic sunflower oil
200 g/7 oz seitan, cut into strips
1 large onion, cut into fine strips
1 large carrot, cut into fine strips
4 juniper berries
175 ml/¾ cup good red wine
1 tsp vegetable stock powder
4 squares vegan dark chocolate (45–55 % cocoa solids)
2 tsp tomato purée
250 ml/1 cup water
1 dried bay leaf
2 unsulphured dried apricots, cut into strips (see tip)
8–10 sprigs fresh rosemary
1–2 dash(es) lemon juice
5 g/1 tsp white almond butter (optional)
About ¾ tsp salt
Freshly ground pepper
Heat the oil over a high heat in a non-stick frying pan and sear the seitan for 4 minutes. Add the onion strips and fry for 2 minutes. Top up the oil if necessary.
Add the rest of the ingredients, reduce the heat to low–medium and half cover the pan. Cook for 20–30 minutes, stirring occasionally, until the carrots are soft. Season with salt and freshly ground pepper.
This dish can be served with rösti, sautéed potatoes or pasta.
### Tip
Unsulphured dried apricots can be distinguished by their characteristic brown colour. More aromatic than the bright orange, sulphured apricots, they don't just make a healthy snack, but also work well in savoury dishes.
1. Cover
2. Title
3. Copyright
4. Contents
5. Foreword
6. Introduction
1. About the recipes
2. Some important ingredients
7. Chocolate: Essentials, Production, and Products
1. The correct way to heat and melt chocolate
8. Recipes
1. Spreads and Jams
2. Biscuits, Muffins, Cupcakes and Brownies
3. Cakes, Tarts and Cheesecakes
4. Chocoholic Favourites
5. No-Bake Chocolate Treats
9. ... the savoury versions
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Erste Bank der oesterreichischen Sparkassen AG o, simplemente, Erste Bank es una caja de ahorros de Austria y uno de los consorcios bancarios más importantes de dicho país.
Esta institución posee subsidiarias en varios países de los Balcanes, Europa Central y Oriental; aunque en otros países también está presente como la empresa matriz de varias instituciones bancarias como Česká spořitelna (República Checa), Slovenská sporiteľňa (Eslovaquia) y Banca Comercială Română (Rumanía).
Historia
Este banco fue fundado en 1819 a iniciativa de Johann Baptist Weber, párroco de St.Leopold en Viena. Su nombre original era Verein der Ersten österreichischen Spar-Casse (Asociación de la primera caja de ahorros de Austria).
Tras la anexión alemana de Austria, el nombre del banco cambió a Die Erste, omitiendo el uso del adjetivo österreichisch (austríaco).
El banco actual es el resultado de la fusión en 1997 del antiguo Erste österreichische Spar-Casse con otro llamado GiroCredit.
El Erste Bank cotiza en la Bolsa de Viena y en la de Praga desde noviembre de 1997 y octubre de 2002, respectivamente. Forma parte de los índices bursátiles austríaco ATX y el checo PX.
CaixaBank participa en un 10% de su capital.
Referencias
Empresas de Austria
Bancos de Austria
Economía de Viena
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,415
|
Insects of Keewatin and Mackenzie
Douglas C. Currie1, Donna Giberson2, and Brian V. Brown3
1Centre for Biodiversity and Conservation Biology, Royal Ontario Museum,
100 Queen's Park, Toronto, ON, M5S 2C6
2Department of Biology, University of Prince Edward Island,
550 University Ave., Charlottetown, PEI C1A 4P3
3Entomology Section, Natural History Museum of Los Angeles County,
900 Exposition Boulevard, Los Angeles, CA, 90007, U.S.A.
The Canadian north remains among the most inadequately surveyed areas in North America in terms of its insect fauna. During the late 1940's and early 1950's the Departments of Agriculture and National Defence collaborated on a project called The Northern Insect Survey. Although the resulting collections provided valuable insights about insect diversity in the far north, relatively few sites were sampled and material collected is unsuitable for modern analysis (e.g., cytology, DNA sequence data). Problems associated with lack of access and infrastructure continue to hinder efforts to document insect diversity in much of northern Canada.
With completion of the Insects of the Yukon book, it seems appropriate to direct attention towards the inadequately surveyed territory between the Mackenzie River and Hudson Bay. This area, which constitutes mainland Northwest Territories and Nunavut, corresponds with the Districts of Mackenzie and Keewatin. Scientific Committee members Doug Currie (Royal Ontario Museum and University of Toronto) and Donna Giberson (University of Prince Edward Island) are leading a multiyear initiative to survey representative areas throughout the region. Given the short collecting season and logistical constraints, the focus of the survey reflects the interests of participating scientists. However, mass collecting techniques (e.g., Malaise traps, sweep netting, aquatic kick sampling) provide representatives of many 'non-target' organisms. The survey is expected to generate valuable new insights about the diversity and biogeography of northern insects, as well as to provide information about energetics and food web dynamics.
The Horton River
The Horton River was selected as the venue for the first year of the project because of its close proximity to the eastern boundary of Beringia. It is situated mainly within the Northern Interior Plain region of the Mackenzie Lowlands and is characterized by hilly topography marked with numerous lakes and small streams. Originating north of Great Bear Lake, the river flows in a northwesterly direction for approximately 700 km before emptying into Franklin Bay on the Beaufort Sea. There are no settlements along the Horton River and even the Inuit of Paulatuk (situated some 100 km east on the arctic coast) rarely travel that far. The water is clear in the upper reaches and flows over limestone-dominated sedimentary rock through a series of cobble riffles and long deep pools. The water here is characterised by generally high pH (8.0-8.5) and low conductivity (160-200 Fs/cm). The surrounding vegetation consists of sedge- and shrub tundra with scattered black spruce. Spruce, willow and alder are confined mainly to the valley and south-facing slopes. Approximately 180 km downstream of Horton Lake the river enters an area with sheer limestone cliffs; and over the next 150 km it flows through a series of three canyons, each characterised by deep bedrock pools and whitewater rapids. Following the canyon section the geology changes and the surrounding landscape is characterised by eroding hillsides and muddy tributaries, which significantly increase the turbidity of the river. The final 100 km of the Horton River passes through sparsely vegetated badlands dominated by vast deposits of lignite and sulphur. Lignite spontaneously combusts when exposed to oxygen and the area is aptly named the Smoking Hills. Tributaries in the Smoking Hills are muddy and typically highly acidic, reducing the pH of the Horton River to 6.5-7.5.
The 2000 Horton River Expedition
The process of obtaining a Scientific Research Licence proved to be exceedingly cumbersome and time consuming. Because our proposal involved two different first nations settlement regions, approval was needed from both Sahtu and Inuvialuit authorities (i.e., various Hunters and Trappers Associations and Renewable Resources Councils within the two regions). The NWT Environmental Impact Screening Committee then reviewed our proposal, taking into consideration the views and comments of various stakeholders. Finally, a Land Use Permit from the Inuvialuit Land Administration was needed before a licence could be issued. It arrived in the mail less than a week before departure.
Our team of 5 entomologists convened in Norman Wells on July 17. Doug Currie and Donna Giberson were joined by Peter Adler (Clemson University), Brian Brown (Natural History Museum of Los Angeles), and Malcolm Butler (North Dakota State University). The five of us, along with our guide Tim Gfeller (Wilderness Adventure Company), boarded a chartered Twin Otter for the hour and a half flight to Horton Lake. We carried all our gear and enough food to support our expedition for one month. Given the absence of roads and the high cost of air transportation, our plan was to travel 620 km by canoe from Horton Lake to the Beaufort Sea. Although strenuous, this approach gave us access to a wide variety of microhabitats along the way. The route also provided a south-to-north transect from the High Subarctic Ecoclimatic Region to the Low Arctic Ecoclimatic Region.
Although insects were the focus of our expedition the Horton River proved equally favourable for viewing wildlife. Numerous species of birds were observed including Arctic loon, ptarmigan, peregrine and gyrfalcons, bald and golden eagles, mergansers, scoters, plovers, and jaegers, to name a few. Hundreds of caribou were also seen throughout the journey, along with occasional sightings of muskoxen, moose, fox, wolf, and grizzly bear. Arctic grayling, lake trout and burbot 'collected' from the Horton River were welcome additions to our larder.
Twenty four days were needed to travel the entire length of the river with only 2 days of respite from paddling. The weather ranged from stifling heat (30�C+) during the first part of the trip to uncomfortably cool (4�C) towards the end. On August 9 a chartered Twin Otter retrieved us from a gravel bar near Franklin Bay and flew us to Inuvik.
Preliminary results
Black Flies (Diptera: Simuliidae)
Doug Currie and Peter Adler made collections at 52 sites along the Horton River and its tributaries. Immature stages were collected from watercourses that ranged in width from a few centimetres to more than 100 metres. Larvae were collected mainly into Carnoy's fixative to facilitate cytological study; selected larvae and adults were fixed in 95% ethanol to facilitate molecular analysis. Adults were collected through a combination of Malaise trapping, aspirating from team members and the insides of tents, and rearing of pupae. Identification at the morphospecies level revealed a total of 18 taxa in 3 genera: Simulium, Metacnephia, and Cnephia. Although the actual number of species will undoubtedly be higher following cytological screening, it is clear that the simuliid fauna is depauperate relative to that of similar drainages in Alaska and the Yukon Territory. This probably reflects, in part, the short period that the Horton River and its valley has been deglaciated. Interestingly, the Horton drainage includes species that are sparsely represented or absent from Beringia. The biogeographical implications of this pattern are a focus of study. Larvae are currently being studied chromosomally by Peter Adler. Doug Currie is curating the adults and his University of Toronto graduate student, Miranda Smith, is analysing molecular sequence data from selected species.
Chironomus (Diptera: Chironomidae)
Mac Butler focused on lentic Chironomidae, especially Chironomus. This genus is well-studied cytogenetically, and karyotypes are necessary to confirm identification of most species. In collaboration with an international group of colleagues, he has recently been investigating biogeographic patterns of genetic variation of Chironomus from Europe, Siberia, and North America. Twenty lentic habitats were sampled along the 620 km route: ten pools or ponds (<1m depth), and a like number of small lakes. Chironomus larvae were found in eight of these sites, and pupal exuviae only were collected from an additional lake. A number of other lentic Chironomidae larvae were collected as well, primarily Tanypodinae and Tanytarsini. Chironomus was quite rare in shallow habitats south of the coastal tundra, but larvae were found in all three ponds sampled in the Smoking Hills. All Chironomus larvae were fixed for karyotype analysis, but the material has not yet been examined. It is estimated that at least half a dozen species will be present, perhaps more. In many cases only a few larvae were collected at each site, and not all specimens are likely to provide good-quality karyotypes. Material collected at three sites may be sufficient for characterizing populations in terms of prevailing inversion frequencies, and hopefully to make comparisons with populations from other parts of the Holarctic Region. Nonetheless, the simple knowledge of what cytologically-defined species live in this part of the Arctic is a sufficient outcome from the expedition.
Phorid flies (Diptera: Phoridae)
Phorid flies were collected using mostly Malaise traps. A total of 28 Malaise trap samples were collected, most of which were overnight samples; a few traps were left up for two days worth of collecting. The samples vary widely in the number of insects they contain, due mostly to variation in the weather: warm sunny days produced large diverse catches whereas cold overcast days led to sparse collections. The higher flies (Brachycera) are being removed from the Malaise samples by Brian Brown, after which the residues will be forwarded to Doug Currie to extract lower flies and Donna Giberson will look at the mayflies, stoneflies and caddisflies. The phorid fauna has not yet been analyzed in detail, but it is much more diverse than the literature on northern insects indicates. At least two genera, Megaselia and Triphleba, were collected, with the former predominating in numbers and species richness. It was interesting to see that phorid flies were abundant in samples collected on the tundra. Malaise traps were set on tundra sites using canoe paddles as poles, when the already sparse black spruce stands along the river disappeared.
Energetics and Food Web Ecology of the Horton System
Donna Giberson took samples representing different aquatic insect feeding guilds at approximately 20-30 km intervals along the river. Stable isotope (carbon and nitrogen) analysis of the insect samples, along with potential food sources (e.g., detritus, fine particulate organic matter, biofilms) should provide information about food sources and feeding patterns of the dominant taxa. Stable isotope analysis is a chemical analysis of food sources that takes advantage of the fact that different isotopes of common elements are sequestered by the body in different, but predictable ways, allowing researchers to trace food sources and trophic levels by analysing the ratios of the isotopes in the body tissues. Donna also collected water samples along the river to determine basic water chemistry variables (DO, pH, conductivity) and primary productivity, in collaboration with Joseph Culp at the National Hydrology Research Institute. Donna will also be identifying the Ephemeroptera, Plecoptera, and Trichoptera collected during the trip.
Discussions concerning 2001 field season are now underway. We plan to undertake a similar expedition in Nunavut or northern Manitoba, although a specific destination has yet to be identified. The Thelon, Kazan, and Seal Rivers have been discussed as possible venues, but further information is needed about logistical problems associated with each river. Regardless of the destination chosen, the resulting collections will provide a basis for comparison with collections made along the Horton River in 2000. A number of western species attain their eastern limit before Hudson Bay; and other species (e.g., Simulium giganteum) are known in the Nearctic Region only from the vicinity of Hudson Bay. A west-to-east transect along any of the three rivers should provide more detailed information about the distribution and composition of northern insects. Such data are fundamental to developing sound biogeographical hypotheses. Anyone interested in participating in this project is encouraged to contact us at the addresses given above.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,781
|
Czarnów ( Tschernow, 1936–1945 Schernow) ist ein Dorf in Polen in der Gemeinde Górzyca in der Woiwodschaft Lebus.
Geographie
Geographische Lage
Czarnów liegt in der ehemaligen Brandenburger Neumark an der Warthebruchniederung nah zum Fluss Warthe unweit von Küstrin.
Geschichte
Auch in Tschernow wurde Mitte des 19. Jahrhunderts ein Fort für den äußeren Verteidigungsring der Festung Küstrin erbaut.
Bis 1945 war Schernow im Landkreis Weststernberg, Regierungsbezirk Frankfurt.
Einwohnerentwicklung
Um 1800 lebten im Ort neben dem Lehnschulzen elf Bauern, 24 Kossäten, zehn Büdner und sieben Einlieger.
Wirtschaft und Infrastruktur
Verkehr
Die 1905 gegründete Kleinbahn "Küstrin–Hammer" hatte einen Haltepunkt mit Wartehalle an der Chaussee beim Ort. Weiterhin liegt der Ort an der ehemaligen Reichsstraße 114 (heute DK 22).
Söhne und Töchter der Stadt
Theodor Heinsius (1770–1849), deutscher Pädagoge und Lexikograf
Literatur
Heinz W. Linke: Chroniken der Orte Schernow und Säpzig. Kreis Weststernberg, Regierungsbezirk Frankfurt/Oder. Books on Demand, Nordstedt 2002, ISBN 3-8311-3430-8 (books.google.de).
Einzelnachweise
Ort der Woiwodschaft Lebus
Gmina Górzyca
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,812
|
Q: Windows USB mass storage - Garmin Alpha 200i mounted as a 'Device' but not a 'Drive' We have a python program that scans the mounted drive letters (or volumes, for Linux) for a certain file that indicates a Garmin handheld GPS. But, the Garmin Alpha 200i is mounted by windows as a 'Device' and not as a 'Drive', and therefore it has no drive letter and you can't get to it from Windows batch or Powershell in the standard C:/Folder notation.
How can we go about accessing the files on the 'Device' from python (or batch or PowerShell)?
It's definitely a mass storage device and has a directory structure - just not sure how to get to it programmatically:
Thinking that this is a Windows or python question, rather than a Garmin question. This is the first Garmin handheld GPS model we've encountered that mounts as a 'Device' instead of a 'Drive'.
The Garmin manual says that the handheld should be recognized as one or two removable drives, but, that is not the case. Earlier GPS models do mount up as two drives - one for the handheld's internal storage and another for its memory card if any.
A: Modern devices use Media Transfer Protocol (MTP) instead of USB Mass storage.
This protocol is intentionally nerved, however, and cannot provide drive letters.
You can try one of the Python wrappers for LibMTP.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,614
|
With many years of experience, our business faithfully serves any and all areas where tornadoes may strike. Our installation technicians are experts in the field; they have the knowledge and experience to install your storm shelter in less than four hours in most cases!
We require a 10% down payment at the time of signing your contract to officially schedule the installation.
Please check with your individual city offices for permit requirements/costs.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,997
|
WHY GIVE TO THE SYCAMORE FUND?
For more than 30 years, Sycamore School students have benefitted from those who have given generously of their time, talent, and treasure. The vibrant life of our school is made possible by the ongoing philanthropic support and volunteerism of all community members including parents, board members, staff, grandparents, alumni, parents of alumni, foundations, corporations, and friends.
The Sycamore Fund is the school's highest fundraising priority. As with almost all independent schools, Sycamore relies on our community to give annually to help us fund projects and programs that make us the unique school we are. Your tax-deductible donation goes to work immediately, benefiting students in this school year.
Our unique mission of providing an accelerated and enriched education to gifted students and our successful track record of helping students reach their potential differentiate us from other schools. However, a Sycamore School education is not sustained by tuition alone. Generous donations each year make a significant difference. With the continued support of our community, Sycamore will continue to be one of the top independent schools for gifted students in the country.
We encourage you to make a meaningful gift commensurate with your family's personal circumstances. Every gift makes an immediate impact, is deeply valued and goes towards the exceptional educational experience that Sycamore provides. All staff, alumni, and families are asked to make a gift to the Sycamore Fund.
Sycamore School is a registered 501(c)3 organization. Your gift is tax-deductible to the extent you itemize on your tax return.
*Donors at the Head of School Circle and above are invited to attend a reception with the Head of School and Board of Trustees each spring to acknowledge their generosity.
Q: Can my employer match my gift?
Some employers will match your gift. Simply submit a matching gift form, often available on your company's website or from the Human Resources Department along with your gift or pledge. Our Advancement Office will handle the processing. Your company's name will be listed in our Annual Report and you will be recognized for the value of the gift.
Q: How do I give gifts of securities and stock?
Gifts of appreciated securities offer you a two-fold tax saving. First, you do not pay capital gains tax on the increased value of the securities. Second, you are entitled to a tax deduction equal to the fair market value of the securities on the date the gift was given. Directions on how to give a gift of appreciated securities are available from Holly Lee, Director of Advancement, at 317-202-2504 or lee.holly@sycamoreschool.org.
Q: How does planned giving help Sycamore?
As you choose where to invest your estate funds, we hope you will consider a gift to Sycamore School. The Sycamore Society recognizes and honors individuals who provide support for Sycamore in their wills, trusts, life insurance designations, and other planned gifts. Your investment funds the Financial Aid Endowment, which attracts and retains gifted students at Sycamore, despite their family's ability to pay.
Q: Can I make a tribute gift?
Making a gift to the Sycamore Fund in honor or in memory of a loved one or cherished teacher is a wonderful way to pay tribute to him/her. We will send an acknowledgment to the honoree or family of the honoree as well as provide you with a receipt for your tax-deductible contribution. When giving online, in the comment box below, tell us about your tribute/in honor of/in memory of gift.
Thanks for contributing to the Sycamore Fund with an online gift. If you would like to make a tribute gift, please list the name(s) of the honoree(s) in the comment box below.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,048
|
CASK_APP_LIST="macdown atom alfred iterm2 google-chrome firefox virtualbox vagrant postgres"
BREW_APP_LIST="colordiff tree homebrew/versions/maven31 gradle groovy jmeter node npm rbenv gcal"
JAVA_VERSIONS="java java7"
NVM_VERSION="v0.32.1"
# create symlinks for system settings
echo
echo "* App installations. This will take a little while."
echo "* Installing Homebrew..."
ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)"
echo "* Installing Homebrew Cask..."
brew install caskroom/cask/brew-cask
echo "* Setting up the previous version caskroom"
brew tap caskroom/versions
echo "* Installing terminal apps via brew."
brew install $BREW_APP_LIST
echo "* Now we'll install some binary apps: $CASK_APP_LIST"
#brew cask install --force --appdir=/Applications $CASK_APP_LIST
brew cask install --appdir=/Applications $CASK_APP_LIST
echo "* Installing Java: $JAVA_VERSIONS"
#brew cask install --force $JAVA_VERSIONS
brew cask install $JAVA_VERSIONS
echo "* Installing nvm"
curl -o- https://raw.githubusercontent.com/creationix/nvm/$NVM_VERSION/install.sh | bash
echo "* Done."
echo
echo "Symlink Setup"
echo " Create symlink bash_profile."
rm -f ~/.bash_profile; ln -s ~/my-env/bash_profile.sh ~/.bash_profile
echo " Create symlink .gitconfig."
rm -f ~/.gitconfig; ln -s ~/my-env/conf/git/gitconfig ~/.gitconfig
echo " Create symlink ssh config."
rm -f ~/.ssh/config; ln -s ~/my-env/conf/ssh/config ~/.ssh/config
echo " Create maven repo symlink."
rm -Rf ~/.m2; ln -s ~/my-env/conf/m2 ~/.m2
echo " Create gradle symlink."
rm -Rf ~/.gradle; ln -s ~/my-env/conf/gradle ~/.gradle
echo " Create Intellij IDEA 'idea.vmoptions' symlink."
rm -f ~/Library/Preferences/IntelliJIdea13/idea.vmoptions
ln -s ~/my-env/conf/intellij-idea-14-ce/idea64.vmoptions ~/Library/Preferences/IdeaIC14/idea64.vmoptions;
echo " Done."
echo
echo "Alfred 2 Setup"
echo " Quitting Alfred 2 ..."
echo " rsync Alfred 2 prefs ..."
#Alfred 2 prefs cannot be symlinks
osascript -e 'quit app "Alfred 2"'
mkdir -p ~/my-env/conf/alfred2/Alfred\ 2
rsync -r ~/Library/Application\ Support/Alfred\ 2/ ~/my-env/conf/alfred2/Alfred\ 2
rm -Rf ~/Library/Application\ Support/Alfred\ 2;
cp -R ~/my-env/conf/alfred2/Alfred\ 2 ~/Library/Application\ Support/Alfred\ 2;
echo " Starting Alfred 2 ..."
open -a /Applications/Alfred\ 2.app/Contents/MacOS/Alfred\ 2
echo " Done."
echo
echo "Sourcing '~/.bash_profile'"
source ~/.bash_profile
echo " Done."
echo
echo "Install Fonts"
# Add the custom fonts
install_fonts ~/my-env/opt/fonts
echo " Done."
echo
echo "Initialize Git Submodules"
git updatesubs
echo "Done"
echo
echo
echo "Setup complete. yay."
echo
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,942
|
Q: rails : type search with data address use elasticsearch chewy gem I use chewy gem elasticsearch .
I have LocationsIndex, mapping :
class LocationsIndex < Chewy::Index
settings analysis: {
analyzer: {
folding: {
tokenizer: "standard",
filter: [ "lowercase", "asciifolding" ]
},
sorted: {
tokenizer: 'keyword',
filter: ['lowercase', 'asciifolding']
}
}
}
define_type Location do
field :name, type: 'string', analyzer: 'standard' do
field :folded, type: 'string', analyzer: "folding"
end
field :address, type: 'string', analyzer: 'standard' do
field :address, type: 'string', analyzer: 'folding'
end
field :locations, type: 'geo_point', value: ->{ {lat: lat, lon: lon} }
end
end
when i query:
LocationsIndex::Location.query(
multi_match: {
query: keyword,
fields: ["address", "address.folded" ,"name", "name.folded"]
}
)
data sample :
{ "took" : 2, "timed_out" : false, "_shards" : {
"total" : 5,
"successful" : 5,
"failed" : 0 }, "hits" : {
"total" : 10,
"max_score" : 1.0,
"hits" : [ {
"_index" : "locations",
"_type" : "location",
"_id" : "131",
"_score" : 1.0,
"_source":{"name":"Việt Nam","address":"Việt Nam","locations":{"lat":16.9054,"lon":106.576}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "136",
"_score" : 1.0,
"_source":{"name":"Quan truong Ngo Mon","address":"23/8, Thừa Thiên Huế, Việt Nam","locations":{"lat":16.4669,"lon":107.58}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "132",
"_score" : 1.0,
"_source":{"name":"Thừa Thiên Huế","address":"Thừa Thiên Huế, Việt Nam","locations":{"lat":16.4674,"lon":107.591}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "137",
"_score" : 1.0,
"_source":{"name":"Phu Van Lau","address":"23/8, Thừa Thiên Huế, Việt Nam","locations":{"lat":16.4655,"lon":107.581}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "133",
"_score" : 1.0,
"_source":{"name":"Ha Noi","address":"Ha Noi, Việt Nam","locations":{"lat":16.4674,"lon":107.591}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "138",
"_score" : 1.0,
"_source":{"name":"Cau gia Vien","address":"Le Duan, Thừa Thiên Huế, Việt Nam","locations":{"lat":16.4601,"lon":107.571}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "134",
"_score" : 1.0,
"_source":{"name":"TP Ho Chi Minh","address":"TP Ho Chi Minh, Việt Nam","locations":{"lat":16.4674,"lon":107.591}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "139",
"_score" : 1.0,
"_source":{"name":"Chua thien Pagoda","address":"Kim Long, Thừa Thiên Huế, Việt Nam","locations":{"lat":16.4537,"lon":107.545}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "130",
"_score" : 1.0,
"_source":{"name":"Việt Nam","address":"Việt Nam","locations":{"lat":16.9054,"lon":106.576}}
}, {
"_index" : "locations",
"_type" : "location",
"_id" : "135",
"_score" : 1.0,
"_source":{"name":"Dai Noi Hue","address":"23/8, Thừa Thiên Huế, Việt Nam","locations":{"lat":16.4698,"lon":107.577}}
} ] } }
when i run query with keyword = "viet nam"
result :
_id = [131,132,,133,134,135,136,137,138,139,130] # => OK working
but i when run query with keywork = "thua thien hue"
result :
_id = [132,135,139] # => Don't working ???, should have been: _id = [132,135,136,137,138,139]
Same with keywork = "hue"
result :
_id = [132,135] # => Don't working ???, should have been: _id = [132,135,136,137,138,139]
how search results that contain the above word (add type, or do anything)
A: You have an error when declaring field address. You are declaring address.address instead of address.folded. In other words, the declaration should be:
field :address, type: 'string', analyzer: 'standard' do
field :folded, type: 'string', analyzer: 'folding'
end
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,788
|
{"url":"http:\/\/tillahoffmann.github.io\/2016\/03\/31\/variational-linear-regression-in-tensorflow.html","text":"# Variational Linear Regression In Tensorflow\n\nWe discussed some of the problems with the variational mean-field optimisation algorithms in a previous post, and showed that the optimisation problem can be solved using the machine learning library theano. Let\u2019s try to do the same using tensorflow, which provides a nice optimisation interface.\n\nHere are some simulated data.\n\n%matplotlib inline\nimport numpy as np\nfrom matplotlib import pyplot as plt\nfrom matplotlib import rcParams\nrcParams['figure.figsize'] = (9, 6)\n# This module contains a few helper functions\nfrom linear_regression import *\n\n# Generate data and plot it\nX, y, theta = generate_data(theta=[1, 2])\nplot_data(X, y, theta)\n\n# Use same starting position for all optimisers\nn, p = X.shape\nmu0 = np.random.normal(0, 1, p)\n\n\nRecall that the evidence lower bound (ELBO) is given by\n\nLet\u2019s construct a symbolic computation graph in tensorflow to compute the ELBO.\n\nimport tensorflow as tf\n\nwith tf.Graph().as_default() as graph:\n# Define variables\nt_X = tf.placeholder(tf.float32, name='X')\nt_y = tf.placeholder(tf.float32, name='y')\nt_mu = tf.Variable(mu0.astype(np.float32), name='mu')\n\n# Compute the linear predictor\nt_predictor = tf.reduce_sum(t_X * t_mu, [1])\n\n# Compute the elbo\nt_elbo = -0.5 * tf.reduce_sum(t_predictor ** 2) + tf.reduce_sum(t_predictor * t_y)\n\n# Create an optimiser\noptimizer = tf.train.MomentumOptimizer(1e-2 \/ n, .3)\ntrain_step = optimizer.minimize(-t_elbo)\n\n\ntensorflow makes use of fast libraries such as blas behind the scenes, and stores all data in a session that is decoupled from the python kernel. We first need to start such a session and then feed it the data. We repeatedly call the train operation of the optimiser we defined above to improve the fit of the model.\n\n# Start a session\nwith tf.Session(graph=graph) as sess:\n# Initialise variables\nsess.run(tf.initialize_all_variables())\n\n# Run the optimisation\ntrace = []\nelbos = []\nfor step in range(50):\n# Call the optimisation and evaluate the current values of mu and the elbo\n_, mu, elbo = sess.run([train_step, t_mu, t_elbo], {t_X: X, t_y: y})\ntrace.append(mu)\nelbos.append(elbo)\n\nplot_trace(elbos, trace, theta)\npass\n\n\nLet\u2019s plot the trajectory of the optimisation through parameter space.\n\nplot_trajectory(X, y, trace, theta, levels=np.logspace(4, 4.45, 20))\n\n\ntensorflow and theano are very similar packages and are both aimed at large-scale machine learning. tensorflow still has a bunch of annoying pecularities (such as missing a standard implementation of the dot product), but promises to develop into a great product. It also provides a range of optimisers out of the box. theano is somewhat faster and more stable but optimisation is sometimes tedious because optimisers are not provided.","date":"2017-08-18 08:40:51","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\": 0.5460687875747681, \"perplexity\": 3914.6471763930194}, \"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-2017-34\/segments\/1502886104631.25\/warc\/CC-MAIN-20170818082911-20170818102911-00360.warc.gz\"}"}
| null | null |
{"url":"https:\/\/bodheeprep.com\/cat-quant-practice-problems\/735","text":"# CAT Quant Practice Problems\n\nQuestion: A circle with radius 2 is placed against a right angle. Another smaller circle is also placed as shown in the adjoining figure. What is the radius of the smaller circle?\n\n1. $3 - 2\\sqrt 2$\n2. $3 - 2\\sqrt 2$\n3. $3 - 2\\sqrt 2$\n4. $3 - 2\\sqrt 2$\n\nCorrect Option:4\n\n## CAT Quant Questions with Video Solutions\n\nCAT Quant Practice Problems","date":"2019-03-22 03:47:34","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\": 0.9239150881767273, \"perplexity\": 2114.112888332965}, \"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-2019-13\/segments\/1552912202628.42\/warc\/CC-MAIN-20190322034516-20190322060516-00209.warc.gz\"}"}
| null | null |
Lan Wei (China, 7 de mayo de 1968) es un clavadista o saltador de trampolín chino especializado en trampolín de 1 metro, donde consiguió ser subcampeón mundial en 1994.
Carrera deportiva
En el Campeonato Mundial de Natación de 1994 celebrado en Roma (Italia), ganó la medalla de plata en el trampolín de 1 metro, con una puntuación de 375 puntos, tras el zimbabuense Evan Stewart (oro con 382 puntos) y por delante del estadounidense Brian Earley (bronce con 351 puntos).
Referencias
Enlaces externos
Saltadores de la República Popular China
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,946
|
Luisa Ulrica de Prusia (Berlín, 24 de julio de 1720-Svartsjö, 16 de julio de 1782) fue reina de Suecia por su matrimonio con Adolfo Federico de Suecia. Era hija del rey Federico Guillermo I de Prusia y de Sofía Dorotea de Hannover.
Biografía
La princesa Luisa Ulrica se comprometió en Berlín el con el entonces heredero a la corona de Suecia, Adolfo Federico de Holstein-Gottorp. La celebración del matrimonio se llevó a cabo en Suecia, en el Palacio de Drottningholm, el . Su marido había sido elegido como heredero gracias en parte a la influencia rusa en el parlamento.
Luisa Ulrica intervino muy pronto en las actividades políticas. Como su hermano Federico II de Prusia, era admiradora de la cultura francesa y buscó alianzas entre Adolfo Federico y el partido de los sombreros, proclives a la política francesa. El acercamiento con los sombreros motivó disgustos entre el cuerpo diplomático ruso, y ello a su vez fomentó que los sombreros, celosos de la independencia rusa, fortalecieran sus lazos con el heredero.
Desde sus años de princesa en Suecia, Luisa Ulrica llevó una activa vida social, llena de actividades artísticas y culturales. Residía en fastuosos palacios, de los cuales su favorito era el de Drottningholm. Sus lujos y su despilfarro resultaron muy caros para las finanzas públicas.
Cuando Adolfo Federico se convirtió en rey de Suecia, en 1751, los sombreros se negaron a aumentar la influencia de la monarquía en la política, lo que derivó en una abierta enemistad entre los reyes y el parlamento. En esa coyuntura, se rumoreó que Luisa Ulrica había empeñado sus joyas para financiar una revolución que trajera de vuelta la monarquía absolutista. El rumor propició un inventario de las joyas de la corona. Con todo, la reina llevó sus ambiciones al extremo y conspiró en los preparativos de una revolución. El complot fue descubierto en 1756, con la consecuente ejecución de varios partidarios de la reina. Luisa Ulrica recibió una amonestación por parte del clero, que la encontró responsable de las muertes. A partir de entonces el poder de los reyes descendió como nunca en la historia de la Suecia contemporánea.
La reina cobró importancia nuevamente tras el estallido de la guerra de los Siete Años, en la que Suecia tomó parte en la coalición contra Prusia. El partido de los sombreros perdió prestigio con la guerra y buscó a la reina a fin de pactar la paz con Prusia, aludiendo al vínculo familiar que Luisa Ulrica mantenía con ese país.
Perdidas sus esperanzas de entablar alianzas sólidas con sus adversarios en el parlamento, Luisa Ulrica encontró un poderoso colaborador en su hijo, el príncipe Gustavo (posteriormente Gustavo III), cuya actividad política lo llevó a escalar rápidamente en el partido de la corte, el instituto político representante de los intereses monárquicos en el parlamento. Las relaciones con su hijo se estropearon súbitamente, luego de que la reina se negara a aceptar el liderazgo de aquel por encima del propio.
En 1777, fue obligada a abandonar Drottningholm con todo su acervo artístico, que pasó entonces a resguardo del Estado. En 1778, cuando nació el primogénito de Gustavo, Luisa Ulrica despertó al parecer el rumor de que el príncipe no era hijo del rey. El rumor violentó la relación con Gustavo, y fue obligada a hacer una aclaración pública en la que desmentía el rumor. A partir de entonces, a Luisa Ulrica se le prohibió presentarse en la corte, y tuvo que vivir en una especie de exilio, en la soledad de los castillos de Fredrikshov o de Svartsjö. En su lecho de muerte se reconcilió con su hijo. Falleció en el castillo de Svartsjö el 16 de julio de 1782.
Ancestros
Referencias
Reinas consortes de Suecia del siglo XVIII
Princesas de Prusia del siglo XVIII (Casa de Hohenzollern)
Consortes de miembros de la casa de Holstein-Gottorp
Nacidos en Berlín
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 1,803
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Project: Springfield Rise
Springfield Rise
In partnership with Divaworks, a smart functional display centre and real estate sales office was fitted out in Brisbane's busiest growth corridor.
This interesting and challenging brief was for a shop-fitting project to be carried out with our industry partner Divaworks. The design called for the complete renovation of one 100m² Land Sales office and the fit out of the adjoining 60m² Real Estate sales office. Further to the internal requirements, there was also need for various styles of smart outdoor signage and large window graphics to be on display around the precinct.
The Land Sales office was a modular European style cassette building that had been moved from another site close by. We were charged with completely renovating the interior while also introducing new data networks, electrical services, plumbing and air-conditioning upgrades. All of the internal walling was renovated with complete new spaces created in plaster and glass while existing walls were repaired, reconfigured, painted and in some cases, covered in laminated timber look panels. A new reception and other cabinet work were specified with composite stone tops along with a staff sales office. New signage throughout consisted of a mixture of backlit fabric, 3D logos, printed window graphics and cut vinyl signs. New flooring was installed and the entire space was given a comprehensive makeover. The large table model of the building development was renovated and positioned under new feature lighting.
Following the Divaworks design, the building came to life looking smart, fresh and functional, while the outdoor signage was an absolute home run.
The Real Estate sales office was a much less demanding task where we were able to fit out the existing layout of a new building and once again icatchers had the opportunity to innovate with a decorative feature wall area to highlight the client's brand in a stylish manner.
The essence of this project was to display the building development in a highly professional and appealing way to the prospective new home buyers drawn to the estate. Divaworks had to reflect the quality of life that was marketed and icatchers had the task of bringing this to a reality in a very high standard of finish. A vast array of different materials and surfaces were used to blend into the appeal and diversity of the different architectural designs being offered for this new and vibrant suburb.
Share this project:
BRISBANE | SYDNEY | MELBOURNE
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© 2020 ICATCHERS, ABN 84 144 875 187, QBCC LIC NUMBER 1050570. ALL RIGHTS RESERVED. WEBSITE BY WEBSITE QUICK FIX.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 367
|
Q: Sharepoint 2013 JSLINK template with view grouping rendering I'm looking for a tutorial or documentation on how to create a JSLINK rendering template that will perform view grouping rendering. I need to use this in connection with the list view webpart, where I have configured its view to group items by a given field. Any assistance would be greatly appreciated.
A: I found the way to make it work:
overrideCtx.Templates.Group = CustomGroup;
function CustomGroup(ctx, group, groupId, listItem, listSchema, level, expand) {
var html = '<div style="float:left">' + listItem[group] + ' : </div>';
return html;
}
A: Martin Hatch has a series on how to use JSLink
The following items are or will be covered
*
*Part 1 – Overview, URL Tokens and Applying JSLink to objects
*Part 2 – Changing how individual fields display
*Part 3 – Creating a custom editing interface
*Part 4 – Validating user input
*Part 5 – Creating custom List Views
*Part 6 – Creating View Templates and Deployment Options
*Part 7 – Code Samples
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 1,616
|
import urllib from 'urllib';
class HackerNewsService {
constructor(ctx) {
this.ctx = ctx;
this.serverUrl = 'https://hacker-news.firebaseio.com/v0';
this.pageSize = 20;
}
async request(api, options) {
let data = null;
if(options.cacheKey) {
data = await this.ctx.cache.get(options.cacheKey);
if(data) {
return data;
}
}
const opts = {
dataType: 'json',
timeout: 10000,
...options
};
const url = `${this.serverUrl}/${api}`;
const request = await urllib.request(url, opts);
data = request.data;
if (options.cacheKey && data) {
await this.ctx.cache.set(options.cacheKey, data, options.cacheExpire);
}
return data;
}
async getTopStories(page) {
page = page || 1;
const result = await this.request('topstories.json', {
data: {
orderBy: '"$key"',
startAt: `"${this.pageSize * (page - 1)}"`,
endAt: `"${this.pageSize * page - 1}"`
},
cacheKey: `news_ids_${page}`,
cacheExpire: 300
});
const ids = Object.keys(result).map(key => result[key]);
return ids;
}
async getItem(id) {
const item = await this.request(`item/${id}.json`, {
cacheKey: `news_item_${id}`,
cacheExpire: 1800
});
return item;
}
async getUser(id) {
const result = await this.request(`user/${id}.json`);
return result;
}
}
export default HackerNewsService;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,623
|
Free standing emergency department about to open in Wichita
Bob Weeks April 7, 2018 April 7, 2018 0 Comments5 Mins Read
A project in Wichita received substantial subsidy from taxpayers. How have public policy issues been reported?
Free standing emergency rooms are a recent trend in medical care. This is a facility that has equipment, personnel, and capability like a traditional hospital emergency room, but is not connected to a hospital. The first in Wichita is nearly ready to open, on Twenty-first Street east of Webb Road.
Regarding the Wichita facility, the Wichita Business Journal quoted Malik Idbeis, chief information officer for Kansas Medical Center: "We see a lot of patients from the northeast side of Wichita. We thought it'd be nice to bring our style of care closer to them. There are a lot of neighborhoods and families in that area." 1
Here, the spokesman is promoting that facility is located convenient to (affluent) families in northeast Wichita. That wasn't the argument made to the Wichita City Council last year when the facility applied for tax relief through the Industrial Revenue Bonds program. At that time, the facility was pitched as an attraction that would serve many out-of-town customers. City documents reported: "The current Economic Development Policy requires medical facilities to attract at least 30% of patients from outside the Wichita MSA. Kansas Medical Center reviewed the location of patients utilizing the emergency room in Andover, which revealed that 37% come from outside the Wichita MSA." 2
It seems a stretch to assume that the demographic characteristics of a hospital in Andover would also apply to an emergency room in Wichita, but the city council accepted this reasoning.
Aside from this, the Wichita Business Journal article contains problems in its reporting of public policy issues. The reporter wrote: "Last summer, the Wichita City Council authorized issuing industrial revenue bonds for the project to help finance land and construction costs. With IRBs, the city serves as a pass-through entity for developers to obtain a lower interest rate on projects. IRBs require no taxpayer commitment." (For background on IRBs in Kansas, see Industrial revenue bonds in Kansas.)
It's not likely the facility will save on interest costs with IRBs. It might save if the bonds were non-taxable, but these bonds are taxable, according to the agenda packet for this item. The article is correct in that IRBs require no taxpayer commitment — at least superficially. Here, I believe the reporter is letting readers know that the city makes no guarantee as to the bond repayment. If the city guaranteed repayment, that would help the borrower obtain a lower interest rate. But there is no guarantee.
Instead, the benefit of the IRB program is lower taxes. The city estimates the first-year property tax savings to be $61,882, allocated this way: City of Wichita: $17,226. State of Kansas: $792. Sedgwick County: $5,520. USD 259 (Wichita public school district): $28,345. Savings like this would be realized for five years, plus another five years if employment commitments are met.
This property tax forgiveness is, in many ways, a "taxpayer commitment." If we don't recognize that, then we must reconsider the foundation of local tax policy.
In Wichita, as in most cities, the largest consumers of property tax dollars are the city, county, and school district. All justify their tax collections by citing the services they provide: Law enforcement, fire protection, education, etc. It is for providing these services that we pay local taxes.
But through the Industrial Revenue Bond program, properties don't pay property tax. (In the case of this facility, the property tax abatement is limited to 88 percent of the full tax burden.)
Yet, this new facility will undoubtedly demand and consume the services local government provides — law enforcement, fire protection, and education. But it won't be paying property tax to support these services (except for the 12 percent not abated). Others will have to pay this cost.
We're left with an uncomfortable and awkward circumstance. City officials tell us that we must pay property tax so the city can provide services. (In fact, last year the Wichita city manager recommended increasing property taxes to pay for more police officers.)
At the same time, however, the city creates special classes of people who use services but don't pay for them.
Often the justification for economic development incentives is economic necessity, that is, the project could not be built without the incentive. That argument was not made for this project.
Free standing emergency rooms
Free standing emergency departments are controversial. The notes to this article hold references to news articles and academic studies looking at the costs and usage of these facilities. 3 4 5 6 7 8 9 10
Researchers note that the emergency rooms are much more expensive than traditional doctor offices or urgent care facilities, yet many of the diagnoses made at the ERs are the same as made at non-emergency facilities.
Heck, Josh. Medical group sets opening date for free-standing ER. Wichita Business Journal. Available at https://www.bizjournals.com/wichita/news/2018/04/06/medical-group-sets-opening-date-for-free-standing.html. ↩
City of Wichita. Request for Letter of Intent for Industrial Revenue Bonds (E Wichita Properties, LLC/Kansas Medical Center, LLC). City Council agenda packet for June 6, 2017. ↩
NBC News. You Thought It Was An Urgent Care Center, Until You Got the Bill. Available at https://www.nbcnews.com/health/health-care/you-thought-it-was-urgent-care-center-until-you-got-n750906. ↩
Carolyn Y. Johnson. Free-standing ERs offer care without the wait. But patients can still pay $6,800 to treat a cut. Washington Post, May 7, 2017. Available at http://wapo.st/2pUCskD?tid=ss_tw&utm_term=.21bb76a447aa. ↩
Rice, Sabriya.Texans overpaid for some medical services by thousands, study says. Dallas Morning News. Available at https://www.dallasnews.com/business/health-care/2017/03/23/texans-overpaid-medical-services-thousands-study-said. ↩
Blue Cross Blue Shield of Texas. Rice U. Study: Freestanding Emergency Departments In Texas Deliver Costly Care, 'Sticker Shock'. Available at https://www.bcbstx.com/company-info/news/news?lid=j0s5sm9d. ↩
Alan A. Ayers, MBA, MAcc. Dissecting the Cost of a Freestanding Emergency Department Visit. Available at https://c.ymcdn.com/sites/ucaoa.site-ym.com/resource/resmgr/Alan_Ayers_Blog/UCAOA_Ayers_Blog_FSED_Pricin.pdf. ↩
Michael L. Callaham. Editor in Chief Overview: A Controversy About Freestanding Emergency Departments. Annals of Emergency Medicine, Volume 70, Issue 6, 2017, pp. 843-845. Available at http://www.annemergmed.com/article/S0196-0644(17)31505-6/fulltext. ↩
Ho, Vivian et al. Comparing Utilization and Costs of Care in Freestanding Emergency Departments, Hospital Emergency Departments, and Urgent Care Centers. Annals of Emergency Medicine, Volume 70 , Issue 6 , 846 – 857.e3. Available at http://www.annemergmed.com/article/S0196-0644(16)31522-0/fulltext. ↩
Jeremiah D. Schuur, Donald M. Yealy, Michael L. Callaham. Comparing Freestanding Emergency Departments, Hospital-Based Emergency Departments, and Urgent Care in Texas: Apples, Oranges, or Lemons? Annals of Emergency Medicine, Volume 70, Issue 6, 2017, pp. 858-861. Available at http://www.annemergmed.com/article/S0196-0644(17)30473-0/fulltext. ↩
Economic developmentFeaturedIndustrial Revenue BondsSubsidyWichita city councilWichita city government
Previous PostWichita baseball consulting contract extension may have no discussion
Next PostSorting and polarizing in America
Wichita property tax rate: Up, just a little
The City of Wichita property tax mill levy rose very slightly for 2022. (more…)
Century II Roof Needs Repair
Bob WeeksMay 4, 2022
Wichita has delayed maintaining Century II's roof, and it needs repair. (more…)
Information Technology in Wichita
While Wichita economic development leaders have tried to turn Wichita into a hub of information technology-related industry, it has not worked so far. (more…)
Wichita since the start of the pandemic (and the past decade)
Bob WeeksDecember 30, 2021
How has Wichita fared since the start of the pandemic compared to other metropolitan areas? (more…)
The City of Wichita property tax mill levy rose slightly for 2021. (more…)
|
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{"url":"https:\/\/buboflash.eu\/bubo5\/show-dao2?d=1759786700044","text":"On a typical cosmetic repositioning, expect to spend on interior and exterior repairs between $1,000 and$4,000 per unit. Above \\$4,000, it\u2019s becoming a major repair job, and the risk is going up. The more repairs, the more unknowns you may uncover. That means you could end up spending more, taking longer, or both.\nIf you want to change selection, open document below and click on \"Move attachment\"\n\n#### pdf\n\nowner: Redemption - (no access) - [David_Lindahl]_Multi-Family_Millions_How_Anyone_(b-ok.org).pdf, p121","date":"2021-10-25 09:55:23","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\": 0.19548572599887848, \"perplexity\": 5736.4493508003825}, \"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-2021-43\/segments\/1634323587659.72\/warc\/CC-MAIN-20211025092203-20211025122203-00129.warc.gz\"}"}
| null | null |
\section{Introduction}
A convex geometry is an abstract generalization of the notion of convexity
on a finite set of points in Euclidean space. We study an achievement
game where two players take turns selecting previously unselected
points of a convex geometry until the convex closure of the jointly
selected points contains all the points of a given winning set. The
winner of the game is the last player able move. That is, the winner
is the first player to make the convex hull of the jointly selected
points a superset of the winning set.
This game is a version of a group generating game introduced by Anderson
and Harary \cite{anderson.harary:achievement} and further developed
in \cite{Barnes,BeneshErnstSiebenSymAlt,BeneshErnstSiebenDNG,BeneshErnstSiebenGeneralizedDihedral,ErnstSieben}.
Our game is played on a different kind of mathematical object. It
is also a generalization since we introduce a winning set that can
be different from the base set of the mathematical object. The key
tool for studying these generating games is structure equivalence
introduced in \cite{ErnstSieben}. Structure equivalence is an equivalence
relation on the game positions that is compatible with the option
structure of the positions. Taking the quotient of the game digraph
by structure equivalence provides significant simplifications. We
develop a structure theory for our generalization. This allows us
to determine the nim number of our games for several classes of convex
geometries, including one-dimensional affine geometries, vertex geometries
of trees, and games with a winning set consisting of extreme points.
The structure of the paper proceeds as follows. In Section~\ref{sec:Preliminaries}
we recall some basic terminology of impartial games and convex geometries.
In Section~\ref{sec:achievementGame} we describe the convex closure
achievement game in detail and provide some general results. In Section~\ref{sec:structureTheory}
we introduce structure equivalence and structure diagrams. In Section~\ref{sec:extremeCharacterizations}
we determine the nim numbers of games in which the goal set is a subset
of the extreme point set of the ground set. In Section~\ref{sec:treeCharacterization}
we determine the nim numbers of games played on vertex geometries
of trees. This allows us to easily determine the nim numbers of games
played on affine geometries in $\mathbb{R}$ in Section~\ref{sec:affineCharacterization}
since these affine geometries are isomorphic to vertex geometries
of paths. We conclude with some further questions in Section~\ref{sec:FurtherQuestions}.
\section{Preliminaries\label{sec:Preliminaries}}
If $f:X\to Y$ and $A\subseteq X$, then we often use the standard
$f(A):=\{f(a)\mid a\in A\}$ notation for the image of $A$. The cardinality
of a set $A$ is denoted by $|A|$, and we write $\pty(A):=|A|\mod2$
for the \emph{parity of a set}.
\subsection{Impartial games}
We recall the basic terminology of impartial combinatorial games.
Our general references for the subject are \cite{albert2007lessons,SiegelBook}.
An \emph{impartial} \emph{game} is a finite set $\mathcal{P}$ of
\emph{positions} accompanied with a starting position and a collection
$\Opt(P)\subseteq\mathcal{P}$ of \emph{options} for each position
$P$. In every move of the game, the current position $P$ becomes
an option of $P$ chosen by the player to move. Every game must finish
in finitely many steps. In particular, no position can be reached
twice. We can think of it as a finite acyclic digraph with the positions
as vertices, where there is an arrow from a position to every option
of that position. Game play is moving a token from one vertex to another
along the arrows. The game ends when the token reaches a sink of the
graph. The last player to move is the winner of the game.
The \emph{minimum excludant} $\mex(A)$ of a set $A$ of non-negative
integers is the smallest non-negative integer that is not in $A$.
The \emph{nim number} $\nim(P)$ of a position $P$ of a game is defined
recursively as the minimum excludant of the nim numbers of the options
of $P$. That is,
\[
\nim(P):=\mex(\nim(\Opt(P))).
\]
The nim number of the game is the nim number of the starting position.
A position is \emph{terminal} if it has no options. A terminal position
$P$ has nim number $\nim(P)=\mex(\nim(\emptyset))=\mex(\emptyset)=0$.
A position $P$ is losing for the player about to move ($P$-position)
if $\nim(P)=0$ and winning ($N$-position) otherwise. The winning
strategy is to always move to an option with nim number $0$ if available.
This places the opponent into a losing position. The nim number is
a central object of interest for impartial games. In addition to determining
the outcome of the game, it also makes it easy to compute the nim
number of sums of games.
\begin{figure}
\begin{tikzpicture}
\node (p2) at (0,0) {$*2$};
\node (p1) at (1,0) {$*1$};
\node (p0) at (2,0) {$*0$};
\draw[->] (p2) -- (p1);
\draw[->] (p1) -- (p0);
\draw[->] (p2) to[out=25,in=155] (p0);
\end{tikzpicture}
\caption{\label{fig:star2}The acyclic digraph representing the nimber $*2$.}
\end{figure}
\begin{example}
The \emph{nimber} $*n$ is the game with options $\Opt(*n):=\{*0,*1,\ldots,*(n-1)\}$.
The only terminal position of this game is the position $*0$. Induction
shows that $\nim(*n)=n$ since
\[
\begin{aligned}\nim(*n) & =\mex(\nim(\Opt(P)))=\mex(\nim(\{*0,\ldots,*(n-1)\}))\\
& =\mex(\{\nim(*0),\ldots,\nim(*(n-1))\})=\mex\{0,\ldots,n-1\}=n.
\end{aligned}
\]
The acyclic digraph representation of $*2$ is shown in Figure~\ref{fig:star2}.
In this game the first player can move into position $*0$ and win.
If the first player moves into position $*1$ instead, then the second
player moves into position $*0$ and win.
\end{example}
\subsection{Convex geometries}
We recall some facts about convex geometries from \cite{BjornerZiegler,EJ,KorteLovaszSchrader}.
\begin{defn}
A \emph{convex geometry} is a pair $(S,\mathcal{K})$, where $S$
is a finite set and $\mathcal{K}$ is a family of subsets of $S$
satisfying the following properties:
\begin{enumerate}
\item $S\in\mathcal{K}$;
\item $K,L\in\mathcal{K}$ implies $K\cap L\in\mathcal{K}$;
\item $S\ne K\in\mathcal{K}$ implies $K\cup\{a\}\in\mathcal{K}$ for some
$a\in S\setminus K$.
\end{enumerate}
\noindent The sets in $\mathcal{K}$ are called \emph{convex}.
\end{defn}
The third condition is called \emph{accessibility}. Following \cite{BjornerZiegler,Dietrich},
we do not require that the empty set and singleton sets are convex.
Note that the family of complements of the convex sets in a convex
geometry forms an \emph{anti-matroid}.
A convex geometry determines a \emph{convex closure} operator $\tau:2^{S}\to2^{S}$
defined by
\[
\tau(A):=\bigcap\{K\in\mathcal{K}\mid A\subseteq K\}.
\]
A point $a$ of a subset $A$ of a convex geometry is called an \emph{extreme
point} of $A$ if $a\not\in\tau(A\setminus\{a\})$. The set of extreme
points of $A$ is denoted by $\Ex(A)$.
\begin{prop}
The convex closure operator of a convex geometry satisfies the following
properties:
\begin{enumerate}
\item $A\subseteq\tau(A)$;
\item $A\subseteq B$ implies $\tau(A)\subseteq\tau(B)$;
\item $\tau(\tau(A))=\tau(A)$;
\item $a,b\not\in\tau(A)$ and $a\ne b\in\tau(A\cup\{a\})$ implies $a\not\in\tau(A\cup\{b\})$;
\item $A$ is convex if and only if $\tau(A)=A$;
\item $\tau(A)=\tau(\Ex(A))$.
\end{enumerate}
\end{prop}
The fourth property is called the \emph{anti-exchange property}. So
$\tau$ is a closure operator satisfying the anti-exchange property.
Note that the convex closure of the empty set might not be the empty
set since we do not require the empty set to be convex.
\begin{figure}
\begin{tikzpicture}[scale=.6]
\draw[color=blue!10,fill=blue!10] (0,0) -- (1,3) -- (5,2) -- cycle;
\node[inner sep=-10,label=200:$a$] (b1) at (0,0) {$\bullet$};
\node[inner sep=-10,label=100:$b$] (b3) at (1,3) {$\bullet$};
\node[inner sep=0,label=0:$c$] (b4) at (5,2) {$\bullet$};
\node[color=white,inner sep=0] at (2,2) {$\bullet$};
\node[inner sep=0,label=0:$e$] (b5) at (2,2) {$\bullet$};
\node[color=white,inner sep=0] at (1,2) {$\bullet$};
\node[inner sep=0,label=270:$g$] (b6) at (1,2) {$\bullet$};
\node[color=white,inner sep=0] (b7) at (2.5,1) {$\bullet$};
\node[inner sep=-10,label=182:$f$] (b7) at (2.5,1) {$\bullet$};
\node[inner sep=-20,label=290:$d$] (b8) at (4,1) {$\bullet$};
\end{tikzpicture}
\caption{\label{fig:hullVertexNonvertex}A point set in $\mathbb{R}^{2}$.
The points in the shaded region belong to $\tau\{a,b,c,e\}$ in the
affine convex geometry.}
\end{figure}
\begin{example}
Let $S$ be a finite subset of $\mathbb{R}^{n}$. The convex subsets
of $S$ are the intersections of $S$ with convex subsets of $\mathbb{R}^{n}$.
The collection $\mathcal{K}$ of convex subsets of $S$ forms a convex
geometry on $S$ called the \emph{affine convex geometry}. The convex
closure operator is $\tau(A)=S\cap\Conv(A)$, where $\Conv(A)$ is
the convex hull of $A$.
\end{example}
\begin{example}
Figure~\ref{fig:hullVertexNonvertex} shows an example of a point
set $S=\{a,b,\ldots,f\}$ in $\mathbb{R}^{2}$. The set of extreme
points of $S$ in the affine convex geometry is $\Ex(S)=\{a,b,c,d\}$.
The set of extreme points of $A=\{a,b,c,e,f,g\}$ is $\Ex(A)=\{a,b,c\}$,
so $A=\tau(\{a,b,c\})$. We also have $A=\tau(\{a,b,c,e\})=\tau(\{a,b,c,f\})$.
\end{example}
We define deletions by generalizing the definition in \cite{ConvBeta}
and the notion of minor in \cite{affineRep}.
\begin{defn}
Let $(S,\mathcal{K})$ be a convex geometry and $D$ be a subset of
$S$. The \emph{deletion} of $\mathcal{K}$ by $D$ is the collection
\[
\mathcal{K}\setminus D:=\{K\subseteq S\setminus D\mid K\cup D\in\mathcal{K}\}.
\]
\end{defn}
\begin{prop}
If $(S,\mathcal{K})$ is a convex geometry and $D\subseteq S$, then
$(S\setminus D,\mathcal{K}\setminus D)$ is a convex geometry.
\end{prop}
\begin{proof}
We see that $S\setminus D\in\mathcal{K}\setminus D$ since $(S\setminus D)\cup D=S\in\mathcal{K}$.
If $K,L\in\mathcal{K}\setminus D$ then $K\cap L\in\mathcal{K}\setminus D$
since $(K\cap L)\cup D=(K\cup D)\cap(L\cup D)\in\mathcal{K}$. Now
assume $S\setminus D\ne K\in\mathcal{K}\setminus D$. Since $S\ne K\cup D\in\mathcal{K}$,
accessibility implies that $(K\cup D)\cup\{a\}\in\mathcal{K}$ for
some $a\in S\setminus(K\cup D)$. So $a\in(S\setminus D)\setminus K$
and $K\cup\{a\}\in\mathcal{K}\setminus D$.
\end{proof}
Note that we do not require $D$ to be convex. This is not necessary
since the empty set does not need to be convex.
\begin{figure}
\begin{tabular}{ccccccccc}
& & $\circ\bullet\bullet\,\circ$ & ~ & $\circ\bullet\circ\,\bullet$ & ~ & $\circ\bullet\bullet$ & ~ & $\bullet\,\bullet$\tabularnewline
$\mathcal{K}$ & & $\{\{0,1\}\}$ & & $\{\{0\},\{0,1\}\}$ & & $\{\emptyset,\{0\},\{0,1\}\}$ & & $\{\emptyset,\{0\},\{1\},\{0,1\}\}$\tabularnewline
$D$ & & $\{-1,2\}$ & & $\{-1/2,1/2\}$ & & $\{-1\}$ & & $\emptyset$\tabularnewline
$\Ex(S)$ & & $\emptyset$ & & $\{1\}$ & & $\{1\}$ & & $\{0,1\}$\tabularnewline
\end{tabular}
\caption{\label{fig:convexGeom2}The convex geometries up to isomorphism on
a two-element set $S=\{0,1\}$ represented as deletions of affine
convex geometries on a point set $T$ in $\mathbb{R}$. The points
of $S$ are shown as bullets, while the deleted points are shown as
empty circles.}
\end{figure}
\begin{example}
The convex geometries on a two-element set $S=\{0,1\}$ up to isomorphism
are shown in Figure~\ref{fig:convexGeom2}. Each of these convex
geometries can be represented as a deletion of an affine convex geometry
on a subset $T=S\cup D$ of $\mathbb{R}$. The points in $S$ are
shown with bullets. The deleted points in $D$ are shown as empty
circles. The last row of the table shows the extreme points of $S$.
\end{example}
\begin{rem}
\label{rem:repConj}The main result of \cite{affineRep} is that every
convex geometry that contains the empty set can be represented as
a deletion of an affine convex geometry by a convex set. We conjecture
that a version of this representation result holds even for convex
geometries in which the empty set is not closed. In this conjectured
representation, the deleted set does not need to be convex.
\end{rem}
\begin{figure}
\begin{tabular}{ccccccccc}
& & \begin{tikzpicture}[scale=.6]
\node at (.7,0) {$\bullet$};
\node at (1.4,0) {$\bullet$};
\node at (2.1,0) {$\bullet$};
\node at (0,0) {$\circ$};
\node at (2.8,0) {$\circ$};
\end{tikzpicture} & ~ & \begin{tikzpicture}[scale=.3,rotate=-30]
\node at (0:.7) {$\bullet$};
\node at (120:.7) {$\bullet$};
\node at (-120:.7) {$\bullet$};
\node at (20:2) {$\circ$};
\node at (-140:2) {$\circ$};
\end{tikzpicture} & ~ & \begin{tikzpicture}[scale=.3,rotate=0]
\node at (0:0) {$\circ$};
\node at (0:-1.4) {$\bullet$};
\node at (0:1.4) {$\bullet$};
\node at (0:2.8) {$\bullet$};
\node at (0:-2.8) {$\circ$};
\end{tikzpicture} & ~ & \begin{tikzpicture}[scale=.3,rotate=90]
\node at (0:0) {$\bullet$};
\node at (0:-1.4) {$\circ$};
\node at (0:1.4) {$\circ$};
\node at (90:1.4) {$\bullet$};
\node at (-90:1.4) {$\bullet$};
\end{tikzpicture}\tabularnewline
$\mathcal{K}$ & & $\{\{a,b,c\}\}$ & & $\{\{a,b\},$ & & $\{\{a\},\{a,b\},$ & & $\{\{a\},\{a,b\},$\tabularnewline
& & & & $\{a,b,c\}\}$ & & $\{a,b,c\}\}$ & & $\{a,c\},\{a,b,c\}\}$\tabularnewline
$\Ex(S)$ & & $\emptyset$ & & $\{c\}$ & & $\{c\}$ & & $\{b,c\}$\tabularnewline
\end{tabular}
\caption{\label{fig:convexGeom3}Convex geometries up to isomorphism on a three-element
set $S=\{a,b,c\}$ represented as deletions of affine convex geometries
on a point set $T$ in $\mathbb{R}^{2}$. The points of $S$ are shown
as bullets, while the deleted points are shown as empty circles.}
\end{figure}
\begin{example}
There are four convex geometries up to isomorphism with three points
in which the empty set is not convex. They can be represented as deletions
of affine convex geometries on a point set $T=S\cup D$ of $\mathbb{R}^{2}$,
as shown in Figure~\ref{fig:convexGeom3}. For example, the second
convex geometry in the table can be represented as $S=\{a,b,c\}$
with $a=(1,0)$, $b=(2,0)$, $c=(1.5,1)$ and $D=\{x,y\}$ with $x=(0,0)$,
$y=(3,0)$.
\end{example}
\section{Convex closure achievement game\label{sec:achievementGame}}
We now provide a detailed description of the impartial convex closure
achievement game $\GEN(S,W)$ played on a convex geometry $(S,\mathcal{K})$
with a nonempty winning subset $W$ of $S$. In this game, two players
take turns selecting previously unselected elements of $S$. In turn
$k$ the next player picks $p_{k}$ from $S\setminus\{p_{1},\ldots,p_{k-1}\}$.
The game ends when the convex closure $\tau(P)$ of the jointly selected
points $P=\{p_{1},\ldots,p_{k}\}$ contains $W$. The last player
to make a move wins $\GEN(S,W)$. We call $P$ the current position
of the game. The set of options for a nonterminal position $P$ is
$\Opt(P)=\{P\cup\{s\}\mid s\in S\setminus P\}$.
An important special case is when $W=S$. For this game we use the
notation $\GEN(S)$ instead of the more precise $\GEN(S,S)$.
\begin{example}
\label{exa:3InLine}Consider $\GEN(S)$ on the affine convex geometry
with $S:=\{-1,0,1\}\subseteq\mathbb{R}$ shown in Figure~\ref{fig:TwoSamples}.
If the first player selects $-1$, then the second player can select
$1$ and win the game because $S=\tau\{-1,1\}$. So this first move
is a mistake for the first player.
If the first player selects 0, then without loss of generality we
can assume the second player selects $1$. Then the first player can
win by selecting $-1$. So with best play the first player can win
this game.
Figure~\ref{fig:3InLine} shows the full game digraph of $\GEN(S)$.
The nimbers $*0,*1,*2$ below the sets of chosen points indicate the
nim numbers of the positions.
\end{example}
\begin{figure}
\begin{tabular}{ccccccc}
$\bullet\bullet\bullet$ & $\to$ & $\circ\bullet\bullet$ & $\to$ & $\circ\bullet\circ$ & & \tabularnewline
$\emptyset$ & & $\{-1\}$ & & $\{-1,1\}$ & & \tabularnewline
\end{tabular}
~
\begin{tabular}{ccccccc}
$\bullet\bullet\bullet$ & $\to$ & $\bullet\circ\bullet$ & $\to$ & $\bullet\circ\circ$ & $\to$ & $\circ\circ\circ$\tabularnewline
$\emptyset$ & & $\{0\}$ & & $\{0,1\}$ & & $\{-1,0,1\}$\tabularnewline
\end{tabular}
\caption{\label{fig:TwoSamples}Two samples of game play for $\protect\GEN(S)$
with $S=\{-1,0,1\}$.}
\end{figure}
\begin{figure}
\begin{tikzpicture}[scale=1.1]
\draw[draw=white,fill=black!10] (-0.3,0.4) rectangle (.3,-1.4);
\draw[draw=white,fill=black!10] (-0.6,-2.1) rectangle (0.6,-3.9);
\draw[draw=white,fill=black!10] (-3,-0.6) rectangle (-2,-2.4);
\draw[draw=white,fill=black!10] (2.9,-0.6) rectangle (2.1,-2.4);
\draw[draw=white] (-.5,-3) rectangle (.5,-3);
\node (p0) at (0,0) {$\emptyset \atop {*1}$};
\node (p1) at (0,-1) {$\{0\} \atop {*0}$};
\node (p5) at (-2.5,-1) {$\{-1\} \atop {*2}$};
\node (p6) at (2.5,-1) {$\{1\} \atop {*2}$};
\node (p2) at (-2.5,-2) {$\{-1,0\} \atop {*1}$};
\node (p3) at (0,-3.5) {$\{-1,0,1\} \atop {*0}$};
\node (p4) at (2.5,-2) {$\{0,1\} \atop {*1}$};-
\node (p7) at (0,-2.5) {$\{-1,1\} \atop {*0}$};
\draw[->] (p0) -- (p1);
\draw[->] (p1)-- (p2);
\draw[->] (p2) -- (p3);
\draw[->] (p1) -- (p4);
\draw[->] (p4) -- (p3);
\draw[->] (p0) -- (p5);
\draw[->] (p5) -- (p2);
\draw[->] (p0) -- (p6);
\draw[->] (p6) -- (p4);
\draw[->] (p5) -- (p7);
\draw[->] (p6) -- (p7);
\end{tikzpicture}
\caption{\label{fig:3InLine}Game digraph of $\protect\GEN(S)$ with $S=\{-1,0,1\}$.}
\end{figure}
\begin{prop}
\label{prop:WConvWVerW}The nim numbers of $\GEN(S,\Ex(W))$, $\GEN(S,W)$,
and $\GEN(S,\tau(W))$ are the same.
\end{prop}
\begin{proof}
Let $P\subseteq S$. Since $\Ex(W)\subseteq W\subseteq\tau(W)$, $\tau(W)\subseteq\tau(P)$
implies $\Ex(W)\subseteq\tau(P)$. On the other hand, $\Ex(W)\subseteq\tau(P)$
implies $\tau(W)=\tau(\Ex(W))\subseteq\tau(\tau(P))=\tau(P)$. Hence
the positions in all three games are exactly the same.
\end{proof}
The following is an immediate consequence.
\begin{cor}
The nim numbers of $\GEN(S)$ and $\GEN(S,\Ex(S))$ are the same.
\end{cor}
\section{\label{sec:structureTheory}Structure theory}
\subsection{Structure equivalence}
Structure equivalence is an equivalence relation on the set of game
positions. This relation is compatible with the option structure and
hence the nim numbers of the positions. This allows us to use a smaller
quotient of the game digraph to compute nim numbers.
\begin{defn}
Consider the achievement game $\GEN(S,W)$ and let $M\subseteq S$.
If $W\subseteq\tau(M)$ then $M$ is called a \emph{generating} \emph{set.
}Otherwise, $M$ is called a \emph{non-generating set.} If $M$ is
non-generating and $N$ is generating for all $M\subset N\subseteq S$,
then $M$ is called \emph{maximally non-generating.} We denote the
set of maximally non-generating subsets by $\mathcal{M}$.
\end{defn}
\begin{example}
Consider the affine convex geometry on $S=\{0,1,2,3\}\subseteq\mathbb{R}$
and let $W=\{1,2\}$. Then $\mathcal{M}=\{\{0,1\},\{2,3\}\}$.
\end{example}
\begin{prop}
A maximally non-generating set $M$ is convex.
\end{prop}
\begin{proof}
The closure $\tau(M)$ of $M$ is a non-generating superset of $M$
since $W\not\subseteq\tau(M)=\tau(\tau(M))$ and $M\subseteq\tau(M)$.
Hence $M=\tau(M)$ since $M$ is maximally non-generating.
\end{proof}
\begin{defn}
We let
\[
\mathcal{I}:=\{\bigcap\mathcal{N}\mid\mathcal{N}\subseteq\mathcal{M}\}
\]
be the set of \emph{intersection subsets}. The smallest intersection
subset is the \emph{Frattini subset} $\Phi$, which is the intersection
of all maximally non-generating subsets.
\end{defn}
Note that with $\mathcal{N}=\emptyset\subseteq\mathcal{M}$ we get
$S=\bigcap\emptyset=\bigcap\mathcal{N}\in\mathcal{I}$. So $S$ is
always an intersection subset.
\begin{example}
Consider the convex geometry $\mathcal{K}=\{\{0\},\{0,1\}\}$ on $S=\{0,1\}$.
If $W=\{0\}$ then $\mathcal{M}=\emptyset$, $\mathcal{I}=\{S\}$,
$\Phi=S$, and $\nim(\GEN(S,W))=0$. Note that in this game there
is only one game position $\emptyset$, and this position is both
a starting and a terminal position. If $W=\{1\}$ then $\mathcal{M}=\{\{0\}\}$,
$\mathcal{I}=\{\{0\},S\}$, and $\Phi=\{0\}$.
\end{example}
\begin{example}
Consider the convex geometry $\mathcal{K}=\{\emptyset,\{0\}\}$ on
$S=\{0\}$. If $W=\{0\}$ then $\mathcal{M}=\{\emptyset\}$, $\mathcal{I}=\{\emptyset,S\}$,
and $\Phi=\emptyset$.
\end{example}
\begin{defn}
For a position $P$ of $\GEN(S,W)$ let
\[
\mathcal{M}_{P}:=\{M\in\mathcal{M}\mid P\subseteq M\},\qquad\lceil P\rceil:=\bigcap\mathcal{M}_{P}.
\]
Two game positions $P$ and $Q$ are \emph{structure equivalent} if
$\lceil P$$\rceil=\lceil Q\rceil$. The \emph{structure class} $X_{I}$
of $I\in\mathcal{I}$ is the equivalence class of $I$ under this
equivalence relation.
\end{defn}
Note that $\lceil I\rceil=I$ for all $I\in\mathcal{I}$ and that
$I\mapsto X_{I}$ is a bijection from $\mathcal{I}$ to the set of
structure classes.
\begin{example}
\label{exa:3InLineB}Let $W=S=\{-1,0,1\}$ as in Example~\ref{exa:3InLine}.
Then
\[
\begin{aligned}\mathcal{M} & =\{\{-1,0\},\{0,1\}\},\\
\mathcal{I} & =\{\{0\},\{-1,0\},\{0,1\},S\},
\end{aligned}
\]
and $\Phi=\{0\}$. We also have
\[
\begin{aligned}X_{\{0\}} & =\{\emptyset,\{0\}\}, & X_{\{-1,0\}} & =\{\{-1\},\{-1,0\}\}\\
X_{\{0,1\}} & =\{\{1\},\{0,1\}\}, & X_{S} & =\{\{-1,1\},\{-1,0,1\}\},
\end{aligned}
\]
indicated with the gray boxes in Figure~\ref{fig:3InLine}. So $\lceil\{-1\}\rceil=\{-1,0\}$
and $\lceil\emptyset\rceil=\{0\}$ . Notice the lack of arrow from
$\{-1,1\}$ to $\{-1,0,1\}$.
\end{example}
\begin{prop}
If $\lceil P\rceil=\lceil Q\rceil$ then $\mathcal{M}_{P}=\mathcal{M}_{Q}$.
\end{prop}
\begin{proof}
For a contradiction, assume $\mathcal{M}_{P}\neq\mathcal{M}_{Q}$.
Then without loss of generality, there exists an $M\in\mathcal{M}_{P}\setminus\mathcal{M}_{Q}$.
That is, $M\in\mathcal{M}$ such that $P\subseteq M$ but $Q\not\subseteq M$.
Then there exists $q\in Q\setminus M$. This is impossible since $q\in Q\subseteq\lceil Q\rceil=\lceil P\rceil=\bigcap\mathcal{M}_{P}\subseteq M$.
\end{proof}
\begin{prop}
\label{prop:optStruct}Let $P,Q\in X_{I}\neq X_{J}$. If $\Opt(P)\cap X_{J}\neq\emptyset$
then $\Opt(Q)\cap X_{J}\neq\emptyset$.
\end{prop}
\begin{proof}
Assume $\Opt(P)\cap X_{J}\neq\emptyset$. Then there is an $r\in S\setminus P$
such that $P\cup\{r\}\in X_{J}$. We show that $Q\cup\{r\}\in X_{J}$.
We have
\[
\begin{aligned}\mathcal{M}_{P\cup\{r\}} & =\{M\in\mathcal{M}\mid P\cup\{r\}\subseteq M\}\\
& =\{M\in\mathcal{M}\mid P\subseteq M\text{ and }r\in M\}\\
& =\{M\in\mathcal{M}_{P}\mid r\in M\}
\end{aligned}
\]
and similarly $\mathcal{M}_{Q\cup\{r\}}=\{M\in\mathcal{M}_{Q}\mid r\in M\}$.
Hence
\[
\begin{aligned}\lceil Q\cup\{r\}\rceil & =\bigcap\mathcal{M}_{Q\cup\{r\}}\\
& =\bigcap\{M\in\mathcal{M}_{Q}\mid r\in M\}\\
& =\bigcap\{M\in\mathcal{M}_{P}\mid r\in M\}\\
& =\bigcap\mathcal{M}_{P\cup\{r\}}\\
& =\lceil P\cup\{r\}\rceil=J
\end{aligned}
\]
since $\mathcal{M}_{P}=\mathcal{M}_{Q}$.
\end{proof}
\begin{defn}
We say $X_{J}$ is an \emph{option} of $X_{I}$ if $X_{J}\cap\Opt(I)\ne\emptyset$.
The set of options of $X_{I}$ is denoted by $\Opt(X_{I})$.
\end{defn}
The following Lemma was proved in \cite{ErnstSieben}.
\begin{lem}
\label{lem:mex}If $A$ and $B$ are sets containing non-negative
integers such that $\mex(A)\in B$, then $\mex(A\cup\{\mex(B)\})=\mex(A)$.
\end{lem}
\begin{prop}
\label{prop:mainThm}If $P,Q\in X_{I}$ and $\pty(P)=\pty(Q)$, then
$\nim(P)=\nim(Q)$.
\end{prop}
\begin{proof}
Let
\[
Z:=\{(P,Q)\mid\lceil P\rceil=\lceil Q\rceil\text{ and }\pty(P)=\pty(Q)\}.
\]
We say $(P,Q)\succeq(M,N)$ when $P\subseteq M$ and $Q\subseteq N$.
Then $(Z,\succeq)$ is a partially ordered set with minimum element
$(S,S)$. We proceed by structural induction on $Z$. Let $(P,Q)\in Z$.
The statement clearly holds if $P=Q$. In particular, it holds if
$(P,Q)=(S,S)$. Otherwise we let $I:=\lceil P\rceil=\lceil Q\rceil$
and consider several cases.
First, assume $P\neq I\neq Q$. Then both $P$ and $Q$ have options
in $X_{I}$. In fact, $P\cup\{s\}\in\Opt(P)\cap X_{I}$ for each $s\in I\setminus P$.
If $M$ and $N$ are options of $P$ and $Q$ in $X_{I}$ respectively,
then $\pty(M)=\pty(N)$. Hence $\nim(M)=\nim(N)$ by induction since
$(P,Q)\succ(M,N)$ in $Z$. If $P$ has an option $M$ in some $X_{J}\neq X_{I}$,
then $Q$ also has an option $N$ in $X_{J}$ by Proposition~\ref{prop:optStruct}.
Since $\lceil M\rceil=J=\lceil N\rceil$ and $\pty(M)=\pty(N)$, we
have $(P,Q)\succ(M,N)$. Hence $\nim(M)=\nim(N)$ by induction. This
proves that $\nim(\Opt(P))=\nim(\Opt(Q))$. Thus $\nim(P)=\nim(Q)$.
Now, assume $P\ne I=Q$. In this case $Q$ does not have any options
in $X_{I}$. We still have $\nim(\Opt(Q))\subseteq\nim(\Opt(P))$
by Proposition~\ref{prop:optStruct}. Let $M$ be an option of $P$
in $X_{I}$. Since $\pty(M)\ne\pty(I)$, $M$ is different from $I$.
So $M$ has an option $R\in X_{I}$. Then $\pty(R)=\pty(I)=\pty(Q)$
and $\lceil R\rceil=I=\lceil Q\rceil$, so $(P,Q)\succ(R,Q)$. Hence
$\nim(R)=\nim(Q)$ by induction. This implies
\[
\mex(\overbrace{\nim(\Opt(Q))}^{A})=\nim(Q)=\nim(R)\in\overbrace{\nim(\Opt(M))}^{B},
\]
where $A:=\nim(\Opt(Q))$ and $B:=\nim(\Opt(M)$. Thus, by Lemma
\ref{lem:mex},
\[
\mex(\overbrace{\nim(\Opt(Q))}^{A}\cup\{\overbrace{\nim(M)}^{\mex(B)}\})=\mex(\overbrace{\nim(\Opt(Q))}^{A}).
\]
If $N\in\Opt(P)$ then either $N\not\in X_{I}$ or $N\in X_{I}$.
In the first case, Proposition~\ref{prop:optStruct} implies that
$Q$ has an option $L$ with $\lceil L\rceil=\lceil N\rceil$. Since
$\pty(L)=\pty(N)$, $\nim(N)=\nim(L)$ by induction. So $\nim(N)=\nim(L)\in\nim(\Opt(Q))$.
In the second case $\nim(M)=\nim(N)$ by induction since $(P,Q)\succ(M,N)$.
Hence
\begin{align*}
\nim(P) & =\mex(\nim(\Opt(P)))\\
& =\mex(\nim(\Opt(Q))\cup\{\nim(M)\})\\
& =\mex(\nim(\Opt(Q))=\nim(Q)
\end{align*}
since $\nim(\Opt(P))=\nim(\Opt(Q))\cup\{\nim(M)\}$.
\end{proof}
\begin{defn}
The \emph{type }of the structure class $X_{I}$ is
\[
\type(X_{I}):=(\pty(I),\nim_{0}(X_{I}),\nim_{1}(X_{I})).
\]
If $I=S$ then $\nim_{0}(X_{S}):=0$ and $\nim_{1}(X_{S}):=0$. If
$I\ne S$ then
\[
\begin{aligned}\nim_{\pty(I)}(X_{I}) & :=\mex(\nim_{1-\pty(I)}(\Opt(X_{I})),\\
\nim_{1-\pty(I)}(X_{I}) & :=\mex(\nim_{\pty(I)}(\Opt(X_{I}))\cup\{\nim_{\pty(I)}(X_{I})\})
\end{aligned}
\]
are defined recursively. We call the recursive computation of types
using the options of structure classes \emph{type calculus}.
\end{defn}
Note that $X_{S}$ is the only structure class without options.
\begin{example}
Assume that $\Opt(X_{I})=\{X_{J},X_{K}\}$ with $\type(X_{J})=(0,0,3)$
and $\type(X_{K})=(1,2,0)$. If $\pty(I)=0$ then
\[
\begin{aligned}\nim_{0}(X_{I}) & =\mex(\{\nim_{1}(X_{J}),\nim_{1}(X_{K})\})=\mex(\{3,0\})=1,\\
\nim_{1}(X_{I}) & =\mex(\{\nim_{0}(X_{J}),\nim_{0}(X_{K}),\nim_{0}(X_{I})\})=\mex(\{0,2,1\})=3
\end{aligned}
\]
and $\type(X_{I})=(0,1,3)$. If $\pty(I)=1$ then
\[
\begin{aligned}\nim_{1}(X_{I}) & =\mex(\{\nim_{0}(X_{J}),\nim_{0}(X_{K})\})=\mex(\{0,2\})=1,\\
\nim_{0}(X_{I}) & =\mex(\{\nim_{1}(X_{J}),\nim_{1}(X_{K}),\nim_{1}(X_{I})\})=\mex(\{3,0,1\})=2
\end{aligned}
\]
and $\type(X_{I})=(1,2,1)$.
\end{example}
The type of a structure class $X_{I}$ encodes the parity of $I$
and the nim numbers of the positions in $X_{I}$.
\begin{prop}
If $P\in X_{I}$ then $\nim(P)=\nim_{\pty(P)}(X_{I})$.
\end{prop}
\begin{proof}
We use structural induction on the positions, together with Propositions~\ref{prop:optStruct}
and \ref{prop:mainThm}. The statement is clearly true for $I=S$.
Any option $Q$ of position $I$ is in $X_{J}$ for some $X_{J}\in\Opt(X_{I})$.
On the other hand, if $X_{J}\in\Opt(X_{I})$ then $X_{J}$ contains
an option $Q$ of $I$. The parity $\pty(Q)=1-\pty(I)$ of $Q$ is
the opposite of the parity of $I$. Hence $\nim(I)=\mex(\nim(\Opt(I)))=\nim_{\pty(I)}(X_{I})$
by induction.
First assume that $P$ is a position in $X_{I}$ such that $\pty(P)=\pty(I)$.
Then $\nim(P)=\nim(I)$ by Proposition~\ref{prop:mainThm}. Thus
$\nim(P)=\nim(I)=\nim_{\pty(I)}(X_{I})=\nim_{\pty(P)}(X_{I}).$
Now assume that $P$ is a position in $X_{I}$ such that $\pty(P)=1-\pty(I)$.
Then the options of $P$ have parity $\pty(I)$. Since $P$ is strictly
smaller than $I$, $P$ must have an option in $X_{I}$. Every $Q$
in $\Opt(P)\cap X_{I}$ satisfies $\nim(Q)=\nim_{\pty(Q)}(X_{I})=\nim_{\pty(I)}(X_{I})$
by induction. Proposition~\ref{prop:optStruct} implies that every
option of $P$ that is not in $X_{I}$ must be a position in $X_{J}$
for some $X_{J}\in\Opt(X_{I})$. On the other hand, Proposition~\ref{prop:optStruct}
also implies that if $X_{J}\in\Opt(X_{I})$ then $X_{J}$ contains
an option of $P$. Every $Q$ in $\Opt(P)\cap X_{J}$ with $X_{J}\in\Opt(X_{I})$
satisfies $\nim(Q)=\nim_{\pty(Q)}(X_{J})=\nim_{\pty(I)}(X_{J})$ by
induction. Thus
\[
\begin{aligned}\nim(P) & =\mex(\nim(\Opt(P)))\\
& =\mex(\nim((\Opt(P)\cap\bigcup\Opt(X_{I}))\cup(\Opt(P)\cap X_{I}))\\
& =\mex(\nim_{\pty(I)}(\Opt(X_{I}))\cup\{\nim_{\pty(I)}(X_{I})\})\\
& =\nim_{1-\pty(I)}(X_{I})=\nim_{\pty(P)}(X_{I}).
\end{aligned}
\]
\end{proof}
Note that $X_{I}=\{I\}$ is possible. In this case $X_{I}$ contains
no position with parity $1-\pty(I)$. Also note that the nim number
of the game is the nim number of the starting position $\emptyset$.
So $\nim(\GEN(S,W))=\nim(\emptyset)=\mex_{0}(X_{\Phi})$ is the second
component of $\type(X_{\Phi})$.
\subsection{Structure diagrams\label{sec:structureDiagrams}}
The \emph{structure digraph} of $\GEN(S,W)$ has vertex set $\{X_{I}\mid I\in\mathcal{I}\}$
and arrow set $\{(X_{I},X_{J})\mid X_{J}\in\Opt(X_{I})\}$. We visualize
the structure digraph with a \emph{structure diagram} that also shows
the type of each structure class. Within a structure diagram, a vertex
$X_{I}$ is represented by a triangle pointing up or down depending
on the parity of $I$. The triangle points down when $\pty(I)=1$
and points up when $\pty(I)=0$. The numbers within each triangle
represent the nim numbers of the positions within the structure class.
The first number is the common nim number of all even positions in
$X_{I}$, while the second number is the common nim number of all
odd positions in $X_{I}$.
We call an automorphism $\alpha$ of the structure digraph \emph{size
preserving} if $|I|=|J|$ for all $\alpha(X_{I})=X_{J}$. It is often
useful to work with the quotient of the structure digraph with respect
to orbit equivalence determined by the size preserving automorphisms
of the structure digraph. We call the corresponding structure diagram
the \emph{orbit quotient structure diagram}. We draw these quotient
diagrams using representative structure classes. Since orbit equivalent
structure classes have the same type, the nim number of the game can
be determined using the orbit quotient structure diagram.
\begin{figure}
\begin{tabular}{ccc}
\begin{tikzpicture}[scale=1.2]
\node (p1) at (0,-1) {$X_{\{0\}}$};
\node (p2) at (-1,-2) {$X_{\{-1,0\}}$};
\node (p3) at (0,-3) {$X_{\{-1,0,1\}}$};
\node (p4) at (1,-2) {$X_{\{0,1\}}$};
\draw[->] (p1) -- (p2);
\draw[->] (p2) -- (p3);
\draw[->] (p1) -- (p4);
\draw[->] (p4) -- (p3);
\end{tikzpicture} & \includegraphics[scale=0.5]{D1oee} & \includegraphics[scale=0.5]{D1oeeSimp}\tabularnewline
\end{tabular}
\caption{\label{fig:3InLine-1}Structure digraph, structure diagram, and orbit
quotient structure diagram of $\protect\GEN(S)$ with $S=\{-1,0,1\}$.}
\end{figure}
\begin{example}
The structure digraph and structure diagrams of $\GEN(S)$ with the
affine convex geometry on $S=\{-1,0,1\}$ considered in Examples~\ref{exa:3InLine}
and \ref{exa:3InLineB} appear in Figure~\ref{fig:3InLine-1}. We
demonstrate type calculus by verifying that $\nim(\GEN(S))=\nim(\emptyset)=\nim_{0}(X_{\{0\}})=1$.
We have
\[
\begin{aligned}\nim_{1}(X_{\{0\}}) & =\mex(\{\nim_{0}(X_{\{-1,0\}}),\nim_{0}(X_{\{0,1\}})\})=\mex(\{1\})=0,\\
\nim_{0}(X_{\{0\}}) & =\mex(\{\nim_{1}(X_{\{-1,0\}}),\nim_{1}(X_{\{0,1\}})\}\cup\{\nim_{1}(X_{\{0\}})\})\\
& =\mex(\{2\}\cup\{0\})=1.
\end{aligned}
\]
\end{example}
\begin{rem}
If we make a move in the achievement game on an affine convex geometry,
then the rest of the game is essentially played on a convex geometry
that was created from the original convex geometry by the deletion
of the selected point. Since the selected points do not necessarily
form a convex set, the deletion of these points may result in a convex
geometry in which the empty set is not convex.
\end{rem}
\begin{figure}
\begin{tabular}{cccc}
\begin{tikzpicture}
\node[label={[label distance=-2mm]180:$3$}] at (-1,0) {$\bullet$};
\node[label={[label distance=-2mm]$2$}] at (0,0) {$\scriptstyle\otimes$};
\node[label={[label distance=-2mm]0:$4$}] at (1,0) {$\bullet$};
\node[label={[label distance=-2mm]$1$}] at (0,1) {$\scriptstyle\otimes$};
\end{tikzpicture} & \begin{tikzpicture}
\node at (-1,0) {$\bullet$};
\node at (0,0) {$\scriptstyle\otimes$};
\node at (1,0) {$\bullet$};
\node at (0,1) {$\circ$};
\end{tikzpicture} & \begin{tikzpicture}
\node at (-1,0) {$\bullet$};
\node at (0,0) {$\circ$};
\node at (1,0) {$\bullet$};
\node at (0,1) {$\scriptstyle\otimes$};
\end{tikzpicture} & \begin{tikzpicture}
\node at (-1,0) {$\circ$};
\node at (0,0) {$\scriptstyle\otimes$};
\node at (1,0) {$\bullet$};
\node at (0,1) {$\scriptstyle\otimes$};
\end{tikzpicture}\tabularnewline
$\{\{1,3\},\{1,4\},\{2,3,4\}\}$ & $\{\{3\},\{4\}\}$ & $\{\{3,4\}\}$ & $\{\{1\},\{2,4\}\}$\tabularnewline
& & & \tabularnewline
\includegraphics[scale=0.5]{delNone} & \includegraphics[scale=0.5]{del1} & \includegraphics[scale=0.5]{del2} & \includegraphics[scale=0.5]{del3}\tabularnewline
\end{tabular}
\caption{\label{fig:deleted}Structure diagram of the achievement game on an
affine convex geometry and some of its deleted convex geometries.
The second row contains the maximally non-generating sets. Deleted
points are shown with $\circ$ and the elements of the goal set are
shown with ${\scriptstyle \otimes}$. }
\end{figure}
\begin{example}
Figure~\ref{fig:deleted} shows the structure diagrams of the achievement
game on an affine convex geometry and the effect of deletion on the
structure diagrams. Note that deletion of a point produces a structure
diagram that is a sub-diagram of the original diagram. This sub-diagram
starts at an option of $X_{\Phi}$ and it has reversed parities and
nim numbers.
\end{example}
\section{Winning subsets containing only extreme points \label{sec:extremeCharacterizations}}
In this section we characterize the nim number of $\GEN(S,W)$ where
$W\subseteq\Ex(S)$.
\begin{prop}
\label{prop:maxConvex}If $W\subseteq\Ex(S)$ then the set $\mathcal{M}$
of maximally non-generating subsets of $\GEN(S,W)$ is $\{M_{v}\mid v\in W\}$,
where $M_{v}:=S\setminus\{v\}$.
\end{prop}
\begin{proof}
We show that $M_{v}$ for $v\in W$ is a maximally non-generating
set. Since $v$ is an extreme point of $S$, $v\not\in\tau(M_{v})$.
So $M_{v}$ is a non-generating set. If $M_{v}\subset N\subseteq S$
then $N=S$ is a generating set. So $M_{v}$ is maximally non-generating.
We show that every maximally non-generating set is $M_{v}$ for some
$v\in W$. Let $M$ be a maximally non-generating set. Since $M$
is non-generating, there must be a $v$ in $W\setminus M$. We clearly
have $M\subseteq M_{v}$, which means $M_{v}$ a non-generating superset
of $M$. Thus $M=M_{v}$ since $M$ is maximally non-generating.
\end{proof}
\begin{cor}
\label{cor:intersectSub}If $W\subseteq\Ex(S)$ then the set of intersection
subsets of $\GEN(S,W)$ is $\mathcal{I}=\{S\setminus V\mid V\subseteq W\}$.
\end{cor}
\begin{prop}
If $W=\{w\}\subseteq\Ex(S)$, then
\[
\nim(\GEN(S,W))=\begin{cases}
1, & \pty(S)=1\\
2, & \pty(S)=0.
\end{cases}
\]
\end{prop}
\begin{proof}
We have $\mathcal{M}=\{M\}$ and $\mathcal{I}=\{M,S\}$, where $M=S\setminus\{w\}$.
If $|S|$ is odd, then $\nim(\GEN(S,W))=1$ since $M$ is even and
the structure diagram is the one shown in Figure~\ref{fig:diagGenSW}(c).
If $|S|$ is even, then $\nim(\GEN(S))=2$ since $M$ is odd and the
structure diagram is the one shown in Figure~\ref{fig:diagGenSW}(a).
\end{proof}
\begin{figure}
\begin{tabular}{ccccccc}
\includegraphics[scale=0.5]{D1o} & & \includegraphics[scale=0.5]{hullC4I0_gen_dot} & & \includegraphics[scale=0.5]{D1e} & & \includegraphics[scale=0.5]{hullC4I1_gen_dot}\tabularnewline
\multicolumn{3}{c}{$|S|$ is even} & $\quad$ & \multicolumn{3}{c}{$|S|$ is odd}\tabularnewline
$|W|=1$ & & $|W|=4$ & & $|W|=1$ & & $|W|=4$\tabularnewline
(a) & & (b) & & (c) & & (d)\tabularnewline
\end{tabular}
\caption{\label{fig:diagGenSW}Orbit quotient structure diagrams for $\protect\GEN(S,W)$,
where $W\subseteq\protect\Ex(S)$.}
\end{figure}
\begin{defn}
Let $W\subseteq\Ex(S)$. For $I\in\mathcal{I}$ we define $\delta(I):=|S\setminus I|$
to be the number of goal points missing from $I$.
\end{defn}
\begin{prop}
If $W\subseteq\Ex(S)$ with $|W|\ge2$, then
\[
\nim(\GEN(S,W))=\begin{cases}
0, & \pty(S)=0\\
1, & \pty(S)=1.
\end{cases}
\]
\end{prop}
\begin{proof}
First consider the case when $|S|$ is even. We are going to use type
calculus and structural induction on the elements of $\mathcal{I}$
to show that
\[
\type(X_{I})=\begin{cases}
(0,0,0), & \delta(I)=0\\
(1,2,1), & \delta(I)=1\\
(0,0,1), & \delta(I)>1\text{ and }\delta(I)\equiv_{2}0\\
(1,0,1), & \delta(I)>1\text{ and }\delta(I)\equiv_{2}1
\end{cases}
\]
for all $I\in\mathcal{I}$, as shown in Figure~\ref{fig:diagGenSW}(b).
Then we will have
\[
\nim(\GEN(S,W))=\nim(\emptyset)=\nim_{0}(X_{\Phi})=0
\]
because $\delta(\Phi)=|W|\geq2$. If $\delta(I)=0$ then $X_{I}=X_{S}$
and $\type(X_{S})=(0,0,0)$. If $\delta(I)=1$ then $I=S\setminus\{w\}$
for some $w\in W$, so $\pty(I)=1-\pty(S)=1$. Then $\nim(I)=\mex(\nim(\Opt(I)))=\mex(\nim(S))=\mex\{0\}=1$.
Furthermore, for any $P\in X_{I}$ satisfying $\pty(P)=0$, we have
$\nim(P)=\mex(\nim(\Opt(P)))=\mex\{0,1\}=2$. Hence $\type(X_{I})=(1,2,1)$.
Suppose $\delta(I)\ge2$. We have two cases to consider. If $\pty(I)=0$
then $\delta(I)=|S\setminus I|\equiv_{2}0$, and so $\delta(J)=|S\setminus J|\equiv_{2}1$
with $\type(J)\in\{(1,2,1),(1,0,1)\}$ for all $J\in\Opt(I)$ by induction.
Hence
\begin{align*}
\type(X_{I}) & =(0,\nim_{0}(X_{I}),\nim_{1}(X_{I}))\\
& =(0,\mex(\nim_{1}(\Opt(I))),\mex(\nim_{0}(\Opt(I)))\cup\{\nim_{0}(X_{I})\})\\
& =(0,\mex(\{1\}),\mex(A))=(0,0,1),
\end{align*}
where $A=\{0,2\}$ or $A=\{0\}$.
If $\pty(I)=1$ then $\delta(I)\equiv_{2}1$, and so $\delta(J)=|S\setminus J|\equiv_{2}0$
with $\type(J)=(0,0,1)$ for all $J\in\Opt(I)$ by induction. Hence
\begin{align*}
\type(X_{I}) & =(1,\nim_{0}(X_{I}),\nim_{1}(X_{I}))\\
& =(1,\mex(\nim_{1}(\Opt(I))\cup\{\nim_{1}(X_{I})\}),\mex(\nim_{0}(\Opt(I))))\\
& =(1,\mex(\{1\}),\mex(\{0\}))=(1,0,1).
\end{align*}
Now consider the case when $|S|$ is odd. One can show that
\[
\type(X_{I})=\begin{cases}
(1,0,0), & \delta(I)=0\\
(0,1,2), & \delta(I)=1\\
(1,1,0), & \delta(I)>1\text{ and }\delta(I)\equiv_{2}1\\
(0,1,0), & \delta(I)>1\text{ and }\delta(I)\equiv_{2}0
\end{cases}
\]
for all $I\in\mathcal{I}$, as show in Figure~\ref{fig:diagGenSW}(d).
The argument is essentially the same as the one we used in the previous
case but with reversed parities. Hence
\[
\nim(\GEN(S,W))=\nim(\emptyset)=\nim_{0}(X_{\Phi})=1
\]
because $\delta(\Phi)=|W|\geq2$.
\end{proof}
The essence of the previous proof is captured in Figures~\ref{fig:diagGenSW}(b)
and (d). The parity of $|S|$ determines the direction of the triangles,
while $|W|$ determines the height of the diagram.
\section{Convex geometries from vertices of trees\label{sec:treeCharacterization}}
Consider a tree graph $T$ with vertex set $S$. The vertex sets of
connected subgraphs of $T$ form a convex geometry on $S$. We call
this the \emph{vertex geometry} of $T$. In this section we study
the achievement game $\GEN(S,W)$ on vertex geometries of trees.
We use the standard $N(v)$ notation for the set of vertices adjacent
to a vertex $v$.
\begin{example}
The vertex geometry of the path graph $P_{3}$ with vertex set $S=\{1,2,3\}$
is $\mathcal{K}=2^{S}\setminus\{\{1,3\}\}$.
\end{example}
\begin{figure}
\begin{centering}
\begin{tabular}{cc}
\begin{tabular}{c}
\includegraphics{K13}\tabularnewline
\end{tabular} & %
\begin{tabular}{c}
\includegraphics[scale=1.3]{K13affine}\tabularnewline
\end{tabular}\tabularnewline
\end{tabular}
\par\end{centering}
\caption{\label{fig:K1,3}The complete bipartite graph $K_{1,3}$ and the vertex
geometry of $K_{1,3}$ represented as a deletion of an affine convex
geometry.}
\end{figure}
\begin{example}
Let $R:=\{{\bf 0,i,j,k}\}$ and $D:=\{{\bf -i-j},{\bf -i-k},{\bf -j-k}\}$
be subsets of $\mathbb{R}^{3}$. The vertex geometry of the complete
bipartite graph $K_{1,3}$ can be represented as a deletion of the
affine convex geometry on $S=R\cup D$ by $D$. The graph $K_{1,3}$
and the point set $S$ are shown in Figure~\ref{fig:K1,3}. The set
of convex sets is $\mathcal{K}=2^{R}\setminus\{\{{\bf i},{\bf j}\},\{{\bf j},{\bf k}\},\{{\bf i},{\bf k}\},\{\mathbf{i},\mathbf{j},\mathbf{k}\}\}$.
\end{example}
Removing a vertex $w$ of a tree $T$ creates a forest that we denote
by $T\setminus w$.
\begin{prop}
Let $(S,\mathcal{K})$ be the vertex geometry of a tree $T$ and $W\subseteq S$.
Then $w\in\Ex(W)$ if and only if the elements of $W\setminus\{w\}$
are vertices in a single connected component of $T\setminus w$.
\end{prop}
\begin{proof}
For each $v\in N(w)$ let $V_{v}$ be the vertex set of the connected
component of $T\setminus w$ containing $v$. The map $v\mapsto V_{v}$
is a bijection from $N(w)$ to the collection of vertex sets of the
connected components of $T\setminus w$.
First assume $w_{1}$ and $w_{2}$ are elements of $W\setminus\{w\}$
that are vertices in different connected components of $T\setminus w$.
Then $w$ is a vertex of every connected subgraph of $T$ that contains
both $w_{1}$ and $w_{2}$. Hence $w\in\tau(\{w_{1},w_{2}\})\subseteq\tau(W\setminus\{w\})$,
so $w\not\in\Ex(W)$.
Now assume $W\setminus\{w\}$ is contained in the vertex set of a
single connected component $V_{v}$ of $T\setminus w$. Since $V_{v}$
is convex and contains $W\setminus\{w\}$, $\tau(W\setminus\{w\})\subseteq V_{v}$.
So $w\not\in\tau(W\setminus\{w\})$ since $w\not\in V_{v}$. Thus
$w\in\Ex(W)$.
\end{proof}
\subsection{Winning subsets with more than one vertex}
First we consider the case when $W$ has at least two vertices. If
$w\in\Ex(W)$ then $T\setminus w$ has exactly one component that
contains some elements of $W$. We denote the vertex set of this component
by $M_{w}$. It is clear from the construction that $w\mapsto M_{w}:\Ex(W)\to\mathcal{M}$
is a bijection. We use the notation $V_{w}:=S\setminus M_{w}$. Note
that $\{V_{w}\mid w\in\Ex(W)\}$ is a collection of pairwise disjoint
sets.
There is a bijective correspondence between the subsets of $\Ex(W)$
and the intersection subsets given by $A\mapsto M_{A}:2^{\Ex(W)}\to\mathcal{I}$,
where
\[
M_{A}:=\bigcap\{M_{w}\mid w\in A\}=S\setminus\bigcup\{V_{w}\mid w\in A\}.
\]
Note that $S=M_{\emptyset}$ and $\Phi=M_{\Ex(W)}$. We define the
\emph{deficiency} of a structure class $M_{A}$ by $\delta(M_{A}):=|A|$.
We also define the \emph{signature} of a structure class $X_{M_{A}}$
to be $\sigma(X_{M_{A}}):=(e,o)$, where $e$ is the size of $\{a\in A\mid\pty(V_{a})=0\}$
and $o$ is the size of $\{a\in A\mid\pty(V_{a})=1\}$. The signature
satisfies $\delta(M_{A})=e+o$. The \emph{signature of the game} is
$\sigma(X_{\Phi})$.
\begin{example}
\label{exa:tree}
\begin{figure}
\begin{tabular}{ccc}
\raisebox{1cm}{\includegraphics{graph}} & \begin{tikzpicture}[yscale=.71]
\node (M) at (0,0) {$M_\emptyset$};
\node (M6) at (-1.5,2) {$M_{\{3\}}$};
\node (M3) at (0,2) {$M_{\{8\}}$};
\node (M8) at (1.5,2) {$M_{\{6\}}$};
\node (M36) at (-1.5,4) {$M_{\{3,8\}}$};
\node (M68) at (0,4) {$M_{\{3,6\}}$};
\node (M38) at (1.5,4) {$M_{\{6,8\}}$};
\node (M368) at (0,6) {$M_{\{3,6,8\}}$};
\draw (M) -- (M6);
\draw (M) -- (M3);
\draw (M) -- (M8);
\draw (M36) -- (M6);
\draw (M36) -- (M3);
\draw (M38) -- (M3);
\draw (M38) -- (M8);
\draw (M68) -- (M6);
\draw (M68) -- (M8);
\draw (M368) -- (M36);
\draw (M368) -- (M68);
\draw (M368) -- (M38);
\end{tikzpicture} & \includegraphics[scale=0.5]{graphGen}\tabularnewline
\end{tabular}
\caption{\label{fig:tree}Tree graph, lattice of intersection subsets, and
structure diagram of $\protect\GEN(S,W)$ with $W=\{1,3,6,8\}$. Dotted
arrows correspond to a signature change of $(0,-1)$, while solid
arrows correspond to a signature change of $(-1,0)$. The label $a$
on an arrow indicates the change from $M_{A}$ to $M_{A\setminus\{a\}}$.}
\end{figure}
Figure~\ref{fig:tree} shows a tree graph with vertex set $S=\{0,\ldots,11\}$,
the lattice of intersection subsets, and the structure diagram of
$\GEN(S,W)$ with winning set $W=\{1,3,6,8\}$. The set of extreme
points of $W$ is $\Ex(W)=\{3,6,8\}$. The set of maximal non-generating
sets is $\mathcal{M}=\{M_{3},M_{6},M_{8}\}$ with $V_{3}=\{3,4,5\}$,
$V_{6}=\{6,7\}$, and $V_{8}=\{8,9,10,11\}$, so that the Frattini
subset is $\Phi=M_{\{3,6,8\}}=\{0,1,2\}$. Any directed path in the
structure diagram from $X_{M_{A}}$ to $X_{S}=X_{M_{\emptyset}}$
corresponds to the elements of $A$. The signature of $X_{M_{A}}$
can be computed by counting the solid and the dotted arrows along
any such directed path. For example $\sigma(X_{M_{\{3,8\}}})=(1,1)$.
Any directed path from $X_{\Phi}$ to $X_{S}$ contains one dotted
and two solid arrows since the signature of the game is $\sigma(X_{\Phi})=(2,1)$.
\end{example}
\begin{prop}
\label{prop:vertexMore}Let $(S,\mathcal{K})$ be the vertex geometry
of a tree $T$ and $W$ be a subset of $S$ with $|W|\ge2$. If the
signature of the game is $(e,o)$, then
\[
\nim(\GEN(S,W))=\begin{cases}
1, & (e,o)=(1,0)\\
2, & (e,o)\in\{(0,1),(1,2)\}\\
3, & (e,o)\in\{(1,1),(2,1)\}\\
0, & \text{otherwise}
\end{cases}
\]
when $\pty(S)=0$, and
\[
\nim(\GEN(S,W))=\begin{cases}
0, & (e,o)\in\{(0,0),(1,1)\}\\
2, & (e,o)\in\{(1,0),(2,0)\}\\
1, & \text{otherwise}
\end{cases}
\]
when $\pty(S)=1$.
\end{prop}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[scale=0.5]{even} & \includegraphics[scale=0.5]{odd}\tabularnewline
\end{tabular}
\caption{\label{fig:structInd}Type calculus computation of the possible structure
class types with deficiency less than 5. A solid arrow represents
no change in parity while a dotted arrow represents a parity change.}
\end{figure}
\begin{proof}
Let $X_{J}$ be an option of $X_{I}$ and $I=M_{A}$. Then $I\cup\{v\}\in X_{J}$
for some $v\in J\setminus I$. So there is a unique $a\in A$ such
that $v\in V_{a}$ and $J=M_{A\setminus\{a\}}$. So every arrow of
the structure diagram corresponds to a unique element $a$ of $\Ex(W)$.
This means $\delta(J)=|A\setminus\{a\}|=|A|-1=\delta(I)-1$. Roughly
speaking this tells us that there are no arrows between structure
classes that are at the same directed distance from $X_{S}$.
Assume $I=M_{A}$ and $\sigma(X_{I})=(e,o)$. If $e,o\ge1$ then there
are $a,b\in A$ such that $\pty(V_{a})=0$ and $\pty(V_{b})=1$. Hence
\[
\sigma(\Opt(X_{I}))=\begin{cases}
\{(e-1,0)\}, & o=0\\
\{(0,o-1)\}, & e=0\\
\{(e-1,o),(e,o-1)\}, & e,o\ge1.
\end{cases}
\]
The result now follows from type calculus and structural induction
on the structure classes. The details of the type calculus computation
are shown in Figure~\ref{fig:structInd}. The signature of a structure
class is $(e,o)$ where $e$ is the number of solid arrows and $o$
is the number of dotted arrows from the structure class to the bottom
structure class along a directed path. Starting at deficiency $4$
the set of possible types remains fixed by induction.
\end{proof}
\subsection{Winning subsets with only one vertex}
Now we consider the case when $W=\{w\}$ is a singleton set. For each
$v$ in the set $N(w)$ of neighbors of $w$, there is a unique connected
component of $T\setminus w$ that contains $v$. The vertex set $M_{v}$
of this component is a maximal non-generating set. In fact, $\mathcal{M}=\{M_{v}\mid v\in N(w)\}$.
Note that $\mathcal{M}$ is a collection of disjoint subsets. The
\emph{signature} of the game is $(e,o)$, where $e$ is the size of
$\{v\in N(w)\mid\pty(M_{v})=0\}$ and $o$ is the size of $\{v\in N(w)\mid\pty(M_{v})=1\}$.
\begin{example}
\begin{figure}
\begin{tabular}{ccc}
\raisebox{.3cm}{\includegraphics{graph2}} & ~ & \includegraphics[scale=0.5]{graphGen2}\tabularnewline
\end{tabular}
\caption{\label{fig:tree2}Tree graph and structure diagram of $\protect\GEN(S,W)$
with $W=\{0\}$.}
\end{figure}
Figure~\ref{fig:tree2} shows a tree graph with vertex set $S=\{0,\ldots,9\}$
and the structure diagram of $\GEN(S,W)$ with winning set $W=\{0\}$.
Some of the maximal non-generating sets are $M_{1}=\{1,2,3\}$ and
$M_{5}=\{5\}$, so that the Frattini subset is $\Phi=\emptyset$.
The signature of the game is $(e,o)=(2,3)$.
\end{example}
\begin{figure}
\setlength{\tabcolsep}{10pt}
\begin{tabular}{ccccc}
\includegraphics[scale=0.5]{D1e} & \includegraphics[scale=0.5]{1V20} & \includegraphics[scale=0.5]{D1o} & \includegraphics[scale=0.5]{1V02} & \includegraphics[scale=0.5]{1V11}\tabularnewline
$e\le1$ & $e\ge2$ & $e=0$ & $e=0$ & $e\ge1$\tabularnewline
$o=0$ & $o=0$ & $o=1$ & $o\ge2$ & $o\ge1$\tabularnewline
(a) & (b) & (c) & (d) & (e)\tabularnewline
& & & & \tabularnewline
\end{tabular}
\caption{\label{fig:TreeV1}Orbit quotient structure diagrams for $\protect\GEN(S,\{w\})$
with vertex geometry. The oval shape in subfigures (d) and (e) indicates
that the parity of the structure class could be either $0$ or $1$.}
\end{figure}
\begin{prop}
\label{prop:vertexOne}Let $(S,\mathcal{K})$ be the vertex geometry
of a tree $T$ and $W=\{w\}\subseteq S$. If the signature of the
game is $(e,o)$, then
\[
\nim(\GEN(S,W))=\begin{cases}
1, & o=0\\
2, & e=0\text{ and }o\ge1\\
3, & e,o\ge1.
\end{cases}
\]
\end{prop}
\begin{proof}
Since $\emptyset\in X_{\Phi}$ and $\{w\}\in X_{S}$, $X_{S}\in\Opt(X_{\Phi})$.
Assume $o=0$, so that $\pty(S)=1$. If $e=0$ then the nim number
of the game is $1$ as shown in Figure~\ref{fig:TreeV1}(a) since
$S=\{w\}$, $\mathcal{M}=\{\emptyset\}$, and $\mathcal{I}=\{\emptyset,S\}$.
If $e=1$ then $N(w)=\{v\}$, $\mathcal{M}=\{M_{v}\}$, and $\mathcal{I}=\{M_{v},S\}$.
Hence the structure diagram is shown in Figure~\ref{fig:TreeV1}(a)
and the nim number of the game is $1$. If $e\ge2$ then $\Phi=\emptyset$
and $\mathcal{I}=\{\emptyset,S\}\cup\{M_{v}\mid v\in N(w)\}$, so
the orbit quotient structure diagram is shown in Figure~\ref{fig:TreeV1}(b)
and the nim number of the game is $1$.
Assume $e=0$ and $o\ge1$. If $o=1$ then $N(w)=\{v\}$, $\mathcal{M}=\{M_{v}\}$,
and $\mathcal{I}=\{M_{v},S\}$, so the structure diagram is shown
in Figure~\ref{fig:TreeV1}(c) and the nim number of the game is
$2$. If $o\ge2$ then $\Phi=\emptyset$ and $\mathcal{I}=\{\emptyset,S\}\cup\{M_{v}\mid v\in N(w)\}$,
so the orbit quotient structure diagram is shown in Figure~\ref{fig:TreeV1}(d)
and the nim number of the game is $2$.
Assume $e,o\ge1$, so that $\Phi=\emptyset$ and $\mathcal{I}=\{\emptyset,S\}\cup\{M_{v}\mid v\in N(w)\}$.
So the orbit quotient structure diagram is shown in Figure~\ref{fig:TreeV1}(e)
and the nim number of the game is $3$.
\end{proof}
\begin{cor}
If $(S,\mathcal{K})$ is the vertex geometry of a tree $T$, then
$\nim(\GEN(S,W))\in\{0,1,2,3\}$.
\end{cor}
\section{Affine convex geometries in $\mathbb{R}$\label{sec:affineCharacterization}}
In this section we study the games played on affine convex geometries
in $\mathbb{R}$. It is easy to see that if $S=\{s_{1},\ldots,s_{n}\}\subseteq\mathbb{R}$
then the affine convex geometry on $S$ is isomorphic to the vertex
geometry of a path graph $P_{n}$ with $n$ vertices.
\begin{prop}
Let $S=\{s_{1},\ldots,s_{n}\}\subseteq\mathbb{R}$ such that $s_{i}<s_{i+1}$
for all $i$ and $W=\{s_{i_{1}},\ldots,s_{i_{k}}\}$ such that $i_{1}<i_{2}<\cdots<i_{k}$
and $k\ge2$. Then
\[
\nim(\GEN(S,W))=\begin{cases}
0, & i_{1}\equiv_{2}i_{k}+1\\
1, & i_{1}\equiv_{2}1\equiv_{2}i_{k}\text{ and }n\equiv_{2}1\\
2, & i_{1}\equiv_{2}0\equiv_{2}i_{k}\text{ and }n\equiv_{2}1\\
3, & i_{1}\equiv_{2}i_{k}\text{ and }n\equiv_{2}0.
\end{cases}
\]
\end{prop}
\begin{proof}
We consider the equivalent game on the vertex geometry of the path
graph $P_{n}$ with vertex set $\{1,2,\ldots,n\}$ and $W=\{i_{1},i_{2},\ldots,i_{k}\}$.
The set of extreme points of the winning set is $\Ex(W)=\{i_{1},i_{k}\}$.
We have $|V_{i_{1}}|=i_{1}$ and $|V_{i_{k}}|=n-i_{k}+1$. So the
signature of this game is
\[
\sigma(X_{\Phi})=\begin{cases}
(1,1), & i_{1}\equiv_{2}i_{k}+1\text{ and }n\equiv_{2}1\\
(2,0), & i_{1}\equiv_{2}0\equiv_{2}i_{k}+1\text{ and }n\equiv_{2}0\\
(0,2), & i_{1}\equiv_{2}1\equiv_{2}i_{k}+1\text{ and }n\equiv_{2}0\\
(0,2), & i_{1}\equiv_{2}1\equiv_{2}i_{k}\text{ and }n\equiv_{2}1\\
(2,0), & i_{1}\equiv_{2}0\equiv_{2}i_{k}\text{ and }n\equiv_{2}1\\
(1,1), & i_{1}\equiv_{2}i_{k}\text{ and }n\equiv_{2}0.
\end{cases}
\]
The result now follows from Proposition~\ref{prop:vertexMore}.
\end{proof}
\begin{prop}
Let $S=\{s_{1},\ldots,s_{n}\}\subseteq\mathbb{R}$ such that $s_{i}<s_{i+1}$
for all $i$ and $W=\{s_{k}\}$. Then
\[
\nim(\GEN(S,W))=\begin{cases}
1, & n\equiv_{2}1\equiv_{2}k\\
2, & n\equiv_{2}1\equiv_{2}k+1\\
2, & n\equiv_{2}0\text{ and }k\in\{1,n\}\\
3, & n\equiv_{2}0\text{ and }k\in\{2,\ldots,n-1\}.
\end{cases}
\]
\end{prop}
\begin{proof}
We consider the equivalent game on the vertex geometry of the path
graph $P_{n}$ with vertex set $V=\{1,2,\ldots,n\}$ and $W=\{k\}$.
The neighborhood of $k$ is $N(k)=\{k-1,k+1\}\cap V$. So the signature
of this game is
\[
\sigma(X_{\Phi})=\begin{cases}
(0,0), & n=1=k\\
(1,0), & n\equiv_{2}1\equiv_{2}k\text{ and }k\in\{1,n\}\\
(2,0), & n\equiv_{2}1\equiv_{2}k\text{ and }k\in\{2,\ldots,n-1\}\\
(0,2), & n\equiv_{2}1\equiv_{2}k+1\\
(0,1), & n\equiv_{2}0\text{ and }k\in\{1,n\}\\
(1,1), & n\equiv_{2}0\text{ and }k\in\{2,\ldots,n-1\}.
\end{cases}
\]
The result now follows from Proposition~\ref{prop:vertexOne}.
\end{proof}
Deleted affine convex geometries can be solved using the techniques
of this section.
\begin{example}
Consider the deleted affine convex geometry on the set $S=\{0,1,\ldots,6\}$
with deleted set $D=\{3,5\}$. The achievement game with winning set
$W=\{2,6\}$ is equivalent to $\GEN(S\setminus D,W)$, so it has nim
number $1$. The achievement game with winning set $W=\{1,2\}$ is
equivalent to $\GEN(\{1,2,4,6\},\{1\})$, so it has nim number $2$.
\end{example}
\section{\label{sec:FurtherQuestions}Further directions}
\begin{figure}
\begin{tikzpicture}[scale=.4]
\node at (-4,-3) {$\bullet$};
\node at (-4,0) {$\bullet$};
\node at (-4,4) {$\bullet$};
\node at (-3,-4) {$\bullet$};
\node at (-3,-1) {$\bullet$};
\node at (-2,2) {$\bullet$};
\node at (-1,-3) {$\bullet$};
\node at (0,0) {$\scriptscriptstyle\otimes$};
\node at (0,1) {$\bullet$};
\node at (0,2) {$\bullet$};
\node at (2,-4) {$\bullet$};
\node at (2,2) {$\bullet$};
\node at (3,-3) {$\bullet$};
\end{tikzpicture}
\caption{\label{fig:nim6}Affine convex geometry with $\protect\nim(\protect\GEN(S,\{w\}))=6$.
The winning point $w$ is indicated by $\otimes$.}
\end{figure}
We provide some comments and propose some questions for further study.
\begin{enumerate}
\item The conjecture in Remark~\ref{rem:repConj} probably can be proved
by adjusting the approach of \cite{affineRep}.
\item Our results may suggest that the nim number of $\GEN(S,W)$ is in
the set $\{0,1,2,3\}$. The point set in Figure~\ref{fig:nim6} provides
an example in $\mathbb{R}^{2}$ where the nim number is $6$. The
companion web site \cite{WEB} provides an example in $\mathbb{R}^{3}$
where the nim number is $8$. What are the possible nim numbers for
convex geometries? What are the possible nim numbers for affine convex
geometries? Does the answer depend on the dimension of the space?
\item The definition of vertex geometry can be generalized to forest graphs.
Our results might generalize to this setting.
\item Consider a tree graph $T$ with edge set $S$. The edge sets of connected
subgraphs of $T$ form a convex geometry on $S$. What is the nim
number of $\GEN(S,W)$ played on these \emph{edge geometries}?
\item There are several ways to build a convex geometry from a partially
ordered set. What can we say about $\GEN(S,W)$ played on these convex
geometries?
\item What can we say about $\GEN(S,W)$, where $W$ is the Frattini set
of $\GEN(S)$?
\item The original games introduced by \cite{anderson.harary:achievement}
and played on a group can be generalized by allowing a winning set
that is a subset of the group. This may produce interesting results
for special subgroups as the winning set.
\item It might be easier to determine the possible nim values of the avoidance
game $\DNG(S,W)$ played on convex geometries. The avoidance version
was also introduced by \cite{anderson.harary:achievement}. In this
game it is not allowed to select an element that creates a set whose
convex closure contains the winning set. In the group version of the
game, Lagrange's Theorem is a powerful restriction on the possible
structure diagrams \cite{BESspectrumArxive}. Without this restriction,
the spectrum of nim numbers in the convex geometry version could be
all nonnegative integers.
\item Does the Tutte polynomial of the anti-matroid associated with a convex
geometry have any information about $\GEN(S,W)$? Some connection
is expected since both deletion and contraction on a convex geometry
have meaningful game theoretic interpretations.
\end{enumerate}
\section*{}
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,026
|
The show committee is urgently seeking a new secretary, who is willing and able to co-ordinate all aspects of staging the show, which is held each year in September. This includes arranging & documenting meetings, publicity, attracting sponsorship, booking judges, producing the show schedule and co-ordinating events on show day. No experience necessary, just enthusiasm, good organisation and a desire to help stage this very popular event.
Please contact Barbara on 07740 698844.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,595
|
La Clase Durandal fue la primera clase de verdaderos contratorpederos construida para la Marina Nacional Francesa, entre los años 1899 y 1900.
Los buques y su concepción
Cuatro buques formaban la clase: Durandal, Espingole, Fauconneau y Hallebarde; Todos ellos fueron construidos en los astilleros Augustin Normand de Le Havre. Dos unidades fueron botadas en 1899 y las otras en 1900.
Esta clase de contratorpederos era semejante a la Clase Havock británica.
Su cubierta principal era del tipo turtledeck (cubierta de tortuga), coronada de una cubierta volante de madera para circular.
Historia operacional
El Espingole, estando en servicio en la Escuadra del Mediterráneo, se hundió en la rada de Hyères, el 4 de febrero de 1903, tras haber chocado contra una roca, cerca del Cabo Lardier.
El Durandal, el Fauconneau y el Hallebarde participaron en la Primera Guerra Mundial, durante la cual, los buques, sirvieron en diferentes frentes.
Los tres contratorpederos sobrevivieron a la Gran Guerra. Fueron dados de baja entre 1919 y 1921, y desguazados en Cherbourg y Tolón.
Los buques
Enlaces internos
Anexo:Clases de destructores
Enlaces externos
(en inglés) classe Durandal (site battleships.cruisers.co.uk)
FRA Durandal- site navalstory.flixco.info (caractéristiques techniques)
French destroyers - site naval-histoty.net
Type Durandal - site page 14-18
Referencias
Conway's All the World's Fighting Ships (1860-1905)
Durandal
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section{Introduction}
Side-channel attacks exploit information leakage through physical behavior of computing devices. The physical behavior depends on the processed data. The resulting data-dependent patterns in physical signals such as power consumption, electromagnetic emanations or timing behavior can be analyzed to extract secrets such as cryptographic keys~\cite{kocher1996timing,brumley2005remote,mangard2008power,genkin2015stealing}. Despite the physical proximity requirement for most physical attacks, there exist remotely exploitable side channels such as
microarchitectural attacks~\cite{ge2016survey}.
Microarchitectural attacks exploit shared hardware features such as cache~\cite{bernstein2005cache,percival2005cache,osvik2006cache}, branch prediction unit (BPU)~\cite{aciiccmez2007power}, memory order buffer (MOB)~\cite{moghimi2017memjam} and speculative execution engine~\cite{Kocher2018spectre} to extract secrets from a process executed \emph{on the same system}. These attacks can be mounted remotely or locally on systems where untrusted entities can execute code on a shared hardware, either because the system is shared or untrusted code is executed. Scenarios include but are not limited to cross-VM attacks in the cloud environment~\cite{irazoqui2015s,liu2015last}, drive-by JavaScript trojans inside the browser sandbox~\cite{lipp2017practical}, attacks originating from untrusted mobile applications~\cite{lipp2016armageddon} and system-adversarial attacks against Intel Software Guard eXtensions (SGX)~\cite{moghimi2017cachezoom,van2017telling}. Microarchitectural leakage can be used to break software implementations of cryptographic schemes where the adversaries recover the secret key by combining the leaked partial information from key-dependent activities ~\cite{pereida2016make,yarom2014recovering,benger2014ooh}. These side channels can be further exploited to violate user's privacy through activity profiling~\cite{gulmezoglu2017perfweb}, or to steal user's keystrokes~\cite{gruss2015cache}. Memory protections such as Address Space Layout Randomization (ASLR) can be bypassed by exploiting microarchitectural side-channel leakages~\cite{evtyushkin2016jump}.
Defense against microarchitectural side channels have been proposed based on new hardware design~\cite{costan2016sanctum,kayaalp2017ric}, systematic mitigation~\cite{liu2016catalyst} and activity monitoring~\cite{zhang2016cloudradar,briongos2018cacheshield}. However, the most widely-used protection against microarchitectural leakage is software hardening using constant-time programming techniques~\cite{hamburg2009accelerating,brickell2006mitigating}. In this context, constant-time programming implies using microarchitectural resources in a secret-independent fashion. Therefore, timing, or trace-based leakages~\cite{disselkoen2017prime} in the hardware would not reveal any information about the secret. These techniques depend on the underlying microarchitecture and side-channel knowledge, i.e. software implementations are hardened to follow a constant-time behavior based on published attacks on the target microarchitecture. Consequently, a novel microarchitectural attack demands new changes to these software protections. While true constant-time code avoids such problems, manual verification of the software implementation for constant-time behavior is an error-prone task, and it requires extensive, and ever growing knowledge of side channels. Besides, what we observe in the source code is not always what is executed on the processor~\cite{simon2018you}, and there are leakages in the program binary that remain unobserved in the source code~\cite{kaufmann2016constant}. The state of art tools and techniques for automated finding of side-channel leakages in software binaries fall short in practice, particularly when the source code is not available. As a result, commercial cryptographic products such as \emph{Microsoft Cryptography API Next Generation (CNG)}, which is used everyday by millions of users, have never been externally audited for side-channel security.
\subsection{Our Contribution}
We propose a leakage detection technique, and develop a framework named \tool~to locate leakages within software binaries. We apply \tool~to analyze two commercial closed-source cryptographic libraries hardened toward constant-time protections and report previously unknown vulnerabilities, in summary:
\begin{itemize}
\item We propose a technique based on Dynamic Binary Instrumentation (DBI) and Mutual Information (MI) Analysis to locate memory based and control-flow based microarchitectural leakages in software binaries.
\item We develop the \tool~framework to perform automated leakage testing and quantification based on our technique. Our framework can be extended to locate other and new types of microarchitectural leakages.
\item We demonstrate the ease-of-use of \tool~by showing how it significantly eases the analysis of binary code even in cases where source code is not accessible to the analyst.
\item We apply \tool~to cryptographic schemes implemented in \emph{Microsoft CNG} and \emph{Intel IPP}, which are both widely used, yet closed source crypto libraries. Our results include previously unknown leakages in these libraries.
\item We perform analysis and quantification of the critical leakages, and discuss the security impact of these leakages on the relevant cryptographic schemes.
\end{itemize}
\subsection{Analysis Setup and Targeted Software}
Our machine for analysis is a Dell XPS 8920 machine with Intel(R) Core i7-7700 processor, 16\,GB of RAM and a traditional hard disk drive running \emph{Microsoft Windows 10}. The \tool~Framework uses \emph{Pin} v3.6 as the DBI backend, and \emph{IDA Pro} v6.95 for binary visualization and leakage analysis. The tested cryptographic modules are \emph{Microsoft bcryptprimitives.dll} v10.0.17134.1 as part of \emph{Microsoft CNG}, and \emph{Intel IPP} v2018.2.185.
\section{Background}
\subsection{Dynamic Binary Instrumentation}
Dynamic program analysis is more accurate compared to static analysis due to availability of real system states and data~\cite{nethercote2004dynamic}. Dynamic analysis requires instrumentation of the program binary, and it analyzes the program when it executes. The instrumentation code is added to the program binary without changing the normal logic and execution flow of the program under analysis, and it contains minimal instructions and subroutines for collecting metadata and measurements. The instrumentation code and the instrumented code execute at the same time following each other. Indeed, adding instrumentation is easier during the compilation phase and when the source code is available~\cite{lattner2004llvm}, but source code is not always available, and the analysis would not be as accurate due to compiler transformations. Thanks to Dynamic Binary Instrumentation (DBI) frameworks such as Pin~\cite{luk2005pin}, it is possible to instrument program binaries without source code.
\emph{Pin} is a DBI framework based on just-in-time (JIT) compilation. In general, JIT compilers transform a source language to executable binary instructions at runtime. ~\autoref{fig:pin} shows how an embedded JIT engine is part of \emph{Pin} to recompile the binary instructions at runtime and combine the program's instruction with instrumentation codes, named \emph{Pintools}. To avoid the performance pitfall of JIT compilation, \emph{Pin} uses a \texttt{code cache} that stores the combined code, and re-execution of the same basic blocks occur from the \texttt{code cache}. Binary instrumentation using \emph{Pintools} gives us an easy to use interface to collect runtime metadata about program states such as the accessed memory addresses, targets of indirect branches and memory allocations. \emph{Pin} makes sure the instrumentation is transparent, i.e., it preserves the original application behavior~\cite{luk2005pin}. These events can be measures as accurate as they occur on the OS and the processor and as it would be an uninstrumented execution. In terms of microarchitectural analysis, we can observe the program behavior and resource usage as they appear on the hardware, and this gives us the ability to model a known microarchitectural leakage based on the observation of states from a real system.
\begin{figure}[tp]
\includegraphics[width=.9\columnwidth]{pin}
\caption{Pin: The JIT compiler combines application and instrumentation codes, and it stores the transformed binary in code cache. The virtual machine maintains and tracks program states, while it executes from the code cache.}
\label{fig:pin}
\end{figure}
\subsection{Microarchitectural Leakage} \label{sec:micro}
Modern microarchitectures feature various shared resources, and these resources are distributed among malicious and benign processes with different permissions. A malicious process, sharing the same hardware, can cause resource contention with a victim and measure the timing of either the victim or herself to learn about the victim's runtime. In a cache attack, the adversary accesses the same cache set that the victim's security-critical memory accesses are mapped to, and she measures the memory accesses' timing. A slow memory access reveals some information about the address bits of the victim's memory access. As motivated by cache attacks on AES~\cite{osvik2006cache,bernstein2005cache}, knowledge of secret-dependent memory accesses such as S-Box operations leaks information about the internal runtime state, and this information can be used for cryptanalysis and secret key recovery. In cache attacks, the size of each cache block is 64\,B which stops adversaries from gaining information about the $\log_2(64)=6$ least significant address bits. While some constant-time software countermeasures assume that the adversary cannot leak these bits, there are microarchitectural attacks on \texttt{cache banks} and \texttt{MOB} that leak beyond this assumption~\cite{yarom2017cachebleed,moghimi2017memjam}. In this work, we consider all secret-dependent memory accesses and treat them as memory-based leakages disregarding their spatial resolution.
Memory operations are not the only source of leakage. A conditional statement, or a processing loop that depends on a secret to choose an execution path can leak information about the secret. Each unique execution path operates on a different set of instructions, and it consumes the shared resources uniquely. Shared resources such as \texttt{instruction cache} and \texttt{BPU} leak information about the state of branches~\cite{aciiccmez2010new,aciiccmez2007predicting}. \autoref{fig:secretbranch} resembles a classical side-channel leakage in RSA Montgomery modular exponentiation. This algorithm processes a secret exponent one bit at a time, and it performs an additional arithmetic operation when the secret bit is one. An adversary who is able to track the execution of the left branch is able to determine the secret value that affected the conditional jump decision. We treat all the attacks that are triggered due to secret-dependent branches as control-flow based attacks.
\begin{figure}[tp]
\centering
\begin{minipage}{.4\columnwidth}
\begin{lstlisting}[label={lst:xxxx}]
// Square and Multiply
// Pseudocode
while(i>0){
r = (r * r)
if( key[--i] == 1 ){
r = (r * x)
}
}
\end{lstlisting}
\end{minipage}\hfil
\begin{minipage}{.5\columnwidth}
\includegraphics[width=\columnwidth]{secretbranch}
\end{minipage}
\caption{Montgomery \texttt{Square and Multiply} operations can leak information about the secret exponent. While \texttt{r9} points to the exponent in memory, comparison of a value from the exponent determines if the left jump should occur which leaves a key-dependent microarchitectural footprint.}
\label{fig:secretbranch}
\end{figure}
\subsection{Mutual Information Analysis}
Mutual information (MI) measures the mutual dependence of two random variables, and it can be used to quantify the average amount of obtainable information about one variable through observation of the second variable~\cite{guiau1977information}. Mutual information using Shannon entropy is defined as
\begin{equation*}
I(X,Y)=\sum_{x\in X}\sum_{y\in Y}p(x,y)\log_2\left(\frac{p(x,y)}{p(x)p(y)}\right),
\end{equation*}
where $p(x)$ and $p(y)$ are the probability distributions of random variables $X$ and $Y$ respectively. $p(x,y)$ is the joint probability of $X$ and $Y$, and $I(X,Y)$ tells us the average amount of dependent information in bits\footnote{$\log_2$ measures the MI in \emph{bit} unit.} between the variables $X$ and $Y$. MI has been utilized to quantify side-channel security~\cite{standaert2009unified,zhang2014new,bayrak2015automatic,Irazoqui16Leak}, or to mount side-channel attacks~\cite{gierlichs2008mutual}. Redefining MI in the side-channel context, we can define variable $X$ as the secret and variable $Y$ as an internal physical state of a system leaked through a side channel. $I(X,Y)$ will measure the average amount of leakage from secret $X$, through observing the side-channel information $Y$.
\subsection{Signing Algorithms} \label{sec:signing}
\subsubsection{DSA}
\emph{Digital Signing Algorithm (DSA)}~\cite{pub1993digital} is a signature scheme based on the discrete logarithm problem (DLP)~\cite{kevin1990discrete}. Choosing a prime $p$, another prime $q$ divisor of $p-1$, the group generator $g$, a secret key $x$, the public key $y=g^x \bmod p$, and the hash of the message to be signed $z$, the \emph{DSA} signing operation is defined as
\begin{gather*}
k\leftarrow~RANDOM\ |\ 1<k<q\\
r = (g^k \bmod p) \bmod q,\
s = k^{-1}(z + r\cdot x) \bmod q
\end{gather*}
where $(r, s)$ are the output signature pairs.
\subsubsection{ECDSA}
\emph{Elliptic-Curve DSA (ECDSA)}, as an analogue of \emph{DSA}, is a signature scheme based on elliptic curves~\cite{johnson2001elliptic}, in which the subgroup of a prime $p$ is replaced by the group of points on an elliptic curve over a finite field. Choosing an elliptic curve, a point on the curve $G$, the integer order $n$ of $G$, a secret key $d_A$, the public key $Q_A=d_A \times G$, and the hash of message to be signed $z$, the ECDSA signing operation is defined as
\begin{gather*}
k\leftarrow~RANDOM\ |\ 1<k<n-1 \\
(x_1, y_1) = k \times G\\
r=x_1 \bmod n,\
s=k^{-1}(z+r\cdot d_A) \bmod n
\end{gather*}
Both \emph{DSA} and \emph{ECDSA} use an ephemeral secret $k$ that needs to be chosen randomly for each operation.
\subsubsection{Modified Elliptic Curve Signature}
\emph{Elliptic-Curve Nyberg-Rueppel (ECNR)}~\cite{NybergRueppel} and \emph{SM2}~\cite{sm2BaiZhang}, a standard signature scheme, are modified schemes based on \emph{ECDSA} that allow signatures with message recovery. \emph{ECNR} and \emph{SM2} are widely used, and they are both supported by \emph{Intel IPP}. Public parameters, the private/public key pair and the ephemeral secrets are chosen similar to \emph{ECDSA}. The pair $(x_1, y_1)$ is also calculated similarly, but the signature generation for \emph{ECNR} is defined as
\begin{gather*}
r=x_1+z,\
s=k-r\cdot d_A,
\end{gather*}
and the signature generation for \emph{SM2} is defined as
\begin{gather*}
r=x_1+z,\
s=(1+d_A)^{-1}(k-r\cdot d_A)
\end{gather*}
\section{\tool~Analysis Technique}\label{sec:mi_technique}
\tool~aims to find microarchitectural leakages in software binaries. A binary implementation is vulnerable to microarchitectural side-channel attacks when there is a dependency between a secret and internal computation states observable through the side channels. We expose such relationships and quantify the amount of observable leakage in these implementations. This helps security analysts \textbf{1)} to reveal whether an implementation has leakages, \textbf{2)} to locate the exact location of each leakage in the binary, and \textbf{3)} to measure the dependency between the secret and the internal state, i.e., it can give some confidence value on the severity of the leakage. In contrast to side-channel analysis model, we are able to perform this analysis in a white-box model.
\subsection{Leakage Analysis Model}
We assume a strong adversary with full access to runtime events such as memory accesses, execution path and even register values. Further, the adversary can choose and modify any secret input of the system. This strong adversary can define any
internal computation state such as addresses of memory accesses and register values as a potential leakage vector, based on her knowledge of a category of side-channel attacks, e.g., memory based attacks (\autoref{sec:micro}). The adversary executes the system under her full control, and feeds the system with arbitrary secrets while collecting runtime traces for the defined leakage vector. ~\autoref{fig:analysismodel} compares our leakage analysis model with the side-channel analysis model. As an example, if we try to analyze a binary implementation of AES, we need to define certain operations as our leakage vectors. Based on cache attacks, an adversary defines memory accesses as a leakage vector, and she collects all memory accesses during the execution of AES using arbitrary secret keys. If there is a dependency between different secret keys and the variation of memory accesses, the adversary can locate which instructions relate to any secret-dependent memory accesses, and identify potential leakages.
\begin{figure}[tp]
\includegraphics[width=.87\columnwidth]{analysismodel}
\caption{Left: side-channel analyst finds relationship between a real leakage such as cache access pattern and secrets such as cryptographic keys. Right: \tool~follows a white-box model where the security analyst has full access to runtime states such as memory accesses, and she can find dependencies between arbitrary secrets and internal states.}
\label{fig:analysismodel}
\vspace{-1.5ex}
\end{figure}
\subsection{Capturing Internal States}
We choose two common sources of leakage as our leakage vectors: \textbf{1)} execution path and \textbf{2)} memory accesses. A true constant-time implementation follows a linear execution path for any given secret input; each time a software performs a secret-dependent conditional branch, it leaks some amount of information about the secret. Defining execution path as a leakage vector helps us to check whether for any secret input the same operations are performed. The second common source of leakage are memory accesses. A constant-time implementation should follow a secret-independent memory access pattern. If, for example, an implementation of a cryptographic algorithm does key-dependent table look ups which can be exploited by measuring cache timings, an attacker will be able to extract parts of the secret key; we ensure that memory accesses are either invariant, or at least uncorrelated to the input (e.g. blinding in RSA~\cite{kocher1996timing}).
To be able to detect these two types of leakages, we need to collect the internal state for all memory accesses and branch operations. First of all, we generate a set of arbitrary inputs for a chosen secret. These inputs can be either random (e.g. plain texts for encryption) or have a special structure with some random components (e.g. private keys or ephemeral secrets). We then execute the target binary on each input and log the following events:
\begin{itemize}
\item memory allocations
\item branches, calls and returns
\item memory reads and writes
\item stack operations.
\end{itemize}
Absolute memory addresses may vary even for constant-time programs, e.g., due to ASLR and dynamic heap allocation. We use the trace of memory allocations and stack operations to compute relative memory addresses; our meta data then consists of a list of relative addresses for memory accesses and the branches to, from and within the code we are analyzing. Note that one can define other leakage sources based on the underlying microarchitecture and collect the state of relevant instructions for analysis, e.g., the \texttt{multiplication} on some ~\emph{ARM} platforms leaks information~\cite{armCortexM3}.
\subsection{Preparing State Variables}
We do not make any assumptions on the leakage granularity; compared to similar techniques, that stop at cache line level, we keep this parameter freely configurable. This has the advantage that the analysis can be restricted to leakage sizes that are actually relevant to the analyst: For example, as of writing this paper, on Intel processors the finest known attack has a leakage granularity of 4 bytes~\cite{moghimi2017memjam}. Applying our technique in 1-byte mode will give all positions where a leakage might occur, but if one only expects 4-byte leakages to be exploitable, this may yield some false positives. Instead, the security analyst can choose the leakage granularity that fits to the desired spatial resolution. After applying the chosen leakage granularity of $g\in\mathbb{N}$ bytes by discarding the lower $\log_2g$ bits of each address, we can acquire an efficient representation of a specific execution state by computing a hash value of all or a subset of the trace entries; a truly constant-time program should have identical hashes of the full trace for every secret input. If we are only interested in analyzing individual instructions, e.g., memory access leakages of a specific subroutine, we can as well just compute the hash for the subset of traces for a single instruction.
\subsection{Leakage Analysis}
Our approach identifies any variations resulting from unique inputs and captured internal states per input. A naive approach is to compare the collected traces and divide them into classes. Observing more than one class informs us about secret-dependent operations. One can also compare raw traces sequentially which outlines all positions where the program behaves input-dependent and thereby allows to isolate the problematic sections. In addition to these simple approaches, we use MI to detect/locate these leakages, and to quantify the observable information.
To simplify MI analysis, we assume that $X$ is a set of unique uniformly distributed input test cases, which trigger deterministic behavior of the investigated program. If the program makes use of randomization (e.g. blinding in RSA~\cite{kocher1996timing}), the test cases $x\in X$ should contain the corresponding sources of randomness too.
Let $Y$ be a set of possible internal states (e.g. hashes of execution traces). We then define the execution state $T_i\subset X\times Y$ of the analyzed program at time point $i$ as
\begin{equation*}
(x,y)\in T_i\land(x,y')\in T_i\Rightarrow y=y',
\end{equation*}
i.e. each test case $x\in X$ appears at most once in $T_i$. The probability of one observed state $y\in Y$ is
\begin{equation*}
p_i(y)=\frac{\left|\{(x',y')\in T_i\,|\,y=y'\}\right|}{\left|T_i\right|}.
\end{equation*}
For the probability of pairs $(x,y)\in X\times Y$ we get
\begin{equation*}
p_i(x,y)=\left\{
\begin{matrix}
\frac{1}{\left|X\right|} & \text{if } (x,y)\in T_i,\\
0 & \text{else,}
\end{matrix}
\right.
\end{equation*}
since each input and therefore each input/state tuple occur exactly once: $\left|T_i\right|=\left|X\right|$.
With this knowledge we can finally compute the mutual information between test cases $X$ and the set of all occurring states $Y_i\coloneqq\{y\,|\,(x,y)\in T_i\}$:
\begin{align*}
I_i(X,Y_i)&=\sum_{(x,y)\in T_i}\frac{1}{\left|X\right|}\log_2\left(\frac{\frac{1}{\left|X\right|}}{\frac{1}{\left|X\right|}\cdot\frac{\left|\{(x',y')\in T_i\,|\,y=y'\}\right|}{\left|T_i\right|}}\right)\\
&=\sum_{(x,y)\in T_i}\frac{1}{\left|X\right|}\log_2\left(\frac{\left|T_i\right|}{\left|\{(x',y')\in T_i\,|\,y=y'\}\right|}\right).
\end{align*}
\subsection{Interpretation of MI Score}
As mentioned before, we can compute the MI for the entire trace, or a single instruction. A non-zero score for whole-trace MI tells us that an implementation has leakages, but it cannot locate the leakage point, and an implementation that has multiple leakage points over the execution period will have an aggregated MI value. The MI for single instructions is more precise, in which we can locate the instructions with positive score. The MI score $I_i(X_i,Y_i)$ is bounded by the amount of input bits $\log_2\left|X\right|$, and (for instruction MI) by the operand size: For example, an instruction that once accesses memory depending on 8 bits of the input will generate MI $\min\{8,\log_2\left|X\right|\}$. If we only execute $\left|X\right|=128$ test cases, we get MI score 7; for 256 or more test cases we get MI score 8.
%
The analyzed MI score is an estimate of the average leakage over the given test cases. MI is the appropriate metric in cases where the analyzed inputs are not under the attackers control and commonly used in leakage quantification. Alternatively, the \emph{worst case} leakage for any attacker-chosen input is given by the \emph{min entropy}, which only considers the most likely guess. The use of min entropy instead of MI in \tool\ is recommended if the adversary has full control over the inputs and specific high-leakage inputs exist~\cite{minentropy2009}.
\begin{figure*}[!t]
\centering
\subfloat{\includegraphics[width=.75\linewidth]{Pipeline}%
\label{l}}
\hfil
\caption{The \tool~ pipeline: Given the software binary under test, the framework generates test cases using the selected source, that are then used to produce execution traces. These traces need to be preprocessed to extract important information. The resulting trace files can then be analyzed for leakages, which are shown to the user in the visualization stage. Each stage can be easily modified to add further functionality, that is used either interchangeably or in addition to existing features.}
\label{fig:pipeline}
\end{figure*}
\section{\tool~Framework}
The \tool~framework is built as a pipeline with separate stages for test case generation, tracing and analysis (Figure \ref{fig:pipeline}). This modular design reduce the complexity, leading to easier extensibility: If one wants to implement additional analysis techniques apart from the ones that we already provide, she can directly add a new analysis stage, without needing to touch other parts like the trace generation. We will continue explaining these stages in more detail.
\subsection{Investigated Binary}
Although we are only interested in analyzing a specific function within a binary, we have to instrument and collect traces for the entire setup stage of the application before reaching to the analysis point, including the target library or executable itself, and parts of dependencies like system components. This process leads to an enormous decrease in analysis speed. A more efficient approach is to load and instrument the setup code only once, and then process the incoming test cases in a controlled loop. For libraries, we create a wrapper executable that executes an interface in a loop with new test cases. For executable applications, we can adopt in-memory fuzzing techniques where we inject hooks at the beginning and end of the target function and control the execution of the function to reset to the beginning with new test cases~\cite{bekrar2011finding}. To separate traces of the different test cases and avoid that the loop code causes false positives, we place calls to two instrumented dummy functions \texttt{PinNotifyTestcaseStart} and \texttt{PinNotifyTestcaseEnd}, which mark the start and the end of the analyzed section. A similar approach is taken by some fuzzers like \texttt{WinAFL}~\cite{WinAFL}, which use a built-in functionality of \texttt{DynamoRIO}~\cite{DynamoRIO} to exchange the argument list of \texttt{main} or a similar function.
\subsection{Input Generation}
The \tool~framework utilizes cryptographically secure pseudorandom number generators to create random test cases of any specified length. This performs well when analyzing cryptographic code, e.g. decrypting random ciphertexts. If a special input format is required, the test case generation code can be easily extended to produce such inputs, e.g. cryptographic keys in PEM format; this way, parts of the input can be kept constant while other parts are randomized, allowing to isolate the parameters which cause non-constant time behavior. Further, the framework supports passing a directory containing already generated inputs of any format.
\subsection{Trace Generation}
To trace the execution of individual test cases, we create a custom so-called \emph{Pintool}, which is a client library making use of \texttt{Intel Pin}'s dynamic instrumentation capabilities. In summary, our \emph{Pintool} logs the following events in a custom binary format on disk:
\begin{itemize}
\item module loads and the respective start and end addresses;
\item calls to dummy functions in the instrumented executable, to identify start and end of a test case execution;
\item sizes and addresses of allocated memory blocks through heap allocation functions such as \texttt{malloc} and \texttt{free} (Platform dependent), for resolving relative memory addresses;
\item Stack pointer modifications, for resolving relative addresses;
\item branches, calls and returns to and from all involved modules;
\item memory reads and writes in investigated modules.
\end{itemize}
\subsubsection{Instruction Emulation} \label{sec:emul}
Several cryptographic libraries use the \texttt{CPUID} instruction to detect the supported instructions for the respective processor and select a fitting implementation (that e.g. makes use of \texttt{AES-NI}). we enabled the \emph{Pintool} to change the output of this instruction. This allows to test arbitrary subsets of the instruction set that is available on the computer running \emph{Pin}.
As mentioned in~\autoref{sec:mi_technique}, cryptographic implementations might use randomization techniques like blinding to hide correlations between secret inputs and execution, or use ephemeral secrets. Some of these rely on the \texttt{RDRAND} instruction, which provides random numbers seeded with hardware entropy~\cite{RDRAND}. We provide an option to override the output of this instruction with arbitrary fixed values to control the randomization of the program under investigation.
\subsection{Trace Preprocessing}
The resulting raw trace files now need some preprocessing: First we add the common trace prefix that is generated before running the first test case, and which contains allocation data from the setup phase. In a second step, we calculate relative offsets of memory addresses. This involves associating branch targets with instruction offsets in the respective libraries, and identify offsets of memory accesses, such that traces generated using the same test case but during different runs of the \emph{Pintool} still match, regardless of the usage of randomized virtual addresses, e.g., \emph{ASLR}. For accesses to heap memory, we need to maintain a list of all currently allocated blocks: We use a stack to match the allocation size with their respective returned memory addresses, since in some implementations the heap allocator tends to call itself to reserve memory for internal bookkeeping. Finally the resulting preprocessed trace file is much smaller than the raw one (which can be discarded after this step), saving disk space and speeding up the following analysis stage.
\subsubsection{Applying Leakage Granularity}
We apply the leakage granularity immediately before the analysis starts; this way the preprocessed trace files are not modified, so the analysis can be performed on the same traces with different parameters. An analysis granularity of $g=2^b$ bytes ($b\in\mathbb{N}$) is introduced by discarding the $b$ least significant bits of each relative address.
\subsection{Leakage Analysis}
We implemented three different analysis methods in our framework:
\subsubsection{Analysis 1: Trace Comparison}
The first analysis method implements the trace comparison technique; given two preprocessed traces, we compare them entry by entry to check whether they differ at all. This performs well for leakage detection of particularly small algorithms such as symmetric ciphers. Optionally, the user can use trace diffs to manually inspect varying sections.
\subsubsection{Analysis 2: Whole-trace MI}
For leakage detection of an entire logic and calculation of the average amount of input bits that might leak over arbitrary parts of the execution (assuming that the attacker has full access to the trace), we provide an option to estimate the MI between input data and resulting trace. Given a set $X$ of unique test cases, we need to determine matching outputs for each trace prefix. Since we can compute the final MI only after waiting for completion of all test cases, it would be inefficient to store the entire trace; instead we reduce the trace data by encoding information like relative memory accesses and branch targets into 64-bit integers, and then compress them into one 64-bit integer $y\in\{0,\ldots,2^{64}-1\}$ using a hash function. We store the resulting tuples of inputs and hashes in sets $T_i\subset X\times\{0,\ldots,2^{64}-1\}$ for each prefix length $i$. We then apply the methods from~\autoref{sec:mi_technique} to measure the trace leakage.
\subsubsection{Analysis 3: Single-instruction MI}
The average amount of bits leaked by a single memory instruction is calculated analogously to the trace prefixes: Here, for a specific instruction $i$, $T_i\subset X\times\{0,\ldots,2^{64}-1\}$ contains hashes of the accessed memory addresses for each input $x$. These hashes change when the accessed addresses, their amount or their order vary, thus we get the maximum amount of information that is leaked by the respective instruction.
\subsection{Manual Inspection and Visualization}
To be able to manually inspect the preprocessed traces, the program has an option to convert binary traces into a readable text representation. If \texttt{MAP} files with function names are available (exported by some compilers or disassemblers), these can be used to symbolize memory addresses. We also created an IDA python plugin to import our single-instruction MI results as disassembly annotations. This helps further analysis on which parts of functions and loops leak.
Further we developed an experimental visualization tool, that renders function names and then draws an execution path. It also provides an option to render two traces simultaneously and highlight all sections where they have differences. This gives a quick overview of potential leakages and their structure.
\section{Case Study I: Intel IPP}
Intel's \emph{Integrated Performance Primitives (IPP)} cryptographic library aims to provide high performance cryptographic primitives that are compatible with various generations of Intel's processor~\cite{intelIPP}. \emph{Intel IPP} supports symmetric operations such as AES, as well as asymmetric signature and encryption schemes such as ECDSA. Intel IPP is used as the cryptographic backend for many of Intel's security products such as Intel SGX. Each of the implemented schemes in this library comes in variants optimized for different processors~\cite{inteldispatch}. The dynamic library checks the supported instruction set at runtime and chooses the most optimized implementation. However, developers can statically link toward a specific implementation by choosing the proper architecture code, e.g., \texttt{n8\_ippsAESInit} rather than \texttt{ippsAESInit}. In this case study, we test implementations for the variant optimized for processors supporting \emph{Intel® Advanced Vector Extensions 2 (Intel® AVX2)} with architecture code \texttt{l9}.
\subsection{Applying \tool~MI Analysis to IPP}
To be able to test \emph{Intel IPP} cryptographic implementations, we prepared wrappers that perform encryption and signing operations. For each tested implementation, we configured the wrappers for testing multiple test case scenarios: \textbf{1)} randomized plaintexts/ciphertexts to be encrypted/decrypted, or the message to be signed, \textbf{2)} randomized symmetric keys or private asymmetric keys, and \textbf{3)} random ephemeral secrets, when it is applicable, e.g., \emph{DSA} and \emph{ECDSA} as the input to MI Analysis. As suggested by chosen plaintext/ciphertext attacks, attacks on the cipher key and lattice attacks on ephemeral secrets~\cite{benger2014ooh}, using these scenarios, we are able to detect leakages that are dependent on various types of secrets.
\begin{table}[t!]
\begin{center}
\captionof{table}{Singe-instruction MI Analysis of Intel IPP cryptographic implementations v2018.2.185. All implementations are chosen from the \emph{l9} architecture code.} \label{tab:ippmi}
\begin{tabular}{ | p{1.2cm} | c | p{1.35cm} | p{1.05cm} | p{.9cm} | }
\hline
\textbf{\small{Scheme}} & \textbf{\small{Interfaces}} & \textbf{\small{Executed / Unique Instructions}} & \textbf{\small{Analysis Time (ms)}} & \textbf{\small{Leakage Found}}\\ \hline
\small{3DES/ECB} & \makecell{\scriptsize{ippsDESInit}\\\scriptsize{ippsTDESDecryptECB}} & \small{$4074613$ / $70205$} & $11921$ & $0$ \\ \hline
\small{SM4/ECB} & \makecell{\scriptsize{ippsSMS4Init}\\\scriptsize{ippsSMS4EncryptECB}} & \small{$4085517$ / $68221$} & $10004$ & $0$ \\ \hline
\small{AES/CTR} & \makecell{\scriptsize{ippsAESInit}\\\scriptsize{ippsAESEncryptCTR}} & \small{$2138799$ / $49181$} & $27289$ & $2$ \\ \hline
\small{DSA}~\scriptsize{(512)} & \makecell{\scriptsize{ippsDLPGenKeyPair}\\\scriptsize{ippsDLPSignDSA}} & \small{$12245281$ / $57423$} & $1735153$ & $2$ \\ \hline
\small{RSA}~\scriptsize{(512)} & \makecell{\scriptsize{ippsRSA\_Decrypt}} & \small{$43987943$ / $55167$} & $275090$ & $1$ \\ \hline
\small{ECDSA} \scriptsize{(SECP256R1)} & \makecell{\scriptsize{ippsECCPGenKeyPair}\\\scriptsize{ippsECCPSignDSA}} & \small{$4085155$ / $63785$} & $358373$ & $3$ \\ \hline
\small{ECDSA} \scriptsize{(BN256)} & \makecell{\scriptsize{ippsECCPGenKeyPair}\\\scriptsize{ippsECCPSignDSA}} & \small{$5383210$ / $63699$} & $750188$ & \footnotesize{*} \\ \hline
\small{ECDSA} \scriptsize{(SM2)} & \makecell{\scriptsize{ippsECCPGenKeyPair}\\\scriptsize{ippsECCPSignDSA}} & \small{$5158607$ / $63741$} & $353435$ & \footnotesize{*} \\ \hline
\small{ECNR} \scriptsize{(SECP256R1)} & \makecell{\scriptsize{ippsECCPGenKeyPair}\\\scriptsize{ippsECCPSignNR}} & \small{$4028592$ / $62447$} & $281937$ & $2$ \\ \hline
\small{SM2} & \makecell{\scriptsize{ippsECCPSignSM2}} & \small{$6021005$ / $64273$} & $554035$ & $3$ \\
\hline\hline
\multicolumn{2}{|c|}{\textbf{\small{Total}}} & \small{$91208722$ / $618142$} & $73$~\scriptsize{minutes} & $13$ \\ \hline
\multicolumn{4}{l}{\footnotesize{* Different curves did not change the results for ECDSA.}}
\end{tabular}
\end{center}
\end{table}
\autoref{tab:ippmi} shows the single-instruction MI analysis results, where symmetric ciphers: \emph{(Triple) DES}, \emph{AES} and \emph{SM4} and asymmetric ciphers: \emph{DSA}, \emph{RSA}, \emph{ECDSA}, \emph{ECNR} and \emph{SM2} have been tested. On our analysis setup, the total computational time to analyze $10$ different implementations with about $92$ million total instructions is $73$ minutes of CPU time, highlighting the efficiency of our method. Note that we performed analysis with input size $2^7=128$ (7-bit \emph{MI}) and input size $2^{10}=1024$ (10-bit \emph{MI}), for analysis of symmetric and asymmetric operations respectively. Although analysis with more iterations is possible, state-of-the art side-channel attacks on these implementations suggest that the random secret should show leakage behavior after this number of iterations. \emph{Intel IPP} uses two separate interfaces for the key schedule, and ephemeral secret generation for most implementations (\autoref{tab:ippmi}).
\emph{(Triple) DES}, \emph{AES} and \emph{SM4} are block ciphers that use table-based S-Box operations. The results suggest that these implementations are heavily protected against memory-based leakages. Our target architecture code uses the \texttt{AES-NI} instruction set for \emph{AES} and \emph{SM4} operations. \texttt{AES-NI} is inherently secure against known attacks. However, testing the \emph{CTR mode} reveals some leakages. All asymmetric ciphers suffer from at least one leakage. For schemes that are based on elliptic curves such as \emph{ECDSA}, \emph{ECNR} and \emph{SM2}, Intel IPP supports various standard curves. As some developers optimize curve arithmetic differently for various standard elliptic curves, we tested the \emph{ECDSA} signing operation with three different curves: \texttt{SECP256R1}, \texttt{BN256} and \texttt{SM2}. However, the \emph{MI} analysis results are exactly the same for different choices of elliptic curves. We found a total of $13$ leakages in \emph{Intel IPP}, while some of these leakages are triggered through calling the same subroutine, e.g., both \emph{ECDSA} and \emph{SM2} use the leaky subroutine for \texttt{scalar multiplication}. We will discuss these subroutines in more detail.
\subsection{Discovered leakages in Intel IPP}
We have found $7$ different subroutines that have leakages, i.e., perform data-depended memory accesses or branch decisions (\autoref{tab:ippleakage}). We performed an initial analysis of these leakages using our visualization tool and \emph{IDA Pro}. The subroutine \texttt{gfec\_MulBasePoint} performs scalar multiplication of a scalar and point on the elliptic curve, as a common operation in all curve-based signature schemes: \emph{ECDSA}, \emph{ECNR}, \emph{SM2}. As defined by the signing algorithms~\autoref{sec:signing}, \emph{gfec\_MulBasePoint} leaks information about the ephemeral secret. This leakage occurs due to the dependability of the number of times the window-based multiplier loop processes the ephemeral secret. Further leakages exist in the curve operations after the scalar multiplication: The subroutine \emph{alm\_mont\_inv} leaks information during the mapping of $x$ coordinate of computed public point. As $(x_1, y_1)$ are not secrets in the signing operation, this leakage is not critical, and we refrain from further root cause analysis. Similarly, the subroutine \emph{cpModInv} has leakages with a relatively high MI score that is due to the secret-dependent loop count. \emph{cpModInv} performs a modular inversion operation using Extended Euclidean Algorithm (EEA). In \emph{ECDSA}, $k^{-1}$ leaks information about the secret ephemeral, and in \emph{SM2}, $(1+d_A)^{-1}$ leaks information about the secret signing key. \emph{ECNR} does not perform any modular inversion and is safe from leakages due to this subroutine. The existing leakage in \emph{cpModInv} subroutine also applies to \emph{DSA} where a modular inversion on ephemeral secret, $k^{-1}$ can leak.
\begin{table}[t!]
\begin{center}
\captionof{table}{Discovered leakage subroutines within Intel IPP cryptographic implementations v2018.2.185. Some of the subroutines expose critical and potentially exploitable leakages.} \label{tab:ippleakage}
\begin{tabular}{ | l | p{1.4cm} | l | l | }
\hline
\textbf{\small{Subroutine}} & \textbf{\small{Affected}} & \textbf{\small{MI}} & \textbf{\small{Leakage Source}} \\ \hline
\small{gfec\_MulBasePoint}& \small{ECDSA, ECNR, SM2}& \small{$0.86$ / $10$} & \small{Conditional Loop} \\ \hline
\small{cpMontExpBin}& \small{DSA}& \small{$3.73$ / $10$} & \small{Conditional Loop} \\ \hline
\small{cpModInv}& \small{DSA, SM2, ECDSA}& \small{$3.88$ / $10$} & \small{Conditional Loop} \\ \hline
\small{ExpandRijndaelKey}& \small{AES/CTR}& \small{$7.00$ / $7$} & \small{Memory Lookup} \\ \hline
\small{ippsAESEncryptCTR}& \small{AES/CTR}& \small{$0.13$ / $7$} & \small{Conditional Loop} \\ \hline
\small{gsMontExpWin}& \small{RSA}& \makecell{\small{$1.12$ / $10$}\\\small{$3.11$ / $10$}} & \makecell{\small{Conditional Loop}\\\small{Memory Lookup}} \\\hline
\small{alm\_mont\_inv}&\small{ECDSA, ECNR, SM2}& \makecell{\small{$5.33$ / $10$}\\\small{$9.98$ / $10$} } & \makecell{\small{Conditional Loop}\\\small{Memory Lookup}} \\ \hline
\end{tabular}
\end{center}
\end{table}
\emph{Intel IPP} supports two distinct functions for performing Montgomery exponentiation. Exponentiation of big numbers is a common operation in schemes such as \emph{RSA} and \emph{DSA}. The \emph{RSA} algorithm uses the \emph{gsMonthExpWin} subroutine which is a window-based implementation of the Montgomery exponentiation. This function has leakages based on both memory lookup and conditional loop. The second Montgomery exponentiation subroutine~\texttt{cpMontExpBin} is a protected binary implementation that has leakage due to the conditional loop count. \emph{DSA} uses the latter, which leaks information about the ephemeral secret during computation of $(g^k \bmod p)$.
The only leakage exposed during testing of symmetric ciphers are due to \emph{AES} key generation subroutine \texttt{ExpandRijndaelKey}, and calculation of the nonce length in \emph{CTR mode}. \texttt{ExpandRijndaelKey} is called every time the \texttt{ippsAESInit} is used. As the high \emph{MI} score shows, \emph{AES} key schedule used during the CTR mode has full leakage. This leakage can be considered critical in scenarios such as the SGX environment where an adversary has a high resolution side channel~\cite{moghimi2017cachezoom,wang2017leaky}. When the symmetric key is passed to the AES key schedule, a high resolution adversary can steal the secret key before any encryption/decryption. While \emph{AES/CTR} encryption uses \texttt{AES-NI}, there is a loop within this implementation where calculating the length of nonce leaks about the leading zero bits.
\subsubsection{Leakage of Scalar Multiplication}
Scalar multiplication in \emph{Intel IPP} uses a fixed-window algorithm with a window size of $5$: for a 256-bit ephemeral secret, as defined by \texttt{SECP256R1}, the algorithm performs $51$ iterations of the window operation. However, our dynamic analysis of the algorithm with various random ephemeral secrets shows that \texttt{gfec\_MulBasePoint} skips the leading zero bits and applies fewer windows if there are leading zero bits in the beginning, as the multiple of the window size. In this case, the main loop performs $50$ times for $2$, $49$ for $7$ and $48$ times for $12$, etc, leading zero bits. \emph{CacheQuote}~\cite{dall2018cachequote} exploits a similar vulnerability used by \emph{Intel EPID} signature scheme, but \emph{EPID} uses a different function of \emph{Intel IPP} for scalar multiplication~\texttt{cpEcGFpMulPoint}. As our discovery suggests, this was a common issue in \emph{Intel IPP} that was existed among other curve implementations. Although this implementation has countermeasure based on Scatter-Gather technique~\cite{brickell2006mitigating}, this vulnerability can easily be exploited in high resolution settings using a lattice attack~\cite{dall2018cachequote}.
\begin{algorithm}[t]
\caption{Bitmasked Montgomery Exponentiation}
\label{alg:montexp}
\begin{algorithmic}[1]
\Procedure{BinExp}{base $g$, exponent $k$}
\State $A \gets R \bmod p$
\State $\widetilde{g} = \mathrm{MontMul}(g, R^2 \bmod p)$
\State $m \gets 0$
\State $i = 1$
\While{$i < (\mathrm{BitLength}(k) \bmod 64)$}
\State $t \gets A \BitAnd \BitNeg m \BitOr \widetilde{g} \BitAnd {m}$
\State $A \gets \mathrm{MontMul}(A, t)$
\State $m = \BitNeg m \BitAnd k_i$
\State $i = i + 1 - m$
\EndWhile
\For{$j \gets 1$ \textbf{to} $\mathrm{BitLength}(k) / 64$}
\State perform the same operations as above.
\EndFor
\State \Return $A$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\subsubsection{Bitmasked Montgomery Exponentiation}
The Montgomery exponentiation in \emph{Intel IPP} follows a bit-by-bit operation based on the Montgomery Reduction technique~\cite{ha1998common}. However, the implementation is protected by obfuscating the conditional statements as bit-masked operations. Therefore, the subroutine always executes the same \texttt{Montgomery multiplication (MontMul)} subroutine disregarding the value of the exponent bits. However, the exponent bits are used as a mask to choose the operand of the \texttt{MontMul} and to execute the \texttt{MontMul} two times with two different operands when the exponent bit is one. Although this implementation looks secure at first sight, the exponent bits are used \textbf{1)} to calculate the exponent bit length, i.e., leading zero-bit leakage, and \textbf{2)} to decide the number of iterations of the loop. Based on Algorithm~\ref{alg:montexp}, the main loop executes two times if an exponent bit is one and once if the exponent bit is zero. This leaks the Hamming weight of the ephemeral secret to a microarchitectural adversary.
Further, the algorithm performs a similar operation with separate instructions for different parts of the key. For example, for a 160-bit DSA exponent, the algorithm first processes the first 32 bits, and then another code section processes the remaining 128 bits of exponent. This gives an adversary a local Hamming weight leakage of the first 32-bit of the secret exponent.
\section{Case Study II: Microsoft CNG}
The \emph{Cryptography API: Next Generation (CNG)} is the cryptography platform supplied with every Windows system beginning with Windows Vista, and replaces the older \emph{CryptoAPI} as the default cryptographic stack. It includes many common algorithms, including \emph{RSA}, \emph{AES}, \emph{ECDSA}. While the public API for \emph{Microsoft CNG} resides in the \texttt{BCrypt.dll} system file, its cryptographic implementations themselves are located in another library file, \texttt{BCryptPrimitives.dll}. Microsoft does provide neither source code nor documentation for the internal functionality, but one can download \texttt{PDB} symbol files from Microsoft's symbol server, which contain most of the internal function names, helping to reduce the reverse engineering effort.
\subsection{Applying \tool~MI Analysis to CNG}
As we did with IPP, we again created wrapper executables to call the respective library functions of \emph{RSA}, \emph{DSA}, \emph{ECDSA} and \emph{AES/ECB}. For \emph{AES}, the library uses the \texttt{CPUID} instruction to choose between two different implementations, one that uses \texttt{AES-NI} vector instructions, and a plain T-table based implementation. We tested both implementations by emulating the \texttt{CPUID} instruction, as explained in~\autoref{sec:emul}. The results are shown in~\autoref{tab:cngmi}. We analyzed a total of $21$ million instructions in $31$ minutes of CPU time, finding four different leakage points. For \emph{RSA}, we discovered that Microsoft's implementation behaves truly constant-time. \emph{ECDSA} and \emph{DSA} implementations both suffer from leakage due to calling the same subroutine for modular inversion.
\begin{table}[t!]
\begin{center}
\captionof{table}{Singe-instruction MI Analysis of some of the bcryptprimitives.dll v10.0.17134.1 cryptographic implementations. } \label{tab:cngmi}
\begin{tabular}{ | p{.95cm} | p{2.4cm} | p{1.4cm} | p{.95cm} | p{.85cm} | }
\hline
\textbf{\small{Scheme}} & \textbf{\small{Interfaces}} & \textbf{\small{Executed / Unique Instructions}} & \textbf{\small{Analysis Time (ms)}} & \textbf{\small{Leakage Found}}\\ \hline
\small{AES/ECB} & \scriptsize{SymCryptAesEcbEncrypt} & \small{$2384298$ / $55451$} & $17546$ & $0$ \\ \hline
\small{AES/ECB} & \scriptsize{SymCryptAesEcbEncryptAsm} & \small{$2324391$ / $63179$} & $26211$ & $2$ \\ \hline
\small{DSA}~\scriptsize{(512)} & \scriptsize{MSCryptDsaSignHash} & \small{$3586162$ / $63748$} & $223356$ & $1$ \\ \hline
\small{RSA}~\scriptsize{(1024)} & \scriptsize{MSCryptRsaDecrypt} & \small{$8073605$ / $66454$} & $760450$ & $0$ \\ \hline
\small{ECDSA} \scriptsize{(SECP256R1)} & \scriptsize{MSCryptEcDsaSignHash} & \small{$4764783$ / $64732$} & $831136$ & $1$ \\ \hline\hline
\multicolumn{2}{|c|}{\textbf{\small{Total}}} & \small{$21133239$ / $313564$} & $31$~\scriptsize{minutes} & $4$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Discovered leakages in Microsoft CNG}
Analyzing the aforementioned algorithms yielded two leakage candidates (see Table \ref{tab:cngleakage}); the first one resides within the modular inversion function of \emph{DSA} and \emph{ECDSA} and is used for all processors. The MI returns full leakage for the modular inversion leakage, implying that the implementation is heavily unprotected. The second one is in the encryption function of \emph{AES} and only used by processors not supporting \texttt{AES-NI}. As it is a table-based implementation, the leakage is expected.
\begin{table}[t!]
\begin{center}
\captionof{table}{Discovered leakage subroutines within bcryptprimitives.dll v10.0.17134.1 cryptographic implementations.} \label{tab:cngleakage}
\begin{tabular}{ | l | p{1.0cm} | p{0.5cm} | p{1.3cm} | }
\hline
\textbf{\small{Subroutine}} & \textbf{\small{Affected}} & \textbf{\small{MI}} & \textbf{\small{Leakage Source}} \\ \hline
\small{SymCryptFdefModInvGeneric}& \small{DSA, ECDSA}& \small{$10.00$ / $10$} & \small{Conditional Loop} \\ \hline
\small{SymCryptAesEncryptAsmInternal}& \small{AES}& \small{$7.96$ / $10$} & \small{Memory Lookup} \\ \hline
\end{tabular}
\end{center}
\end{table}
\subsubsection{Leakage of Modular Inversion}
The modular inversion function that is used for \emph{DSA} and \emph{ECDSA} gives full \emph{MI} on 1024 signing operations for random ephemeral secrets with fixed key and plaintext. This subroutine does not have any constant-time protection. However, while this is a non-constant time behavior and suggests that the ephemeral leaks, we considered this as not exploitable; Microsoft protects this implementation through a masking countermeasure. The masking countermeasure for modular inversion works as follow:
\begin{enumerate}
\item A mask value $m$ is generated randomly.
\item The ephemeral secret $k$ is multiplied by $m$ before the modular inversion: $s = (k\cdot m)^{-1}(z + x) \bmod q$
\item Then the signature $s$ is multiplied again with $m$ to produce the correct signature:
\begin{gather*}
s = sm = (k\cdot m)^{-1}(z + x)\cdot m
= k^{-1}(z + x)
\end{gather*}
\end{enumerate}
Thus, the implementation leaks $k\cdot m$, where $m$ is a random per-signature generated mask, effectively preventing extraction of useful information. Leakage of ephemeral keys is exploitable~\cite{benger2014ooh}, the randomized product of ephemeral key and a random value is not.
\subsubsection{Leakage of AES T-table Lookup}
The non-vector version of \emph{AES} uses a common lookup table implementation, where four so-called \emph{T-tables} combine the steps \texttt{SubBytes}, \texttt{ShiftRows} and \texttt{MixColumns}. Each round consists of four of such lookups per table, leading to $16\cdot r$ memory accesses per encryption, where $r\in\{10,12,14\}$ is the number of rounds. The 8-bit indices used for the table accesses depend on the plaintext and the key; since the \emph{MI} is $7.96$ for $1024$ measurements, these indices can be considered fully leaking. Each table entry has $4$ bytes size, thus each T-table has $1024$ bytes, and therefore takes $16$ cache lines on an Intel processor; such implementations have already been shown to be exploitable with cache attacks \cite{bernstein2005cache}.
\section{Related Work}
\textbf{Programming languages} can support constant-time code generation and verification~\cite{sinha2017compiler,cauligi2017fact,bond2017vale}. The general approach is to support annotation of security-critical variables and to generate instructions that operate obliviously on annotated secrets. Annotated secrets can be verified for constant-time behavior using SMT-based techniques~\cite{bond2017vale}. Constant-time behavior can be enforced for some operations by using primitives such as oblivious RAM (ORAM)~\cite{stefanov2013path} and obfuscated execution~\cite{rane2015raccoon}. Language-based approaches are not widely used, and annotation is an error-prone task.
\textbf{Black-box testing} approaches use statistical methods to quantify leakages of physical channels~\cite{coron2000statistics}. In particular, \emph{Dudect}~\cite{reparaz2017dude} performs black-box timing analysis, in which the timing of a target system with different inputs will be analyzed using the \emph{t-test}~\cite{welch1947generalization}, but these black-box techniques do not scale to microarchitectural attacks with a gray-box model. With an abstract model of the leakage channel, methods based on~\textbf{Static Program Analysis} are proposed to analyze program code and to quantify leakages~\cite{kopf2012automatic,almeida2013formal,barthe2014system,almeida2016verifying,blazy2017verifying}. Similar to language-based approaches, these techniques are limited to correct annotation of the source code. While some of these approaches are limited to the source code and cannot find leakages that are potentially introduced by the compiler~\cite{almeida2013formal}, others perform the analysis on the lower level LLVM bitcode~\cite{almeida2016verifying} or the annotated machine code~\cite{barthe2014system,blazy2017verifying}. However, they rely on the availability of the source code. \emph{CacheAudit}~\cite{doychev2015cacheaudit,doychev2017rigorous} is based on Static Binary Analysis (SBA). SBA approaches need to initially reconstruct the original basic blocks and control flow graph. Precise reconstruction of the program semantic and control flow graph is infeasible without the runtime information, by just using static disassembly~\cite{andriesse2016depth}. As a result, while they give formal guarantees on the absence of leakages, they do not scale to accurately analyze large program binaries, e.g., \emph{CacheAudit} approach has only been tested on rather simple algorithms such as sorting and symmetric encryption. Other proposals based on \textbf{Symbolic execution} quantify side-channel leakage by determining symbolic secret inputs that affect the runtime behavior~\cite{pasareanu2016multi,chattopadhyay2017quantifying}. However symbolic execution is an expensive approach, and the proposed methods require access to the source code.
In this work, we leverage \textbf{Dynamic Program Analysis} techniques to accurately locate microarchitectural leakage in software binaries, as they execute on the processor. \emph{ctgrind}~\cite{langley2010ctgrind} based on LLVM memcheck can check all branches and memory accesses to make sure that they do not have dependency on secret data. Irazoqui et al.\ instrument the source code to obtain and analyze cache traces using MI~\cite{Irazoqui16Leak}. Sensitive code sections are identified by taint analysis. On binary-only approaches, \emph{CacheD}~\cite{wang2017cached} analyzes binaries based on symbolic execution and constraint solving. They initially use DBI to get execution traces for a set of input values; then, given the information which input values are considered secret, a taint analysis extracts all instructions that work with secrets, either directly or indirectly. These instructions are then analyzed using symbolic execution to detect whether cache leakages exist. In comparison, our method aims at maximum performance without too much loss of accuracy by only storing necessary information and using hash compression to get small execution states. The symbolic execution approach introduces a large bottleneck, as their analysis time suggest. This saving of computation time allows us to detect also other types of leakages like differing loop counts or byte-level memory access differences. Also, since \tool~is designed as a modular open source framework, one can implement arbitrary analysis stages for other types of leakages. Zankl et al.~\cite{zankl2016automated} use DBI to collect traces for instruction based leakage detection. They use \emph{t-tests} for leakage analysis and only test for execution flow leakages. \emph{STACCO}~\cite{xiao2017stacco} is focused on differential trace analysis for Bleichenbacher attacks ~\cite{bleichenbacher1998chosen}. Independently, \emph{DATA}~\cite{WeiserDATA} follows a similar approach based on DBI. They use trace differentiation and \emph{t-tests} for leakage analysis. As of our knowledge, our work is the first that has been tested on actual closed-source binaries.
\section{Conclusion}
The lack of efficient and practical tools for leakage analysis of binaries leave the reliability of these untested deployed implementations a mystery. To be able to analyze the compiler outputs and closed-source libraries, we have created an extensible framework that supports various types of microarchitectural leakages based on \texttt{instruction and data cache}, \texttt{MOB}, \texttt{BPU}, etc. \tool\ can be extended to analyze other and future side channels. Our framework leverages \emph{DBI} to collect the internal state of a program under test, and it applies multiple analysis techniques based on trace comparison and \emph{MI}. \tool~is open source and is publicly accessible: \url{https://github.com/UzL-ITS/Microwalk}. We used this framework to thoroughly analyze two widely used closed-source libraries, \emph{Intel IPP} and \emph{Microsoft CNG}. The tested implementations are optimized for the current generation of Intel processors. Our report shows that side-channel countermeasures for these implementations are still not fully leakage-free, e.g., all the curve-based signature schemes in \emph{Intel IPP} suffer from at least one vulnerability. We have identified several leakages in symmetric and asymmetric ciphers, and reported them to the respective vendors. Our analysis shows that despite the existing efforts on protecting these implementations, some of them still suffer from security-critical leakages.
\subsection{Future Work}
\subsubsection{Coverage-based Fuzzing}
We use random test cases to get a uniform random distribution of potential memory accesses and execution paths; while this works well with cryptographic implementation, it would not scale to targets such as protocols or data structures. Coverage-based \emph{Fuzzing}\cite{mcnally2012fuzzing} is a technique to generate test cases with the aim of achieving maximum code coverage; while it was originally developed to find software bugs, e.g., memory corruption, the same approach can be applied for finding side-channel leakages, e.g., leakage in the \emph{JPEG} library~\cite{xu2015controlled}. We have already implemented an experimental support for using \texttt{WinAFL}\cite{WinAFL} as a test case generator; in that setting \texttt{AFL} helps to generate samples with higher coverage, while at the same time the test cases are sent to our framework for further processing. It is desirable to enhance this experimental feature and apply it to non-cryptographic implementations that are critical in terms of side-channel security.
\subsubsection{Distinguishing leakages in call graph}
We observed that in some cases control flow leakages in the higher level algorithm residing at the top of the call chain hide leakages in the subroutines invoked in deeper levels. Also, if separate functions use a common subroutine, a positive MI result in this subroutine can not easily be assigned to its root cause. We therefore propose to add an option to \tool\ to take the call graph into account when computing mutual information.
\medskip\noindent{\bf Responsible Disclosure}
We have informed the \emph{Intel Product Security Incident Response Team (PSIRT)} and \emph{Microsoft Security Response Center (MSRC)} of our findings. \emph{MSRC} has not responded. After the initial report, we noticed that Intel have already patched \texttt{gfec\_MulBasePoint} in Intel IPP v2018.3.240. Intel have acknowledged the receipt for the remaining vulnerabilities. Here is the time line for the responsible disclosure:
\begin{itemize}
\item \textbf{06/22/2018:} We informed our findings to the Intel Product Security Incident Response Team (Intel PSIRT) and the Microsoft Security Response Center.
\item \textbf{06/25/2018:} Intel PSIRT acknowledged the receipt.
\item \textbf{07/31/2018:} Intel PSIRT confirmed a work-in-progress patch for IPP 2018 update 4 (CVE-2018-12155, CVE-2018-12156).
\end{itemize}
\medskip\noindent{\bf Acknowledgements}
This work is supported by the National Science Foundation, under grant CNS-1618837.
\bibliographystyle{ACM-Reference-Format}
\citestyle{acmnumeric}
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{"url":"http:\/\/www.ck12.org\/book\/Basic-Speller-Student-Materials\/r1\/section\/12.5\/","text":"<meta http-equiv=\"refresh\" content=\"1; url=\/nojavascript\/\">\nYou are reading an older version of this FlexBook\u00ae textbook: Basic Speller Student Materials Go to the latest version.\n\n12.5: Sometimes Long < e > is Spelled < i > or < y >\n\nDifficulty Level: At Grade Created by: CK-12\n\nSometimes Long <e> is Spelled $<\\text{i}>$ or <y>\n\n1. Two other very important spellings of [\u0113] are $<\\text{i}>$ and <y>. The $<\\text{i}>$ spelling of [\u0113] usually occurs in the V.V pattern and sometimes in the VCV pattern. It only occurs in the V# pattern in foreign words recently brought into our language, such as broccoli, spaghetti, macaroni. The V# pattern is the one in which the <y> spelling of $[\\bar{\\text{e}}]$ always occurs. Both the $<\\text{i}>$ and the <y> spellings often occur in weakly stressed syllables. Underline the $<\\text{i}>$s and <y>s that are spelling [\u0113] in the following words:\n\n$& \\text{ability} && \\text{gasoline} && \\text{champion} && \\text{angry} && \\text{community} \\\\& \\text{curiosity} && \\text{enthusiasm} && \\text{machine} && \\text{dignity} && \\text{glorious} \\\\& \\text{magazine} && \\text{fiery} && \\text{guardian} && \\text{medium} && \\text{police} \\\\& \\text{gloomy} && \\text{obedience} && \\text{obvious} && \\text{period} && \\text{library} \\\\& \\text{variety} && \\text{reality} && \\text{piano} && \\text{routine} && \\text{various} \\\\& \\text{jolliest} && \\text{chocolaty} && \\text{ingredient} && \\text{polliwog} && \\text{encyclopedia}$\n\n2. Sort the words into the following two groups. One word goes into both groups:\n\n3. Now sort the words with [\u0113] spelled $<\\text{i}>$ into the following two groups:\n\n4. In what pattern does the <y> spelling of [\u0113] always occur? _______\n\n5. Five words in the list in Item 1 that contain [\u0113] spelled <e> are . . .\n\nWord Alchemy. Hundreds of years ago alchemy was the ancestor of modern chemistry. The alchemists worked hard trying to change lead into gold. In the puzzle below you can change the word lead into the word gold. Here are the rules:\n\n1. Any shaded square must contain the same letter as the square directly above it.\n2. Any unshaded square must contain a different letter from the square directly above it.\n3. Every row must contain an English word.\n\nHints: Since you know that the two shaded squares in row 2 must contain the same letters as the two squares directly above them, you know that they must contain <e> and $<\\text{a}>$. And since you know that the two shaded squares in row 4 contain the same letters as the two squares directly above them, you know that the word in row 3 must end with the letters <ld>. You should write the <ea> and <ld> into rows 2 and 3. You won't know what the shaded square in row 3 contains until you know the word that goes in row 2, so you can't write in the first letter in row 3 yet. That gives you the following:\n\nYour job now is to find two words that fit into rows 2 and 3. Each must contain four letters. Because of rule number one above, you know that the first word must have <ea> in the middle; the second must end in <ld>, and they must both start with the same letter. Because of rule number two, you also know that the word in row 2 cannot start with <l> or end with <d> above, and the word in row 3 cannot have $<\\text{go}>$ as its first two letters. The two words beat and bald would work. So would meat and mild. There are other workable pairs.\n\nHere are some more Word Alchemies for you to solve:\n\nGrades:\n\n1 , 2 , 3 , 4 , 5\n\nFeb 23, 2012\n\nLast Modified:\n\nJan 16, 2015\nFiles can only be attached to the latest version of section\n\nReviews\n\nPlease wait...\nPlease wait...\nImage Detail\nSizes: Medium | Original\n\nCK.ENG.ENG.SE.1.Basic-Speller.12.5","date":"2015-10-06 15:19:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 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\": 10, \"texerror\": 0, \"math_score\": 0.46659424901008606, \"perplexity\": 1516.3482633555034}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-40\/segments\/1443736678818.13\/warc\/CC-MAIN-20151001215758-00033-ip-10-137-6-227.ec2.internal.warc.gz\"}"}
| null | null |
Aphthona tristis is een keversoort uit de familie bladkevers (Chrysomelidae). De wetenschappelijke naam van de soort werd in 1950 gepubliceerd door Har. Lindberg.
tristis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,147
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<!DOCTYPE html>
<html>
<head>
<title>Sacache test runner</title>
</head>
<body>
<h4>Sacache test runner</h4>
<p>Please open browser console to see the output :)</p>
<script src="../Sacache.js"></script>
<script src="TestDuck.js"></script>
<script src="sacache_test_suite.js"></script>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,301
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#ifndef _GST_INTER_SUB_SRC_H_
#define _GST_INTER_SUB_SRC_H_
#include <gst/base/gstbasesrc.h>
#include "gstintersurface.h"
G_BEGIN_DECLS
#define GST_TYPE_INTER_SUB_SRC (gst_inter_sub_src_get_type())
#define GST_INTER_SUB_SRC(obj) (G_TYPE_CHECK_INSTANCE_CAST((obj),GST_TYPE_INTER_SUB_SRC,GstInterSubSrc))
#define GST_INTER_SUB_SRC_CLASS(klass) (G_TYPE_CHECK_CLASS_CAST((klass),GST_TYPE_INTER_SUB_SRC,GstInterSubSrcClass))
#define GST_IS_INTER_SUB_SRC(obj) (G_TYPE_CHECK_INSTANCE_TYPE((obj),GST_TYPE_INTER_SUB_SRC))
#define GST_IS_INTER_SUB_SRC_CLASS(obj) (G_TYPE_CHECK_CLASS_TYPE((klass),GST_TYPE_INTER_SUB_SRC))
typedef struct _GstInterSubSrc GstInterSubSrc;
typedef struct _GstInterSubSrcClass GstInterSubSrcClass;
struct _GstInterSubSrc
{
GstBaseSrc base_intersubsrc;
GstInterSurface *surface;
char *channel;
int rate;
int n_frames;
};
struct _GstInterSubSrcClass
{
GstBaseSrcClass base_intersubsrc_class;
};
GType gst_inter_sub_src_get_type (void);
G_END_DECLS
#endif
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,297
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Q: Installation failed: 504 - Gateway Timeout When installing the big plugins I am getting this error - WooCommerce. This problem does not happens with small plugins. It takes long time to install the plugin.
Installation failed: <!DOCTYPE html> 504 - Gateway Timeout * { box-sizing: border-box; -moz-box-sizing: border-box; -webkit-tap-highlight-color: transparent; } body { margin: 0; padding: 0; height: 100%; -webkit-text-size-adjust: 100%; } .fit-wide { position: relative; overflow: hidden; max-width: 1240px; margin: 0 auto; padding-top: 60px; padding-bottom: 60px; padding-left: 20px; padding-right: 20px; } .background-wrap { position: relative; } .background-wrap.cloud-blue { background-color: #b0e0e9; } .background-wrap.white { background-color: #fff; } .title { position: relative; text-align: center; margin: 20px auto 10px; } .title--regular { font-family: 'Roboto', Arial, sans-serif; } .title--size-large { font-size: 36px; line-height: 46px; } .title--size-semimedium { font-size: 20px; line-height: 28px; } .title--weight-normal { font-weight: 400; } .title--weight-bold { font-weight: 700; } .title--subtitle { padding-bottom: 40px; font-family: 'Open Sans', Arial ,sans-serif; text-align: center; } .error h1, .error p { color: #226d7a; } .error--bg__cover { position: absolute; width: 100%; } .error--shape__clouds svg { display: block; } .abstract-half-dot--circle { z-index: 0; position: absolute; left: 15em; right: 0; width: 300px; height: 300px; margin: 0 auto; } .icon-graphic { z-index: 1; position: relative; width: 300px; height: 300px; margin: 0 auto; } @media screen and (max-width: 767px) { .error--bg__cover { display: none; } .abstract-half-dot--circle { left: 0; } } bg_error_lines circle_dots .st0{fill:#fff;} .st1{fill:#226D7A;} .st2{fill:#22B8D1;} 504 - Gateway Timeout This is a temporary error. Please try again later. clouds_shape
an another error that i faced is :
cURL error 18: transfer closed with 1075636 bytes remaining to read
This is with default clean wordpress and zero plugins
Is this an hosting error with CURL? Does wordpress downloads plugin with curl request?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 251
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After quick bottom-up-swipe followed by another quick bottom-up to close the pane, accompanied by some clumsiness, I left with a screen where notifications were at the background and active, and the home screen with all the running applications was running on top. Only faint hint of familiar shadows showed behind the top layer, where the control was.
After the initial confusion, re-swiping from bottom-up returned the notifications pane at the top.
Seen this a couple of times too, but haven't been able to consistently reproduce. From what I remember the home screen is underlayed by the notifications (i.e. home screen on top) but pulley actions are for the notifications underneath, so the whole thing looks like a mess. Blanking the display doesn't fix it, but as @sjtoik swiping the notifications up does.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,534
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\section{Introduction}
\label{i}
Contrast between the simplicity of constituent
quark model (CQM) classification scheme for
hadrons in particle data tables \cite{PDG} and the
complexity of QCD, exists since quarks were
proposed to explain hadrons \cite{Zweig} and QCD
was proposed as a theory of strong interactions
\cite{ WeinbergGaugeHistory, GellMannQCD, Dubna}.
The contrast can be illustrated by the simplicity
of attempts to build relativistic quark models, in
which one introduces harmonic oscillator
potentials~\cite{Feynman}, in comparison with the
complexity of QCD sum rules \cite{SVZ}, in which
even the state without any hadron, i.e., vacuum,
is meant to contain a complex structure. The
complex structure is reflected in the concept of
condensates.
The vacuum structure in quantum field theory (QFT)
eludes theorists \cite{DiracDeadWood} despite that
they have a complete command of perturbative
calculations. For example, in the case of QCD,
theorists can use asymptotic freedom \cite{af1,af2}
to calculate quark and gluon scattering amplitudes
but they also have to use a parton model
\cite{partonmodel} to relate their calculations to
observables. The point is that the mechanism of
binding of partons continues to be a mystery. The
mystery is understood in principle as related to
the complex features of the theory that lie beyond
reach of perturbative approaches. Regarding these
complex features, palpable progress in calculating
hadronic masses is achieved using lattice
formulation of QCD \cite{Durr,Wilczek}. However,
so far it is not clear how to use QCD for
producing quark and gluon wave functions of
hadrons in the Minkowski space and thus provide
a picture of hadrons of comparable precision to
the picture of atoms achieved in QED.
In this situation, it is interesting to note that
there exists a formulation of QCD which uses
Dirac's light-front (LF) form of Hamiltonian
dynamics \cite{Dirac,DiracQuantum} and does not
introduce any complex vacuum structure
\cite{Wilsonetal}. How could then the LF
formulation of QCD produce the effects that in
other approaches are associated with the concept
of a complex vacuum state, including effects
associated with spontaneous symmetry breaking or
the gluon condensate? Ref. \cite{Wilsonetal}
points out that an effective Hamiltonian expected
to result from a renormalization group analysis of
the canonical LF Hamiltonian of QCD with all
counterterms that are necessary for stabilizing
the analysis, may contain new terms that provide
such effects.
Here it is argued that the renormalization group
procedure for effective particles (RGPEP, see
Section \ref{effectiveparticles}) provides a new
way of thinking about light quarks in LF QCD. In
this new way, the harmonic oscillator potentials,
previously associated in Ref. \cite{GlazekSchaden}
with a gluon condensate in vacuum, can be
interpreted as coming from the gluon content of a
hadron rather than a vacuum. Previous reasoning
in Ref. \cite{GlazekSchaden} was developed using
the instant form \cite{Dirac} of dynamics, i.e., the
standard Hamiltonian evolution in time. Now, the
possibility of considering harmonic oscillator
potentials for light quarks at distances
comparable with the characteristic size of
hadrons, meant here to be equivalent to the
distances on the order of $1/\Lambda_{QCD}$ in the
RGPEP scheme, is pointed out using a decomposition
of hadronic states into the LF Fock space components,
i.e., states obtained from empty vacuum using
creation operators defined in the LF quantization
of fields rather than the standard canonical quantization
of the instant form. These Fock components contain various
numbers of virtual effective particles whose
interactions depend on the renormalization group
scale in the RGPEP scheme. The scale is denoted by
$\lambda$. One can think about $\lambda$ as having dimension
of momentum. Low-energy features of light states in
QCD made of light quarks are expected describable
using an effective Hamiltonian that corresponds
to $\lambda$ comparable with $\Lambda_{QCD}$.
Readers interested in details of RGPEP that are
already known in the case of heavy quark dynamics,
can consult Ref. \cite{GlazekMlynik}, where an
approximate QCD calculation of masses of heavy
quarkonia is described. Here only the case of light
quarks is discussed.
According to Ref. \cite{BrodskyShrock}, if gluon
condensate effects are coming from the gluon
content of hadrons rather than vacuum\footnote{The
quark condensate has also been suggested to
originate in hadronic content instead of vacuum
\cite{BrodskyRobertsShrockTandy}.}, one can avoid
the problem of an excessively large vacuum energy
density in cosmology. However, the new reasoning
presented here does not deal with cosmological
issues. This article concerns only the particular
mechanism that was described in Ref.
\cite{GlazekSchaden}. In the mechanism in Ref.
\cite{GlazekSchaden}, the gluon condensate meant
to be in vacuum provided harmonic oscillator
potentials for constituent quarks in light mesons
and baryons. The potentials agreed with known
phenomenological CQMs~\cite{CQM1,CQM2,CQM3} when
the expectation value of the gluon field strength
squared was equated to the vacuum gluon condensate
value used in the QCD sum rules \cite{SVZ}. It is
argued below that it was not necessary to assume,
as it was done in Ref. \cite{GlazekSchaden}, that
the gluonic expectation value came from the empty
vacuum. The point of view developed here is that
the gluon expectation value might be coming from
the content of a hadron itself, and the relevant
content can in principle be identified in LF QCD
using RGPEP. Potentially broad implications of
such change in interpretation are not discussed
here. The hypothesis put forward in this article
requires a considerable amount of work to verify
and it would be premature to draw broad
conclusions before relevant RGPEP calculations are
completed.
The calculations described here arrive at similar
harmonic potentials to those found in Ref.
\cite{GlazekSchaden} but in a different way and with
a significantly different interpretation. Namely,
instead of starting from the non-relativistic (NR)
Schr\"odinger equation and gluon vacuum
condensate, one starts from the LF QCD Hamiltonian
eigenvalue equation for mesons and baryons.
Canonical quark and gluon field operators are
formally transformed using RGPEP to effective
operators at the running cutoff scales that are
comparable with $\Lambda_{QCD}$. Using color gauge
invariance principle, together with a mean-field
approximation that still needs justification and is
not verified yet by a rigorous {\it ab initio}
calculation, one arrives at the eigenvalue
equations in which expectation values of effective
gluon fields are evaluated in a gluonic component
of hadronic states rather than in the vacuum. The
gluonic component is postulated to be universal.
Then, using new relative momentum variables, one
obtains LF wave functions that describe the
relative motion of hadronic effective constituents
in a boost-invariant way. In summary, the reasoning
described here implies a relativistic picture of
hadrons in which the same harmonic oscillator potentials
that agree with CQMs phenomenology may result from the
gluon content of hadrons in LF QCD rather than from the
vacuum per se.
It should be clarified that harmonic potentials
among effective constituents in lightest hadrons
apply in a situation where color charges are close
to each other. If they were separated by a
distance considerably larger than the size of a
light hadron, their invariant mass would increase
considerably due to the potential. Interactions
other than the potential term alone would be
activated, able to create additional particles and
changing the dynamics, perhaps including a string
of effective gluons.
The article is organized in the following way.
Section \ref{effectiveparticles} describes the
concept of effective particle in the context of
RGPEP and explains why the vertex form factors
(denoted by $f_\lambda$) eliminate changes of the
number of massive virtual effective particles.
This feature is required for thinking that a
hadron can be represented by a convergent
expansion in the basis of the Fock space. The
basis that counts is not built in terms of
canonical quark and gluon operators but in terms
of the effective ones, corresponding to relatively
small scale parameter $\lambda$. The convergence
is not proven, but it is deemed not excluded given
the exponential falloff of the RGPEP form factors
$f_\lambda$ as functions of constituent invariant
masses. Section \ref{RGPEPstates} outlines the
representation of states of light hadrons in the
effective particle basis in the Fock space.
Section \ref{structureW} discusses a perturbative
expansion for the unitary transformation
$U_\lambda$ that connects current, or canonical
quarks and gluons with their CQM counterparts, and
for the transformations $W_{\lambda_1 \lambda_2}$
that connect effective particles that correspond
to different values of the scale $\lambda$. The
eigenvalue problem for light hadrons is considered
in Section \ref{eigenvalue}. This Section
describes our derivation of LF invariant mass
operators for quarks including their minimal
coupling to the gluons condensed in a hadron. Next
Section \ref{Phenomenology} shows how the RGPEP
can be applied in phenomenology of hadrons, with
emphasis on form factors, structure functions, and
connection with AdS/QCD ideas. Section \ref{c}
concludes the paper. Finally, four appendices are
provided in order to support the claim that RGPEP
can be used to verify our reinterpretation of the
gluon condensate in QCD. Appendix
\ref{RGPEPuniversalityappendix} outlines how CQMs
can be viewed as limited representatives of the
same universality class that QCD belongs to.
Appendix \ref{AppendixW} describes details
concerning perturbative calculations of RGPEP
transformations. Presentation of RGPEP beyond
perturbation theory is given in Appendix
\ref{NPRGPEP}. The last Appendix
\ref{OverlappingSwarmsModel} offers a
visualization of the RGPEP scale dependence of
hadronic structure.
\section{ Effective particles and vertex form factors }
\label{effectiveparticles}
Reasoning described in the next sections requires
elements of RGPEP \cite{RGPEP}.\footnote{Appendix
\ref{RGPEPuniversalityappendix} describes RGPEP in
the context of asking how QCD can be transformed
into an effective theory that resembles CQMs as
approximate representatives of the same
universality class.} This section focuses on the
concept of effective particles and the role of
vertex form factors that RGPEP introduces in the
interactions of effective particles, starting from
QCD.
In QCD, RGPEP begins with the canonical LF
Hamiltonian in which the dynamically independent
quark and gluon quantum fields are expanded into
their Fourier components. These components are the
operators that create or annihilate single bare
quarks or gluons of definite momentum. For
brevity, creation and annihilation operators will
be commonly called particle operators, in order
distinguish them where necessary from quark and
gluon field operators.
The canonical LF Hamiltonian contains singular
interaction terms. It must be regulated to avoid
infinities. This is done by introducing
regularization factors in interaction terms. These
factors are constructed to limit relative motion
of particles that participate in interactions. For
example, let the total momentum of all particles
in an interaction term have components $P^+$ and
$P^\perp$. The notation means: $P^+ = P^0 + P^z$,
$z$-axis is the direction distinguished in the
definition\footnote{LF is a hyper-plane in
space-time that is swept by a wave front of a
plane wave of light. In standard notation, the
frame of reference of the inertial observer who
develops a quantum theory is set up so that the
plane wave moves against the $z$-axis in the
chosen frame.} of the LF, $\perp$ denotes
transverse components, i.e., the components $x$
and $y$ that are transverse to the $z$-axis. Let a
selected particle have momentum components $p^+ =
xP^+$ and $p^\perp = xP^\perp + k^\perp$. Then,
the regulating factor for this particle that is
sufficient for taming logarithmic divergences in
QCD can be of the form $x^\delta
\exp{(-k^\perp\,^2 /\Delta^2)}$~\cite{gluons},
where $\delta \rightarrow 0$ and $\Delta
\rightarrow \infty$ are the small-$x$ and
ultraviolet regularization parameters,
respectively. Initially, one also introduces an
absolute infinitesimal lower bound $\epsilon^+ <
p^+ $ for all particle operators in order to
eliminate the LF zero modes. This step amounts,
however, to saying that all particles must have
positive momentum component $p^+$. This
requirement eliminates complex vacuum.
Subsequently, one demands that for every particle
in every interaction term $x > \epsilon$, with an
infinitesimal number $\epsilon$. The number
$\epsilon$ is a fraction of momentum. It is not
related to $\epsilon^+$. Then, when one works in
the limit $\epsilon \, x^{-\delta} \rightarrow 0$
point-wise in $x$ when $\epsilon$ is sent to 0,
the factor $x^\delta$ takes over the regulatory
function from $\epsilon$. As a result, the
variables $x$ range from 0 to 1 and there are
regulating factors $x^\delta$ and $(1-x)^\delta$
at the end points in all interaction vertices. The
variable $x = p^+/P^+_h$ for a particle of
momentum $p$ in a hadron of momentum $P_h$
coincides with the parton model longitudinal
momentum fraction in a hadron in the infinite
momentum frame (IMF). Variable $k^\perp$ is equal
to the transverse momentum of a parton in the IMF
when the parent hadron of the parton moves
precisely along $z$-axis. Note, however, that LF
QCD is formulated in terms of the same variables
$x$ and $k^\perp$ no matter how hadrons move,
i.e., the LF Hamiltonian theory is explicitly
invariant with respect to the 7 Poincar\'e
transformations that preserve the LF. This means
that the theory of a hadron structure in a rest
frame of the hadron has the same form as in the
IMF.
When the regularization is being removed, $\delta
\rightarrow 0$ and $\Delta \rightarrow \infty$,
RGPEP establishes the required ultraviolet
counterterms \cite{RGPEP}. The small-$x$
regularization drops out from dynamics of
colorless states because they cannot produce
long-distance interactions along the LF.
Let the regularized LF Hamiltonian of QCD with
counterterms be denoted by $H$ and let $b$ denote
bare particle operators. RGPEP introduces effective
particles of scale $\lambda$ through a unitary
transformation,
\begin{eqnarray}
\label{blambda}
b_\lambda & = & U_\lambda \, b \, U_\lambda^\dagger \, .
\end{eqnarray}
The corresponding Hamiltonian operator,
\begin{eqnarray}
\label{H}
H_\lambda(b_\lambda) & = & H(b) \, ,
\end{eqnarray}
is a combination of products of operators
$b_\lambda$ with coefficients $c_\lambda$ that are
different from coefficients $c$ of corresponding
products of operators $b$ in the canonical
Hamiltonian with counterterms, $H(b)$. Since
$\lambda$ is related to an upper limit on momentum
transfers in interactions, the operators $b$
correspond to $\lambda = \infty$. For the infinite
$\lambda$, we have $b_\infty = b$ and $H(b) =
H_\infty(b_\infty)$.\footnote{Where it is unlikely
to lead to a confusion, $H_\lambda(b_\lambda)$ is
abbreviated to $H_\lambda$. Later, also operators
such as $H_{\lambda_1} (b_{\lambda_2})$ occur,
meaning an operator that is a combination of
products of particle operators $b_{\lambda_2}$
with coefficients $c_{\lambda_1}$ instead of
$c_{\lambda_2}$.}
RGPEP provides differential equations that produce
expressions for the coefficients $c_\lambda$ in
$H_\lambda$. The operator $U_\lambda$ is
calculable in RGPEP order-by-order in an effective
coupling constant \cite{RGPEP} in the form of
normal-ordered products of particle operators
$b_\lambda$. Appendix \ref{AppendixW} illustrates
how it is done in lowest orders. Appendix
\ref{NPRGPEP} shows how non-perturbative
calculations can be attempted. Appendices
\ref{RGPEPuniversalityappendix}, \ref{AppendixW},
and \ref{NPRGPEP}, provide a formal background for
the entire discussion that follows.
The key feature of $H_\lambda(b_\lambda)$ that
emerges from RGPEP is that all coefficients
$c_\lambda$ contain a form factor $f_\lambda$. The
form factor prevents changes of the total
invariant mass of effective particles undergoing
interaction by more than $\lambda$. The RGPEP
parameter $\lambda$ plays thus the role of a
momentum-space width of vertex form factors in all
interaction terms. In fact, RGPEP is designed to
work this way. When $\lambda$ is small, the form
factors suppress all interactions among massive
particles besides the terms one can call
potentials, i.e., the interaction terms that do
not change the number of massive particles (see
below). In other words, the RGPEP transformation
$U_\lambda$ is designed to identify the dynamical
relationship between nearly massless current
quarks and their field-theoretic interactions at
formally infinite $\lambda$ with massive
constituent quarks and their interactions through
potentials at $\lambda$ so small that potential
models may apply as an approximation to solutions
of the whole theory for states of smalles masses.
If one so desires, transformation $U_\lambda$ can
be kinematically complemented with the Melosh
transformation \cite{Melosh}, in order to express
the spinors typically used for description of
constituent quarks in the constituents
frame\footnote{The CRF frame is an inertial
reference frame in which the total momentum of
constituents treated as free particles of definite
masses has only time component different from
zero. The concept of CRF in LF dynamics is
different from a similar concept in the instant
form of dynamics, because conservation of $P^+$ in
interactions makes the CRF differ from the
center-of-mass system (CMS) of a hadron in the LF
dynamics, while the CRF and CMS are the same in
the instant form of dynamics, where $P^z$ is
conserved by interactions. $P^\perp$ is preserved
in interactions in both forms of dynamics
equally.} (CRF) with the spinors one can
conveniently use in the IMF.\footnote{Ref.
\cite{GellMannQCD} is of interest here because the
last sentence in it suggests one might perhaps use
some collective coordinates as a satisfactory
method for truncating QCD, instead of ``the
brute-force lattice gauge theory approximation!''
While Ref. \cite{GellMannQCD} is quite new to the
author at the time of writing this article, it
should be noted that RGPEP can be viewed as an
attempt of this type. Namely, the effective quarks
and gluons, as constituents of size $\lambda^{-1}
\sim \Lambda_{QCD}^{-1}$, describe collective
modes in dynamics of many small quarks, anti-quarks,
and gluons that nevertheless may be individually
active in virtual processes characterized by large
invariant-mass changes, allowed in a single
interaction only when $\lambda \gg \Lambda_{QCD}$.}
It follows from its definition that the form
factor $f_\lambda$ prevents the number of
effective particles from changing if their masses
$m_\lambda$ exceed $\lambda$. For example, suppose
that a virtual particle of mass $m_\lambda$ emits
another virtual particle with mass $m_\lambda$.
The change of invariant mass of particles in the
interaction is at least $m_\lambda$. The form
factor $f_\lambda$ is designed to quickly tend to
0 for invariant-mass changes greater than
$\lambda$. So, if $m_\lambda$ exceeds $\lambda$,
the form factor eliminates the emission. In
particular, if effective gluons are assumed to
have effective masses $m_\lambda$, the Hamiltonian
$H_\lambda$ will not include interactions that can
change the number of effective gluons when
$m_\lambda > \lambda$. A sizeable mass of
effective gluons in $H_\lambda$ is what one may
expect to happen at small values of $\lambda$ in
QCD because such effective gluon mass is a
candidate for explaining the absence of small
spacing in the spectrum of hadronic masses. There
should be small spacing in the presence of
massless gluons, as it happens in atoms described
by QED with massless photons. In contrast to atoms,
hadrons do not exhibit such small spacing.
Note that a smooth form factor $f_\lambda$ must
allow for some small, transitional range of values
of $\lambda < m_\lambda$. In the transitional
range, interactions that change the number of
effective gluons gradually disappear when
$\lambda$ is lowered below $m_\lambda$. In the
reversed RGPEP evolution in $\lambda$ from small
to large values, gluons will gradually appear in
the dynamics as $\lambda$ increases and $m_\lambda$
decreases so that at some point $\lambda$ becomes
greater than $m_\lambda$.
\section{ RGPEP representation of states of light hadrons }
\label{RGPEPstates}
If QCD is to explain precisely the success of
classification of light hadrons in terms of
constituent quarks with masses on the order of 1/3
of a nucleon mass, a clearly defined procedure
must explain how the masses $m_u$ and $m_d$ of the
lightest quarks increase from their standard model
(SM) values of order 5 MeV in a local gauge theory
to their common constituent value of order
$\Lambda_{QCD}$ in a corresponding effective
theory.\footnote{ How RGPEP is able to relate a
local canonical theory to a non-local effective
theory is explained in \cite{NonlocalH}.} It is
not known yet if RGPEP leads to such increase of
effective masses of light quarks in QCD when the
parameter $\lambda$ is lowered toward
$\Lambda_{QCD}$, or even below $\Lambda_{QCD}$. We
assume here that RGPEP does lead to such
result.\footnote{Quark masses increase as momentum
scale is lowered in perturbation theory developed
in terms of the Feynman diagrams
\cite{pdgQCDmass}. Similar results are obtained
from Dyson-Schwinger equations \cite{Roberts}. LF
constituent models can incorporate running masses
\cite{GlazekNamyslowski}.} This assumption cannot
be rigorously verified yet because the domain of
$\lambda \sim \Lambda_{QCD}$ cannot be reached
using low orders of perturbation theory and
non-perturbative solutions to RGPEP equations are
still not known in QCD.
When $\lambda$ is large in comparison to
$\Lambda_{QCD}$, so that the effective coupling
$g_\lambda$ is small and perturbation theory
applies, the eigenvalue problem involves many Fock
sectors with many quarks of small Lagrangian
masses. In these circumstances, it is hard to
analyze the eigenvalue equations precisely. When
$\lambda$ is lowered, the number of necessary Fock
sectors is expected to decrease because of the
vertex form factors $f_\lambda$ but the coupling
constant $g_\lambda$ increases and it is hard to
evaluate $H_\lambda$ precisely. A way out of this
situation is to calculate $H_\lambda$ in terms of
successive approximations.\footnote{This includes
successive approximations for solutions of the
non-perturbative RGPEP equations in Appendix
\ref{NPRGPEP}.}
A workable RGPEP method for building successive
approximations has been outlined including results
for hadron masses and wave functions in the case
of heavy quarkonia \cite{GlazekMlynik}, where the
number of heavy quarks is nearly constant, equal 2.
In that case, one can take advantage of the NR
approximation for relative motion of quarks, based
on smallness of the effective coupling constant
$g_\lambda$ when $\lambda$ is comparable with heavy
quark masses that are much larger than $\Lambda_{QCD}$.
However, no counterpart for such approximation
scheme is available yet for light quarks in LF QCD
using RGPEP. The reason, stated already above, is
that, in distinction from the case of heavy
quarks, the light quark masses are so much smaller
than $\Lambda_{QCD}$ that many light quarks can a
priori be created or annihilated in the
interactions that limit virtual energy changes by
about $\Lambda_{QCD}$. Therefore, one needs to
approximate reasonably well a great deal of
relativistic dynamics of virtual particles that
are constantly created and annihilated by strong
interactions. It is suggested below that a
reinterpretation of the gluon condensate as a part
of a hadron provides a guideline for moving in that
direction.
It is a conceptual jump that requires a connection
between constituent quarks and structure functions
(see Section \ref{structurefunctions}), but one
can observe here that parton distribution
functions at small $Q^2$ are typically very stable
as functions of $Q^2$ and quarks always carry only
about half of the nucleon momentum
\cite{pdMSR1994, pdMSR1995, pdGRV1995, pdGRV1998,
DurhamPDF}. The gluon component of hadrons, which
plays the role of a gluon condensate in the
picture described below, may only be responsible
for a part of the other half of hadron momentum.
Namely, the constituent quarks also contain
gluons. The issue of how heavy is the part of a
hadron that is made of effective quarks at $
\lambda \lesssim \Lambda_{QCD} $ and how heavy is
the corresponding gluon condensate component, is
discussed in Section \ref{Phenomenology} and
Appendix \ref{OverlappingSwarmsModel}. Firm
conclusions require RGPEP calculations that have
not been done yet.
In the picture described below, the masses of
effective quarks and gluons at some scale $\lambda
\lesssim \Lambda_{QCD}$ are assumed greater than
$\lambda$ itself. All interactions that change the
number of the effective particles are absent and
the only interactions left in $H_\lambda$ with
$\lambda$ near and below $\Lambda_{QCD}$ are
potentials. These potentials are expected to match
potentials used in the CQMs. Our reinterpretation
of the gluon condensate is associated with a definite
result for the potentials that act among the
constituent quarks. In summary, the LF Hamiltonian
of QCD with some $\lambda = \lambda_c \lesssim
\Lambda_{QCD}$ is assumed to describe quarks of
mass $m_c \gtrsim \lambda_c$. These quarks interact
only through potentials, i.e., interactions that do
not change the number of constituents. The subscript
$c$ refers to the word constituent.
An eigenstate $|\psi \rangle$ of the effective
Hamiltonian $H_{\lambda_c} (b_{\lambda_c})$ that
represents a hadron of momentum $P_h$, is a
superposition of effective-particle basis states
($n=2$ for mesons and $n=3$ for baryons)
\begin{eqnarray}
| 1...n \rangle_{\lambda_c} & = & \prod_{i=1}^n
b^\dagger_{i \lambda_c} | 0 \rangle \, , \\
|\psi \rangle & = & \sum_{1...n}
\psi_{\lambda_c}(1,...,n)| 1...n \rangle_{\lambda_c} \, .
\label{psi}
\end{eqnarray}
The arguments 1 to $n$ of the wave function
provide a shorthand notation for momenta, spins,
flavors, and colors of the corresponding
particles. For example, 2 as an argument of a wave
function stands for the three-momentum, spin
projection on $z$-axis, flavor, and color of the
particle number 2. Summation over numbers 1 to $n$
is a shorthand notation for summation over the
quantum numbers of the corresponding particles
including integration over their momenta. In this
abbreviated notation, the eigenvalue equation for
a mass and a wave function of a hadron with momentum
$P_h^+$ and $P_h^\perp$ built from $n$ constituents
of scale $\lambda_c$, takes the form
\begin{eqnarray}
\label{eqpsi}
_{\lambda_c}\langle 1...n | (P_h^+H_{\lambda_c} - P_h^\perp\,^2) |
\psi \rangle
& = &
M^2 \,\,
_{\lambda_c}\langle 1...n | \psi \rangle \, .
\end{eqnarray}
This eigenvalue equation with $n=2$ or $n=3$ is
meant to correspond to the CQM picture of hadrons
as built from 2 or 3 constituent quarks.
Note that the vacuum state $|0\rangle$ is not
changed when $\lambda$ changes. This is a unique
feature of LF dynamics. In the standard, instant
form of dynamics, a change in particle operators
must be accompanied with a change of the
corresponding vacuum state.
One can write the same eigenvalue equation for the
same states using other values of $\lambda$ than
$\lambda_c$. The eigenstate $|\psi\rangle$ is the
same for all values of $\lambda$ but the basis
states and corresponding wave functions depend on
$\lambda$. For $\lambda \gg \Lambda_{QCD}$, the
state $|\psi \rangle$ may contain a giant number of
multi-particle Fock components built by acting with
creation operators $b^\dagger_\lambda$ on the
vacuum, while when $\lambda = \lambda_c$, the entire
state contains only 2 or 3 constituent quarks.
Using RGPEP, one can express constituent quarks at
scale $\lambda_c$ in terms of effective quarks and
gluons corresponding to $\lambda > \lambda_c$,
\begin{eqnarray}
\label{blq}
b_{\lambda_c} & = & W \, b_\lambda \, W^\dagger \, , \\
\label{W}
W & = & U_{\lambda_c}U_\lambda^\dagger \, .
\end{eqnarray}
Momenta, spins, and izospins of the effective particles
are the same, irrespective of the change in parameter
$\lambda$. Since $W|0\rangle = W^\dagger |0\rangle = |0\rangle$,
one has
\begin{eqnarray}
| 1...n \rangle_{\lambda_c} & = & W \prod_{i=1}^n
b^\dagger_{i \lambda} | 0 \rangle \, ,
\end{eqnarray}
and the same eigenstate can be written as
\begin{eqnarray}
\label{psilambda}
|\psi \rangle & = &
\sum_{1...n}
\psi_{\lambda_c}(1,...,n) \, W \, | 1...n \rangle_\lambda \, .
\end{eqnarray}
The result of action of $W$ on the basis state
with $n$ constituents corresponding to scale
$\lambda$, with $\lambda > \lambda_c$, is a
coherent slew of Fock sectors with various
particle numbers, with the number of quarks
grater or equal to the minimal constituent
number $n$ for a hadron. The momentum
space wave functions of the resulting Fock
components are determined by the bound-state
eigenvalue condition of $H_\lambda$.
Although a precise structure of $H_\lambda$ is beyond
insight of perturbation theory for $\lambda \sim
\Lambda_{QCD}$ and $W$ at such scales also contains
non-perturbative dynamics, some generic features
of the Fock wave functions in Eq. (\ref{psilambda})
can be assessed on the basis of generic features of
$W$ implied by properties of operators $U_{\lambda_c}$
and $U_\lambda$ that are visible already in
perturbation theory \cite{GlazekMaslowski}.
Operator $W$ conserves momentum. $W$ can replace
a single effective particle with a bunch of other
particles which carry together the same momentum
as the replaced particle. $W$ can also annihilate
a whole set of particles and create a new set with
the same momentum.
Hence, the colorless meson (M) and baryon (B) states
$|\psi\rangle$ that correspond to the CQM picture
(color factors are written explicitly),
\begin{eqnarray}
\label{Mlambdaq}
|\psi \rangle_M
& = &
\sum_{12} \psi_{\lambda_c}(1,2)
\, {\delta^{ab}\over \sqrt{3}} \,
b_{1 \lambda_c}^{a \dagger}
d_{2 \lambda_c}^{b \dagger}
|0\rangle \, , \\
\label{Blambdaq}
|\psi \rangle_B
& = & \sum_{123} \psi_{\lambda_c}(1,2,3)
\, {\epsilon^{abc}\over \sqrt{6}} \,
b_{1 \lambda_c}^{a \dagger}
b_{2 \lambda_c}^{b \dagger}
b_{3 \lambda_c}^{c \dagger}
|0\rangle \, ,
\end{eqnarray}
are equal to, respectively,
\begin{eqnarray}
\label{Mlambda}
|\psi \rangle_M
& = &
\sum_{12} \psi_{\lambda_c}(1,2)
\, {\delta^{ab}\over \sqrt{3}} \,
\, W \,
b_{1 \lambda}^{a \dagger}
d_{2 \lambda}^{b \dagger}
|0\rangle \, , \\
\label{Blambda}
|\psi \rangle_B
& = &
\sum_{123} \psi_{\lambda_c}(1,2,3)
\, {\epsilon^{abc}\over \sqrt{6}} \,
\, W \,
b_{1 \lambda}^{a \dagger}
b_{2 \lambda}^{b \dagger}
b_{3 \lambda}^{c \dagger}
|0\rangle \, .
\end{eqnarray}
The representation of meson and baryon states in
Eqs. (\ref{Mlambdaq}) and (\ref{Blambdaq}) in
terms of only 2 or 3 effective constituents at
scale $\lambda_c$, is thus transformed into the
representation of the same states in Eqs.
(\ref{Mlambda}) and (\ref{Blambda}) that are built
from effective particles of scale $\lambda$. The
resulting Fock space wave functions for effective
particles of size $\lambda^{-1}$ involve functions
$\psi_{\lambda_c}(1,2)$ and
$\psi_{\lambda_c}(1,2,3)$ through convolutions
that emerge from the sums in Eqs. (\ref{Mlambda})
and (\ref{Blambda}) and the structure of $W$. This
is how the complex Fock-space structure of QCD
hadrons is supposed to contain information about a
simple CQM picture.
\section{ Structure of $W$ }
\label{structureW}
In order to provide an illustration of how $W$ may act
on quark and gluon states, this Section explains
the structure of $W$ obtained just in first-order
perturbation theory in the effective coupling
constant $g_\lambda$. The perturbative expression
is obtained starting from Eqs. (\ref{blambda}) and
(\ref{differentialeq}), which produce together
\begin{eqnarray}
{d \over d\lambda} {\cal H}_\lambda
& = &
\left[ {\cal H}_\lambda , {\cal T}\right] \, ,
\end{eqnarray}
where ${\cal T} = {\cal U}_\lambda^\dagger {d \over d\lambda}
{\cal U}_\lambda$ and ${\cal U}_\lambda = U_\lambda(b_\infty)$.
This means that the functional $ {\cal F} _\lambda$ in Eq.
(\ref{differentialeq}) is set to have the form
\begin{eqnarray}
F_\lambda \left[ {\cal H}_\lambda \right]
& = &
\left[ {\cal H}_\lambda , {\cal T}\right]
\, ,
\end{eqnarray}
and the key to RGPEP is the dependence of $\cal T$
on ${\cal H}_\lambda$ \cite{RGPEP}. This
dependence is designed keeping in mind that the
resulting Hamiltonian $H_\lambda(b_\lambda)$
should contain the form factor $f_\lambda$ in each and
every interaction term. Suitable notation for this
condition is provided by writing $H_\lambda =
f_\lambda G_\lambda$, where $H_\lambda$ contains
the form factor $f_\lambda$ in interaction
vertices while $G_\lambda$ does not. If an
operator $G_\lambda$ contains a product of
creation and annihilation operators $b_\lambda$
with a coefficient $c_\lambda$, the operator
$H_\lambda = f_\lambda G_\lambda$ contains exactly
the same product with coefficient $f_\lambda
c_\lambda$, where $f_\lambda$ depends on the
difference between the invariant mass squared of
particles annihilated by annihilation operators in
the product and the invariant mass squared of
particles created by creation operators in the
product. Both invariant masses are calculated
using the particle kinematical momentum variables
and eigenvalues of part $H_{0 \lambda} = G_{0
\lambda}$ of the Hamiltonian $H_\lambda$.
Thus, using ${\cal H}_\lambda = H_\lambda(b)$
and omitting $\lambda$, one writes
\begin{eqnarray}
{\cal H} & = & f {\cal G} \, = \, {\cal G}_0 + f {\cal
G}_I \, .
\end{eqnarray}
Differentiating $\cal H$ with respect to $\lambda$
one arrives at the equation that defines $\cal T$
in such a way that it must vanish when interactions
vanish (this guarantees that $\cal T$ is expandable
in powers of the coupling constant) and that its
matrix elements between eigenstates of ${\cal G}_
{0\lambda}$ vanish when the differences between
the corresponding eigenvalues vanish \cite{RGPEP},
\begin{eqnarray}
\label{Tau}
{[} {\cal T}, {\cal G}_0 {]}
& = &
\left[ (1-f){\cal G}_I \right]' \, .
\end{eqnarray}
Prime denotes differentiation with respect
to $\lambda$. The solution is denoted by
\begin{eqnarray}
{\cal T}
& = &
\left\{ \left[ (1-f){\cal G}_I \right]'
\right\}_{{\cal G}_0} \, ,
\end{eqnarray}
where the curly braces with subscript ${\cal G}_0$
indicate the energy difference (difference of
eigenvalues of ${\cal G}_0$) in denominator that
results from the commutator of $\cal T$ with
${\cal G}_0$ on the left-hand side in Eq. (\ref{Tau}).
Then $\cal U$ satisfies
\begin{eqnarray}
\label{U}
{\cal U}'
& = &
{\cal U}
\left\{ \left[ (1-f){\cal G}_I \right]'
\right\}_{{\cal G}_0} \, ,
\end{eqnarray}
with condition ${\cal U}_\infty = 1$. Eq. (\ref{U})
can be expanded in powers of the coupling constant.
\subsection{ Perturbative expansion for $U_\lambda$ }
\label{PTforU}
Let us assume here that ${\cal G}_0$ does not
depend on the coupling constant to all orders of
perturbation theory. This means that perturbative
self-interaction terms are included in ${\cal
G}_I$ and ${\cal G}_0$ is actually independent of
$\lambda$.\footnote{ \label{ConformalWindow} This way
of proceeding is not necessary, but it considerably
simplifies the discussion that follows because there
is no need to expand the form factor $f_\lambda$ and
the energy denominators in powers of the coupling
constant.} While this condition may seem quite
restrictive, note that it allows for appearance of
mass terms that are proportional to positive powers
of $\Lambda_{QCD}$, since $\Lambda_{QCD}$ vanishes to
all orders of perturbation theory. Writing expansions
in the bare coupling constant,
\begin{eqnarray}
{\cal U}_\lambda & = & 1 + g u_{\lambda 1}
+ g^2 u_{\lambda 2} + ... \, , \\
{\cal G}_{I\lambda} & = & g {\cal G}_{I\lambda 1} +
g^2 {\cal G}_{I\lambda 2} + ... \, ,
\end{eqnarray}
one obtains equations (for convenience of notation,
the prime is now put outside the curly braces,
which is justified because eigenvalues of ${\cal G}_0$
are independent of $\lambda$)
\begin{eqnarray}
\label{u1'}
u'_{\lambda 1}
& = &
\left\{ (1-f) {\cal G}_{I\lambda 1}
\right\}_{{\cal G}_0}' \, , \\
u'_{\lambda 2}
& = &
u_{\lambda 1} \left\{ (1-f) {\cal G}_{I\lambda 1}
\right\}_{{\cal G}_0}'
+
\left\{ (1-f) {\cal G}_{I\lambda 2}
\right\}_{{\cal G}_0}' \, ,
\end{eqnarray}
etc., with solutions of the form
\cite{GlazekMaslowski}
\begin{eqnarray}
\label{u1}
u_{\lambda 1}
& = &
\left\{ (1-f) {\cal G}_{I\lambda 1}
\right\}_{{\cal G}_0} \, , \\
\label{u2}
u_{\lambda 2}
& = &
{1 \over 2}
u_{\lambda 1}^2
+
{1 \over 2}
\int_\infty^\lambda ds \, [u_{s1}, u_{s1}']
+
\left\{ (1-f) {\cal G}_{I\lambda 2}
\right\}_{{\cal G}_0} \, ,
\end{eqnarray}
etc. This expansion can be rewritten in terms of
the effective coupling $g_\lambda$ for the purpose
of eliminating ultraviolet divergences involved in
the definition of the bare coupling constant and
the corresponding counterterm. However, for $g =
g_\lambda + O(g_\lambda^3)$, one can simply
replace $g$ by $g_\lambda$ in the terms explicitly
listed in Eqs. (\ref{u1}) and (\ref{u2}). In addition,
$U_\lambda$ is unitary by construction. Therefore,
$ {\cal U}_\lambda = U_\lambda (b_\infty) =
U_\lambda(b_\lambda)$ and one can freely replace
$b_\infty$ by $b_\lambda$ in these equations,
obtaining thus first two terms in expansion of
$U_\lambda$ in powers of $g_\lambda$.
\subsection{ Perturbative expansion for $W$ }
\label{PTforW}
Now consider $\lambda_c$ and $\lambda \ge \lambda_c$.
Eq. (\ref{W}) implies $W(b_\infty) =
U_{\lambda_c}(b_\infty)
U^\dagger_\lambda(b_\infty)$, while in Eqs.
(\ref{Mlambda}) and (\ref{Blambda}) the operator
$W$ acts on states created from the vacuum by
operators $b_\lambda$. It is therefore convenient
to use an equivalent expression
\begin{eqnarray}
W(b_\infty)
& = &
U^\dagger_\lambda(b_\lambda) \,
U_{\lambda_c}(b_\lambda) \\
& = &
\left[ 1 + g_\lambda u^\dagger_{\lambda 1} + g_\lambda ^2
u^\dagger_{\lambda 2} + ... \right]
\left[ 1 + g_\lambda u_{\lambda_c 1} + g_\lambda^2 u_{\lambda_c 2}
+ ... \right] \, ,
\end{eqnarray}
where $b_\infty$ in every $u$ on the right-hand side is replaced
by $b_\lambda$. Thus,
\begin{eqnarray}
\label{W12}
W
& = &
1 + g_\lambda W_1 + g_\lambda^2 W_2 + ... \, , \\
\label{W1}
W_1
& = &
\left\{ (f_\lambda - f_{\lambda_c} )
{\cal G}_{I\infty 1} \right\}_{{\cal G}_0} \, , \\
W_2
& = &
{1 \over 2}
\left( u_{\lambda 1} - u_{\lambda_c 1} \right)^2
+
{1 \over 2} \left[ u_{\lambda_c 1}, u_{\lambda 1}
\right]
+
{1 \over 2}
\int_{\lambda_c}^\lambda ds \, [u_{s1}, u_{s1}']
\nn \ps
\label{W2}
\left\{ (1-f_{\lambda_c}) {\cal G}_{I\lambda_c 2}
\right\}_{{\cal G}_0}
-
\left\{ (1-f_\lambda) {\cal G}_{I\lambda 2}
\right\}_{{\cal G}_0} \, ,
\end{eqnarray}
where $b_\infty$ is replaced by $b_\lambda$
everywhere on the right-hand sides.
The perturbative expansion can be similarly
carried out to higher orders. The term of
focus here is $W_1$, as an illustration of
how $W$ acts on quark and gluon states.
This illustration is valid when the coupling
constant $g_\lambda$ is very small, which
means that $\lambda \gg \Lambda_{QCD}$ in the
RGPEP scheme.
To some extent, $W_1$ is also useful as an
indicator of how $W$ acts on states when $\lambda$
approaches $\lambda_c \sim \Lambda_{QCD}$. In this
case, $g_\lambda$ is relatively large. Although
the size of contribution of terms containing $W_k$
with $k > 1$ in comparison to the size of
contribution of the term with $W_1$ for such
small $\lambda$ is not known, it is certain that
all these terms together contribute 0 when
$\lambda = \lambda_c$. They vanish no matter how
large is $g_\lambda$ because integrals in them
effectively range from $\lambda_c$ to $\lambda$
and thus vanish when $\lambda$ tends to
$\lambda_c$. On the other hand, the higher power
of $g_\lambda$ the larger number of integrals
involved. All these integrals must effectively
range from $\lambda_c$ to $\lambda$, so that they
vanish when $\Delta \lambda = \lambda - \lambda_c$
tends to 0 as the appropriate power of $\Delta
\lambda$. The first term in the expansion in
$\Delta \lambda$ must be linear in $\Delta
\lambda$ and $W_1$ is such.
Regarding the size of $g_\lambda$, the range of
$\lambda$ right above $\lambda_c$ is a region
where $g_\lambda$ in the RGPEP scheme in QCD may
be limited in size, instead of having large values
that one may expect on the basis of a
straightforward extrapolation of low-order
expressions obtained in the region of large
$\lambda$ \cite{alphac}. Such limitation of
$g_\lambda$ from above is certainly observed in
simple mathematical models that include asymptotic
freedom and bound states in a renormalization
group procedure for Hamiltonians \cite{AFandBS}.
Similar limitation of $g_\lambda$ is also
suggested in phenomenology of non-perturbative
running couplings based on AdS/QCD ideas and LF
holography \cite{gAdS}. New theoretical
information about the size of $g_\lambda$ at small
$\lambda$ is expected to follow from a
non-perturbative RGPEP equations described in
Appendix \ref{NPRGPEP}.
\subsection{ $W_1$ for quarks and gluons }
\label{W1inQCD}
According to Eq. (\ref{W1}), the first-order
term in $W = 1 + g_\lambda W_1 + O(g_\lambda^2)$,
is given by
\begin{eqnarray}
\label{W1example}
W_1
& = &
\left\{ (f_\lambda - f_{\lambda_c} )
{\cal H}_{I\infty 1} \right\}_{{\cal H}_0} \, ,
\end{eqnarray}
where ${\cal H}_0$ is the term independent of the
bare coupling constant $g$ and ${\cal H}_{I\infty 1}$
is a regulated interaction term proportional to the
first power of $g$ in the canonical LF QCD Hamiltonian,
${\cal H}_\infty = {\cal H}_{0\infty} + g {\cal H}_
{I\infty 1} + O(g^2)$, derived using gauge $A^+=0$ from
the Lagrangian for QCD,
$ {\cal L} = \bar \psi(i\hspace{-4pt}\not\!\!D - m)
\psi - {1\over 2}Tr F^{\mu\nu}F_{\mu\nu}$. Thus,
\begin{eqnarray}
\label{calHinfty}
{\cal H}_\infty & = & \int dx^- d^2 x^\perp
\left[ h_{0\infty} + g h_{I\infty 1} +
O(g^2)\right] \, ,
\end{eqnarray}
where the Hamiltonian density terms independent of $g$ are
\begin{eqnarray}
\label{hinfty0}
h_{0\infty}
& = &
{1\over 2} \bar \psi \gamma^+ {-\partial^{\perp \, 2} +
m^2 \over i\partial^+} \psi
\, - \,
Tr A^\perp (\partial^\perp)^2 A^\perp \, ,
\end{eqnarray}
and the interaction terms order $g$ are
\begin{eqnarray}
\label{hinfty1}
h_{I\infty 1}
& = &
\bar \psi \hspace{-4pt}\not\!\!A \psi
+
2i Tr \partial_\alpha A_\beta [A^\alpha,A^\beta] \, ,
\end{eqnarray}
with $SU(3)$ color notation $ A = A^a t^a$,
$[t^a,t^b] = i f^{abc} t^c$, $Tr(t^a t^b) =
{1\over 2} \delta^{ab}$. The calculation of $W_1$
starts with the above expressions and proceeds as
described in Appendix \ref{AppendixW}.
Appendix \ref{AppendixW} also argues that $W$ for
$\lambda \gtrsim \lambda_c \sim \Lambda_{QCD}$
contains all the terms in ${\cal H}_\lambda$ that
change the number of particles. This means that
$W$ can create a coherent slew of the effective
Fock components at scale $\lambda$ in Eqs.
(\ref{Mlambda}) and (\ref{Blambda}) that are
implied by the 2 or 3 constituent quark wave
functions at $\lambda_c$ in Eqs. (\ref{Mlambdaq})
and (\ref{Blambdaq}). In turn, this means that the
result of action of $W$ is fully encoded through
the Hamiltonian in the wave functions
$\psi_{\lambda_c}(1,2)$ and
$\psi_{\lambda_c}(1,2,3)$ that correspond to the
CQM. These wave functions are solutions to the
eigenvalue problem for $H_{\lambda_c}$, which does
not change the number of constituents. But the
larger $\lambda$ the closer $H_\lambda$ to the
canonical LF Hamiltonian of QCD and the more
complex the states generated by $W$ in Eqs.
(\ref{Mlambda}) and (\ref{Blambda}).
\section{ Eigenvalue problem for light hadrons }
\label{eigenvalue}
According to previous sections, the same
eigenvalue problem for light hadrons can be
written in different ways when one uses different
Hamiltonian expressions that correspond to
different values of $\lambda$. Three values are
distinguished in further discussion: $\lambda =
\infty$, $\lambda \gtrsim \lambda_c$, and $\lambda
= \lambda_c$. The corresponding eigenvalue
problems are equivalent if RGPEP equations are
solved exactly. However, since exact solutions are
not available, one is forced to guess plausible
candidates for the Hamiltonian expressions at
different values of $\lambda$.
At large $\lambda$, insight comes from the
perturbative expansion of RGPEP, using regulated
canonical LF QCD Hamiltonian with counterterms to
start with at $\lambda = \infty$ and taking
advantage of asymptotic freedom. At small
$\lambda$, one has to guess a first approximation.
Phenomenology suggests that some form of a
Hamiltonian for constituent quarks bound in a
potential well is a good candidate for a first
approximation to the QCD Hamiltonian with $\lambda
= \lambda_c$ for hadron states of smallest masses.
These two extreme regions of $\lambda$ need to be
connected to each other within the RGPEP framework
in terms of some interpolation.\footnote{A similar
reasoning applies also in the case of Yukawa theory,
which is not asymptotically free, when one
considers a limited range of scales and effective
coupling constant is sufficiently small from
theoretical point of view to use perturbative
RGPEP \cite{GlazekWieckowski}.} The size of
calculable corrections to any first approximation
constructed this way will eventually tell us how
far from a true RGPEP solution such first
approximation can be. The candidate for the first
approximation that is developed in this article
suggests that the concept of a gluon condensate
may be interpreted in a new way.
\subsection{ Three ways of writing the eigenvalue problem }
\label{3ways}
The first of the three ways, which uses the
regulated canonical Hamiltonian for LF QCD with
all due counterterms, $H_\infty$, amounts to
formal writing of the eigenvalue problem in terms
of Fock states created by products of operators
$b^\dagger_\infty$, $d^\dagger_\infty$, and
$a^\dagger_\infty$ from the vacuum state
$|0\rangle$ in the limit of regularization being
removed,
\begin{eqnarray}
\label{hadroninfty}
H_\infty |\psi \rangle
& = &
P_h^- |\psi \rangle \, .
\end{eqnarray}
In this equation, a reasonably accurate
description of the eigenstate $|\psi \rangle$
presumably requires a very large number of wave
functions for many significant bare-particle Fock
components. The number of such components is
expected to grow when the regularization is
lifted. Although the full structure of $|\psi
\rangle$ is not known precisely, one can assume
some model for one part of it, such as a Fock
sector with a smallest possible number of bare
particles in a meson or a baryon, and try to
determine another part, such as a component with
one additional gluon, using perturbation theory.
Such approach is useful in description of
exclusive processes that involve large momentum
transfers \cite{LepageBrodsky}.\footnote{A
different way from RGPEP to attack the eigenvalue
problem of a LF Hamiltonian for QCD head on is to
use discretized light-cone quantization (DLCQ) in
which one introduces a finite minimal unit of
$p^+$ momentum, $\epsilon^+$, which is the inverse
of the size of a periodicity box for fields as
functions of the position variable $x^-$ on the
LF. Since the total $P^+$ of an eigenstate of
$H_\infty$ is conserved, the unit $\epsilon^+$
naturally limits the number of bare particles in
an eigenstate from above by the ratio
$P^+/\epsilon^+$ \cite{PauliBrodsky1,
PauliBrodsky2, BrodskyPauliPinsky}. Additional
cutoffs need to be imposed in DLCQ in order to
limit transverse momenta of constituents that may
appear in an eigenvalue problem with some
simultaneously fixed eigenvalues of the total
momentum components $P^+$ and $P^\perp$.}
The second of the three ways is designed for the
opposite end of the scale for $\lambda$ in RGPEP,
i.e., when $\lambda = \lambda_c$. The same
Hamiltonian is expected to be expressible there
in terms of operators for constituent quarks. In
this case, one has the eigenvalue equation of the form
\begin{eqnarray}
\label{hadronlambdac}
H_{\lambda_c} |\psi \rangle
& = &
P^-_h |\psi \rangle \, ,
\end{eqnarray}
in which the state $| \psi \rangle $ of a single
light hadron is represented by Eq.
(\ref{Mlambdaq}) for mesons and Eq.
(\ref{Blambdaq}) for baryons. This means that the
light hadron eigenstates are described by only one
wave function, $\psi_c(1...n)$, with $n=2$ or
$n=3$. These wave functions appear in the Fock states
built using creation operators $b^\dagger_{\lambda_c}$
and $d^\dagger_{\lambda_c}$. There are no components
with effective gluons or additional quark-anti-quark
pairs. In other words, the complexity of structure of
light hadrons in QCD is encapsulated in the structure
of constituent quarks. This means that the operators
$b^\dagger_{\lambda_c}$ and $d^\dagger_{\lambda_c}$
in Eqs. (\ref{Mlambdaq}) and (\ref{Blambdaq}) are
complex combinations of products of operators
$b^\dagger_\infty$, $d^\dagger_\infty$, $a^\dagger_\infty$
and their conjugates.
Using Eqs. (\ref{eqpsi}), (\ref{Mlambdaq}) and
(\ref{Blambdaq}), the eigenvalue problem of Eq.
(\ref{hadronlambdac}) for the $n=2$ or $n=3$
constituents is obtained in the form
\begin{eqnarray}
\label{Vcn}
\left( \sum_{i = 1}^n p_i^- + V_{cn}/P_h^+ \right)
\,
{_{\lambda_c}\langle 1...n | \psi \rangle } & = & P_h^- \,
{_{\lambda_c}\langle 1...n | \psi \rangle } \, .
\end{eqnarray}
The unknown element in this form is the
interaction potential term $V_{cn}$. In order to
derive the structure of $V_{cn}$ in mesons and
baryons, we shall employ the third way of writing
the same eigenvalue problem using RGPEP. The third
way suggests the reinterpretation of gluon
condensate that is the subject of this article.
When all gluons considered in the third way are
incorporated in the constituent quarks at
$\lambda_c$, one is left only with the potential
$V_{cn}$, see Appendix
\ref{OverlappingSwarmsModel}.
The third way of writing the same eigenvalue
problem uses $H_\lambda$ with $\lambda \gtrsim
\lambda_c$. The size of $\lambda$ is assumed to be
such that gluons are already somewhat active as
participants in the dynamics while the effective
quarks are still heavy enough for a NR analysis to
apply to their relative motion in a light hadron.
The NR approximation is thinkable for a slowly
moving hadron according to previous Sections,
since interactions in the effective Hamiltonian at
scale $\lambda \gtrsim \lambda_c$ contain form
factors. The effective particles of sizable masses
cannot change their momenta through interactions
by large amounts. Thus, an eigenvector
corresponding to a slowly moving light hadron can
be expected to be described by the wave functions
that have significant values only for small
momenta of the constituents.
The third picture at some scale $\lambda \gtrsim
\lambda_c$ is considered a candidate for a first
approximation to hadrons in RGPEP. The
corresponding Hamiltonian will be proposed below
using ideas motivated by gauge symmetry in its NR
form. Difference between this approximate NR form
and the interactions that can be systematically
calculated using RGPEP in LF QCD, is to be treated
as a perturbation.\footnote{This strategy is
similar to the one described in Ref.
\cite{Wilsonetal} with an artificial linear
potential. RGPEP introduces new elements: the
dynamical transformation from bare to effective
particles available at different scales,
construction of corresponding effective fields,
possibility of using NR approximation at $\lambda
\gtrsim \lambda_c$ for identifying interactions
through gauge symmetry as indicated later in the
text (instead of introducing an artificial
potential), extension beyond perturbation theory
(see Appendix \ref{NPRGPEP}), and invariance with
respect to 7 kinematical LF symmetries for the
resulting interaction terms and wave functions
\cite{NonlocalH}. As a result, the first
approximation potential is proposed to be not a
linear but a quadratic function of a relative
distance between color charges (see next
Sections). The quadratic potential is expected to
cause creation of additional particles when a
distance between two colored particles increases
and the net linear increase of energy with
distance must ultimately result from the energy of
particles created along a line that connects the
two widely separated color charges.} The
approximate picture at $\lambda \gtrsim \lambda_c$
is described in more detail in the next Sections.
Here we only suggest that RGPEP provides a scheme
that may be used in future calculations of
$V_{cn}$ in Eq. (\ref{Vcn}) at $\lambda_c$ by
evolving the third-way picture at $\lambda \gtrsim
\lambda_c$ down to $\lambda_c$ and taking
advantage of the RGPEP reinterpretation of the gluon
condensate that is described in the next Sections.
\subsection{ Light hadron states at $\lambda \gtrsim \lambda_c$ }
\label{states}
When $\lambda \gtrsim \lambda_c$, the meson and
baryon states are described by Eqs. (\ref{Mlambda})
and (\ref{Blambda}), respectively. In these equations,
the operator $W$ acts on the Fock components with
2 or 3 quarks of scale $\lambda$ and creates additional
components. The meson state is simpler than the
baryon state and it will be discussed first.
The meson state is
\begin{eqnarray}
\label{mesonWork}
|\psi_M\rangle & = &
\sum_{12} \psi_{\lambda_c}(1,2) \,
\, W \,
b_{1 \lambda}^\dagger
d_{2 \lambda}^\dagger
|0\rangle \, .
\end{eqnarray}
From Eqs. (\ref{W1exact}), (\ref{WY1ra}) and
(\ref{WY1a}), it is clear that $W$ shares
properties of the interaction Hamiltonian.
For example, if $H_{\lambda I}$ changes a
group of particles to another group, a similar
change with additional factors results from
action of $W$. So, out of 2 effective quarks
in a meson (3 in a nucleon) more quarks and
gluons are created. In order to use local gauge
symmetry to propose the approximate form of
the state that results from action of $W$,
it is useful to represent the state of
Eq. (\ref{mesonWork}) in position space.
This is done using quantum fields built from
effective particle operators at scale $\lambda$.
The effective quantum fields are constructed
using Eq. (\ref{quantumfieldpsi}) for quarks and
Eq. (\ref{quantumfieldA}) for gluons and replacing
operators $b$, $d$ and $a$ with $b_\lambda$,
$d_\lambda$ and $a_\lambda$, respectively. Thus,
the operator $\psi_\lambda(x)$ on the LF is built
in the same way from $b_\lambda$ and $d^\dagger_
\lambda$ as the canonical operator $\psi(x)$ is
built in Eq. (\ref{quantumfieldpsi}) from the
operators $b$ and $d^\dagger$ that are equal
$b_\infty$ and $d^\dagger_\infty$, respectively;
\begin{eqnarray}
\label{quantumfieldpsilambda}
\psi_\lambda(x)
& = &
\sum_{\sigma c f} \int [k]
\left[ \chi_c u_{\lambda fk\sigma} b_{k\sigma c f \lambda } e^{-ikx} +
\chi_c v_{\lambda fk\sigma} d^\dagger_{k\sigma c f \lambda } e^{ikx}
\right] \, .
\end{eqnarray}
The spinors $u_{\lambda fk\sigma}$ and $v_{\lambda
fk\sigma}$ include corresponding vectors in flavor
space. Spinors might depend on $\lambda$ if the
effective quark masses in the field expansion are
allowed to depend on $\lambda$.\footnote{Note that
the independent field components, $\psi_+ = \Lambda^+
\psi$, $\Lambda^+ = (1/2)\gamma^0 \gamma^+$, are
independent of the quark masses, and inclusion of
effective masses in spinors is merely a way of
useful notation for some effects of interactions.}
Similarly, operator $A^\mu_\lambda(x)$ on the LF
is built from $a_\lambda$ and $a^\dagger_\lambda$
as the canonical operator $A^\mu(x)$ is built in
Eq. (\ref{quantumfieldA}) from the operators $a$
and $a^\dagger$ that are equal $a_\infty$ and
$a^\dagger_\infty$, respectively. Gluon
polarization vectors do not depend on $\lambda$.
\begin{eqnarray}
\label{quantumfieldAlambda}
A_\lambda^\mu (x)
& = &
\sum_{\sigma c} \int [k] \left[ t^c \varepsilon^\mu_{k\sigma}
a_{k\sigma c \lambda } e^{-ikx} + t^c \varepsilon^{\mu *}_{k\sigma}
a^\dagger_{k\sigma c \lambda } e^{ikx}\right] \, .
\end{eqnarray}
As a consequence, the dynamically independent
components, $\psi_\lambda = \Lambda^+
\psi_\lambda$ and $A^\perp_\lambda$ in $A^+=0$ gauge
on the LF $x^+=0$ have the same commutation relations
as in a canonical theory.
With particle operators at $\lambda = \lambda_c$, an
effective constituent quark field operator is
constructed in the same way. When this field is
used to describe a slowly moving meson, it is useful
to write the field as
\begin{eqnarray}
\psi_{\lambda_c}
& = &
\left[ \begin{array}{c} U_{\lambda_c}(\vec x\,) \\
V_{\lambda_c}(\vec x\,)
\end{array} \right] \, ,
\end{eqnarray}
where the upper two-component field $U_{\lambda_c}$
annihilates quarks and the lower two-component
$V_{\lambda_c}$ creates anti-quarks.\footnote{The
three-vector notation $\vec x$ or $\vec k$ refers
here to the LF co-ordinates $(x^-, x^\perp)$ in position
space. A similar notation is sometimes also used for
momentum variables $(k^+,k^\perp)$. However, when we
proceed later to the Schr\"odinger eigenvalue problem
for effective Hamiltonians for light hadrons, the same
notation will be adopted also for three-vectors built
from $\perp$ and $+$ or $-$ components in such a way
that the standard three-dimensional notation respecting
rotational symmetry will be natural.}
Thus, by inverting the Fourier transforms through
integration over the LF hyperplane and using
conventions explained in Appendix \ref{AppendixW},
the state in Eq. (\ref{Mlambdaq}) for a meson of
momentum $P_h$ can be written as
\begin{eqnarray}
\label{mesonpositionc}
|\psi \rangle_M
& = &
{1 \over \sqrt{3}}
\int { d^3x_1 \, d^3x_2 \over 4m^2}
\int[12] \,
16\pi^3 P_h^+ \delta^3(P_h-k_1-k_2) \,
\nn \ts
e^{-i (k_1 x_1 + k_2 x_2)} \,
U^\dagger_{\lambda_c}(x_1) \,
\psi_{2\times 2}(\vec k_{12}) \,
V_{\lambda_c}(x_2)
|0\rangle \, ,
\end{eqnarray}
where $\psi_{2\times 2}(\vec k_{12})$ denotes the
$2\times 2$ matrix wave function of relative
motion of the quarks. The three-vector $\vec
k_{12}$ can be defined as a relative momentum of the
quarks in the CRF. Details of the definition of
$\vec k_{12}$ are not essential at this point
but later discussion will include relevant details.
In order to obtain analogous position representation
of the meson state in Eqs. (\ref{Mlambda}) or
(\ref{mesonWork}), one needs to apply $W$ to the
expression in Eq. (\ref{mesonpositionc}). The result is
\begin{eqnarray}
\label{meson71}
|\psi \rangle_M
& = &
{1 \over \sqrt{3}}
\int { d^3x_1 \, d^3x_2 \over 4m^2}
\int[12] \,
16\pi^3 P_h^+ \delta^3(P_h-k_1-k_2) \,
\nn \ts
e^{-i (k_1 x_1 + k_2 x_2)} \,
W \, U^\dagger_\lambda(x_1) \,
\psi_{2\times 2}(\vec k_{12}) \,
V_\lambda(x_2)
|0\rangle \, .
\end{eqnarray}
A similar expression is generated in terms of
three quark fields and $W$ for baryons.
Integration over momenta in these expressions
generates position wave functions as coefficients
in the expansion of meson or baryon states into
basis states that are created from the LF vacuum
by action of a product of two or three quark
fields at scale $\lambda$ and $W$.
Continuing with the meson states, the central
hypothesis about action of $W$ is that gauge
symmetry forces the result of action of $W$
on a state of two quarks at scale $\lambda$ to
have the form
\begin{eqnarray}
&& W \, \int d^3x_1 d^3x_2 \, U^\dagger_\lambda(x_1) \,
\psi(x_1,x_2) \,
V_\lambda(x_2)
|0\rangle \\
& = &
\int d^3x_1 d^3x_2 d^3 x_0 \,
U^\dagger_\lambda(x_1) \, W(x_1, x_2, x_0) \,
V_\lambda(x_2) |0\rangle \, ,
\nonumber \\
\end{eqnarray}
where the operator $W(x_1, x_2, x_0)$ is responsible
for creating components generated by $W$. These
components can be of two kinds. One kind is formed
by operators that carry the color of initial two
quarks. For example, if $W$ generates a gluon from
a quark, the generated gluon and the emerging quark
together carry the color of the initial quark.
The other kind is formed by colorless operators.
For example, $W$ may cause emission of two gluons
by a quark and the gluons may form a color
singlet.
These two kinds of contributions are encapsulated
in $W(x_1, x_2, x_0)$ by writing
\begin{eqnarray}
\label{mesonW}
W(x_1, x_2, x_0)
& = &
\psi(x_1,x_2,x_0) \, T(x_1, x_2) \, G^\dagger(x_0) \, ,
\end{eqnarray}
where $\psi(x_1,x_2,x_0)$ denotes a new wave
function at scale $\lambda \gtrsim \lambda_c$,
$G^\dagger(x_0)$ denotes the colorless component
of the state, and
\begin{eqnarray}
\label{mesonT}
T(x_1, x_2) & = &
P \exp \left[-ig \int_{x_2}^{x_1} dx^\mu A_\mu(x)
\right]\, ,
\end{eqnarray}
is the color-transport factor along a straight
line between quarks that maintains local gauge
symmetry by bringing in required gluon
fields.\footnote{In the LF gauge $A^+=0$, one
may expect only transverse separation between
quarks to count. However, the dependent (constrained)
components of fields, $\psi_-$ and $A^-$, contribute
to the effective interactions in a non-trivial
way and one has to keep in mind that the complete
effective theory at small $\lambda$ should have
full rotational symmetry restored. Therefore, a
complete RGPEP expression for $W$ must account
for the dynamics along $x_1^- -x_2^-$ as well as
along $x_1^\perp - x_2^\perp$. This issue will be
addressed later by introducing effective interactions
that respect rotational symmetry inside slowly
moving hadrons through a new definition of
three-dimensional relative momenta of constituents
to which the minimal gauge coupling rule can
be applied in a rotationally symmetric way.}
This factor is constructed in analogy with Ref.
\cite{GlazekSchaden}.
The operator that generates the colorless
component of a hadron state, $G^\dagger(x_0)$,
will provide the contribution in dynamics of
quarks that is associated with gluon condensation
in hadrons, rather than in a vacuum. Namely,
instead of the vacuum expectation values of
operators considered in Ref. \cite{GlazekSchaden},
such as $\langle \Omega | A^i A^j | \Omega
\rangle$ with $i,j = 1,2$, the Schr\"odinger
equation for the wave function $\psi(x_1,x_2,x_0)$
will involve expectation values
\begin{eqnarray}
\label{expectation}
\langle A^i A^j \rangle_G
& = &
{ \langle G | A^i A^j | G \rangle \over
\langle G | G\rangle } \, , \\
|G\rangle & = & G^\dagger(x_0) |0\rangle \, .
\end{eqnarray}
A similar reasoning is followed regarding baryons.
In the case of baryons, one has to deal with three
color-transport factors that are constructed in
analogy to Ref. \cite{GlazekSchaden}. The operator
$G^\dagger$ in baryons generates the state
$G^\dagger|0\rangle$ that plays the same role that
the vacuum state $|\Omega \rangle$ played in Ref.
\cite{GlazekSchaden}. It is assumed that the
colorless components of mesons and baryons are
approximately the same. The issue of universality
of expectation values such as in Eq. (\ref{expectation})
for mesons will be further discussed below when
we come to the construction of the effective
Hamiltonian.
In summary, the claim of gauge symmetry regarding
the colorless basis states of quarks and gluons
at scale $\lambda$ from which mesons and baryons
are made, is that they are of the form (all fields
are effective at scale $\lambda \gtrsim \lambda_c$)
\begin{eqnarray}
\label{Mnewbasisx0}
|\vec x_1, \vec x_2, \vec x_G \rangle
& = &
\sum_{ab} {\delta^{ab} \over \sqrt{3}} \,
(u_{1\lambda}^\dagger T_1)^a \,
(T_2^\dagger v_{2\lambda})^b \,
G^\dagger |0\rangle \, , \\
\label{Bnewbasisx0}
|\vec x_1, \vec x_2, \vec x_3, \vec x_G \rangle
& = &
\sum_{abc} {\epsilon^{abc} \over \sqrt{6} } \,
(u_1^\dagger T_1)^a\,
(u_2^\dagger T_2)^b\,
(u_3^\dagger T_3)^c\,
G^\dagger |0\rangle \, ,
\end{eqnarray}
where
\begin{eqnarray}
\label{generalT}
T_i
& = &
e^{-ig \int_{\underline{x}}^{x_i} dx_\mu A^\mu } \, ,
\end{eqnarray}
and $\underline{x} = (\vec x_1 + \vec x_2)/2$ in
a meson and $\underline{x} = (\vec x_1 + \vec x_2
+ \vec x_3)/3$ in a baryon. The factor $T$ is
defined here along a straight path in such a way
that it reproduces the color-transport factor in
mesons in Eq. (\ref{mesonT}). The path dependence
is ignored as an unnecessary complication at the
level of mean-field approximation. The operator
$G$ with argument $\vec x_G$ represents the white
energy density background called glue. It is
assumed to be a scalar boson field with
corresponding commutation relations.
There is a trouble on the LF with non-locality of
the boson commutation relations in the direction
of $x^-$, which requires a solution. However, the
effective dynamics that will be constructed in
next Sections will explicitly circumvent this
difficulty by using an operator $G$ that is a
function of a three-dimensional position space
variables associated with slowly moving hadronic
constituents in a slowly moving hadron. Thus, the
problem ultimately requiring a solution is not so
much the non-locality of a scalar field but how LF
QCD can generate a rotationally symmetric
effective theory for light hadrons. The
construction offered in next Sections proposes to
treat $G$ as a field depending on a suitable
three-dimensional position variables in which it
can be local in a way that respects rotational
symmetry.
The quanta of $G$ are meant to represent
excitations of the states of gluons condensed
inside hadrons. The quanta of $G$ are not
point-like. Their size is characterized by
$1/\lambda$ and can be considered roughly on the
order of $1/\Lambda_{QCD}$, i.e., the quantum
extends over the volume of an entire
hadron.\footnote{ The effective glue degree of
freedom represents contributions of all white Fock
sectors of effective particles that share the
hadron momentum to a varying degree as $\lambda$
varies.} There exists a possibility to associate
vector or even higher spin nature with $G$,
contributing to the hadron spin, but there is no
need to do so here.
The basis states are orthonormal in the sense that, for
all quarks being different,
\begin{eqnarray}
\langle
\vec x_1, \vec x_2, \vec x_G |
\vec x_{1'}, \vec x_{2'}, \vec x_{G'}
\rangle
& = &
\delta_{11'} \delta_{22'} \delta_{GG'} \, , \\
\langle
\vec x_1, \vec x_2, \vec x_3, \vec x_G |
\vec x_{1'}, \vec x_{2'}, \vec x_{3'}, \vec x_{G'}
\rangle
& = &
\delta_{11'} \delta_{22'} \delta_{33'} \delta_{GG'} \, ,
\end{eqnarray}
and $\delta_{kk'}$ includes $\delta^3(\vec x_k -
\vec x_{k'})$.\footnote{With the qualification
that locality in the $x^-$ direction requires
proper definition of $z$-components in a
rotationally invariant effective theory to be
discussed in next Sections.} If quark quantum
numbers besides color are not different (this
comment concerns only the baryon case), one
has to adjust the normalization of basis states
by including all permutations that contribute
to the scalar products.
In the abbreviated notation used in the remaining
part of the article,
\begin{eqnarray}
|\vec x_1, \vec x_2, \vec x_G \rangle
& = &
|12G\rangle \, , \\
\label{Bnewbasisx}
|\vec x_1, \vec x_2, \vec x_3, \vec x_G \rangle
& = &
|123G\rangle \, .
\end{eqnarray}
In these abbreviated notation, the
meson and baryon states read:
\begin{eqnarray}
\label{mesonab}
|\psi \rangle_M
& = &
\sum_{12G} \psi(12G) \,
|12G\rangle \, , \\
\label{baryonab}
|\psi \rangle_B
& = &
\sum_{123G} \psi(123G) \,
|123G\rangle
\, .
\end{eqnarray}
\subsection{ Preliminaries concerning Hamiltonian
density at $\lambda \gtrsim \lambda_c$ }
\label{dynamicsl}
Using the concept of effective quark and gluon
fields, one can propose that the LF Hamiltonian
at $\lambda \gtrsim \lambda_c $ be expressible,
by analogy to the canonical theory, in terms of
an integral over the LF of a density that is a
function of the fields. Thus,
\begin{eqnarray}
\label{Hlambda}
H_\lambda & = & \int_{LF} \, \left( h_{0\lambda} +
\, h_{I\lambda} \right)
\, ,
\end{eqnarray}
where the density $h_{0\lambda}$ is bilinear in
the fields and renders single-particle operators
while the density $h_{I\lambda}$ describes
interactions in terms of products of at least
three fields (the interactions involve
at least three effective particles).
Using power-counting \cite{Wilsonetal}, the
bilinear density at scale $\lambda$ can
be assumed in the form (all fields at $x^+=0$)
\begin{eqnarray}
\label{hlambda0}
h_{0\lambda}
& = &
{1\over 2} \bar \psi_\lambda \gamma^+
{-\partial^{\perp \, 2} + m^2
\over i\partial^+} \psi_\lambda \, + \,
Tr A^\perp_\lambda (- \partial^\perp \,^2
+ m_g^2) A^\perp_\lambda + CT_{I\lambda}\, ,
\end{eqnarray}
where the masses correspond to $\lambda$. The
symbol $CT_{I\lambda}$ denotes mass-like terms
needed to make the mass parameters $m$ and $m_g$
to count as masses in the eigenvalue equations
for light mesons and baryons. In other words,
$CT_{I\lambda}$ are by construction cancelled by
self-interactions in light colorless states.\footnote{
There is no need to specify these terms further
here. Such terms can be identified explicitly in
perturbation theory, e.g., proceeding in a similar
way to how it is done in the eigenvalue problem
for heavy quarkonia in Ref. \cite{GlazekMlynik}.}
The interaction density $h_{I\lambda}$ in Eq.
(\ref{Hlambda}) must be non-local in the sense
that it involves products of fields at different
points.\footnote{The non-locality discussed here
is due to the RGPEP vertex form factors of width
$\lambda$ in momentum space. The RGPEP-induced
non-locality must not be confused with the
canonical non-locality of LF Hamiltonians in which
the dependent parts of fields, such as $\psi_- =
\Lambda_- \psi$ and $A^-$, involve inverse powers
of $i\partial^+$, which is a non-local integral
operator.} In particular, the minimal coupling
between quarks and gluons, which in the
canonical gauge theory is of the form
\begin{eqnarray}
H_{MC\infty}
& = &
\int d^3x \, h_{MC\infty}(x) \, ,
\end{eqnarray}
with
\begin{eqnarray}
\label{hiinfty}
h_{MC\infty}(x)
& = &
g \, \bar \psi(x) \not\!\!A(x) \, \psi(x) \, ,
\end{eqnarray}
in the effective theory must take a non-local form
that in a lowest-order approximation is of the
type \cite{NonlocalH}
\begin{eqnarray}
\label{HpsiApsi}
H_{MC\lambda}
& = &
\int d^3x_1 \, d^3 x_2 \, d^3 x_3 \,
h_{MC\lambda}(x_1, x_2, x_3)
\, ,
\end{eqnarray}
with
\begin{eqnarray}
\label{hpsiApsi}
h_{MC\lambda}
(x_1, x_2, x_3)
& = & g_\lambda f_\lambda (x_2-x_1, x_3-x_1) \,
\bar \psi_\lambda(x_1) \, \hspace{-4pt}\not\!\!A_\lambda(x_2) \,
\psi_\lambda(x_3) \, .
\end{eqnarray}
This non-local interaction term appears with
other non-local terms in the Hamiltonian
obtained from RGPEP at scale $\lambda$.
The dependent and independent components of
the fields are grouped in Eq. (\ref{hpsiApsi})
according to a free theory. The dependent
components involve the inverse of $i\partial^+$,
which is a non-local operator. This non-locality
is of the same type as in a canonical theory and
the RGPEP non-locality appears here on top of
the canonical one.
The non-local interaction terms with small $\lambda$
can be simplified considerably if the domain of
action of the Hamiltonian is restricted to slowly
moving effective quarks. Namely, consider
again the case of Eqs. (\ref{HpsiApsi}) and (\ref{hpsiApsi})
and introduce a gradient expansion of the form
\begin{eqnarray}
h_{MC\lambda}
(x_1, x_2, x_3)
& = &
g_\lambda f_\lambda (x_2-x_1, x_3-x_1) \, \bar \psi_\lambda(x_1) \,
\nn \ts
\left[ \not\!\!A_\lambda(x_1) +
\partial \hspace{-1pt}\not\!\!A_\lambda(x_1)(x_2-x_1) + ...\right]
\nn \ts
\left[ \psi_\lambda(x_1) + \partial\psi_\lambda(x_1) (x_3-x_1) +
... \right] \, .
\end{eqnarray}
The three dots indicate terms with higher
derivatives. If all effective particles move
slowly and the derivative terms are small,
the Hamiltonian can be approximated by the
first term in the expansion,
\begin{eqnarray}
H_{I\lambda}
& = &
\int d^3x_1 \, d^3 x_2 \, d^3 x_3 \, h_{I\lambda} \\
& \sim &
g_\lambda \int d^3x \,
\bar \psi_\lambda(x) \not\!\!A_\lambda(x) \, \psi_\lambda(x) \,
\int d^3y \, d^3 z \, f_\lambda (y, z) + ...\, .
\end{eqnarray}
This means that the non-local effective interaction
in a slowly moving, NR system still looks like a local
one except that its strength is determined not solely
by the coupling constant $g_\lambda$ but also by the
integral of the non-local form factor on the LF hyperplane,
\begin{eqnarray}
\tilde g_\lambda
& = &
g_\lambda
\int d^3y \, d^3 z \, f_\lambda (y, z) \, .
\end{eqnarray}
The point is that the effective Hamiltonian
density at $\lambda \gtrsim \lambda_c$ may partly
resemble a Hamiltonian of local gauge theory in
its terms that couple quarks with gluons even though
the actual effective interactions are non-local,
provided that one limits the domain of the
Hamiltonian to slowly moving hadrons. In this
case, construction of approximate candidates for
the LF Hamiltonian density of coupling between
quarks and gluons, before they are corrected using
RGPEP, may proceed in analogy to QED
\cite{BjorkenKogutSoper}. This means that one uses
the gauge $A^+=0$. The derivative
$i\partial^\perp$ in the quark kinetic energy in
Eq. (\ref{hlambda0}) is supplied with the minimal
coupling addition of $g A^\perp$ (the product
of derivatives is separated by the inverse of
$i\partial^+$, which is not altered). In addition,
one includes terms dictated by constraints that imply
the result of Eq. (\ref{hiinfty}) for the
quark-gluon coupling.
Much less is understood about the gluon part of
the effective Hamiltonian density for light
hadrons at $\lambda \gtrsim \lambda_c$. The lack
of understanding of the Hamiltonian is reflected
in the lack of understanding of the gluon
components in its eigenstates. One would have to
calculate the operator $H_\lambda$ precisely in order to
uncover the information it contains about how to
build a model approximating light hadrons in LF QCD.
While RGPEP provides tools for such calculations,
the remaining part of the paper is only devoted to
deriving the model that can be treated as a first
approximation.
A simple candidate for an approximate Hamiltonian
for light hadrons in LF QCD is constructed in the
next Section assuming that: (1) the result of
action of $W$ in RGPEP can be represented by
inclusion of the color-transport factors $T$ that
maintain local gauge symmetry of hadronic states,
(2) the operator $G$ represents the condensation
of gluons inside hadrons, (3) a mean field
approximation can be applied to the gluon field
operator $A$ in the minimal coupling of effective
quarks to the gluons condensed in a hadron, and
(4) the effects of quark and gluon binding that
are not treatable in the mean field approximation
can be included by an ad hoc Gaussian approximation
to the wave function of relative motion of
constituent quarks with respect to the condensed
gluons.
\subsection{ Mass squared with minimal coupling
at $\lambda \gtrsim \lambda_c$ }
\label{dynamics2}
The leading idea is that the approximate LF Hamiltonian
for 2 or 3 quarks at scale $\lambda \gtrsim \lambda_c$
in light hadrons should have a form compatible with
minimal coupling between quarks and gluons and it
should respect Poincar\'e symmetry. Realization of
the idea employs the four assumptions listed at
the end of the previous Section in the following
way.
The LF Hamiltonian eigenvalue problem at $\lambda
\gtrsim \lambda_c$ for a light hadron built from
2 or 3 quarks and condensed gluons,
\begin{eqnarray}
H_\lambda |\psi\rangle & = & {M^2 + P_h^{\perp \, 2} \over
P^+_h} |\psi\rangle \, ,
\end{eqnarray}
can be written in terms of the eigenvalue equation
for the effective invariant mass operator of the
hadron, $ {\cal M} ^2_\lambda$, in the form
\begin{eqnarray}
{\cal M} ^2_\lambda |\psi\rangle & = & M^2 |\psi\rangle \, .
\end{eqnarray}
One can think about the operator on the left-hand side
of this equation in terms of the invariant mass squared
of free particles plus interaction terms.
For a free quark of mass $m_1$ and a free anti-quark of
mass $m_2$, one has
\begin{eqnarray}
{\cal M} ^2_{free} & = &
\left( \sqrt{m_1^2 + \vec p_1^{\,2}}
+ \sqrt{m_2^2 + \vec p_2^{\,2}} \right)^2
- (\vec p_1 + \vec p_2)^2 \, .
\end{eqnarray}
For slowly moving particles, neglecting terms smaller than
the ones that are quadratic in momenta, the NR approximation
renders
\begin{eqnarray}
{\cal M} ^2_{free}
& = &
( m_1+ m_2)^2 + {m_1+ m_2 \over m_1} \, \vec p_1^{\,2}
+ {m_1+ m_2 \over m_2} \, \vec p_2^{\,2}
- \vec P^{\,2}_{q \bar q}
\, ,
\end{eqnarray}
where $\vec P$ denotes the total momentum of the
two quarks. Using
\begin{eqnarray}
\beta_1 & = & {m_1 \over m_1 + m_2} \, , \\
\beta_2 & = & {m_2 \over m_1 + m_2} \, ,
\end{eqnarray}
one has
\begin{eqnarray}
{\cal M} ^2_{free}
& = &
( m_1+ m_2)^2 + {1 \over \beta_1} \, \vec p_1^{\,2}
+ {1 \over \beta_2} \, \vec p_2^{\,2} - \vec P^{\,2}_{q \bar q} \, .
\end{eqnarray}
According to the gauge rule of minimal coupling,
the interaction of quarks with the condensed
gluons should be described by
\begin{eqnarray}
\label{minimalM}
{\cal M} ^2_{minimal}
& = &
( m_1+ m_2)^2 + { \left( \vec p_1 - g\vec A_1 \right)^2 \over \beta_1} \,
+ { \left( \vec p_2 - g\vec A_2 \right)^2 \over \beta_2} \,
- \vec P^{\,2}_{q \bar q} \, ,
\end{eqnarray}
where $\vec A_i$ is an abbreviation for $\vec
A(x_i)$. Analogous reasoning in the case of
baryons yields
\begin{eqnarray}
\label{minimalB}
{\cal M} _{minimal}^2
& = &
\left( \sum_{i=1}^3 m_i \right)^2
+
\sum_{j=1}^3 { \left( \vec p_j - g \vec A_j\right)^2 \over \beta_i}
- \vec P^{\,2}_{3q} \, ,
\end{eqnarray}
where $\beta_i = m_i/(m_1 + m_2 + m_3)$.
For all quarks having the same mass $m$,
$\beta_M = \beta_1 = \beta_2 = 1/2$ in mesons and
$\beta_B = \beta_1 = \beta_2 = \beta_3 = 1/3$ in baryons.
Assuming these simplifications, the minimal coupling
terms in Eqs. (\ref{minimalM}) and (\ref{minimalB})
can be interpreted following Ref. \cite{GlazekSchaden}
in the context of standard Hamiltonian dynamics as
resulting from the Hamiltonian operator with a proper
$\beta$, i.e., $\beta_M$ in mesons and
$\beta_B$ in baryons,
\begin{eqnarray}
\label{Hmin}
H_{min} & = & {1 \over \beta}
\int d^3 x \, : \bar \psi(\vec x \,)
[ -i \vec \nabla_x - g \vec A(\vec x\,) ]^2 \psi(\vec x\,) :
\, .
\end{eqnarray}
\subsection{ Reinterpretation of the gluon condensate }
\label{dynamics3}
Using Eqs. (\ref{mesonab}), (\ref{baryonab}),
and (\ref{Hmin}), one can consider the eigenvalue
problems
\begin{eqnarray}
H_{min} \sum_{12G} \psi(12G) \, |12G\rangle
& = &
M^2_{q \bar q} \, \sum_{12G} \psi(12G) \, |12G\rangle \, , \\
H_{min} \sum_{123G} \psi(123G) \, |123G\rangle
& = &
M^2_{3q} \, \sum_{123G} \psi(123G) \, |123G\rangle \, ,
\end{eqnarray}
for quark subsystems in mesons and baryons and
project them on the corresponding basis states.
The calculation proceeds in a similar way to the
one in Ref. \cite{GlazekSchaden}, with three exceptions.
The first difference is that the eigenvalues
$M^2_{q \bar q}$ and $M^2_{3q}$ refer only to the
$q\bar q$ subsystem in mesons, instead of the entire
meson mass squared, and to the $3q$ subsystem in a
baryon, instead of the entire baryon mass squared.
Note also that the eigenvalue equations considered
in Ref. \cite{GlazekSchaden} were for usual energies
while we now consider the invariant mass squared
operators that include the quark free invariant mass
squared and minimal coupling interactions.
The second difference is that the vacuum state
$|\Omega \rangle$ is replaced by the state of
gluons condensed in a hadron,
\begin{eqnarray}
|G\rangle & = & G^\dagger(\vec
x_G)|0\rangle \, .
\end{eqnarray}
The vacuum state $|0\rangle$ does not develop any condensate.
The third difference is that in the mean-field
approximation for the gluon field,\footnote{This
approximation replaces the Schwinger gauge formula
for the background gluon field that reproduces
QCD sum rules for heavy quarkonia in the LF
Hamiltonian formulation of the theory
\cite{SDGcondensates}.}
\begin{eqnarray}
\label{Ameanfield}
\vec A (\vec x) & = & {1 \over 2} \, \vec B(\vec x_G)
\times (\vec x - \vec x_G) \, ,
\end{eqnarray}
one introduces the color magnetic field operator
at the center of the gluon component $G$ of
a hadron, $\vec B(\vec x_G)$, instead of the
vacuum operator $\vec B$ at an arbitrary point
$\vec x_0$. However, these differences do not
change the formal calculation of the matrix
elements. The field $\vec B(\vec x_G)$ is assumed
factorized into a product of vectors, one with space
and the other with color components as in Ref.
\cite{GlazekSchaden}. This assumption is meant
to reflect the assumed lack of correlation between
position space and color space directions in
the mean-field approximation and it renders a
simple, Abelian form of the approximate model
for the effective Hamiltonian. Matrix elements
of terms linear in $A$ are set to zero because
they change under gauge transformations whereas
the white component of a hadron does not.
The matrix elements one obtains for a
$q\bar q$ pair in a meson and $3q$ in
a baryon read\footnote{Eq. (14) in Ref.
\cite{GlazekSchaden} misses the factor 1/3
in front of the condensate term that is
correctly printed here in Eq. (\ref{mM}).}
\begin{eqnarray}
\label{mM}
\langle 12G | H_{min} |\psi \rangle_M
& = &
{1 \over \beta_M}
\sum_{i=1}^2
\left\{ -\Delta_i
+{g^2 \over 3}
\,
Tr
{ \langle G| \left( A_1 - A_2 \right)^2 |G'\rangle
\over \langle G | G' \rangle }
\right\}
\langle 12 G| \psi\rangle \, ,
\nonumber \\
&& \\
\label{mB}
\langle 123G | H_{min} |\psi \rangle_B
& = &
{1 \over \beta_B} \,
\sum_{i=1}^3( -\Delta_i) \langle 123 G| \psi\rangle_B
+
{1 \over \beta_B} \,
\sum_{i=1}^3
{g^2 \over 3}
\nn \ts
Tr
{
\langle G|
\left( A_i - A_{j+k\over 2} \right)^2
+ {1 \over 12} (A_j-A_k)^2 |G'\rangle
\over \langle G | G' \rangle } \,
\langle 123 G| \psi\rangle_B \, . \nonumber \\
\end{eqnarray}
All gluon field matrix elements contain squares of
differences of the effective gluon field operator
at different points. Therefore, the position of the
gluon body, $\vec x_G$, drops out. In the mean-field
approximation, one has
\begin{eqnarray}
\vec A(\vec x\,) - \vec A(\vec y\,)
& = &
{1 \over 2} \, \vec B \times (\vec x - \vec y\,)
\, ,
\end{eqnarray}
where $\vec B = \vec B(\vec x_G)$. Consequently,
\begin{eqnarray}
Tr \langle G| {g^2 \over 4\pi^2}
(\vec A_x - \vec A_y)^2 |G'\rangle
& = &
2 \, (\vec x-\vec y\,)^2 \, C_{glue} \,
\langle G | G' \rangle \, ,
\end{eqnarray}
with
\begin{eqnarray}
\langle G|G'\rangle & = & \delta_{GG'} \, .
\end{eqnarray}
The expectation value $C_{glue}$ plays here
the same role that the vacuum gluon condensate
value $C_{vacuum} = \varphi^2_{vacuum}/96$ with
\begin{eqnarray}
\varphi_{vacuum}^2 & = & \langle \Omega|
(\alpha/\pi) G^{\mu \nu c} G_{\mu \nu}^c |\Omega
\rangle \, ,
\end{eqnarray}
plays in Ref. \cite{GlazekSchaden}. Replacement
of the constant $C_{vacuum}$ by the constant
$C_{glue} = \varphi^2_{glue}/96$, implies our
reinterpretation of the phenomenologically
useful quantity $\varphi_{vacuum}$ on the order of
$\Lambda_{QCD}$ as coming from the quantity
$\varphi_{glue}$ that originates according to
the RGPEP model in the gluon condensation only
inside a hadron instead of the entire space.
Hence,
\begin{eqnarray}
\label{mMf}
\langle 12G | H_{min} |\psi \rangle_M
& = &
{1 \over \beta_M}
(-\Delta_1 - \Delta_2)\, \langle 12 G| \psi\rangle_M
\nn \ps
{1 \over \beta_M} \,
\left( {\pi \varphi_{glue} \over 3}
\right)^2
{ r_{12}^2 \over 2} \,
\langle 12 G| \psi\rangle_M \, , \\
\label{mBf}
\langle 123G | H_{min} |\psi \rangle_B
& = &
{1 \over \beta_B} \,
( -\Delta_1 -\Delta_2 -\Delta_3) \langle 123 G| \psi\rangle_B
\nn \ps
{1 \over \beta_B} \, {5 \over 8} \,
\left( {\pi \varphi_{glue} \over 3 } \right)^2
\left( {r_{12}^2 \over 2} + { 2 r_3^2 \over 3} \right) \,
\langle 123G |\psi \rangle_B \, ,
\end{eqnarray}
where
\begin{eqnarray}
r_{12} & = & x_1 - x_2 \, , \\
r_3 & = & x_3 - (x_1+x_2)/2 \, .
\end{eqnarray}
The results for $H_{min}$ with new interpretation of
the gluon condensate inside hadrons are used in the
next Section to construct candidates for effective
LF Hamiltonians of light mesons and baryons as
first approximations to solutions of RGPEP in QCD.
\subsection{ Approximate LF Hamiltonian for quarks
at $\lambda \gtrsim \lambda_c$ in
light hadrons }
\label{dynamics4}
Candidates for approximate Hamiltonians for light
hadrons can be obtained starting from inclusion of
results in Eqs. (\ref{mM}) and (\ref{mB}), or
(\ref{mMf}) and (\ref{mBf}), in Eqs.
(\ref{minimalM}) and (\ref{minimalB}),
correspondingly. The meson case is simpler than
the baryon case and explains the leading idea of
constructing the effective Hamiltonians using
the minimal coupling dictated by gauge symmetry.
We have
\begin{eqnarray}
{\cal M} ^2_{q \bar q}
& = &
( m_1+ m_2)^2 - (\vec p_1 + \vec p_2)^2
\nn \ps
{1 \over \beta_1} \,
\left[ \vec p_1^{\,2} + \left< {g^2 \over 3} Tr (\vec A_1 - \vec A_2)^2 \right>_G \right]
\nn \ps
{1 \over \beta_2} \,
\left[ \vec p_2^{\,2} + \left< {g^2 \over 3} Tr (\vec A_1 - \vec A_2)^2 \right>_G \right] \, .
\end{eqnarray}
Using the variables
\begin{eqnarray}
\vec P_{12} & = & \vec p_1 + \vec p_2 \, , \\
\vec k & = & \beta_2 \vec p_1 - \beta_1 \vec p_2 \, , \\
\vec p_1 & = & \beta_1 \vec P_{12} + \vec k \, , \\
\vec p_2 & = & \beta_2 \vec P_{12} - \vec k \, ,
\end{eqnarray}
one arrives at
\begin{eqnarray}
\label{mqq}
{\cal M} ^2_{q \bar q}
& = &
( m_1+ m_2)^2
+
{1 \over \beta_1 \beta_2} \,
\left[ \vec k^{\,2} +
\left< {g^2 \over 3} Tr (\vec A_1 - \vec A_2)^2 \right>_G \right] \, .
\end{eqnarray}
The LF counterpart of this result is obtained by
considering three-vectors with $+$ and $\perp$
components instead of $z$ and $\perp$. So, one
writes
\begin{eqnarray}
P_{12}^{+,\perp} & = & p_1^{+,\perp} + p_2^{+,\perp} \, , \\
x_1 & = & p_1^+/P^+_{12} \, , \quad x_2 \, = \, p_2^+/P^+_{12} \, , \\
\kappa^\perp & = & x_2 p_1^\perp - x_1 p_2^\perp \, , \\
p_1^\perp & = & x_1 P^\perp_{12} + \kappa^\perp \, , \\
p_2^\perp & = & x_2 P^\perp_{12} - \kappa^\perp \, .
\end{eqnarray}
These are standard expressions in LF
description of relative motion of two
constituents carrying the total momentum
$P$. Using these variables for two free
particles named 1 and 2, one obtains their
free invariant mass squared in the form
\begin{eqnarray}
{\cal M} ^2_{12}
& = &
{\kappa^{\perp \, 2} + m_1^2 \over x_1}
+
{\kappa^{\perp \, 2} + m_2^2 \over x_2} \\
& = &
(m_1 + m_2)^2
+
{1 \over x_1 x_2}
\left[
\kappa^{\perp \,2} + (m_2 x_1 - m_1 x_2)^2
\right] \, .
\end{eqnarray}
Comparison with Eq. (\ref{mqq}) identifies
the relative momentum $\vec k$,
\begin{eqnarray}
\label{kperp}
k^\perp
& = &
\sqrt{ \beta_1 \beta_2 \over x_1 x_2} \, \kappa^\perp \, , \\
\label{kz}
k^z
& = &
\sqrt{ \beta_1 \beta_2 \over x_1 x_2} \, [ m_2
x_1 - m_1 x_2 ] \, .
\end{eqnarray}
This is a new way of parameterizing relative motion
of two constituents in LF approach to quantum
mechanics and field theory. The new variables
match non-relativistic relative momenta that
include effects of particle masses. On the other
hand, it is known that in the relativistic relative
motion of two constituents the mass parameters
appear not significant. For example, the standard
relative momentum, in which $k^\perp = \kappa^\perp$,
has length
\begin{eqnarray}
\label{standardk}
\vec k_{standard}^{\,2}
& = &
{1 \over 4 {\cal M} ^2_{12} }
\left[ {\cal M} ^2_{12} - (m_1+m_2)^2 \right]
\left[ {\cal M} ^2_{12} - (m_1-m_2)^2 \right] \, ,
\end{eqnarray}
and a relatively complicated expression is
obtained for $k^z$ if one insists on the
identification $k^\perp = \kappa^\perp$. But when
$ {\cal M} _{12}$ is large, one has $\vec k_{standard}^2
= {\cal M} _{12}^2/4$ independently of the quark masses.
At the same time, the variable $k^z_{standard}$
depends on $\kappa^\perp$. The length of the
standard relative three-momentum in the
constituent rest frame differs from the length of
the new one,
\begin{eqnarray}
\vec k^{\,2}_{standard}
& = &
{ \vec k^{\,2} \over 4 \beta_1 \beta_2}
\left[ 1 - {(m_1 - m_2)^2 \over {\cal M} ^2_{12}} \right] \, ,
\end{eqnarray}
and the angular orientations of the three-vectors
$\vec k_{standard}$ and $\vec k$ also differ.
This means that the new choice of LF three-momentum
variables in relativistic cases involves rotation
by some polar angle (the azimuthal angles are the
same). Analysis of rotational symmetry is influenced
in the sense that in order to recover a simple
NR quantum mechanical picture of hadrons from
quantum field theory one is motivated by the effective
picture in RGPEP to use the new variable $\vec k$
rather than $\vec k_{standard}$ as arguments of
potentials for constituent quarks. One should also
remember that when a change of variables is made
in the expressions involving effective quark wave
functions, the new variables produce Jacobian
factors in phase-space integration.
Using Eq. (\ref{mMf}), one obtains
\begin{eqnarray}
\left< {g^2 \over 3} Tr (\vec A_1 - \vec A_2)^2 \right>_G
& = &
{1 \over 2} \left( {\pi \varphi_{glue} \over 3}
\right)^2
{ r_{12}^2 \over 2} \, ,
\end{eqnarray}
where $\vec r_{12}$ should be the relative position
variable that is canonically conjugated to
$\vec k$. This means in quantum mechanics that
\begin{eqnarray}
\vec r_{12} & = & i \, {\partial \over \partial \vec k} \, .
\end{eqnarray}
Thus, the LF mass squared for a constituent quark-anti-quark
pair at scale $\lambda \gtrsim \lambda_c$ interacting in
a gauge minimal way with gluons condensed inside a meson and
treated in a mean-field approximation, has the form
\begin{eqnarray}
\label{mqqfinal}
{\cal M} ^2_{q \bar q}
& = &
( m_1+ m_2)^2
+
{1 \over \beta_1 \beta_2} \,
\left[ \vec k^{\,2} +
{1 \over 2}\left( {\pi \varphi_{glue} \over 3}
\right)^2
{1 \over 2} \left( i \, {\partial \over \partial
\vec k} \right)^2
\right] \, ,
\end{eqnarray}
where the vector $\vec k$ is defined by Eqs.
(\ref{kperp}) and (\ref{kz}).
In particular, the effective quarks $u$ and $d$
are expected to have practically the same mass
$m$ order $\Lambda_{QCD}$ in the RGPEP scheme.
For them, $\beta_1 = \beta_2 = 1/2$ and the
associated Jacobi variables are
\begin{eqnarray}
\vec \rho & = & \vec r_{12} / \sqrt{2} \, , \\
\vec p_\rho & = & \vec k \sqrt{2} \, .
\end{eqnarray}
In terms of these variables,
\begin{eqnarray}
\label{mqqJacobi}
{\cal M} ^2_{q \bar q}
& = &
4m^2
+
2 \, p_\rho^2
+
2 \left( {\pi \varphi_{glue} \over 3} \right)^2 \,
\rho^2 \, .
\end{eqnarray}
For small relative momenta and corresponding
distances, this result approximately matches
the NR oscillator dynamics,
\begin{eqnarray}
\label{mqqJacobiNR}
{\cal M} _{q \bar q}
& = &
2m
+
{ p_\rho^2 \over 2 m }
+
{1 \over 2} \, m \, \left( {\pi \varphi_{glue} \over 3 m} \right)^2 \,
\rho^2 \, ,
\end{eqnarray}
with frequency $\omega_M = \pi
\varphi_{glue}/(3m)$.
On the one hand, this result relates
the RGPEP reasoning to the CQM phenomenology
since the reinterpreted value of the gluon
condensate produces physically reasonable
frequency $\omega_M$ \cite{GlazekSchaden}.
On the other hand, the variables $\vec k$
and $\vec r$ identified here include the
characteristic factor $\sqrt{x(1-x)}$ that
is required in AdS/QCD holographic variables
\cite{FAdS1,FAdS2,FAdS3}. Thus, it becomes
plausible that also the RGPEP reasoning
regarding the $\lambda$-dependence of
effective interactions may provide insight
into the significance of soft wall (SW) models
\cite{SW} of hadron spectrum.
As a result of analogous reasoning in the baryon
case, we obtain
\begin{eqnarray}
{\cal M}_{123}^2 - (m_1 + m_2 + m_3)^2
& = &
{1 \over \beta_3 (1-\beta_3)} \, \vec Q^{\,2}
+ {1-\beta_3 \over \beta_1 \beta_2} \, \vec
K^{\,2} \, ,
\end{eqnarray}
where
\begin{eqnarray}
\label{xi}
x_i & = & p_i^+/P^+_{123} \, , \\
\beta_i & = & {m_i \over m_1 + m_2 + m_3} \, , \\
\label{qperp}
p_3^\perp & = & x_3 P^\perp_{123} + q^\perp \, , \\
\label{kperp2}
p_2^\perp & = & x_2 P^\perp_{123} - {x_2 \over 1-x_3} \, q^\perp - k^\perp \, , \\
\label{kperp1}
p_1^\perp & = & x_1 P^\perp_{123} - {x_1 \over 1-x_3} \, q^\perp + k^\perp \, , \\
\label{Qperp}
Q^\perp & = & \sqrt{ \beta_3 (1-\beta_3) \over x_3 (1-x_3)} \,\, q^\perp \, , \\
\label{Qz}
Q^z & = & \sqrt{ \beta_3 (1-\beta_3) \over x_3 (1-x_3)} \,\,
\left[ (m_1 + m_2) x_3 - m_3(x_1 + x_2)\right] \, , \\
\label{Kperp}
K^\perp & = & \sqrt{ {\beta_1 \beta_2 \over x_1 x_2} {1-x_3 \over 1-\beta_3} } \,\, k^\perp \, , \\
\label{Kz}
K^z & = & \sqrt{ {\beta_1 \beta_2 \over x_1 x_2} {1-x_3 \over 1-\beta_3} } \,\,
\left( m_2 \, {x_1 \over 1-x_3} - m_1 \, {x_2 \over 1-x_3} \right) \, .
\end{eqnarray}
For slow relative motion of constituents, the new
momentum variables $\vec K$ and $\vec Q$ match the
non-relativistic three-momenta used in quark models,
but in fact they apply in the entire range of relativistic
kinematics of motion of constituents inside
baryons, and for arbitrary motion of baryons as a
whole, thanks to the kinematical symmetries of LF
formulation of the theory. However, one needs to
remember that the lengths and angular orientations
of the new momentum variables differ from the
standard LF variables among which, in particular,
$q_{12}^\perp$ and $q_3^\perp$ (see below) could
be thought directly useable for obtaining some
effective constituent picture for baryons from
LF QCD.
The LF mass squared for 3 constituent quarks of
one and the same mass $m$ at scale $\lambda \gtrsim
\lambda_c$ interacting in a gauge minimal way with
gluons condensed inside a baryon in a mean-field
approximation, has the form
\begin{eqnarray}
\label{m3qfinal}
{\cal M}_{3q}^2 & = & 9m^2 + 6 \, \vec K^{\,2}
+ {9 \over 2} \, \vec Q^{\,2}
- 3m^2 \left( {\pi \varphi \over 3 m }
\right)^2 {5 \over 8} (\Delta_K^2/2 + 2 \Delta_Q/3 )
\, .
\end{eqnarray}
We identify relations
\begin{eqnarray}
\vec K
& = &
\vec q_{12} \, = \, \vec p_\rho/\sqrt{2} \, , \\
-i {\partial \over \partial \vec K}
& = &
\vec r_{12} \, = \, \sqrt{2} \, \vec \rho \, , \\
\vec Q
& = &
\vec q_3 \, = \, - \sqrt{2/3} \, \vec p_\lambda \, , \\
-i {\partial \over \partial \vec Q}
& = &
\vec r_3 \, = \, - \sqrt{3/2} \, \vec \lambda \, ,
\end{eqnarray}
in terms of the Jacobi co-ordinates for 3 quarks
with equal masses, $\vec \rho$, $\vec p_\rho$,
$\vec \lambda$, and $\vec p_\lambda$.
One has
\begin{eqnarray}
P_{123} & = & p_1 + p_2 + p_3 \, , \quad
q_{12} = (p_1 - p_2)/2 \, , \\
P_{12} & = & p_1 + p_2 \, , \quad
q_3 = (2 p_3 - P_{12})/3 \, , \\
P_{12} & = & 2P_{123}/3 - q_3 \, , \quad p_3 \, = \, P_{123}/3 + q_3 \, , \\
p_1 & = & P_{12}/2 + q_{12} \, = \, P_{123}/3 + q_{12} - q_3/2 \, , \\
p_2 & = & P_{12}/2 - q_{12} \, = \, P_{123}/3 - q_{12} - q_3/2 \, , \\
p_3 & = & P/3 + q_3 \, .
\end{eqnarray}
\begin{eqnarray}
p_1^2 + p_2^2 + p_3^2
& = &
P_{123}^2/3 + 2 q_{12}^2 + 3q_3^2/2 \, .
\end{eqnarray}
\begin{eqnarray}
R_{123} & = & (x_1 + x_2 + x_3)/3 \, , \quad
r_{12} \, = \, x_1 - x_2 \, , \quad
r_3 \, = \, x_3 - R_{12} \, , \\
x_3 & = & R_{123} + 2r_3/3\, , \quad
R_{12} \, = \, (x_1+x_2)/2 \, = \, R_{123} - r_3/3 \, , \\
x_1 & = & R_{12} + r_{12}/2 \, = \, R_{123} - r_3/3 + r_{12}/2 \, , \\
x_2 & = & R_{12} - r_{12}/2 \, = \, R_{123} - r_3/3 - r_{12}/2 \, .
\end{eqnarray}
\begin{eqnarray}
\sum_{i=1}^3 p_ix_i
& = & P_{123}R_{12} + q_{12} r_{12} + q_3 r_3 \, .
\end{eqnarray}
In the case of the same masses, the relativistic
LF result for three quarks in a baryon reads,
\begin{eqnarray}
\label{b3qJacobi}
{\cal M}_{3q}^2 & = & 9m^2 + 3 \, \vec p_\rho^{\,2}
+ 3 \, \vec p_\lambda^{\,2}
+ 3m^2 \left( {\pi \varphi \over 3 m }
\right)^2 {5 \over 8} (\vec \rho^{\,2} + \vec \lambda^{\,2} )
\, .
\end{eqnarray}
In the NR limit,
\begin{eqnarray}
\label{b3qJacobiNR}
{\cal M}_{3q}
& = &
3m + {\vec p_\rho^{\,2} \over 2m}
+ {\vec p_\lambda^{\,2} \over 2m}
+ {1 \over 2} \, m \, {5 \over 8} \, \left( {\pi \varphi \over 3 m }
\right)^2 (\vec \rho^{\,2} + \vec \lambda^{\,2} ) \, ,
\end{eqnarray}
which matches precisely the result of Ref.
\cite{GlazekSchaden}, and
\begin{eqnarray}
\omega_B^2 & = & {5 \over 8} \, \omega_M^2 \, .
\end{eqnarray}
This numerical relation is in a reasonable
agreement with phenomenology of constituent
quark models \cite{CQM1,CQM2,CQM3}, as noticed
already in Ref. \cite{GlazekSchaden} (see also
Section \ref{AdS}).
\subsection{ Dynamics of quarks and glue in hadrons
at $\lambda \gtrsim \lambda_c$ }
\label{dynamics5}
The constituent models do not, however, include
the gluon component $G$. Model potentials are also
not assigned energy or momentum, while gluons do
carry energy and momentum. Bag models do include a
bag in terms of boundary conditions on quark wave
functions on the bag walls and a constant
contribution of the bag to the total hadron
energy, but the bag is not placed yet in the
context of QCD \cite{bag}. The author does not
know how to interpret results of lattice formulation
of QCD regarding the constituent model and the gluon
component in terms of hadronic wave functions.
Summarizing, it is not clear how the addition of $G$
will affect phenomenology familiar through the
CQMs without $G$, although some foreseeable
changes appear welcome (see next Section).
Regarding the contribution of mass of $G$ to the
mass squared of the whole hadron, one cannot say
anything but hypothesize that the mass of $G$ may
depend on $\lambda$. It can be small, due to
interactions that hold gluons among themselves in
the form of $G$. The mass of $G$ may even decrease
when $\lambda$ drops down to $\lambda_c$. A
graphical picture that makes this idea plausible
as a result of overlapping of quarks of size
$1/\lambda$ (a quark and an anti-quark in a meson
or three quarks in a baryon) and $G$ in position
space, is described in Appendix
\ref{OverlappingSwarmsModel}. However, a simple
concept of a constituent $G$ may be insufficient
in the case of $\pi$-mesons, considering that
$\pi$-mesons may differ from other hadrons as a
consequence of their relationship to chiral
symmetry and its breaking. The symmetry breaking
can be addressed in LF QCD \cite{Wilsonetal} but
it is not addressed in this article. Such
discussion requires reinterpretation of the quark
condensate and $G$ built from gluons alone is not
sufficient.
Regarding the motion of the glue component $G$
inside a hadron, one may observe that if the state
$G^\dagger|0\rangle$ in RGPEP is to correspond to
the vacuum in the instant dynamics way of thinking
and $G$ is associated through RGPEP with a hadron
in an effective theory of scale $\lambda \gtrsim
\lambda_c$, the effective quarks at this scale
should not be free to move farther away from $x_G$
than about $1/\lambda$ (see also Appendix
\ref{OverlappingSwarmsModel}). The mean field
approximation misses the fact that the gluon
component has a size comparable with a hadron. The
Schwinger gauge expression for the gluon field
potential $A_\lambda(x)$ with only one term linear
in $\vec x - \vec x_G$ and $\vec B(\vec x_G)$, Eq.
(\ref{Ameanfield}), must be missing important effects
at distances comparable with the diameter of a
hadron, cf. \cite{SDGcondensates}. This means that
the gauge transport factors $T(x,{\underline x})$
cannot be approximated well in the entire volume
of a hadron using the Schwinger gauge with a
linear term alone. Thus, the mean-field
calculation does not report well on what happens
at the boundary of the glue or the boundary of a
hadron. While one cannot determine in advance what
will eventually result from RGPEP calculations of
effective Hamiltonians and eigenvalue problems for
these Hamiltonians, one should certainly expect
that there will be forces that keep the center of
the quarks and the glue together.
There is no simpler choice for the needed
interaction term between the quarks and $G$ than a
quadratic function of the distance between
$\underline x$ and $x_G$. Such distance is easy
to write in NR quantum mechanics. However, one has
to carefully define its analog in the hadron mass
squared operator on the LF. Solution of this
technical issue described here, is found by looking
at hadrons as built only from two constituents.
One constituent is the effective quarks treated as
one particle of mass resulting from eigenvalues
of $ {\cal M} ^2_{q \bar q}$ in mesons or $ {\cal M} ^2_{3q}$ in
baryons. The other constituent is $G$ treated as a
particle. The mass of $G$ is unknown, may depend on
$\lambda$, and is free to adjust in a process of
defining a first approximation to hadrons at
$\lambda \gtrsim \lambda_c$. With this setup, one
applies the same method for constructing the LF mass
squared operator that we used for a quark and an
anti-quark treated as two constituents in a
harmonically bound state. There is no minimal
gauge coupling to use for the colorless state of quarks
interacting with a white $G$, and a large degree
of ambiguity must be confronted without clear
guidance from QCD.\footnote{It is precisely this type
of ambiguity at small energies that RGPEP is meant
to eventually help in resolving.} But there is no
problem now with introducing the appropriate
oscillator potential operator using the distance
that is quantum-mechanically conjugated to a
relative momentum.
The first-approximation to LF Hamiltonian for light
hadrons at $\lambda \gtrsim \lambda_c$ is suggested
to have the form
\begin{eqnarray}
\label{masshadron}
H_{\lambda \gtrsim \lambda_c}
& = &
{ {\cal M} ^2_{quarks} + P_{quarks}^{\perp \, 2} \over P_{quarks}^+ }
+
{ {\cal M} ^2_G + P_G^{\perp \, 2} \over P_G^+ }
+
{ {\cal M} ^2_{qG} \over P^+_{hadron}} \, .
\end{eqnarray}
In order to define a suitable candidate
for the last term, one can introduce
\begin{eqnarray}
\label{momentahadron}
P_{hadron}^{+ , \perp} & = & P^{+, \perp}_{quarks} + P^{+,
\perp}_G \, , \\
x_q & = & P^+_{quarks}/P^+_{hadron} \, , \\
x_G & = & P_G^+/P^+_{hadron} \, , \\
\kappa_q^\perp & = & x_G P_{quarks}^\perp - x_q P_G^\perp \, , \\
P_{quarks}^\perp & = & x_q P^\perp_{hadron} + \kappa_q^\perp \, , \\
P_G^\perp & = & x_G P_{hadron}^\perp - \kappa_q^\perp \, .
\end{eqnarray}
In terms of these variables, the hadron mass squared
reads
\begin{eqnarray}
\label{mhfinal}
{\cal M} ^2_{hadron}
& = &
{ {\cal M} ^2_{quarks} + \kappa_q^{\perp \, 2} \over x_q }
+
{ {\cal M} ^2_G + \kappa_q^{\perp \, 2} \over x_G }
+
{\cal M} ^2_{qG} \, ,
\end{eqnarray}
where $ {\cal M} ^2_{quarks}$ is now to be identified with the
eigenvalue $M_q^2$ for the mass squared of the quark
subsystem in a hadron and $ {\cal M} _G$ is set to the corresponding
eigenvalue $M_G^2$, which could be considered to have a ground
state value or an excited value, if the condensed gluons were
in an excited state. The constant $M_q^2$ is either $4m^2$ or
$9m^2$ plus an appropriate number times $m\omega$ with
$\omega = \omega_M$ or $\omega = \omega_B$, respectively.
Having introduced
\begin{eqnarray}
\beta_q & = & { M_q \over M_q + M_G} \, , \\
\beta_G & = & { M_G \over M_q + M_G} \, ,
\end{eqnarray}
one can define
\begin{eqnarray}
\label{khperp}
k_h^\perp
& = &
\sqrt{ \beta_q \beta_G \over x_q x_G} \, \kappa_q^\perp \, , \\
\label{kzh}
k_h^z
& = &
\sqrt{ \beta_q \beta_G \over x_q x_G} \, [ M_G
x_q - M_q x_G ] \, .
\end{eqnarray}
Proceeding as in the case of two quarks in a meson,
Eq. (\ref{mqqfinal}), one establishes
\begin{eqnarray}
\label{hMint}
{\cal M} ^2_{qG}
& = &
{1 \over \beta_q \beta_G} \,
{1 \over 2}\left( {\pi \varphi_h \over 3}
\right)^2
{1 \over 2} \left( i \, {\partial \over \partial
\vec k_h} \right)^2 - M_\varphi^2 \, , \\
{\cal M} ^2_{hadron}
& = &
(M_q + M_G)^2 - M_\varphi^2
\nn \ps
\label{hadronmassfinal}
{1 \over \beta_q \beta_G} \,
\left[ \vec k_h^{\,2} +
{1 \over 2}\left( {\pi \varphi_h \over 3}
\right)^2
{1 \over 2} \left( i \, {\partial \over \partial
\vec k_h} \right)^2
\right] \, ,
\end{eqnarray}
where two new parameters, $\varphi_h$ and
$M_\varphi$, are introduced.
The parameter $\varphi_h$ is used in the similar
way to how $\varphi_{glue}$ is used in the
invariant mass of quarks in mesons. However,
$\varphi_h$ serves merely the purpose of notation
for the unknown quantity of binding between quarks
and $G$ whose value must be adjusted in the process
of correcting the candidate for the first approximation.
In RGPEP, $\varphi_h$ can be expected to depend on
$\lambda$. Intuitive arguments are offered in
Appendix \ref{OverlappingSwarmsModel}. This candidate
for a first approximation will only be satisfying
if it turns out in future calculations in RGPEP that
some optimal value of $\varphi_h$ can be adjusted
as a function of $\lambda/\Lambda_{QCD}$ and the
corresponding coupling constant, assuming that the
light quark mass parameters do not matter at small
$\lambda$ where the effective quark masses are on
the order of $\Lambda_{QCD}$ anyway and do not vary
with $\lambda$ so rapidly that no average constituent
picture can correspond to QCD. Assuming that $\varphi_h$
does not vary from hadron to hadron, one can introduce
\begin{eqnarray}
\label{omegah}
\omega_h
& = &
{\pi \varphi_h \over 3 \mu_h} \, , \\
\label{muh}
\mu_h
& = &
{M_q M_G \over M_q + M_G} \, .
\end{eqnarray}
The parameter $\mu_h$ is the reduced
mass for the two-body system quarks-$G$.
It depends on a hadron as far as $M_q$ depends
on a hadron, assuming $M_G$ can be considered
universal in the first approximation. Thus,
the mass-squared eigenvalues for the whole hadron
takes the form
\begin{eqnarray}
\label{hadronspectrum}
M^2_h
& = &
(M_q + M_G)^2 - M_\varphi^2
+
{\mu_h \over \beta_q \beta_G} \,
(n_h + 3/2) \omega_h \, ,
\end{eqnarray}
where $n_h$ denotes the excitation quantum number
in relative motion of quarks with respect to
the gluon condensate in a hadron, being zero
in a ground state. Thus the momentum width of
relative motion o quarks with respect to $G$ is
order $\varphi_h$.
The mass parameter $M_\varphi$ is introduced by
fiat and brings in another considerable degree of
ambiguity. Physical motivation for $M_\varphi$ is
that for the gluon condensation to occur in hadrons
it must be favorable energetically. If one just added
the free energy of $G$ and a harmonic quarks-glue
potential energy to the energy of quarks, the
condensation of gluons would only add energy to
the system of quarks. The effect of condensation
should rather be opposite: the concept of
condensation of gluons inside hadrons implies
thinking that the mass of a hadron state is
lowered by condensation of gluons in comparison
with a state in which condensation is absent.
Since the kinematical minimum of energy of quarks
and glue is $M_q + M_G$, which corresponds to
$\kappa_q^\perp = 0$ and $x_q = \beta_q$, the
interaction mass parameter $M_\varphi$ may be
estimated by inspecting the condition
\begin{eqnarray}
M_q^2 & > & (M_q + M_G)^2 + {3 \over 2}
(M_q + M_G)\omega_h - M_\varphi^2 \, .
\end{eqnarray}
Assuming that the resulting lower bound on
$M_\varphi$ provides an estimate of its
likely magnitude in QCD, we obtain
\begin{eqnarray}
\label{hadronestimate}
M_\varphi^2 & \sim &
M_q M_G \left ( 2 + {M_G \over M_q} + { 3 \omega_h
\over 2 \mu_h} \right) \, ,
\end{eqnarray}
where 3 signifies the number of spatial
dimensions. Assuming that $M_q$ equals nucleon
mass $m_N$, and $M_G \sim \omega_h \sim m_N/3$,
one obtains $M_\varphi \sim 1.2 \, m_N$. If $M_G
\sim M_q \sim m_N$ and $\omega_h \sim m_N$, we
have $M_\varphi \sim 2.4 \, m_N$. This means that
$M_\varphi$ can be adjusted to obtain a hadron
mass as coming mainly from the quark mass
eigenvalue $M_q$ provided that $M_\varphi$ is
quite sizable. Such sizable energy benefit from
gluon condensation eliminates a large contribution
of $G$ to a hadron mass and sustains the possibility
that a simple oscillator quark model without any
glue can reproduce masses of light hadrons assuming
that the effective quark masses are on the order of
$\Lambda_{QCD}$.\footnote{If the condensation mass
advantage, $-M_\varphi^2$, were associated also
with condensation of quark-anti-quark pairs, one
might expect a pronounced reduction in masses of
the hadron states in which a bilinear effective
quark field expectation value plays a significant
role in the dynamics, with $\pi$ mesons being the
primary candidates.}
We wish to stress that Eq. (\ref{hMint}) does not
imply one and the same interaction between quarks
and glue $G$ in all hadrons even if the parameters
concerning $G$ are assumed the same for all
hadrons. Different hadrons are actually having
different Hamiltonians characterized by different
quantum numbers of their quark component, such as
radial and orbital excitations (elementary spin
effects are discussed in the next Section) and the
mass eigenvalue for the quarks, $M_q$, depends on
these quantum numbers. This mass enters the
definition of $\vec k_h$ and thus also the
definition of harmonic potential between the
quarks and $G$ that is defined in terms of
$\partial / \partial \vec k_h$. As a result,
potentials between quarks and $G$ vary from hadron
to hadron in a well-defined pattern: the higher
excitation of quarks, the greater their mass and
the more momentum of a hadron carried by
quarks.\footnote{Provided that the
first-approximation quantity $M_\varphi$ is kept
constant.} For example, quarks in excited nucleons
are expected to carry a larger fraction of the
resonance momentum than quarks carry in a ground-state
nucleon. And vice versa, an excitation of the gluons
condensed in a hadron increases $M_G$ and the
fraction of a hadron momentum they thus carry.
Nevertheless, a CQM for light
hadrons should be viewed as corresponding to
$n_h=0$ when $\lambda \gtrsim \lambda_c$ is
actually lowered to $\lambda_c$. As suggested in
Appendix \ref{OverlappingSwarmsModel}, when $\lambda$
is lowered to $\lambda_c$, the role of $G$ may be
imagined taken over entirely by the content
of extended constituent quarks that overlap
each other heavily, so that no separate glue
component can exist in hadron besides what is
already contained in the extended effective
quarks. Instead of the speculation, however,
the proper problem is what will result from
attempts to solve the RGPEP equation for the
effective Hamiltonian at $\lambda = \lambda_c$.
Strictly speaking, nothing is known currently
about the solution.
\section{ RGPEP in QCD and phenomenology }
\label{Phenomenology}
The discussion that follows is limited to the
case of all effective quarks having the same
mass $m$. The common mass is expected to be a
reasonable approximation for quarks $u$ and $d$
at $\lambda \sim \lambda_c$. Calculational
complications due to differences in mass
between these quarks are not discussed.
Assuming that $\varphi_{glue}$ is on the order of
$\Lambda_{QCD}^2$, which means that it has a
similar value to the values obtained in QCD sum
rules for $\varphi_{vacuum}$, one obtains
phenomenologically attractive values of $\omega_M$
and $\omega_B$ \cite{GlazekSchaden}. Eqs.
(\ref{mqqfinal}), (\ref{m3qfinal}), (\ref{mhfinal}),
(\ref{hMint}), and (\ref{hadronmassfinal}), imply
together eigenvalues and wave functions for light
hadrons in the form of harmonic oscillator solutions
that include the glue component $G$. The generic
oscillator eigenvalue problem for two particles has
the form
\begin{eqnarray}
\left( {\vec p^{\,2} \over 2 m} + {1 \over 2} m
\omega^2 \vec r\,^2 \right) \psi & = & E \, \psi \, ,
\end{eqnarray}
with eigenvalues $E_n = (n+3/2)\omega$ and the
ground-state wave function $\psi_0 = N \exp{[-\vec
p^{\,2}/(2 m \omega)]}$ for $n=0$, where $\vec p$
and $\vec r$ are canonically conjugated variables.
For two effective constituent quarks in mesons,
using Eq. (\ref{mqqJacobiNR}) that explains what
happens in the LF eigenvalue equation for
$ {\cal M} ^2_{q \bar q}$ in terms of a NR approximation
to $ {\cal M} _{q\bar q}$, Eq. (\ref{mqqJacobi}) implies
the first LF approximation of the form
\begin{eqnarray}
M^2_{q \bar q \, n}
& = &
4m^2
+ 4 (n+3/2) m \omega_M \, , \\
\psi_{q \bar q \, 0}
& = & N_{q \bar q \, 0} \exp{[-\vec k^{\,2}/(m\omega_M)]} \\
& = & N_{q \bar q \, 0} \exp{\left\{ -
{ 1 \over 4m\omega_M }
\left[{ \kappa^{\perp \, 2} + m^2 \over x(1-x) } - 4m^2 \right] \right\} } \, ,
\end{eqnarray}
with frequency $\omega_M = \pi \varphi_{glue}/(3m)$
and wave functions of excited states, $\psi_{q \bar q \, n}$,
generated by building harmonic oscillator
excitation operators from the vector $\vec k$ and
gradient $\partial /\partial \vec k$ in a standard way
and applying them to the ground state wave function
$\psi_{q \bar q \, 0}$.
For three effective constituent quarks in baryons,
using Eq. (\ref{b3qJacobiNR}) that explains what
happens in the LF eigenvalue equation for
$ {\cal M} ^2_{3q}$ in terms of a NR approximation
to $ {\cal M} _{3q}$, Eq. (\ref{b3qJacobi}) implies
the first LF approximation of the form
\begin{eqnarray}
M^3_{3q n_1 n_2}
& = &
9m^2
+ 6(n_1+n_2+3)m \omega_B \, , \\
\psi_{3 q \, 0}
& = & N_{3q \, 0} \exp{ \left\{ -{1 \over 6 m\omega_B}
\left[
{9 \over 2} \, \vec Q^{\,2}
+ 6 \, \vec
K^{\,2} \,
\right]
\right\} } \\
& = & N_{3q \, 0}
\exp
\left\{
- {1 \over 6 m\omega_B}
\left[ { (1 - x_3) \, k^{\perp \, 2} \over x_1 x_2 }
+ { q^{\perp \, 2} \over x_3 (1-x_3) }
\right.
\right.
\nn \ps
\left.
\left.
m^2\left( {1 \over x_1} + {1 \over x_2} + {1 \over x_3} - 9 \right) \right] \right\} \, ,
\end{eqnarray}
with baryon oscillator frequency $\omega_B = \sqrt{5/8} \,\, \omega_M$
and wave functions of excited states, $\psi_{3q\, n_1 n_2}$,
generated in a standard way by building harmonic oscillator
excitation operators from the vectors $\vec K$ and
$\vec Q$ and gradients $\partial /\partial \vec K$ and
$\partial /\partial \vec Q$ and applying them to the
ground state wave function $\psi_{3q \, 0}$. The LF
momentum variables are defined in Eqs. (\ref{xi}),
(\ref{qperp}), (\ref{kperp2}) and (\ref{kperp1}).
In summary, the factor in a hadron
ground-state wave function that depends on the
relative motion of $n$ quarks has the form
\begin{eqnarray}
\label{finalGaussian}
\psi_{n q \, 0}
& = &
N_{n q \, 0} \,
\exp{ \left\{ - {1 \over 2 n m
\omega_n} \left[ \left( \sum_{i=1}^n p_i \right)^2
- (nm)^2 \right] \right\} }\, ,
\end{eqnarray}
with $n=2$ for mesons, $n=3$ for baryons, $\omega_2 =
\omega_M$, $\omega_3 = \omega_B$, and $p_i$ the
on-maas-shell four-momentum for quark number $i$.
Exponentials of an invariant mass squared of
quarks are popular as wave functions in
phenomenological studies. Here such exponentials
are related to gauge symmetry and reinterpretation
of the gluon condensate as a part of a hadron.
The spectrum of light hadron masses in Eq.
(\ref{hadronspectrum}), i.e., the spectrum
corresponding to the ground state of relative
motion of quarks with respect to the glue $G$,
becomes equal to the spectrum of masses of the
quark component alone when the estimate
(\ref{hadronestimate}) is adopted for the benefit
of the gluon condensation in a hadron. This result
reproduces success of the constituent quark
models.\footnote{The oscillator functions of
relative quark motion are independent of quark
spins. A Coulomb part of the wave function must
depend on spins. For small $\lambda$, effective
quark spin wave function can be treated as a
separate factor in the oscillator part of the wave
function. Spin factors can be constructed in a
boost-invariant way using LF spinors and following
the example of heavy quarkonia \cite{GlazekMlynik}
in the case of mesons or Ioffe currents
\cite{GlazekNamyslowski} in the case of baryons.
However, in contrast to models not related to QCD,
RGPEP provides a scheme for systematic study of
spin-dependent corrections to the first oscillator
approximation.}
At the same time, when the relative motion of
quarks with respect to the glue part is described
by Eq. (\ref{hadronmassfinal}), the hadron mass is
given by Eq. (\ref{hadronspectrum})
and the corresponding wave function is generated
from the ground-state wave function
\begin{eqnarray}
\label{psihadron}
\psi_h
& = &
\psi_{n q\,0} \, \psi_{qG \, 0} \, , \\
\label{psiqG}
\psi_{qG \, 0}
& = &
N_{qG\,0}
\exp{
\left[ - { (P_q+P_G)^2 - (M_q+ M_G)^2
\over 2(M_q + M_G) \omega_h } \right]} \, ,
\end{eqnarray}
in a standard way for harmonic oscillators,
first for the quark component and then, using
the quark component mass eigenvalue, for the
whole hadron if the quarks are excited in
their motion with respect to $G$. Thus, the
invariant mass squared of quarks and $G$
together, $(P_q+P_G)^2$ is evaluated in a standard
way using their LF momenta and their minus
components calculated using quark mass eigenvalue
$M_q$ and the glue mass $M_G$, respectively.
Normalization factors are fixed by normalizing
probability to 1, or fixing the charge to the
appropriate value (if the charge is not zero
these normalization conditions are the same).
\subsection{ Form factors }
\label{ff}
The relative motion of quarks with respect to the
glue $G$ smears the quark observables that follow
from the quark factor in the hadron wave function
alone. Consider the ground state of quarks' motion
with respect to $G$. Calculation of the baryon
form factor at a small momentum transfer $q$, $q^2
= - Q^2$, $ 0 \le Q^2 \le \Lambda^2_{QCD}$, is
represented graphically in Fig. \ref{fig:FormFactor}.
Meson form factor calculation involves 2 instead of
3 quarks but otherwise proceeds in the same way.
\begin{figure}
\begin{center}
\includegraphics[scale=.6]{FormFactor}
\end{center}
\caption{The RGPEP calculation of a baryon
form factor at $\lambda \sim \lambda_c$.}
\label{fig:FormFactor}
\end{figure}
Using the Breit frame with $q^+=0$, which makes
$q^-$ that depends on masses irrelevant, one has
$Q^2 = q^{\perp \, 2}$ and
\begin{eqnarray}
&& F_h(Q^2) \, = \, \int {dx_q d^2 \kappa^\perp_q \over 16\pi^3 x_q(1-x_q)}
\nn \ts
\psi^*_{qG \, 0}[x_q, \kappa_q + (1-x_q)q^\perp]\,
F_{3q}(Q^2)\,
\psi_{qG \, 0}(x_q, \kappa_q) \, ,
\label{ff1}
\end{eqnarray}
with normalization condition
\begin{eqnarray}
1 & = & \int {dx_q d^2 \kappa^\perp_q \over 16\pi^3
x_q(1-x_q)} \,
|\psi_{qG \, 0}(x_q, \kappa_q)|^2 \, .
\end{eqnarray}
Using the wave function in Eq. (\ref{psihadron})
and changing variables to $x = x_q$ and $\kappa^\perp
= \kappa_q^\perp + (1-x)q^\perp/2$, one obtains
\begin{eqnarray}
\label{ff2}
F_h(Q^2) & = &
F_{3q}(Q^2)\, f(Q) \, , \\
\label{fQ}
f(Q) & = & \int {dx \, d^2 \kappa^\perp \over 16\pi^3 x(1-x)}
|\psi^*_{qG \, 0}(x, \kappa^\perp)|^2\, e^{- {1-x
\over x} {Q^2 \over 4(M_q + M_G) \omega_h} } \, .
\end{eqnarray}
After integration over $\kappa^\perp$, one is
left with
\begin{eqnarray}
f(Q) & = &
{ \int_0^1 dx \,
e^{ - a \left({ M^2_q \over x} + { M_G^2 \over 1-x }
+ {1-x \over x}\,{Q^2 \over 4} \right) }
\over
\int_0^1 dx \,
e^{ - a \left({ M^2_q \over x} + { M_G^2 \over 1-x } \right)}
} \, ,
\end{eqnarray}
where $a^{-1} = (M_q + M_G)\omega_h$. It is now
visible that $f(Q^2)$ is hardly different from
1 for small $Q^2$: the inclusion of the
glue component $G$ does not significantly alter
the result for a form factor calculated as if the
glue component was absent. Moreover, in the picture
discussed in Appendix \ref{OverlappingSwarmsModel},
the glue component may disappear in favor of
overlapping constituent quarks when $\lambda$ is
lowered below $\lambda_c$. The momentum fraction
carried by the quarks, $x=x_q$, becomes very close
to 1 and the factor $f(Q^2)$ becomes 1.\footnote{Even
if the component $G$ contained quark-anti-quark
pairs, and were used to account for the quark
condensate inside hadrons \cite{BrodskyRobertsShrockTandy},
its neutrality would imply a small contribution to
form factors of charged hadrons, such as proton. It
might, however, contribute a detectable piece for
chargeless hadrons, such as neutron.}
At large $Q$, much greater than a hadron mass, the
hadron state needs to be represented in terms of
the Fock components created by quark and gluon
operators at $\lambda$ comparable with $Q$ itself,
in order to obtain a simple picture based on the
smallness of an asymptotically small coupling
constant that determines the strength of
interactions which are responsible for
distributing the large momentum transfer $q$ to a
minimal set of constituents required to build a
hadron. The minimal quark component may carry practically
the whole momentum of a hadron, $x_q \rightarrow 1$.
When this configuration dominates the transition
amplitude, the whole hadron is turned from the total
momentum $P$ to $P' = P+q$ in a combination of two
mechanisms. One is $x_q \rightarrow 1$, and the other
is the distribution of $q$ in the quark component.
The former contributes to soft effects, and the
latter is responsible for hard exclusive processes,
cf. \cite{LepageBrodsky}.
\subsection{ Structure functions }
\label{structurefunctions}
\begin{figure}
\begin{center}
\hspace{-1cm}
\includegraphics[scale=.6]{StructureFunction}
\end{center}
\caption{The RGPEP calculation of a baryon
structure function.}
\label{fig:StructureFunction}
\end{figure}
The transition amplitude for deep inelastic
lepton-hadron scattering is illustrated in Fig.
\ref{fig:StructureFunction}.\footnote{The lepton
is not shown. Also, Fig.
\ref{fig:StructureFunction} ignores the
possibility that the impinging boson is absorbed
by a constituent inside the component $G$, which
may be thought here to be only made of gluons. If
$G$ included quark-anti-quark pairs, it would
contribute to the sea parton distributions as well
as the quark condensate in hadrons, cf.
\cite{BrodskyRobertsShrockTandy}.} A hard photon
or other boson is suddenly absorbed by a
constituent characterized by the scale $Q$. The
new element RGPEP introduces is the possibility of
using the transformation $W_{Q\lambda}$ of Section
\ref{structureW} to calculate the probability
amplitude for finding such constituent in a
hadron. The scale $\lambda$ refers to the particle
operators in terms of which the hadron wave
function is obtained from the eigenvalue problem.
The scale $Q$ refers to the hard boson. The final
state in Fig. \ref{fig:StructureFunction} is made
of many constituents produced by $W_{Q\lambda}$.
Fig. \ref{fig:StructureFunction} does not show
that $W_{Q\lambda}$ actually acts on all
constituents.
The final state in Fig.
\ref{fig:StructureFunction} is in fact a virtual
state whose evolution factor into observable
particles is assumed to amount to unity. The
energy, or $P^-$ of the constituents that appear
in the final state in Fig.
\ref{fig:StructureFunction}, is dominated by the
constituent at scale $Q$ that absorbs the boson.
This situation corresponds to the leading operator
expansion terms, i.e., hand-bag diagrams that are
more important than cat-ears diagrams. One can use
the Hamiltonian $H_Q(b_Q)$ to account for $P^-$ of
the virtual state. This Hamiltonian is actually
the same as $H_\lambda(b_\lambda)$. Therefore,
$H_Q(Q)$ may count the energy of particles in the
final state including the interactions that are
responsible for grouping constituents at scale $Q$
into spectators at scale $\lambda$. Thus, the
inclusive sum over final states may be replaced by
summing over a relatively small number of
spectators at scale $\lambda$ and a potentially
large number of constituents at scales between
$\lambda$ and $Q$. The operator $W_{Q\lambda}$ in
RGPEP is hence expected to describe the evolution
of structure functions with $Q^2$.
The above qualitative reasoning must be verified
by new type of calculations using RGPEP evolution
equations in place of other evolution equations in
$Q^2$ \cite{GribovLipatov,Dokshitzer,AltarelliParisi}.
Since it is known that RGPEP incorporates the well-known
splitting functions in QCD \cite{gluons}, there is no
obvious reason for expecting that RGEPEP evolution will
significantly differ from known results where they apply.
On the other hand, RGPEP provides the framework for
combining the perturbative evolution with a non-perturbative
hadron wave functions obtained by solving eigenvalue
problem for $H_\lambda(b_\lambda)$. The eigenvalue problem
describes saturation at small $\lambda$. But the evolution
parameter $\lambda$ determines dependence of wave functions
on invariant masses of constituent states. The invariant
masses depend on transverse momenta and fractions $x$
of longitudinal momentum in a specific way. One can thus
expect that the evolution in $x$ \cite{BFKL1, BFKL2}
is uniquely related in RGPEP with the evolution in
$Q^2$.
\subsection{ Other processes }
\label{OtherProcesses}
The approximate picture of a hadron at scale $\lambda
\gtrsim \lambda_c$ as built from the effective quark
component and from $G$ treated as a scalar boson
suggests thinking that the dynamics of $G$ can be
further approximated by suitable effective Hamiltonian
interaction terms. These terms would result in
processes in which quanta of type $G$ participate
in strong interactions, softened by RGPEP form factors
in vertices. Nothing can be said at this point about
physical relevance of the approximate concept of
quanta of the type $G$ being exchanged between hadrons.
In particular, it is not excluded that an exchange of
such quanta may contribute to diffractive processes
in hadron-hadron collisions \cite{pomeron}, or even
in photon-hadron scattering \cite{GolecBiernat} if a
photon were allowed to contain its own glue component
coupled to quarks.
\subsection{ Connection with AdS/QCD }
\label{AdS}
The NR form of minimal coupling in LF Hamiltonians
at small $\lambda$ apparently leads to variables
and harmonic oscillator potentials that are
similar to the ones that Brodsky and Teramond
\cite{FAdS1, FAdS2,FAdS3} identified as providing
a correspondence between formulae for hadronic
form factors in LF QCD and in field theory in
5-dimensional AdS space
\cite{WittenHolography,PolczynskiStrassler}. The
new variables $\vec k$ for mesons, Eqs.
(\ref{kperp}) and (\ref{kz}), and $\vec K$ and
$\vec Q$ for baryons, Eqs. (\ref{Qperp}),
(\ref{Qz}), (\ref{Kperp}), (\ref{Kz}), differ from
standard LF relative three-momenta of constituents
in the CRF.\footnote{In LF dynamics, CRF differs
from the bound state rest frame because
conservation of $P^+$ implies that $P^3$ is not
conserved by interaction terms in LF
Hamiltonians.} For example, $\vec k_{standard}$ in
Eq. (\ref{standardk}) differs in length and
direction from $\vec k$. But there is no
difference between these momentum variables when
they are small in comparison to masses. Thus, the
new variables are identical to standard variables
in the NR dynamics at $\lambda \gtrsim \lambda_c$.
In addition, they fully describe a relativistic
relative motion of constituents, always providing
an exact representation of their free invariant
mass, but in a different way than the standard
momentum variables do. Standard variables
$k^z_{standard}$, $K^z_{standard}$, and
$Q^z_{standard}$,\footnote{E.g., see Ref.
\cite{GlazekNamyslowski}.} depend on the
transverse relative momenta. The new variables
$k^z$, $K^z$, and $Q^z$, are defined using the
constituent masses and fractions $x$ of their
total $P^+$, independently of the transverse
relative momenta. At the same time, the transverse
relative momenta are scaled by mass ratios and
$\sqrt{x(1-x)}$. The square-root factor is
precisely the one that appears in Brodsky-Teramond
holography \cite{FAdS3}.
To illustrate the possibility of correspondence
between the reinterpreted gluon-condensate induced
harmonic potential in LF QCD and a soft-wall potential
in Brodsky-Teramond AdS/QCD holography, we use example of a
meson form factor. The discussion is far from
complete and raises many questions regarding
interpretation of simple equations. However, it is
needed for showing similarities and differences
among different approaches.
Let the struck quark and the spectator quark carry
nearly whole hadron momentum. The glue component
$G$ provides additional smearing in the form
factor expression, Eq. (\ref{ff1}), unless it is
totally absorbed in the constituent
quarks.\footnote{See Appendix
\ref{OverlappingSwarmsModel}.} Let us assume here
that the latter option holds and all one needs to
consider is two constituent quarks.
In terms of the new variables,
\begin{eqnarray}
k^\perp & = & \kappa^\perp/\sqrt{4x(1-x)} \, , \\
k^z & = & (x-1/2)\,2m/\sqrt{4x(1-x)} \, ,
\end{eqnarray}
the LF eigenvalue equation for a meson of mass $M$
reads
\begin{eqnarray}
(4m^2 + 4\vec k^{\,2} + \kappa^4 \vec r^{\,2} ) \,
\psi & = & M^2 \, \psi \, ,
\end{eqnarray}
where $\kappa^2 = m\omega_M$ and $\vec r =
i\partial/\partial \vec k $. Using a factorized
wave function, $ \psi(k^\perp, k^z) = N \phi(k^\perp)
\, f_\eta(x)$, where $f_\eta$ is an eigenfunction
of the oscillator in $z$-direction,
\begin{eqnarray}
f_\eta(x)
& = &
H_\eta(k_z)\, e^{-
{k_z^2 \over \kappa^2}}
\, = \,
H_\eta\left[m(x-1/2)\over \kappa \sqrt{x(1-x)}\right]\,
e^{- {m^2 (x-1/2)^2 \over \kappa^2 x(1-x)}} \, ,
\end{eqnarray}
one arrives at the eigenvalue equation for
$\phi(k^\perp)$,
\begin{eqnarray}
\label{BT1}
\left[4m^2 + 4k^{\perp\,2} + \kappa^4 r^{\perp\,2}
+(4\eta + 2)\kappa^2 \right] \,
\phi & = & M^2 \, \phi \, ,
\end{eqnarray}
which is the eigenvalue problem to compare with
the Brodsky-Teramond holographic eigenvalue problem,
e.g., Eq. (10) in \cite{FAdS3}. We denote quantities
used by Brodsky and Teramond with subscript $BT$,
except for their $\zeta$,
\begin{eqnarray}
k^\perp & = & k^\perp_{BT}/2 \, , \\
r^\perp & = & 2 \zeta^\perp \, .
\end{eqnarray}
Eq. (\ref{BT1}) can be written using $\kappa_{BT} = \sqrt{2} \, \kappa$
as
\begin{eqnarray}
\left(k^{\perp\,2}_{BT}
+ \kappa_{BT}^4 \zeta^{\perp\,2} \right) \,
\phi & = & M_{BT}^2 \, \phi \, , \\
M_{BT}^2 & = & M^2 - 4m^2 - (2\eta + 1)\kappa^2_{BT} \, ,
\end{eqnarray}
where $k^{\perp\,2}_{BT} = - \left(\partial/ \partial
\zeta^\perp \right)^2$. For angular momentum around
$z$-axis equal $l_z$, using $\zeta = |\zeta^\perp|$,
one has
\begin{eqnarray}
\left[ - { \partial^2 \over \partial \zeta^2 }
-
{1 \over \zeta} {\partial \over \partial \zeta}
+
{l_z^2 \over \zeta^2}
+
\kappa_{BT}^4 \zeta^2 \right] \,
\phi & = & M^2_{BT} \, \phi \, .
\end{eqnarray}
Defining $ \phi = \phi_{BT}/\sqrt{\zeta}$,
we arrive at
\begin{eqnarray}
\label{BTeigenvalue}
\left[ - { \partial^2 \over \partial \zeta^2 }
-
{1 - 4l_z^2 \over 4\zeta^2}
+
\kappa_{BT}^4 \zeta^2
+ (2\eta + 1)\kappa^2_{BT}
\right] \, \phi_{BT}
& = &
(M^2 - 4m^2)\, \phi_{BT} \, , \nonumber \\
\end{eqnarray}
which by comparison with Eq. (11) in
\cite{FAdS3} for massless quarks renders
\begin{eqnarray}
U(\zeta)
& = &
\kappa_{BT}^4 \zeta^2
+
(2\eta + 1)\kappa^2_{BT}
\, ,
\end{eqnarray}
as a counterpart of the potential $U(\zeta)$ in
Ref. \cite{FAdS3}, with
\begin{eqnarray}
\label{kappaphi}
\kappa_{BT}^2 & = & 2 m \omega_M \, = \, {2 \pi \over 3}
\, \varphi_{glue} \, .
\end{eqnarray}
The role of quark masses on the right-hand side of
Eq. (\ref{BTeigenvalue}) requires explanation. A
sound explanation is currently not available in the
sense that it is not clear why the quark mass is set
to 0 in Eq. (10) and (11) in Ref. \cite{FAdS3}. But
it is plausible for constituent quarks that the
condensate mass-advantage constant $M_\varphi$ in
Eq. (\ref{hadronspectrum}) can be considered a result
of incorporation of $G$ in the constituent quarks
entirely. $M_\varphi$ may be so large that it reduces
the contribution of the quark masses to the hadron
mass and thus reduces the term $-4m^2$ in the
eigenvalue.
On the other hand, the SW model eigenvalue Eq. (12)
in Ref. \cite{SW} reads
\begin{eqnarray}
\label{SWeq}
- \psi'' + \left[ z^2 + {\lambda_z^2 -1/4 \over
z^2}\right] \psi & = & E \psi \, , \\
E \, = \, 4n_z + 2\lambda_z + 2 \, . &&
\end{eqnarray}
By dividing Eq. (\ref{BTeigenvalue}) by $\kappa_{BT}^2$,
and using variable $z = \kappa_{BT}\zeta$, one
obtains
\begin{eqnarray}
- \phi''_{BT}
+
\left[ W(z) + {l_z^2 -1/4\over z^2} \right] \, \phi_{BT}
& = & E \, \phi_{BT} \, , \\
E \, = \, { M^2 - 4m^2 - (2\eta + 1)\kappa_{BT}^2
\over \kappa_{BT}^2 } \, . &&
\end{eqnarray}
Comparing this result with the SW model result, Eq.
(\ref{SWeq}), and identifying $\lambda_z$ with $l_z$,
one obtains
\begin{eqnarray}
W(z) & = & z^2 \, .
\end{eqnarray}
The SW model quadratic potential appears to have
a coefficient given by the gluon condensate inside
hadrons, Eq. (\ref{kappaphi}).
If $\eta = 0$, which means no excitation along the
LF, only $-\kappa_{BT}^2$ enters the eigenvalues.
This is a sizable term (see below) and its
compensation is not guaranteed by $M_\varphi$.
Baryon form factors are described in terms of an
active constituent and spectators. The analysis
proceeds in a similar way to the case of a meson
built from two constituents of different masses.
This simplification is available because the free
LF invariant mass can be written in terms of the
active constituent and the rest of a hadron as if
the rest of a hadron was a single particle with
its mass equal to the invariant mass of the rest of
a hadron. The closest analogy to consider is that
a baryon form factor and eigenvalue problem can
be represented in terms of a quark and a di-quark.
It should be stressed, however, that it is far too
early for drawing a firm conclusion regarding
connection between gluon condensation in hadrons
and SW models on the basis of a pure RGPEP
calculation. The reason can be seen using power
counting \cite{Wilsonetal}. Besides uncertainties
concerning $x^-$ and inverse of $i\partial^+$,
spin-independent quark-anti-quark interaction term
in a LF Hamiltonian density with a quadratic
potential, which is absent in canonical QCD, may
have a coefficient proportional to
$\Lambda_{QCD}^2$, to compensate the dimension of
$x^{\perp \, 2}$. There also exists a possibility
\cite{NonlocalH} that a dimensionless square of a
product $r_\mu P^\mu$ is used instead of $x^{\perp
\, 2}$, where $r$ is the relative position of
arguments of two field operators on the LF,
$r^+=0$, and $P$ is the momentum carried by a
product of two quark fields, i.e., two
constituents interacting by the potential. This
possibility implies that the harmonic potential
does not necessarily requires a factor
$\Lambda_{QCD}^2$ because a gradient cancels
dimension of a distance. Another $\Lambda_{QCD}^2$
may be needed to cancel the dimension of
additional integration measure $d^2x^\perp$.
Still, the potential term must vanish when
$\lambda \rightarrow \infty$. If it vanishes as
$1/\lambda^2$, another $\Lambda^2_{QCD}$ may be
expected in the coefficient. There always exists a
possibility to use quark masses as dimensionful
quantities. Together, one may need powers of
$\Lambda_{QCD}$ as high as 6. Such terms may be
hard to establish quickly in any renormalization
group procedure.
In any case, the above results certainly allow one
to entertain the possibility that the
small-$\lambda$ RGPEP picture of hadrons, at
strong coupling, can actually be the one that
corresponds to the AdS picture, including the SW
model \cite{SW}. On the other hand, even if the SW
model does correspond to the Brodsky-Teramond LF
holographic picture with a harmonic oscillator
potential between an active constituent and
spectators (in transverse directions), and even if
these pictures can be related to the gluon
condensation in hadrons, how is RGPEP to explain
emergence of the 5th dimension in the AdS/QCD
analogy on the QCD side?
It has been pointed out that the 5th dimension in
AdS may correspond to a renormalization group
scale in gauge theory \cite{Maldacena1,Polyakov1}.
More recent calculations, such as
\cite{Akhmedov1,Akhmedov2}, also point in this
direction. We observe in this context that RGPEP
provides an opportunity to explicitly introduce a
5th dimension in QCD with a simple physical
interpretation. Namely, Eqs.
(\ref{quantumfieldpsilambda}) and
(\ref{quantumfieldAlambda}) for effective fields
can be rewritten using the observation that
$\lambda$ corresponds to the inverse size of
effective particles. The effective particles
exhibit this size in their interactions; the
effective interactions contain vertex form factors
whose momentum width is determined by the RGPEP
parameter $\lambda$. It is convenient to think
about the size of the effective particles using
variable $s=1/\lambda$.\footnote{ The particle
size parameter $s$ enters the non-perturbative
RGPEP equations through replacement of the
derivative $d/d\lambda^{-4}$ on the left-hand
side of Eq. (\ref{npRGPEP}) by $d/ds^4$.}
So, instead of labeling the field and particle
operators with $\lambda$, we can label them with
$s$ that corresponds to the size of effective
particles. The effective fields on the LF can be
rewritten according to a rule
\begin{eqnarray}
\label{psixs}
\psi_s(x)
& = & \int \hspace{-15pt}\sum
[k] \, b_s(k) \, e^{ikx}
\, = \,
\psi(x,s) \, .
\end{eqnarray}
In this rule, the 5th argument of the quantum
field is the RGPEP size of effective particles.
This size plays the role of a renormalization
group parameter in RGPEP.
While the AdS dual picture involves its own
renormalization issues \cite{BoerVerlinde1,
SkenderisLecture2002, Skenderis2004,
KiritsisPart2}, and RGPEP methodology may
eventually find applications in these issues, too,
the 5th dimension of particle size makes QCD {\it
a priori} a 5-dimensional theory, with scale
invariance broken by $\Lambda_{QCD}$. The
structure of such a theory may or may not be
similar to an AdS field theory. On the other hand,
RGPEP is certainly available for studying the
5-dimensional QCD.
The issue that emerges immediately is that in
order to evaluate observables in QCD one in
principle can use just one value of $s$ or
$\lambda$. There is no need for integration over
the 5th coordinate. So why would one integrate
over the 5th dimension in the dual picture?
The Brodsky-Terramond LF holography for hadronic
form factors suggests that the integration over
the 5-th dimension in AdS field theory corresponds
to integration over relative transverse momentum
of an active hadronic constituent with respect to
spectators. But RGPEP suggests that the LF wave
functions have the form shown in Eq.
(\ref{psihadron}), which is Gaussian in invariant
mass of the hadronic constituents. Integration
over the transverse relative momenta in form
factors amounts to integration over invariant
mass, with a proper rescaling by $\sqrt{x(1-x)}$.
The invariant mass appears in ratio to a square of
the width parameter $\lambda$ or, equivalently, in
product with the square of the effective particle
size parameter $s = 1/\lambda$. The challenge is
to explain how the integration over $ {\cal M} $ can be
turned into integration over $\lambda$ or
$s=1/\lambda$.
Although we do not provide here the ultimate
response to this challenge, we offer a qualitative
argument that suggests where one can look for the
solution in future studies. Consider a fixed scale
of an observable, such as $Q$ in a form factor.
For a given $\lambda = 1/s$, one can evaluate the
observable in QCD by integrating wave functions
over $ {\cal M} $. But if the wave functions are
functions of $ {\cal M} /\lambda = s {\cal M} $, they are
constant on hyperbolas defined by the condition
that $s {\cal M} $ is fixed. One can consider a plane of
variables $ {\cal M} $ and $s$ and plot a wave function
in a perpendicular direction, forming a 3-dimensional
plot of the wave function. Select a value of $s
= 1/Q$ and draw a profile of the 3-dimensional
plot for the selected value of $s$ as a function
of $ {\cal M} $. Draw another profile as a function of
$s$ for $ {\cal M} = Q$. These two profiles are
identical for all wave functions that only depend
on the variables $ {\cal M} $ and $s$ through their
product $t$. In such circumstances, integration
over relative motion of quarks of fixed size $Q$
is equivalent to integration over all sizes $s$ of
quarks that have fixed invariant mass $Q$. In QCD,
this pure scaling picture for lightest quarks
must be broken by the scale set through
$\Lambda_{QCD}$.
The small-$\lambda$, or large-$s$ picture of
hadrons in RGPEP appears in accordance with
expectations that there exists a low-energy regime
with a stable strong coupling constant
\cite{BrodskyCoupling1}.\footnote{RGPEP may be
used to seek a precise definition of such coupling
constant using quark-gluon, three-gluon, and
inter-quark potential terms in the corresponding
Hamiltonians, cf. \cite{AFandBS,gluons}.} In the
stabilization region, the gluon condensation
parameter $\varphi_{glue}$ inside hadrons is
considered constant as a function of $\lambda$.
The hadronic wave functions are Gaussian in the
invariant mass with the width provided by quark
masses and $\varphi_{glue}$. The overall result is
that at small $\lambda$ the effective parameters
are frozen at values comparable with
$\Lambda_{QCD}$. On the other hand, when $\lambda$
increases, and $s$ decreases, the Hamiltonian
changes and it is expected to eventually simplify
to the QCD canonical form with counterterms at
$\lambda \rightarrow \infty$, i.e., for point-like
quarks and gluons, $s \rightarrow 0$. Therefore,
the wave functions of hadronic eigensolutions
change with $\lambda$, or $s$. One may expect that
when the wave function width in momentum space
increases and the sizes of effective quarks and
gluons decrease, the invariant mass dependence
turns from Gaussian to a power-law behavior at
large virtuality, perhaps as indicated by
perturbation theory for exclusive processes
in a collinear approximation \cite{LepageBrodsky}.
Finally, regarding the Brodsky-Teramond holography
and SW models, we wish to mention that one considers
different values of the SW parameter $\kappa^4$ in
front of $\zeta^2$ in the AdS equations for masses
of mesons (M) and baryons (B) \cite{SWkappa}.
Namely, $\kappa_M \sim 0.54-0.59$ GeV and
$\kappa_B \sim 0.49$ GeV, with
\begin{eqnarray}
{\kappa_M \over \kappa_B } & \sim & 1.15 \pm 0.5 \, .
\end{eqnarray}
In the reinterpreted gluon condensate image for hadrons,
assuming a universal value of mass for $u$ and $d$
constituent quarks, one has
\begin{eqnarray}
{\kappa_M \over \kappa_B }& = & \left({8 \over 5}\right)^{1/4}
\sim 1.125 \, .
\end{eqnarray}
This agrees quite well with the phenomenological
result.
\section{Conclusion}
\label{c}
Reinterpretation of the gluon condensate in RGPEP
has several implications that matter in theory of
hadrons. Instead of the entire vacuum, the gluons
condense only inside hadrons. If this option were
actually realized in QCD, there would be no need
anymore to construct the quantum vacuum state in
Minkowski space that satisfies fundamental
requirements of invariance with respect to
Poincar\'e transformations, a construction that so
far eluded theoreticians. At the same time, the LF
Hamiltonian formulation of QCD would become free
from any obligation to produce a non-trivial
vacuum that is commonly assumed to provide
physically important expectation values. When
reinterpreted, the same expectation values come
from distribution of matter that is limited to the
interior of hadrons. Simultaneously, it becomes
not clear how one should interpret expectation
values that are measured in lattice formulation of
gauge theories; they might also correspond to the
hadron interior rather than a state of infinite
volume. In any case, the reinterpretation is
available within the RGPEP that provides tools for
a close inspection what actually happens. Thus,
the picture of hadrons that is sketched here in
the context of reinterpretation of gluon
condensate can be viewed as providing an idea how
the CQM may be an approximation to QCD and a
method for verifying this idea.
The constituent picture based on gauge symmetry
dictates Gaussian LF wave functions that are
exponentials of free invariant masses of
constituents. The associated oscillator
frequencies are in agreement with quark models
phenomenology when the gluon condensate inside
hadrons is set to the value found in phenomenology
using QCD sum rules. The corresponding eigenvalue
problems appear naturally in terms of variables in
which AdS/QCD models are developed, especially the
SW model. The ratio of baryon and meson oscillator
frequencies in SW models agrees with the ratio
implied by the reinterpreted gluon condensate in
RGPEP.
This article provides no comparable evidence for
reinterpretation of the quark condensate. But one
may hope that RGPEP can generate constituent quark
masses when $\lambda$ is lowered to values
comparable with $\Lambda_{QCD}$ and shed this way
some light on the mechanism of absence of chiral
symmetry in hadronic spectrum and the nature of
$\pi$-mesons as nearly Goldstone particles.
The author's opinion that RGPEP provides a method
for verifying the reinterpretation of the gluon
condensate and solving for hadronic structure in
QCD is mainly based on the fact that RGPEP is now
available in both perturbative and
non-perturbative versions, both being boost
invariant. The step beyond perturbation theory
maintaining boost invariance is seen as a chance
for simultaneously taking advantage of knowledge
about hadrons in the constituent picture and in
the parton picture, while RGPEP appears capable of
generating required scale dependence dynamically.
Seen this way, RGPEP provides an operator calculus
for deriving quark and gluon wave functions of
hadrons in the LF Fock space and unify
calculations of the hadron mass spectrum and
partonic distributions inside hadrons. At the same
time, new variables identified in the
reinterpretation of the gluon condensate, relative
momentum three-vectors $\vec k$ in mesons and
$\vec K$ and $\vec Q$ in baryons, provide a frame
of reference in which LF solutions can be
classified in terms of well-known angular momentum
classification of constituent wave functions. The
same transverse variables occur in LF AdS/QCD
holography. Longitudinal variables are new and
require further testing.
A summary of RGPEP program for verifying the
reinterpretation of gluon condensate is following.
Start with canonical QCD. Set up regulated
$H_{QCD}$ on the LF. Apply RGPEP to lower the
scale parameter $\lambda$ toward $\Lambda_{QCD}$.
Find counterterms in $H_{QCD}$ and evaluate
effective $H_\lambda$. Solve eigenvalue problem of
$H_\lambda$ and see if the gluon-condensate
induced oscillator picture is reproduced by gluon
components in hadronic states. So far, the RGPEP
scheme is known to work only in some crudely
approximate calculations in QCD for heavy
quarkonia. Studies of hadrons built in QCD from
light quarks require breaking an entirely new
ground.
\begin{appendix}
\section{ Universality in RGPEP }
\label{RGPEPuniversalityappendix}
This Appendix explains the RGPEP scheme in which
CQMs can be sought as limited representatives of the
universality class which QCD is thought to belong
to as a quantum field theory with a Hamiltonian
acting in a Hilbert space, cf.
\cite{DiracDeadWood}. The scheme is composed of
two interrelated parts.
In the first part of RGPEP, one constructs a
sequence of rotations $U_\lambda$ for creation and
annihilation operators in order to discover
counterterms in $H_\infty$ and obtain the
effective Hamiltonian $H_\lambda$. The second part
involves solving the eigenvalue problem of
$H_\lambda$. Solutions to the eigenvalue problem
of $H_\lambda$ are needed to fix finite parts of
counterterms in $H_\infty$
\cite{GlazekWilson1,GlazekWilson2}.
The first part of the RGPEP calculation involves
dealing with ultraviolet divergences. Instead of
standard renormalization group procedure based on
Gaussian elimination in linear problems
\cite{Wilson1,Wilson2}, RGPEP is based on a
unitary transformation of field or particle
operators. In terms of matrix elements of a
Hamiltonian, this means that one rotates the basis
states like in the similarity renormalization
group approach (SRG) \cite{GlazekWilson1,GlazekWilson2}.
Formulation of RGPEP in terms of operators for
effective particles organizes otherwise
un-intelligible variety of Hamiltonian matrix
elements in a physically motivated way; one traces
coefficients of specific operators instead of only
calculating matrix elements to which many
operators may contribute. It also provides the
concept of effective quantum field; see Eqs.
(\ref{quantumfieldpsilambda}),
(\ref{quantumfieldAlambda}), and below. The
operator structure of RGPEP results in unitarity
features in the calculated interactions that are
absent in the general formulation of SRG, since
SRG applies also to Hamiltonians that cannot be
written in terms of creation and annihilation
operators. For example, $U_\lambda(b_\infty) =
U_\lambda(b_\lambda)$. The operator structure is
also helpful in demonstrating a connection between
the momentum space formulation of the theory in
terms of quanta (effective particles) and position
space formulation in terms of quantum fields
(effective fields). The interaction terms in
effective Hamiltonian densities are non-local.
RGPEP provides a method for calculating the
effective non-local interactions that correspond
to renormalized canonical theories
\cite{NonlocalH}.
In contrast to the first part of the RGPEP scheme
that is akin to the SRG scheme, the second part
resembles the standard Wilsonian procedure because
it involves Gaussian elimination, in the
mathematical sense as part of solving a linear
problem. However, in sharp distinction from the
standard procedure, the second step of RGPEP does
not involve ultraviolet divergences. These are
eliminated in the first step.
The RGPEP differential equation for $H_\lambda$
in its generic form reads
\begin{eqnarray}
\label{differentialeq}
{d \over d\lambda} \, {\cal H} _\lambda
& = &
{\cal F} _\lambda \left[ {\cal H}_\lambda \right] \, ,
\end{eqnarray}
where ${\cal H}_\lambda$ denotes $H_\lambda(b_\infty)$
and $ {\cal F} _\lambda$ is a suitable functional.\footnote{E.g.,
see \cite{RGPEPform}. Appendix \ref{NPRGPEP} provides
a non-perturbative definition of $ {\cal F} _\lambda[ {\cal H} _\lambda]$.
One needs to adjust the dimension of parameter $\lambda$
to dimensions of the Hamiltonian $ {\cal H} _\lambda$ and
functional $ {\cal F} _\lambda \left[ {\cal H}_\lambda \right]$.
The parameter $\lambda$ is defined in terms of a momentum
scale that plays the role of width in momentum-space vertex
form factors, denoted also by $\lambda$, or by the inverse of
the width. The inverse of $\lambda$ corresponds to the
size of effective particles that they exhibit in the
interactions. The inverse is denoted by letter $s$ that
refers to the word {\it size}.} The solution
is thus also of the generic form
\begin{eqnarray}
\label{allsteps}
{\cal H}_\lambda & = &
{\cal H}_\infty + \int_\infty^\lambda d \lambda' \, {\cal F} _{\lambda'}
\left[ {\cal H}_{\lambda'} \right] \, .
\end{eqnarray}
$H_\lambda(b_\lambda)$ is obtained by
replacing $b_\infty$ in ${\cal H}_\lambda$ by
$b_\lambda$.
Irrespective of the theory one considers, for
$\lambda$ much greater than the regularization
parameter $\Delta^2/\epsilon$ (see Section
\ref{effectiveparticles}) times the largest
allowed number of creation or annihilation
operators in any product of them in the initial
interaction Hamiltonian, $ {\cal F} _\lambda[ {\cal H} _\lambda]$ is
exponentially close to zero. The reason is that
the regulated interactions are exponentially close
to zero when the invariant mass of interacting
particles is grater than allowed by the
regularization.
So, one can always introduce some $\Lambda \gg
\Delta/\sqrt{\epsilon}$ and arrange a series of
cutoffs $\lambda_0 = \Lambda$, $\lambda_1 =
\Lambda/2$, $\lambda_2 = \Lambda/4$, etc., until
one reaches $\lambda = \lambda_n = \Lambda/2^n$
for some large $n$.\footnote{Instead of the factor
2, one can choose $e$ or any other similar number.
The exponential spacing of $\lambda$s in the
sequence of cutoffs is introduced for the purpose
of handling logarithmic singularities. In the case
of simple models in which one can easily introduce
an exponential grid in momentum space, such as
the NR Schr\"odinger equation for one particle in
a $\delta$-function potential, it is most
convenient to use a grid in which the chosen
factor for cutoff reduction, similar to 2 or $e$
above, is also equal to an integer power of the
constant used in the grid, i.e., if $p_k = p_0
a^k$ in the grid, then $\Lambda_n/\Lambda_{n+1} =
a^l$ with an integer $l$ such that $a^l$ equals 2,
$e$, or some other convenient factor chosen for the
cutoff reduction. This relates the value of
$a$ chosen for the exponential grid in momentum
variables with the RGPEP cutoff reduction factor in
terms of an integer $l$.} This is done in analogy
to Refs. \cite{Wilson1, Wilson2} for the purpose
of sequencing the RGPEP procedure of evaluating
$H_\lambda$ into manageable steps that are free
from huge terms resulting from ultraviolet
divergences of the local theory. Namely, step
number $k$ in the sequence that builds up the
integral in Eq. (\ref{allsteps}), contains only
integration over scales between $\lambda_{k+1}$ to
$\lambda_k$. When one evaluates effects of
interactions that allow changes of invariant
masses only in the range between $\lambda_{k+1}$
and $\lambda_k$, the ratio $\lambda_0/\lambda_n
\rightarrow \infty$ does not contribute. Moreover,
one can carry out each and every one of the finite
steps using well-known techniques, such as
perturbation theory or numerical methods.
Although the sequence one obtains is analogous to
the sequence considered in Wilsonian
renormalization group procedure \cite{Wilson2}, it
differs considerably because no states are
eliminated from the domain of the Hamiltonian. The
RGPEP steps are automatically free from the
dependence on eigenvalues that appear in Gaussian
elimination in matrix notation for eigenvalue
problems and there is no need for an additional
(and in principle determined only up to a nearly
arbitrary rotation of basis) procedure required
for maintaining formal hermiticity of effective
Hamiltonians.\footnote{ In Ref. \cite{Wilson2},
the operation chosen to maintain Hermiticity is
denoted by $R$, Eq. (4.1).}
The ratio of subsequent parameters, $\lambda_{k+1}
/ \lambda_k$ does not depend on $k$ and one can
seek regularity in how successive integrations
transform $H_{\lambda_k}$ to $H_{\lambda_{k+1}}$.
Every element in the resulting sequence of
Hamiltonians can be written in terms of
dimensionless momentum variables $y^\perp =
p^\perp / \lambda_k $ instead of momenta
$p^\perp$. Creation and annihilation operators
also require rescaling in order to keep their
commutation relations in terms of variables
$y^\perp$ independent of the step number $k$. In
principle, such rescaling corresponds to
scale-dependent renormalization constants for
fields in standard perturbative formulation of
quantum field theory. The coefficients
$c_{\lambda_k}$ of products of creation and
annihilation operators $b_{\lambda_k}$ in
$H_{\lambda_k}$ can be studied as functions of
$y^\perp$ and $x$. These functions evolve in $k$.
Their evolution can be classified in terms of
characteristic dominant behavior associated with
universality classes, presumably associated with
fixed points, limit cycles, and even chaotic
behavior. Such analysis is also expected to help
in identifying dominant effects due to small-$x$
cutoff parameter $\delta$ in gauge theories
quantized on the LF (see Section
\ref{effectiveparticles}). No farther comments are
offered here regarding the evaluation of
$H_\lambda$, except for stressing that the
establishment of finite parts of ultraviolet
counterterms eventually requires a reference to
the second part of the RGPEP scheme.
The second part involves solving for the spectrum
of $H_{\lambda_n}$. This is facilitated using
techniques similar to the ones used in Ref.
\cite{Universality} with the following qualitative
distinction. The Hamiltonian $H_{\lambda_n} = H_{0
\lambda_n} + H_{I \lambda_n}$ is so narrow on
energy (invariant mass) scale (defined by
eigenvalues of $H_{0 \lambda_n}$), that the wave
functions of its eigenstates are typically
exponentially suppressed outside the range of
energies (invariant masses) that are similar in
size to the observable energies (invariant masses)
one is interested in \cite{lcet}. Therefore, one
is no longer in need to solve the severe
ultraviolet renormalization problem when carrying
out calculation that removes residual
(exponentially small) cutoff effects using a
procedure such as in \cite{Universality}. This
suggests, for example, that the non-linear
operators that produce corrections to scaling in
Wegner's sense \cite{WegnerOperators} only lead to
small corrections to the scaling properties (if
any are obtained) that result already from the
first step of the RGPEP scheme.
The expectation of lack of significant sensitivity
of the second part of RGPEP scheme to the actual
values of smallest eigenvalues can be forecast by
analogy with (and it is confirmed in) an elementary
model of a harmonic oscillator with an additional
potential proportional to the fourth power of
distance (a Higgs-like field dynamics in zero
dimensions) \cite{WojcikLicencjat}. Excited states
of the pure oscillator (excited by multiples of
$\hbar \omega$) serve as a model of the Fock space
basis with different numbers of effective
particles, each of mass $m = \hbar \omega$. The
Hamiltonian with the quartic term models
$H_{\lambda_n}$ after the entire first part of
RGPEP scheme is completed. Namely, the interaction
is narrow on the energy scale because it is able
to change the number of particles only by 0, 2, or
4, which means that the size of energy changes due
to interactions is limited from above by $4\hbar
\omega$, or $4m$. The result of numerical
computation of eigenstates in that model
\cite{WojcikLicencjat} is that for obtaining
smallest eigenvalues accurately it is sufficient
to use Gaussian elimination assuming that the
smallest eigenvalues are simply zero. One can come
down this way to cutoffs on energy of basis states
not much greater than about 10 times $\hbar
\omega$, or $10m$. This holds even for quite large
values of the coupling constant that characterizes
the size of the quartic coupling.
Since the entire RGPEP procedure is in fact a
sequence of successive approximations with an
increasing space of variables when regularization
is lifted, both parts of it, the rotation and
solution, need to be iterated, in the sense of
Wilsonian triangle of renormalization, until a
stable result is established. The iteration
involves finite parts of counterterms. They are
fixed by comparison with data in the solution
part. But the finite parts influence the rotation
in the first part. This is why the two parts of
the procedure are interrelated and neither the
part using unitary rotation of particle operators
nor the solution part with elimination of states
can be distinguished as determining the path of
successive approximations for the Hamiltonian on
the triangle. As a result, the evolution of a
Hamiltonian on the triangle is neither purely
unitary nor a plain solving of an {\it a priori}
specified eigenvalue problem. In other words, the
ultimate self-consistent solution to the triangle
of renormalization defines the theory one is
interested in solving to explain observables.
RGPEP requires extensive studies in order to
verify if it can produce a universal low-energy
theory that explains success of the CQMs starting
from the Lagrangian for QCD. In addition, all of
these models could also be subjected to RGPEP or
SRG as long as they can be written in a
Hamiltonian form.\footnote{This is not immediately
obvious in the case of approaches based on the
Dyson-Schwinger and Bethe-Salpeter equations.}
Therefore, there arises a question if all CQMs that
do reproduce a considerable number of observables for
smallest-mass states in hadronic spectrum actually
converge on a single, universal model with some
specific shape of quark potential. Then, the
question would be if a universal model, if it is
obtained, matches the effective theory obtained
from QCD using RGPEP.
It should be stressed, however, that the procedure
outlined here is not limited to RGPEP for quarks
and gluons in LF QCD. It can be applied to studies
of universality in all Hamiltonians that can be
expressed in terms of creation and annihilation
operators acting in some Fock space and, even more
generally, to all Hamiltonians that require
renormalization and can be subject to the SRG
procedure \cite{GlazekWilson1,GlazekWilson2}. In
particular, one can apply the same procedure when
using beautiful Wegner's flow equation
\cite{Wegner, WegnerReview,Kehrein,Bach}, and the
latter can also be suitably altered in order to
improve weak-coupling expansion that may apply in
the case of asymptotically free theories
\cite{AlteredWegnerPRD,AlteredWegnerAPP}.
\section{ Calculations of $W$ }
\label{AppendixW}
Assuming that $\Lambda_{QCD}$ formally tends to 0
in comparison to quark masses, so that $g_\lambda$
can be treated as extremely small when $\lambda$
is comparable with the masses, $W$ could be
calculated in RGPEP using perturbative formulae
given in Section \ref{W1inQCD} and expanding LF
quantum fields into bare creation and annihilation
operators. In this case, the dominant term besides
1 is $g_\lambda W_1$. However, for a realistic
value of $\Lambda_{QCD}$, which is much larger
than $u$ and $d$ quark mass parameters in the SM,
and for $\lambda \sim \lambda_c$ that is expected
to correspond to the CQMs, perturbative expression
cannot be considered reliable on the basis of
smallness of $g_\lambda$. Nevertheless,
perturbative calculations described here
illustrate the structure of $W$ that carries over
to values of $\lambda$ comparable with $\lambda_c$
for as long as ${\cal H}_\infty$ as an initial
condition in RGPEP equations is replaced by the
corresponding ${\cal H}_\lambda$ and $\lambda$ is
sufficiently near in magnitude to $\lambda_c$ so
that $W$ is not violently different from 1. The
key argument is that the strength of the
interaction is not fully determined by $g_\lambda$
alone. Namely, when the RGPEP form factor $f_\lambda$
is narrow as function of momentum variables, the net
strength of the interaction is suppressed and the
interaction can be weak even if $g_\lambda$ is
sizable. Thus, the calculation given below
illustrates not only how the lowest-order perturbative
$W$ looks like in the case of small $g_\lambda$
but also how the structure of $W$ is related to
the structure of ${\cal H}_\lambda$ that is weak
because $\lambda$ is small, even though ${\cal
H}_\lambda$ is expected to significantly differ
from ${\cal H}_\infty$ when $\lambda$ approaches
$\lambda_c \sim \Lambda_{QCD}$ for light quarks.
For the canonical quark field, we have on the LF
\begin{eqnarray}
\label{quantumfieldpsi}
\psi & = & \sum_{\sigma c f} \int [k]
\left[ \chi_c u_{fk\sigma} b_{k\sigma c f} e^{-ikx} +
\chi_c v_{fk\sigma} d^\dagger_{k\sigma c f} e^{ikx}
\right]_{x^+=0} \, .
\end{eqnarray}
The momentum and spinor notation involves
(e.g., see Ref. \cite{GlazekMlynik})
\begin{eqnarray}
{[}k{]} & = & \theta(k^+) dk^+ d^2 k^\perp/(16\pi^3 k^+) \, , \\
u_{fk\sigma} & = & B(k,m_f) u_{f\sigma} \, , \\
v_{fk\sigma} & = & B(k,m_f) v_{f\sigma} \, , \\
v_{f\sigma} & = & C u^*_{f\sigma}
\, , \\
u_{f\sigma} & = &
\sqrt{2m_f} \,\, \zeta_f \, \chi_\sigma \, ,
\end{eqnarray}
where $\zeta_f$ is a normalized vector in flavor space,
$\chi_\sigma$ is a spinor that corresponds to one of
two spin states of a fermion at rest, both normalized
to 1 and orthogonal to each other, $C$ is the charge
conjugation matrix,
\begin{eqnarray}
B(k,m) & = & {1\over \sqrt{k^+ m}}
[ \Lambda_+ k^+ + \Lambda_- (m + k^\perp \alpha^\perp)]
\end{eqnarray}
is the LF boost matrix, and $\Lambda_\pm = {1\over 2}\gamma_0 \gamma^\pm$.
For the canonical gluon field, we have
\begin{eqnarray}
\label{quantumfieldA}
A^\mu & = & \sum_{\sigma c} \int [k] \left[ t^c \varepsilon^\mu_{k\sigma}
a_{k\sigma c} e^{-ikx} + t^c \varepsilon^{\mu *}_{k\sigma}
a^\dagger_{k\sigma c} e^{ikx}\right]_{x^+=0} \, , \\
\varepsilon^\mu_{k\sigma} & = & (\varepsilon^+_{k\sigma}=0,
\varepsilon^-_{k\sigma} \, = \, 2k^\perp \varepsilon^\perp_\sigma/k^+,
\varepsilon^\perp_\sigma) \, .
\end{eqnarray}
The same expansions can be used for effective
quantum fields in which creation and annihilation
operators correspond to some scale $\lambda$
\cite{NonlocalH}\footnote{In order to relate the above
notation to the one introduced in Ref. \cite{NonlocalH},
one needs to treat $d^\dagger$ with positive $k^+$
as $b$ with negative $k^+$, and $a^\dagger$ with
positive $k^+$ as $a$ with negative $k^+$, taking
into account the well-known constraint relations
for fermion field $\psi_- = \Lambda_- \psi$ and
gauge boson fields $A^-$ \cite{BjorkenKogutSoper,Yan1,Yan2,
spinors2,Wilsonetal}.}, see Eqs. (\ref{quantumfieldpsilambda})
and (\ref{quantumfieldAlambda}).
To evaluate $W_1$ using Eq. (\ref{W1example}), one first
needs to evaluate ${\cal H}_{I\infty 1}$ and ${\cal H}_0
= {\cal H}_{0 \infty}$. According to Eq. (\ref{calHinfty}),
\begin{eqnarray}
{\cal H}_{0\infty} & = & \int dx^- d^2 x^\perp \,
h_{0\infty} \, , \\
{\cal H}_{I\infty 1} & = & \int dx^- d^2 x^\perp \,
h_{I \infty 1} \, .
\end{eqnarray}
Using Eq. (\ref{hinfty0}) and the field expansions, one obtains
the normal-ordered expression for ${\cal H}_0$,
\begin{eqnarray}
{\cal H}_0 & = &
\sum_{\sigma c f} \int [k] \, {k^{\perp \, 2} +
m^2_f \over k^+} \,
\left[b^\dagger_{k\sigma c f}b_{k\sigma c f} +
d^\dagger_{k\sigma c f}d_{k\sigma c f} \right]
\nonumber \\
& + &
\sum_{\sigma c} \int [k] \, {k^{\perp \, 2} \over k^+} \,
a^\dagger_{k\sigma c}a_{k\sigma c} \, .
\end{eqnarray}
Similarly, using Eq. (\ref{hinfty1}), one obtains
${\cal H}_{I \infty 1}$,
\begin{eqnarray}
\label{resultH1}
{\cal H}_{I \infty 1}
& = &
\sum_{123}\int[123] \, 2(2\pi)^3 \delta^3( P_c - P_a)\,
r_{\Delta\delta}(1,2,3) \,
\left[\,\bar u_2 \not\!\varepsilon_1^* u_3 \,
t^1_{23} \, b^\dagger_2 a^\dagger_1 b_3 \right.
\nonumber \\
& - &
\left.
\bar v_3 \not\!\varepsilon_1^* v_2 \, \cdot t^1_{32} \cdot
\, d^\dagger_2 a^\dagger_1 d_3
+
\bar u_1 \not\!\varepsilon_3 v_2 \, \cdot t^3_{12} \cdot
\, b^\dagger_1 d^\dagger_2 a_3
\right.
\nonumber \\
& + &
\left.
Y_{123}\, a^\dagger_1 a^\dagger_2 a_3 + h.c. \right]
\end{eqnarray}
where $P_c$ denotes the total momentum of created
particles, $P_a$ denotes the total momentum of annihilated
particles, $r_{\Delta \delta}(1,2,3)$ denotes the
regularization factors described in Section
\ref{effectiveparticles}, $t^a_{ij} =\chi^\dagger_{ic}
t^a \chi_{jc}$,
\begin{eqnarray}
Y_{123} & = & i f^{c_1 c_2 c_3} \left[
\varepsilon_1^*\varepsilon_2^* \cdot \varepsilon_3\kappa
-
\varepsilon_1^*\varepsilon_3 \cdot \varepsilon_2^*\kappa {1\over x_{2/3}}
-
\varepsilon_2^*\varepsilon_3 \cdot \varepsilon_1^*\kappa {1\over x_{1/3}}
\right] ,
\end{eqnarray}
with $\varepsilon \equiv \varepsilon^\perp$ and
$\kappa = k_1^\perp - k_1^+ k^\perp_3/k_3^+$, and
$h.c.$ denotes Hermitian conjugation of the 4
explicitly written terms in the square bracket
in Eq. (\ref{resultH1}).
The result for ${\cal H}_{I \infty 1}$ needed in Eq.
(\ref{W1example}), can be written in an abbreviated
form
\begin{eqnarray}
{\cal H}_{I \infty 1}
& = &
\sum_{123}\int[123] \, 2(2\pi)^3 \delta^3( P_c - P_a)\,
r_{\Delta\delta}(1,2,3) \,
\left\{1,2,3 \right\} \, ,
\end{eqnarray}
where $\left\{1,2,3 \right\} $ denotes the $4 + h.c.$ terms
with $b_\infty$, $d_\infty$, and $a_\infty$ replaced by
$b_\lambda$, $d_\lambda$, and $a_\lambda$, respectively.
Using this notation, the exact perturbative result for
$W_1$ in RGPEP reads
\begin{eqnarray}
\label{W1exact}
W_1
& = &
\sum_{123}\int[123] \, 2(2\pi)^3 \delta^3( P_c - P_a)\,
r_{\Delta\delta}(1,2,3)
\nonumber \\
& \times &
{ f_\lambda - f_{\lambda_c} \over P_a^- - P_c^- } \,
\left\{1,2,3 \right\} \, ,
\end{eqnarray}
where $P_a^-$ and $P_c^-$ are eigenvalues of ${\cal H}_0$
and $f$ denotes the RGPEP form factor,
\begin{eqnarray}
f_\lambda & = & e^{ -(P_c^2 - P_a^2)^2/\lambda^4} \, ,
\end{eqnarray}
in which the squares of invariant masses of
created particles, ${\cal M}^2_c = P_c^2$, and
annihilated particles, ${\cal M}^2_a = P_a^2$,
are evaluated using eigenvalues of ${\cal H}_0$
associated with the particle momentum components
$k_i^+$ and $k_i^\perp$ for $i = 1, 2, 3$ in
every term \cite{NonlocalH}.
The structure of $\left\{1,2,3 \right\}$ implies
that $W_1$ can replace a quark by two particles:
a quark and a gluon. It can replace a gluon by
a quark-anti-quark pair, or by two gluons.
Reversed changes are also possible.
According to Eq. (\ref{W2}), evaluation of $W_2$
involves action of ${\cal H}_{I\infty 1}$ twice
and ${\cal H}_{I \infty2}$ once. This means that
$W_2$ can replace one virtual particle by 3, or
vice versa, change the number of virtual particles
by 1, or do not change their number at all but
only alter their individual momenta and other
quantum numbers as dictated by ${\cal H}_{I\infty}$.
When one evaluates $W$ for $\lambda$ and
$\lambda_c$ using expansion in powers of
$g_\lambda$, creation and annihilation operators
in Eq. (\ref{W1exact}) being those corresponding
to $\lambda$, the coefficient $(f_\lambda -
f_{\lambda_c}) /(P_a^- - P_c^-)$ only appears with
first power of $g_\lambda$. Higher powers involve
different coefficients and the entire sum of the
series in powers of $g_\lambda$ is not known.
Moreover, a complete answer may contain dependence
on $\Lambda_{QCD}$ that perturbation theory cannot
identify. Therefore, the question is what
structure of $W$ one can expect for $\lambda \sim
\lambda_c \sim \Lambda_{QCD}$. In particular,
seeking a connection between CQMs and QCD, one
needs to estimate the structure of $W$ that is
responsible for changing the number of effective
particles, since CQMs do not include such
interactions.
When $\lambda$ is near $\lambda_c$, one knows that
$W$ is near 1 irrespective of the size of both
$\lambda$s and $g_\lambda$, because $W$ is 1 when
$\lambda = \lambda_c$. However, the deviation of
$W$ from 1 is not the same for $\lambda \sim
\lambda_c \sim \Lambda_{QCD}$ as in the
perturbative case discussed above for two
$\lambda$s that are much greater than
$\Lambda_{QCD}$. The significant change in $W$ in
comparison to the perturbative result for large
$\lambda$s comes from the difference between
${\cal H}_\infty$ and full solution for ${\cal
H}_\lambda$. Nevertheless, if one assumes certain
particle-number changing interaction in ${\cal
H}_\lambda$ for $\lambda$ near $\lambda_c$, $W$
will put this structure of ${\cal H}_\lambda$ to
action in a similar way to the one described
above. The reason is not the smallness of
$g_\lambda$ but the presence of the form factors
$f_\lambda$. They weaken the interaction by
limiting the range of momenta that interacting
particles may have. Thus, one can look at a
candidate for ${\cal H}_\lambda$ for some value of
$\lambda$ near $\lambda_c$ as an initial condition
for integration over a relatively small range of
$\lambda$ near $\lambda_c$ and treat the whole
interaction that changes the number of particles
as weak even if $g_\lambda$ appears large.
The weakness of particle-number changing
interactions with small $\lambda$ is reinforced by
the particle masses as described in Section
\ref{effectiveparticles}. Both quark, $m$, and
gluon, $m_g$, mass parameters in the region of
$\lambda \sim \lambda_c$ may be sizable. Gluon
mass terms for effective gluons are not excluded
by local gauge invariance of Lagrangian for QCD
because the interaction terms in LF QCD
Hamiltonians with finite $\lambda$ are not local
and their non-locality is not of canonical type.
Since regularization of LF QCD Hamiltonian
requires gluon mass counterterms and the latter
have finite parts that are expected to depend on
$\Lambda_{QCD}$, perturbation theory for
scattering in femtouniverse \cite{femtouniverse}
cannot be used to argue that effective gluons
cannot have sizable mass terms for $\lambda \sim
\lambda_c$. The spectrum of masses of lightest
hadrons, which is quite sparse in comparison to
the QED near-threshold atomic spectra with
massless photons, suggests instead that there is a
considerable mass gap involved in excitation of
gluon degrees of freedom. Since QCD is assumed to
describe these data, it seems reasonable to expect
that RGPEP equations in QCD lead to massive
effective gluons, instead of massless ones.
As an illustration of what one might expect of $W$
in QCD in the range of $\lambda \sim \lambda_c
\sim \Lambda_{QCD}$ when one reasons by analogy
with the perturbative solution at large $\lambda$,
consider the structure of local three-gluon
interaction term that originates from the
gauge-invariant Lagrangian density $- Tr F^{\mu
\nu} F_{\mu \nu} /2$ in canonical theory. The
interaction causes splitting of a bare gluon into
two bare gluons. Its structure is given in Eq.
(\ref{resultH1}), as the term with $Y_{123}$. In
distinction from this local canonical term (in
which $\lambda = \infty$), the corresponding term
in an effective Hamiltonian at small $\lambda \sim
\Lambda_{QCD}$, is non-local \cite{NonlocalH}.
Suppose the complete non-local three-gluon vertex
includes the same structure $Y_{123}$ and a vertex
factor $V_\lambda(1,2,3)$ times the form factor
$f_\lambda$. Such structure is known to emerge
through order $g_\lambda^3$ \cite{gluons}. The
corresponding 1-to-2-gluon term in $W$ of lowest
order in the effective interaction reads
\begin{eqnarray}
\label{WY1ra}
W_{Y1 \, 21} & = & \sum_{123}\int[123] \, 2(2\pi)^3
\delta^3( P_c - P_a)\, { f_\lambda - f_{\lambda_c}
\over P_a^- - P_c^- } \nn \ts \label{WY1} Y_{123} \,
V_\lambda(1,2,3) \, \, a^\dagger_{\lambda 1}
a^\dagger_{\lambda 2} a_{\lambda 3} \, ,
\end{eqnarray}
where $V_\lambda(1,2,3)$ plays the role of initial
condition, instead of $V_\infty(1,2,3)$ that
provided the initial condition in ${\cal
H}_\infty$. Note that now the regularization
factor $r_{\Delta\delta}(1,2,3)$ is absent. This
follows since the form factors $f_\lambda$ and
$f_{\lambda_c}$ entirely eliminate the ultraviolet
divergences and they eliminate the small-$x$
divergences when one adopts the assumption that
effective gluons are heavy \cite{Wilsonetal}.
In these circumstances, the regularization
factors are immaterial.
Suppose $\lambda_c$ is smaller than the gluon mass
$m_g$ at $\lambda_c$. In this case, for $\lambda \sim
\lambda_c$, one can approximate $P_a^-$ by $m_g^2
/p_3^+$ and $P_c^-$ by $4 m_g^2/p_3^+$ in the
difference in denominator while the form factors can
be approximated using (see \cite{NonlocalH} for
details)
\begin{eqnarray}
f_\lambda
& = &
e^{ - \left[ {\cal M}_{12}^2 - m_g^2
\right]^2 / \lambda^4 } \, ,
\end{eqnarray}
with ${\cal M}^2_{12} = 4(m_g^2 + \vec k\,^2)$,
rewritten as
\begin{eqnarray}
f_\lambda & = & e^{-(3m_g^2/\lambda^2)^2} \,
e^{ - { \vec k\,^2 \, + \,\, 3m_g^2/2 \over (\lambda/2)^2 } \, {
\vec k\,^2 \over (\lambda/2)^2 } } \, .
\end{eqnarray}
For $\lambda$ smaller than $m_g$, the relative momentum
$|\vec k\,|$ of gluons 1 and 2 is smaller than
$\lambda/2 < m_g/2$ and the form factor can
be very well approximated by Gaussian
\begin{eqnarray}
f_\lambda & = & e^{-9m_g^4/\lambda^4} \,
e^{ - 24 m_g^2 \vec k\,^2 /\lambda^4 } \, .
\end{eqnarray}
The exponential in front causes the form factor to
vanish when $\lambda$ is much smaller that $m_g$,
as discussed in Section \ref{effectiveparticles}.
This exponential is a part of the mechanism by
which the presence of RGPEP form factors
suppresses interactions that change the number of
effective particles. The other part of the
mechanism is the Gaussian factor whose width
decreases when $\lambda$ decreases. This factor
causes the range of interaction in momentum space
to decrease and thus to diminish the net effect of
it. For the form factor suppression mechanism to
work, the coupling constant $g_\lambda$ must not
blow up to huge values and compensate for the
smallness of the form factor. However, such
compensation should not be expected in QCD if the
theory is to explain the phenomenological success
of the CQMs.
For $\lambda_c$ much smaller than $m_g$, one could
literally set $f_{\lambda_c}$ to 0 in Eq. (\ref{WY1}).
The resulting estimate for the structure of $W_{Y1 \, 21}$
reads
\begin{eqnarray}
\label{WY1a}
W_{Y1 \, 21}
& = &
e^{-9m_g^4/\lambda^4} \,
\sum_{123}\int[123] \, 2(2\pi)^3 \delta^3( P_{12} - p_3)\,
{ e^{ - 24 m_g^2 \vec k\,^2 /\lambda^4 } \over 3m_g^2/p_3^+ } \,
\nn \ts
\label{WY2}
Y_{123} \, V_\lambda(1,2,3)
\, a^\dagger_{\lambda 1} a^\dagger_{\lambda 2} a_{\lambda 3} \, .
\end{eqnarray}
This term in $W_1$ replaces a gluon with a pair of
gluons. The relative motion of the two emerging
gluons corresponds to a Gaussian wave function of
the form $e^{ - c \vec k\,^2 /\lambda^2 }$
with $c = 24 m_g^2 /\lambda^2 $. Assuming that
$\lambda$ is comparable with $m_g \sim 1$ GeV,
the width of the Gaussian wave function is about
$\lambda/5 \sim 200 $ MeV, which is not
unreasonable. Nevertheless, one has to remember
that the full vertex contains also the function
$V_\lambda(1,2,3)$ which causes that the
non-locality of $W_{Y1 \, 21}$ is not determined solely
by the Gaussian factor in Eq. (\ref{WY1a}).
Another illustration is provided by consideration
of the first term in $\{1,2,3\}$, i.e., the term
that replaces a quark of scale $\lambda$ by a
quark and a gluon of the same scale. The only new
elements in the analysis of this term, in
comparison to the analysis described above in the
case of gluons, are different values of masses and
spinor factors instead of $Y$. However, the
conclusion is similar: $W$ changes a quark into a
quark and a gluon.
The larger the difference between $\lambda$ and
$\lambda_c$, the more important particle-changing
interactions in $H_\lambda$. The full solution for
$W$ may replace a state of two or three
constituent quarks by a superposition of states
with many different numbers of gluons (and
additional quark-anti-quark pairs created from
additional gluons) at $\lambda$, as dictated by
the interaction terms in $H_\lambda$.
The perturbative calculation of $W$, and calculations
for small difference between $\lambda$ and $\lambda_c$
using assumed simple picture at $\lambda_c$ and guesses
concerning interactions at $\lambda \gtrsim \lambda_c$,
can be replaced by non-perturbative solutions of RGPEP
equations, see Appendix \ref{NPRGPEP}.
\newpage
\section{ RGPEP beyond perturbation theory }
\label{NPRGPEP}
For the principles of renormalization group
procedure for Hamiltonians \cite{GlazekWilson1} to
apply in a boost-invariant LF dynamics, the
right-hand side in Eq. (\ref{differentialeq}) must
guarantee that matrix elements of $H_\lambda$
vanish when changes of the invariant masses exceed
$\lambda$ and become comparable with the invariant
masses themselves. This feature will be called
narrowness in invariant mass of interacting
effective particles of width $\lambda$, or just
narrowness. One way of securing narrowness of width
$\lambda$ in perturbation theory is to work with
vertex form factors in RGPEP as described in
Appendix \ref{AppendixW}. While this option is
welcome in perturbative calculations
\cite{AlteredWegnerAPP}, it involves a derivative
of the Hamiltonian in the definition of functional
$ {\cal F} _\lambda$ and the derivative is not easy to
calculate outside perturbation theory. The
perturbative approach of Ref. \cite{GlazekWilson2}
defines differential equations in which the
functional $ {\cal F} _\lambda$ guarantees narrowness of
width $\lambda$ without depending on derivatives
of the Hamiltonian on the right-hand side, but the
generator of the transformation is again defined
recursively in terms of a commutator of the
generator with the Hamiltonian. The recursion
formally secures an expansion to all orders in the
coupling constants, but it is difficult to solve
beyond perturbation theory.
Non-perturbative formulation of RGPEP for operators in
the Fock space can be defined in the form of a double commutator
\cite{doublecommutator1,doublecommutator2,doublecommutator3}
equation that reads
\begin{eqnarray}
\label{npRGPEP}
{d \over d\lambda^{-4}} \, {\cal H} _\lambda & = &
\left[ [ {\cal H} _{0\lambda}, {\cal H} _{P\lambda} ],
{\cal H} _\lambda \right] \, .
\end{eqnarray}
The Hamiltonian $H_\lambda$ is obtained when
$b_\infty$ in $ {\cal H} _\lambda$ is replaced by
$b_\lambda$. The required transformation $U_\lambda$
is thus meant to be always incorporated with
accuracy dictated by the accuracy of solving
Eq. (\ref{npRGPEP}). Strictly speaking, nothing
is known yet about the accuracy one can actually
achieve this way beyond perturbative RGPEP in QCD.
The structure of Eq. (\ref{npRGPEP}) resembles the
beautiful equation for Hamiltonian matrices that
Wegner proposed for solving theoretical problems
in condensed matter physics
\cite{Wegner,WegnerReview,Kehrein}.\footnote{ To
avoid a misunderstanding, it should be clarified
that in his original work Wegner did not relate
his equation to renormalization group procedures
in condensed matter physics or quantum field
theory.} In Wegner's proposal, $ {\cal H} _{0\lambda}$ is
a diagonal part of the Hamiltonian matrix,
$ {\cal H} _{I\lambda}$ is the off-diagonal part, and
$ {\cal H} _{P\lambda}$ is the matrix $ {\cal H} _{I\lambda}$
itself. Attempts have been made to apply the
Wegner equation to Hamiltonian matrices obtained
from quantum field theory for the purpose of
eliminating matrix elements that involve a change
in the number of virtual particles. Such matrix
elements are found in the initial conditions
set using interaction terms from canonical
Hamiltonians in quantum field theory. The
Tamm-Dancoff truncation to a few Fock components
was used to define a sufficiently small Hamiltonian
matrix for using Wegner's equation \cite{GubankovaWegner,
GubankovaPauliWegner, GubankovaJiCotanch1,
GubankovaJiCotanch2}.
Wegner's equation with matrix $ {\cal H} _{P\lambda} =
{\cal H} _{I\lambda}$ implies that the matrix
$ {\cal H} _\lambda$ with small $\lambda$ is narrow with
a width order $\lambda$ on energy scale (it would
be $P^-$ scale in LF dynamics). However, Wegner's
equation for Hamiltonian matrices does not agree
with kinematical LF symmetries, including the Lorentz
boosts that are essential for explaining a
connection between the constituent picture and the
parton model picture of hadrons. Also, one desires
an equation for a Hamiltonian operator that in
principle may act in the entire Fock space rather
than only in a severely truncated space in a
Tamm-Dancoff approach. These two issues will be
addressed below by defining the operator $ {\cal H} _{P\lambda}$
and showing that the resulting equation leads to
narrowness of $ {\cal H} _\lambda$ that respects all
kinematical LF symmetries and is narrow in the
entire Fock space.
\subsection{ Boost invariance }
\label{boosts}
The origin of the difficulty with LF boost
symmetry in Wegner's equation is that the
left-hand side in his equation is supposed to be a
derivative of a Hamiltonian with respect to a
parameter, while the right-hand side is tri-linear
in a Hamiltonian, an object clearly depending on a
frame of reference in a different way than a
Hamiltonian does. LF power counting
\cite{Wilsonetal} for Hamiltonian terms suggests
that a straightforward application of Wegner's
equation in the SRG scheme may create a host of
complex counterterms required for restoring
kinematical LF symmetries. An attempt to cure the
situation was undertaken \cite{AllenPerry,
KylinAllenPerry} by assuming a double-commutator
equation for matrix elements of the invariant mass
squared rather than a Hamiltonian. It was assumed
that the LF boost invariance could be maintained
because powers of the invariant mass are invariant
with respect to boosts if the mass itself is.
However, invariant masses of Fock states depend on
spectators, resulting in effective interactions
that depend on spectators. This means that the
resulting effective interactions require extra
care to recover the cluster decomposition
principle \cite{cluster} and it is not clear that
they can help in literally solving quantum field
theory where this principle is expected to be
valid in effective theories, including effects of
a complete renormalization procedure.
The issue of LF symmetries in RGPEP was considered
before \cite{RGPEP}. Here, the operator ${\cal
H}_{P\lambda}$ is defined using the Hamiltonian
${\cal H}_\lambda$. Assuming that ${\cal
H}_\lambda$ conserves momentum and is of the form
\begin{eqnarray}
\label{Hstructure}
{\cal H}_\lambda & = &
\sum_{n=2}^\infty \, \int \hspace{-15pt}\sum \,
[1...n] \, h_\lambda(1,...,n) \, \, q^\dagger_1
\cdot \cdot \cdot q_n \, ,
\end{eqnarray}
where $q$ denotes annihilation operators $b_\infty$,
the operator $ {\cal H} _{P\lambda}$ is defined to be of
the form
\begin{eqnarray}
\label{HPstructure}
{\cal H}_{P\lambda} & = &
\sum_{n=2}^\infty \, \int \hspace{-15pt}\sum \,
[1...n] \, h_\lambda(1,...,n) \,\left( {1 \over
2}\sum_{k=1}^n p_k^+ \right)^2 \, q^\dagger_1
\cdot \cdot \cdot q_n \, ,
\end{eqnarray}
and the symbol $\int \hspace{-10pt}\Sigma$ denotes
integration over momenta and summation over
discrete quantum numbers that label creation and
annihilation operators while $p_k^+$ denotes the
$+$-component of momentum that labels the creation
or annihilation operator number $k$. In words,
${\cal H}_{P\lambda}$ differs from ${\cal
H}_\lambda$ by multiplication of its terms by the
square of $+$-component of total momentum carried
by the particles that enter a term, which is the same
as the $+$-component of total momentum carried
by the particles that leave the term. In the
resulting Eq. (\ref{npRGPEP}), both sides behave
in the same way with respect to operations of
kinematical LF symmetries, and the RGPEP width
parameter $\lambda$ is invariant with respect to
boosts of kinematical LF symmetry.
The operator ${\cal H}_{0\lambda}$ in Eq.
(\ref{npRGPEP}) is now set equal to a free-particle
Hamiltonian $ {\cal H} _0$, i.e., an operator built from
products of one creation operator and one
annihilation operator per particle species. The
notation $ {\cal H} _{0\lambda} = {\cal H} _0$ also indicates
that $ {\cal H} _{0\lambda}$ in Eq. (\ref{npRGPEP}) is
chosen here to be independent of $\lambda$.\footnote{This
assumption simplifies the analysis that follows.} It
is a diagonal operator in the Fock space spanned by
states created by operators $b_\infty$ from the LF
vacuum state. Diagonal matrix elements of $ {\cal H} _0$
are not equal to the diagonal matrix elements of the
full Hamiltonian. Typical Hamiltonians include
interactions that contribute to diagonal matrix
elements.\footnote{It is possible to include
interactions in $ {\cal H} _{0\lambda}$ in Eq. (\ref{npRGPEP})
in the form of mass effects (self-interactions) and
beyond (potentials), but these options are ignored
here for simplicity.}
\subsection{ Non-perturbative narrowness in the Fock space }
\label{NonperturbativeNarrowness}
Eq. (\ref{npRGPEP}) is designed for operators that
can act in the entire Fock space (action on every
state is well-defined). Eq. (\ref{npRGPEP}) with a
constant $ {\cal H} _0$ renders $\lambda$-dependent
interaction terms that die out when the change of
the invariant mass squared (evaluated using the
kinematical momenta and eigenvalues of $ {\cal H} _0$) of
particles involved in an interaction exceeds
$\lambda$ (spectators do not influence Hamiltonian
interaction terms). The narrowness feature is
obtained through a universal mechanism for double
commutator evolution equations. The factors of
$P^{+\,2}$ in $ {\cal H} _{P\lambda}$ do not change the
general mechanism.
We start from the observation that Eq.
(\ref{npRGPEP}) can be solved using a method of
successive approximations. An approximation is
defined by how many terms are kept in the
Hamiltonian. In principle, a Hamiltonian with
finite $\lambda$ contains infinitely many terms.
The terms that are kept in an approximation can be
defined by the condition that they contain no more
than a prescribed number $N$ of creation and
annihilation operators in a product. The number
$N$ labels the approximation. Terms that contain
more operators in a product are ignored in the
approximation. Successive approximations are
labeled by increasing numbers $N$.
Suppose one wants $ {\cal H} $ to contain at most $N$
creators and at most $N$
annihilators.\footnote{One can limit the number of
particle operators for different species of
particles differently. In this case $N$ becomes a
vector with natural components corresponding to
species.} Such Hamiltonian will be denoted by
$ {\cal H} _{\lambda N}$. Using notation corresponding to Eq.
(\ref{Hstructure}), this means that
\begin{eqnarray}
\label{Hstructure1}
{\cal H}_{\lambda N} & = &
\sum_{n_c = 1}^N \sum_{n_a = 1}^N \,
\int \hspace{-15pt}\sum \,
[1...n_c+n_a] \, h_{\lambda N n_c n_a}(1,...,n_c+n_a) \, \, t_{n_c n_a} \, ,
\end{eqnarray}
where
\begin{eqnarray}
\label{tac}
t_{n_c n_a} & = & \prod_{k=1}^{n_c} q^\dagger_k \prod_{l= n_c+1}^{n_c+n_a} q_l \, ,
\end{eqnarray}
$n_c$ is the number of creation operators,
$n_a$ is the number of annihilation operators,
and subscripts denote also all relevant quantum
numbers of the operators they label.
Suppose one wants to know the terms in $ {\cal H} _{\lambda N}$
that contribute to the dynamics of a physical state
that certainly contains a specified Fock component
$| \psi \rangle $ of the form
\begin{eqnarray}
\label{psizero}
|\psi \rangle & = &
\int \hspace{-15pt}\sum \,
[1...n] \, \psi_P(1,...,n) \, \,
\prod_{k=1}^{n} q^\dagger_k \, |0 \rangle \, ,
\end{eqnarray}
where $P$ denotes the fixed total kinematical momentum
of the state and otherwise the wave function
$\psi_P(1,...,n)$ is not known. For example, in the
case of mesons one may select the state $|\psi \rangle$
that contains a quark and an anti-quark, and in the case
of a baryon a state $|\psi \rangle$ that contains
three quarks. But one could also consider a state
that contains 1182 quarks and many gluons and
quark-anti-quark pairs in order to describe a
collision of two nuclei of gold neglecting all
interactions but the strong.
Having specified the set of operators that are
included in $ {\cal H} _{\lambda N}$, with $n_c \le N$, $n_a
\le N$, and unknown coefficients $h_{\lambda N n_c
n_a}$, one can enumerate forms of all possible
states that can be generated from $|\psi \rangle$
by action of $ {\cal H} _{\lambda N}$ on it once. All these
states are eigenstates of the total kinematical
momentum operator with one and the same eigenvalue
$P$ that characterizes $|\psi \rangle$. Knowing
the set of states that are obtained from $| \psi
\rangle$ by acting on it with $ {\cal H} _{\lambda N}$ once,
one can construct the set of all states that can
be obtained by acting with $ {\cal H} _{\lambda N}$ on $|
\psi \rangle$ twice, and so on.
We introduce the set of states, denoted by $R$,
that can be generated from $|\psi\rangle$ by
acting with $ {\cal H} _{\lambda N}$ on it $\tau$ times
and are normalized in a limit of infinite volume.
Then we introduce a minimal subspace in the Fock
space (a space of smallest possible basis) that is
sufficient to build the set $R$. This subspace in
the Fock space is denoted by $R_{\tau \psi N}$.
For example, if one starts with a state of 3
quarks, action just one time by the $ {\cal H} _{\lambda
N}$ that is equal to regulated canonical
Hamiltonian of LF QCD, $\tau = 1$ and $N=3$, on
$|3q\rangle$ produces states with 3 quarks, 3
quarks and a gluon, 3 quarks and a
quark-anti-quark pair, and 3 quarks and 2 gluons.
The space $R_{1\, 3q \, 3}$ is built from the Fock
space basis states that are needed to construct
states with these particles. Action twice, $\tau =
2$, generates additional gluons and
quark-anti-quark pairs. The corresponding space
$R_{2\, 3q \, 3}$ is built by adding the Fock
space basis states that are needed to construct the
additional states. When one assumes that
$ {\cal H} _{\lambda N}$ has more terms than the
canonical LF Hamiltonian for QCD
\cite{Wilsonetal}, the space $R_{\tau \psi N}$ is
greater than in the canonical case. States created
from $|\psi \rangle$ with $k$ particles by $\tau$
actions of $ {\cal H} _{\lambda N}$ may contain up to $k
+ \tau(N-2)$ particles.
The set $R$ of states one obtains for any $\tau$
is a priori infinite, because one can have an
arbitrary relative motion of particles in a state
and the range of relative momentum is infinite.
The corresponding subspace in the Fock space
is also unlimited. This happens despite that a
regulated Hamiltonian cannot change a relative
momentum by an arbitrarily large amount. The reason
is that the unknown wave function $\psi_P(1,...,n)$
in Eq. (\ref{psizero}) does not limit the magnitude
of relative momenta. For example, a square-integrable
function, such as Gaussian, quickly falls off as a
function of relative momentum of two particles but
the range of relative momentum is infinite.
In order to introduce a space of states with a
finite range of relative momenta, we impose a
cutoff $\Delta$ on the invariant mass. Namely, we
define the space of states that are in $R_{\tau
\psi N}$ and whose invariant mass $ {\cal M} < \Delta$.
$ {\cal M} $ is calculated using kinematical momenta and
eigenvalues of $ {\cal H} _0$. The resulting space with
this cutoff is denoted by $R_{\Delta \tau \psi N}$,
or shortly $R$. We introduce the projection operator
on the space $R$ and denote this projector also by
$R$. Hamiltonian $H_{\lambda N}$ projected on the
space $R$ is denoted by
\begin{eqnarray}
{\cal H} _{\lambda N R} & = & R \, {\cal H} _{\lambda N} R \, .
\end{eqnarray}
The successive approximation to $ {\cal H} _\lambda$
order $N$ in application to states of the type
$|\psi \rangle$ with accuracy to $\tau$ actions
of $ {\cal H} _\lambda$ on $|\psi \rangle$ with invariant
mass cutoff $\Delta$ is obtained by solving
equation
\begin{eqnarray}
\label{narrowR1}
{\cal H} _{\lambda N R}' & = &
\left[ \, [ {\cal H} _0, R \,\, {\cal H} _{\lambda N P}R \,], \, R \,\, {\cal H} _{\lambda
N} R \, \right] \, ,
\end{eqnarray}
with initial conditions specified by the canonical
Hamiltonian of LF QCD with counterterms.
Structure of the counterterms is found by
demanding that all matrix elements of the
effective Hamiltonian $ {\cal H} _{\lambda N R}$ for a
large ratio $\Delta/\lambda$ among basis states
with invariant masses much smaller than $\Delta$
do not depend on the regularization applied in the
canonical Hamiltonian in the limit of this
regularization being removed \cite{GlazekWilson1}.
The degree of dependence of the counterterms on
$N$ and $\tau$ that are used in defining an
approximation order $N$ to $ {\cal H} _\lambda$ using
$\tau$ actions on various states $|\psi \rangle$
requires studies. Specific choices of states
$|\psi \rangle$ may simplify identification of
structure of counterterms in the canonical LF
Hamiltonian for QCD. However, fixing all finite
parts of the counterterms will require systematic
studies of symmetries that are expected to relate
finite parts of different counterterms, and input
from phenomenology in order to fix mass parameters
as required, in addition to the value of
$\Lambda_{QCD}$. We proceed to showing narrowness
of $ {\cal H} _{\lambda N R}$ that satisfies Eq.
(\ref{narrowR1}).
For notational brevity, Eq. (\ref{narrowR2}) is written as
\begin{eqnarray}
\label{narrowR2}
H' & = &
\left[ [H_0, H_{IP}], H \right] \, ,
\end{eqnarray}
where
\begin{eqnarray}
H & = & R \, \, {\cal H} _{\lambda N } R \, , \\
H_0 & = & R \, \, {\cal H} _0 R \, , \\
H_I & = & H - H_0 \, = \, R \, \, {\cal H} _{I\lambda N } R \, , \\
H_{IP} & = & R \, \, {\cal H} _{I\lambda NP} R \, .
\end{eqnarray}
Evolution in $\lambda$ according to Eq.
(\ref{narrowR1}), or (\ref{narrowR2}),
preserves traces of powers of $H$. Trace is defined by
summing diagonal matrix elements in the set $R$. The
diagonal matrix elements are proportional to $\delta^3(0)$
in momentum space which has interpretation of volume and
can be divided out. The argument 0 corresponds to the
conservation of total momentum.
Using the condition that the trace of $H^2$ does
not depend on $\lambda$, one has
\begin{eqnarray}
\label{TrHsquarex}
0 & = &
Tr \left( H_0^2 + 2 H_0 H_I + H_I^2 \right)' \, .
\end{eqnarray}
Since $H_0$ is fixed,
\begin{eqnarray}
\left( Tr H_I^2 \right)'
& = & - 2 \, Tr \, H_0 H_I' \, .
\end{eqnarray}
Thus, using the basis in space $R$ that is
built from normalized eigenstates $|m\rangle$
of $H_0$, with eigenvalues $E_m$, so that
$H_0 |m\rangle = E_m |m\rangle$ and
$H_{mn} = \langle m | H | n \rangle$, etc.,
we have
\begin{eqnarray}
\label{start2x}
\left( \sum_{mn} |H_{Imn}|^2 \right)'
& = &
- 2 \sum_m E_m H'_{Imm} \, .
\end{eqnarray}
But Eq. (\ref{narrowR2}) implies that
\begin{eqnarray}
\label{start3x}
H'_{Imn} & = &
\left[ [H_0, H_{IP}], H_0 \right]_{mn}
+
\left[ [H_0, H_{IP}], H_I \right]_{mn} \\
& = &
-(E_m - E_n)^2 P_{mn}^{+2} H_{Imn}
\nn \ps
\sum_k \left[ (E_m - E_k)P_{mk}^{+2}
+ (E_n - E_k)P_{kn}^{+2} \right]
H_{Imk} H_{Ikn} \, .
\end{eqnarray}
The momentum $P^+_{ij}$ is the $+$ component of
the total momentum of the particles that undergo
interaction in the relevant matrix element
$\langle i | H_I | j \rangle$. Momenta of the
spectators of the interaction do not contribute
to $P^+_{ij}$. The result needed on the right-hand
side of Eq. (\ref{start2x}) is
\begin{eqnarray}
\label{start4x}
H'_{Imm}
& = &
\sum_k (E_m - E_k) \, \left( P_{mk}^{+2} + P_{km}^{+2} \right) \,
H_{Imk} H_{Ikm} \, .
\end{eqnarray}
Therefore, the sum on the right-hand side of Eq.
(\ref{start2x}) is
\begin{eqnarray}
\sum_m E_m H'_{Imm}
& = &
\sum_{km} (E_k - E_m)^2 |H_{Ikm}|^2 \left( P_{mk}^{+2} + P_{km}^{+2} \right)/2 \, ,
\end{eqnarray}
and
\begin{eqnarray}
\label{start5x}
\left( \sum_{mn} |H_{Imn}|^2 \right)'
& = &
- \sum_{km} (E_k - E_m)^2 |H_{Ikm}|^2 (P_{mk}^{+2}+P_{km}^{+2}) \le 0 \, .
\nonumber \\
\end{eqnarray}
This result leads to the following conclusion:
{\it The sum of moduli squared of all matrix
elements of the interaction Hamiltonian decreases
with $\lambda$ until all off-diagonal matrix
elements of the interaction Hamiltonian between
states with different free invariant masses vanish.}
Thus also: {\it Matrix elements of the interaction
Hamiltonian on the diagonal and between states of
equal invariant masses and between states of small
invariant masses as measured by $H_0$ may stay constant
or even increase when $\lambda$ decreases but only at
the expense of still faster decrease of the
off-diagonal and large invariant mass matrix elements
and only until all off diagonal matrix elements
between non-degenerate states are eliminated.}
Another way of writing relation (\ref{start5x}) is
\begin{eqnarray}
\label{start6x}
\left( \sum_{mn} | {\cal H} _{Imn}|^2 \right)'
& = &
- 2 \sum_{km} ( {\cal M} ^2_{km} - {\cal M} ^2_{mk})^2
| {\cal H} _{Ikm}|^2 \le 0 \, ,
\end{eqnarray}
where $ {\cal M} _{ab}$ denotes an invariant mass of
particles in state $a$ that interact in an
action of $ {\cal H} _I$ once on particles in state
$b$. This form concludes our demonstration that
RGPEP provides effective Hamiltonians which are
narrow in the invariant mass of interacting
particles beyond perturbative calculus. The
invariant masses result from cancellation of
spectator contributions to $P^-$ and multiplication
only by $P^+$ of particles that are involved in
an interaction, i.e., not including spectators.
Initial comparison between the non-perturbative RGPEP
calculus described in this Appendix and perturbative
RGPEP calculus described earlier, can be done by
solving Eq. (\ref{npRGPEP}) using expansion in
powers of a coupling constant, such as $g_\lambda$
in the case of QCD, and observing how results of
one way of calculating compare with the other for
terms order $g_\lambda$ and $g_\lambda^2$. Using
notation introduced earlier (e.g., see
\cite{gluons}, Sec. III.C, or \cite{GlazekScalar},
Sec. II.B), one rewrites Eq. (\ref{npRGPEP}) as
\begin{eqnarray}
\label{EQstep14}
{\cal H} _{ac}' & = &
- ac^2 \, [ {\cal H} _I]_{ac}
+ \sum_b (p_{ab} \, ab + p_{cb} \, cb) \,\left[ {\cal H} _I \, {\cal H} _I \right]_{ac}
\, ,
\end{eqnarray}
where letters $a$, $b$, and $c$, denote configurations of
particles. Using expansion
\begin{eqnarray}
{\cal H} & = & {\cal H} _0 + g {\cal H} _1 + g^2 {\cal H} _2 + ... \, ,
\end{eqnarray}
one obtains for the first two terms equations
\begin{eqnarray}
{\cal H} _{1ac}' & = &
- ac^2 \, {\cal H} _{1ac} \, , \\
{\cal H} _{2ac}' & = &
- ac^2 \, {\cal H} _{2ac}
+ \sum_b (p_{ab} \, ab + p_{cb} \, cb) \,\left[ {\cal H} _1 \, {\cal H} _1 \right]_{ac}
\, .
\end{eqnarray}
Using the particle size parameter $s = 1/\lambda$,
one obtains the solutions
\begin{eqnarray}
{\cal H} _{1ac} & = & e^{-ac^2 s^4} \, {\cal H} _{1ac}(0) \, , \\
{\cal H} _{2ac} & = &
e^{-ac^2 s^4} \, {\cal H} _{2ac}(0)
+
\sum_b \left[ {\cal H} _1(0) \, {\cal H} _1(0) \right]_{ac}
\nn \ts
e^{-ac^2 s^4} \, { p_{ba} \, ba + p_{bc} \, bc \over ba^2 + bc^2 - ac^2 }
\,
\left[ e^{-(ab^2 + bc^2 - ac^2)s^4} - 1 \right]
\, .
\end{eqnarray}
The first-order result is Gaussian in the change
of invariant masses squared, identical to the first-order
result in perturbatively defined RGPEP. The second-order
result matches the perturbatively defined RGPEP,
e.g., see Eq. (2.22) in Ref. \cite{GlazekScalar},
when $ac$ can be neglected in comparison to
$ab$ or $bc$. When $ac$ refers to a dominant Fock
component and $ba$ and $bc$ refer to a component
with additional constituents, this condition may
be satisfied very well if the additional
constituents introduce a significant contribution
to the invariant mass of the subsystem that counts.
Such situation is encountered in the dynamics of
heavy quarkonia \cite{GlazekMlynik}, where the
additional constituent is an effective gluon
while kinetic energies of quarks cannot change
by much because quarks move slowly with respect to
each other and their masses are large in comparison
to $\Lambda_{QCD}$. This means that the non-perturbative
formulation of RGPEP can be directly applied to heavy
quarkonia with arbitrary motion as a bound state in
QCD, and implies that the non-perturbative formulation
can be used to derive corrections to the harmonic
oscillator force in Ref. \cite{GlazekMlynik} beyond
perturbation theory to describe the spectrum of
charmonium and bottomonium states that results from
gluon dynamics neglecting light quarks.
The procedure described here differs from
truncation in powers of a bare coupling constant
$g$, truncation in the number of Fock sectors, and
combinations of both these truncations. Although
we use the Fock space subspace $R$ for solving
Eq. (\ref{narrowR1}), and it is useful to use an
expansion for $ {\cal H} _\lambda$ in powers of the coupling
constant $g_\lambda$ \cite{Wilsonetal}, the solutions
provide in principle not just the matrix elements of
Hamiltonian terms in a limited space of states but
the matrix elements of Hamiltonian terms from which
one extracts the coefficients $h_\lambda$ of particle
operators in the Hamiltonian terms that a priori
act in the entire Fock space. Neither the Fock space
nor perturbation theory limitations prevent us from
seeking a non-perturbative solution for the
coefficients $h_\lambda$.
Some comments are in order here. One can alter the
generator in Eq. (\ref{narrowR1}) by introducing
convergence-improving factors that are required
for perturbative approximations
\cite{AlteredWegnerPRD,AlteredWegnerAPP}. For
bound states of low-mass hadrons alone, the
relative motion of constituents can be described
using other basis functions than plane waves, such
as the oscillator basis. Scattering states of
hadrons may require a more elaborate basis in
coupled channels. Reasoning described in this
Appendix justifies methods used in nuclear physics
with only kinetic energy in SRG generator
\cite{OSU1, OSU2, OSU3, OSU4, OSU5}, including the
impact of bound states \cite{impact}. The impact
can only lead to increase of interaction matrix
elements that are small since the combined
strength of all interaction matrix elements must
only decrease until it reaches a minimum. This
result extends the range of applicability of SRG
with a kinetic energy or a similar term in the
generator. It suggests applications of simpler
flow equations than Wegner's in relevant areas of
physics.
\section{ Visualization of RGPEP scale dependence of
hadronic structure }
\label{OverlappingSwarmsModel}
Quantum fields for effective particles of size $s
= 1/\lambda$ in RGPEP, see Eqs.
(\ref{quantumfieldpsilambda}),
(\ref{quantumfieldAlambda}), and (\ref{psixs}),
are the degrees of freedom from which one can
build effective Hamiltonian densities. An
effective Hamiltonian at scale $\lambda$
describes a hadron only in terms of quarks of
size $s = 1/\lambda$. Therefore, when the scale
parameter $\lambda$ approaches $\lambda_c \sim
\Lambda_{QCD}$, the size of quarks, as measured by
the strong interaction range in interaction
vertices \cite{NonlocalH}, becomes comparable with
$1/\Lambda_{QCD}$. This in turn means that the
effective quarks become as large as the whole
hadron. Such quarks must nearly overlap and this
is how they form a white mixture. There is some
imbalance of color on the boundary, perhaps able
to couple to $\pi$-mesons that form a cloud around
the quarks. Visualization of these circumstances
is provided in Figs. \ref{fig:MQ} and
\ref{fig:NQ}. When $\lambda$ is near but greater
than $\lambda_c$, the constituent quarks become
smaller and part of the color-active medium inside
a hadron is no longer fully overlapped by quarks.
Therefore, when $\lambda \gtrsim \lambda_c$, there
must be additional matter density involved in
filling the volume of a hadron.
One can use an analogy with swarms of bees. For
example, a baryon can be seen at scale $\lambda_c$
as built from three swarms. One swarm is made of
red bees, one of green, and one of blue. One of
the three swarms corresponds to one constituent
quark. There is also a fourth swarm, built from
bi-colored glue bees. But the four swarms overlap
and in nearly every part of the baryon volume one
has one red, one green and one blue bee, or glue
bees. The regions where the three quark swarms are
overlapping are locally white.
When one changes the scale $\lambda=\lambda_c$ to
a nearby $\lambda \gtrsim \lambda_c$ as described
in Section \ref{3ways}, the size $s$ of each of
the three swarms becomes smaller than $s_c =
1/\lambda_c$, but the medium around the quarks of
size $s$ remains filled with the glue and quark
bees that can still balance to white, too. When
one decreases $\lambda$ below $\lambda_c$, the mass
of the glue component may decrease toward 0 or
stabilize while the constituent quarks fully
saturate the volume of a hadron.
\begin{figure}
\begin{center}
\includegraphics[scale=.5]{MesonQualitative}
\end{center}
\caption{The RGPEP image of a meson seen at $\lambda$ near $\lambda_c$.}
\label{fig:MQ}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=.5]{NucleonQualitative}
\end{center}
\caption{The RGPEP image of a nucleon seen at $\lambda$ near $\lambda_c$.}
\label{fig:NQ}
\end{figure}
In distinction from the parton model
\cite{partonmodel} and related scaling pictures
\cite{Wilsonpartonmodel,KogutSusskindscaling}, the
new element of this visualization is that the
constituent swarms are much larger in size than
the distances between their centers. It seems to
the author that this feature of RGPEP deserves a
visualization because none of the visualizations
of hadrons known to the author has this feature.
Namely, common visualizations represent a baryon
as made of three constituent quarks that fly like
apples in a bag, sometimes accompanied by some
strands of glue in some kind of a cavity of
unspecified nature. Images associated with an
increase of momentum scale typically include more
tiny objects in two ways: either as finer pieces
scattered around or as interaction processes among
quarks and gluons (such as on the cover of Ref.
\cite{PeskinSchroeder}). It is hard to visualize
the origin of constituent quarks and CQM
potentials in these ways.
In the context of RGPEP, the analogy with swarms
of colored bees provides the following
visualization for the mechanism of formation of
potentials. We start with a meson built from a
quark and an anti-quark. A quark has opposite color
charge density to anti-quark. When the effective
particles are completely overlapping like two
densities of opposite charges, their state is
locally neutral. This picture oversimplifies the
non-Abelian picture to the Abelian one but
still offers some intuition. Imagine now
that the centers of the initially overlapping
swarms move a little bit apart. It is well
known that the Coulomb force between two
spherically symmetric distributions of opposite
charges grows linearly with the distance between
their centers for as long as this distance is
small in comparison to the individual sizes of
the distributions. Therefore, the potential energy
of two nearly overlapping spheres is described by a
harmonic oscillator potential. It should be
stressed that the distance between the centers of
the swarms is much smaller than the size of
each swarm. In the case of baryons, one needs
to consider three centers of three overlapping
swarms being relatively close to each other.
Of course, this visualization has many drawbacks:
it is classical, Abelian, non-relativistic, and
does not provide any insight concerning the
difference between quarks and gluons. But it does
model the transformation $W$ that increases the
number of quarks and gluons of decreasing size $s$
when $\lambda$ increases. These smaller quarks and
gluons form the medium that provides the
expectation value of the gluon field that is
interpreted in this article as a gluon condensate.
\end{appendix}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,011
|
Pedersens björnbär (Rubus pedersenii) är en rosväxtart som beskrevs av Martensen och H.E.Weber. Enligt Catalogue of Life ingår Pedersens björnbär i släktet rubusar och familjen rosväxter, men enligt Dyntaxa är tillhörigheten istället släktet rubusar och familjen rosväxter. Arten har ej påträffats i Sverige. Inga underarter finns listade i Catalogue of Life.
Källor
Rubusar
|
{
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| 3,502
|
A 2015-ös Formula–E Peking nagydíjat október 24-én rendezték. A pole-pozícióból Sébastien Buemi indulhatott. A futamot Sébastien Buemi nyerte meg.
Időmérő
Futam
Fanboost
Az alábbi három versenyző használhatta a Fanboost-ot.
Futam
A világbajnokság élmezőnyének állása a verseny után
(Teljes táblázat)
Jegyzet: Csak az első 5 helyezett van feltüntetve mindkét táblázatban.
Jegyzetek
További információk
Az időmérő végeredménye
A futam végeredménye
Formula-E nagydíjak
|
{
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}
| 6,111
|
{"url":"http:\/\/mymathforum.com\/algebra\/37812-600-million-revs-minute-18-billion-meters-minute.html","text":"My Math Forum Is 600 million revs\/minute = 18 billion meters\/minute?\n\n Algebra Pre-Algebra and Basic Algebra Math Forum\n\n August 29th, 2013, 02:24 AM #1 Newbie \u00a0 Joined: Aug 2013 Posts: 1 Thanks: 0 Is 600 million revs\/minute = 18 billion meters\/minute? Hi, I have a speed question: If a sphere sized 4 millionth of a meter spins at a rate of 600 million revolutions per minute, does the speed of its revolution linearly equate to 18 billion meters per minute (the speed of light)? I want to know if 600 million revolutions per minute in the context of the article below equals the speed of light. I tried to compute it manually and got values which are equal if the sphere was sized 3 millionth of a meter, but I want to be sure because I'm not good at math at all. http:\/\/www.bbc.co.uk\/news\/uk-scotland-e ... e-23861397\n August 29th, 2013, 03:17 AM #2 Senior Member \u00a0 Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Re: Is 600 million revs\/minute = 18 billion meters\/minute? $1 \\; rev=\\frac {4}{1000000} \\cdot \\pi \\; m$ $1.000.000 \\; rev=4 \\pi \\; m$ $600.000.000 \\; rev=2400 \\pi \\; m$ So 600 million revs\/minute $\\approx$ 7540 meters\/minute\n August 30th, 2013, 03:59 AM #3 Math Team \u00a0 Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Is 600 million revs\/minute = 18 billion meters\/minute? You say a sphere \"size 4 millionth of a meter\". Is that the radius or the diameter of the sphere?\nAugust 30th, 2013, 06:33 AM \u00a0 #4\nSenior Member\n\nJoined: Mar 2012\nFrom: Belgium\n\nPosts: 654\nThanks: 11\n\nRe: Is 600 million revs\/minute = 18 billion meters\/minute?\n\nQuote:\n Originally Posted by HallsofIvy You say a sphere \"size 4 millionth of a meter\". Is that the radius or the diameter of the sphere?\nI checked that in the article and there they say it is the diameter\n\n Thread Tools Display Modes Linear Mode\n\n Similar Threads Thread Thread Starter Forum Replies Last Post Setsuna Physics 1 September 28th, 2012 07:35 AM Gamer23 Academic Guidance 0 July 23rd, 2011 02:57 PM r-soy Calculus 2 November 5th, 2010 10:44 AM symmetry Physics 1 July 30th, 2007 05:02 AM karlaxoganal Elementary Math 1 May 10th, 2007 04:55 AM\n\n Contact - Home - Forums - Cryptocurrency Forum - Top","date":"2019-10-23 07:15:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 4, \"\/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\": 0.43832868337631226, \"perplexity\": 3743.4261959032374}, \"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\/1570987829507.97\/warc\/CC-MAIN-20191023071040-20191023094540-00449.warc.gz\"}"}
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});
self.y1 = ko.computed(function() {
return options.clock.size()/2 + (options.clock.size()/2*Math.sin(Math.PI*(self.pos/60*360-90)/180));
});
self.x2 = ko.computed(function() {
return options.clock.size()/2 + (options.clock.size()/2*self.size()*Math.cos(Math.PI*(self.pos/60*360-90)/180));
});
self.y2 = ko.computed(function() {
return options.clock.size()/2 + (options.clock.size()/2*self.size()*Math.sin(Math.PI*(self.pos/60*360-90)/180));
});
};
var Label = function(options) {
var self = this;
self.pos = options.pos;
self.template = "template-label";
self.size = function() {
return .7 ;
};
self.text = ko.computed(function() {
return self.pos == 0 ? 12 : self.pos;
});
self.x = ko.computed(function() {
return options.clock.size()/2 + (options.clock.size()/2*self.size()*Math.cos(Math.PI*(self.pos/12*360-90)/180));
});
self.y = ko.computed(function() {
return options.clock.size()/2 + (options.clock.size()/2*self.size()*Math.sin(Math.PI*(self.pos/12*360-90)/180));
});
};
var Clock = function() {
var self = this;
self.size = ko.observable(100);
self.x = ko.observable(100);
self.y = ko.observable(100);
self.rotate = ko.observable(0);
self.elements = ko.observableArray();
self.elements.push(new SecondsArrow({ clock: self, size: .95 }));
self.elements.push(new MinutesArrow({ clock: self, size: .9 }));
self.elements.push(new HoursArrow({ clock: self, size: .7 }));
for(var i=0; i<60; i++) {
self.elements.push(new Tick({ clock: self, pos: i }));
}
for(var i=0; i<12; i++) {
self.elements.push(new Label({ clock: self, pos: i }));
}
self.circleX = ko.computed(function() {
return self.size()/2;
});
self.circleY = ko.computed(function() {
return self.size()/2;
});
self.circleR = ko.computed(function() {
return self.size()/2;
});
self.transform = ko.computed(function() {
return "translate({0}, {1}) rotate({2} {3} {3})".format(self.x(), self.y(), self.rotate(), self.size()/2);
});
self.onTick = function() {
//self.rotate(self.rotate()+1);
};
self.tickToken = PubSub.subscribe("tick", self.onTick);
self.template = ko.observable("template-clock");
};
var Vm = function() {
var self = this;
self.screenWidth = ko.observable();
self.screenHeight = ko.observable();
self.points = ko.observableArray();
self.onRemove = function(event, point) {
self.points.remove(point);
};
PubSub.subscribe("remove", self.onRemove);
setInterval(function() { PubSub.publish("tick"); }, 100);
self.dimensions = ko.computed(function() { return self.screenWidth() + "x" + self.screenHeight() + " " + self.points().length; });
var top = $("#top");
self.setDimensions = function() {
self.screenWidth(top.width());
self.screenHeight(top.height());
};
$(window).on("resize", function() {
self.setDimensions();
});
self.setDimensions();
self.onTick = function() {
return;
for(i=0; i<2; i++)
{
var x = Math.round(Math.random()*self.screenWidth());
var y = Math.round(Math.random()*self.screenHeight());
self.points.push(new Point(x, y, "click"));
}
};
self.tickToken = PubSub.subscribe("tick", self.onTick);
self.size = ko.computed({
read: function() { self.points().length > 0 ? self.points()[0].size() : 100; },
write: function(x) { self.points()[0].size(x); }
});
};
var vm = new Vm();
var clock = new Clock();
clock.x(15);
clock.y(15);
var top = $("#top");
clock.size(($(window).width() < $(window).height() ? $(window).width() : $(window).height()) - 30);
vm.points.push(clock);
ko.applyBindings(vm, $("body")[0]);
</script>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,498
|
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