TITLE: What Data Set(s) Would Be Best For Head-To-Head Unified Field Theory Model Selection?
QUESTION [1 upvotes]: I'm seriously considering establishing a prize for unified field theories in-line with my history of prize awards, such as the fusion legislation I drafted with a cofoudner of the AEC's program, the Bowery Award for Amateur Rocketry (that led the Ansari X-Prize) and The Hutter Prize for Lossless Compression of Human Knowledge.
The rules would be identical to The Hutter Prize rules with one exception:
Replace the enwik9 Wikipedia corpus with physics data.
The value of this prize would be the rigorous and objective model selection criteria:
Solomonoff Induction
Hutter wrote a paper titled "A Complete Theory of Everything (will be subjective)" in which he explains the applicability of Solomonoff Induction to physics, including what it takes to select among any set of UFT's, including those with large numbers of free parameters, such as string "theory".  I scare-quote "theory" in the case of "string theory" (and would for any purported UFT with free parameters) because the, somewhat sloppy, vernacular phrase "physical theory" has been taken to mean what the more accurate phrase "physical model" means.  In other words, until a physical interpretation is coupled to a theory it remains in the domain of metaphysics if not pure mathematics.   Any theory with free parameters remains indeterminate, hence "not even wrong" as there is no interpretation.
A reductio ad absurdum of the view that model selection can take place in the presence of free parameters:
Any "parameter" of unspecified precision, can, via arithmetic coding, provide an unspecified amount of information.  A theory with such a free parameter can populate it with an infinite precision floating point number and then, via arithmetic coding, expand it to cover any observations whatsoever.
Expanding on how this would work:
Each school of theoretic thought would create an executable archive of the physics corpus.  That executable archive program would have to run within the resource limitations appropriate to the UFT competition.
For instance, purpose served by The Hutter Prize is about a week of 1 CPU's time using less than 10GB of RAM is permitted.  But other language modeling "competitions" invest millions of dollars of compute resources.  Whatever the compute resource choice, the resulting file must match, bit for bit, the original physics corpus.
The shortest program at any given time would be awarded money from the prize purse according to how much it reduced the size from the prior winner's benchmark.
The physics corpus should be adequate to the prize's purpose.  For example, The Hutter Prize's corpus, enwik9 (1 billion byte snapshot of Wikipedia) is adequate to its purpose of modeling natural language knowledge.
A critical but inadequate corpus would be The NIST's Fundamental Physical Constants "Value" column encoded as variable precision floating point.  An obvious addition to the fundamental constants would be the Ionization Energy column from the NIST Atomic Spectra Database Ionization Energies Data.
I looked around and failed to find any pre-existing such corpus, or list of corpora for testing theories against each other in a rigorously fair manner.
What set of physical data would be adequate as a model selection tool?

REPLY [1 votes]: Comparing theories in this way is not likely to be relevant.
Consider the actual candidates for theory of everything. We have string theory, which is uniquely compelling as a step beyond field theory, but whose empirical application requires searching an ill-understood googol-sized landscape of possibilities for worlds that appear to resemble our own, and then calculating the detailed properties of physics in each such world. Machine learning is being used to conduct the search for qualitatively promising worlds, but the second step, of detailed calculation, requires much more progress in fields like algebraic geometry.
Then we have all the theories that aren't string theory: like, loop quantum gravity coupled to a grand unified field theory, or celebrity theories of everything like those due to Lisi or Weinstein, or dozens of similar but unheralded proposals that have been posted to arxiv. None of these theories can calculate everything either - indeed, most of them can't calculate much at all. They're all "works in progress".
On the other hand, you have extensions of the standard model that include new fields and make slightly different predictions. Like the standard model, these are field theories with numerous free parameters. Conceivably you could rank such theories by the number of bits required to specify them, and the number of bits of the standard model parameters that they would explain; but such theories generally make entirely new predictions too (e.g. proton decay), and how they fare on that front is more important for their ultimate credibility.

REPLY [0 votes]: I think there are some features of the problem of predicting various measured quantities from principles, from a computer science perspective, that make it unlike predicting text from a natural language model. Here are a few (there is some overlap between them):

We never know the ground truth in physics. For natural language processing, there is a clear goal: the text of wikipedia is a snapshot of how people write English today, and can serve as a ground truth for a NLP model. In physics, we don't know many crucial experimental facts that would be needed to distinguish between candidate models at the Planck scale. Is there supersymmetry broken at some scale? If so, which scale? What is the scattering cross section of dark matter with neutrinos? What is the fourth digit of the quartic self-coupling constant of the Higgs Boson? We don't know these quantities, but surely they will be needed to distinguish candidate models. Perhaps there are errors in the current knowledge of fundamental constants, so that a theory that matches the list too well will actually turn out not to be a good one in the end.

Computational complexity is not a proxy for truth. While we want the simplest model that will fit the data, there is no way to put an absolute bound on the computational resources needed for an accurate model. A Newtonian gravity simulation would take far fewer resources than a full GR simulation, but to accurately model gravitational waveforms we surely need the latter. Such a simulation pushes the boundaries of what is capable with supercomputers today, but this should not be taken as a point against GR, compared to a phenomenological fit that could reproduce observed gravitational-wave data. In fact, simulating a quantum mechanical system with $N$ qubits takes a very large number (exponential in $N$) of classical resources.

Not all insights are data driven. MOND does an excellent job of fitting galaxy rotation curves. However, by itself, we know it is not a complete theory because it is not relativistic. There are proposals to make MOND relativistic, and I don't want to get into an argument about whether MOND is correct or not, but the point is that there are criteria beyond comparing to data that guide us to a correct description. I suspect a potpourri of different phenomenological models that fit different data in the corpus could do a very good job fitting a set of physics measurements, but would certainly not be acceptable as a fundamental theory of physics.

The value of physics is to be predictive outside of the scope of the original data. Data-driven approaches like machine learning work well when the training set covers a representative range of cases that cover likely data that will be seen in the future. Said differently, statistical models are good at interpolation but poor at extrapolation. On the other hand, the value of fundamental physics is precisely to make predictions about situations that have never been seen before. Nothing in our everyday experience could make us think it would be reasonable to predict the state of the Universe 3 minutes after the Big Bang, for example. But taking our theories of GR and nuclear physics and extrapolating them in a conservative way, we are able to model this process and make predictions that are later confirmed by experiment. Testing theories by how they extrapolate into new areas is a crucial part of the development of physics, but is something that would not be possible in this kind of approach.


TITLE: Antiderivative in complex
QUESTION [3 upvotes]: may i ask you for a little help about the following problem.
Given is the function $f:\mathbb{C}\rightarrow \mathbb{C}, z\rightarrow Re(z)$. The questions are:

1) Does $f$ have an antiderivative on $\mathbb{C}$?

Here i think the answer is no, because the Cauchy-Riemann differential equations are not satisfied, $Re(z)$ is nowhere complex differentiable, thus not holomorphic.

2) Does $f$ have an antiderivative locally on $\mathbb{C}$?

Here i am confused...According to the definition for an antiderivative locally, we need to show that for every $z_{0}\in \mathbb{C}$ there exist an open neighbourhood $V\subset \mathbb{C}$ with $z_{0}\in V$ s.t $f$ on $V$ has an antiderivative.
But $f$ is nowhere complex differentiable?
Thank you in advance!

REPLY [1 votes]: As Berci said, the answer is no to both questions. A function that has an antiderivative in the complex sense, even locally, is necessarily holomorphic -- and $\operatorname{Re}z$ is not holomorphic. ($\LaTeX$ remark: \operatorname{Re}z looks better than Re(z).)


TITLE: suppose that $f :\Bbb R \to \Bbb R$ be a differentiable function, and that there exists $c>0$ such that $|f'(x)|\le c$ prove $|f(x)-f(y)|\le C|x-y|$
QUESTION [0 upvotes]: suppose that $f : \Bbb R \to \Bbb R$ be a differentiable function, and that there exists $c > 0$ such that $|f'(x)|<=c \forall x \in \Bbb R$.prove that  $|f(x)-f(y)|\le C|x-y| \forall x,y \in $$\Bbb R$

REPLY [0 votes]: For any interval $[a,b]$ in $\mathbb{R}$, we can find a point $x$ between $a$ and $b$ such that
$$
\bigg|\frac{f(b)-f(a)}{a-b}\bigg|=|f'(x)|\leq C
$$
by the mean value theorem and by your assumption. Since this will hold for any interval you choose, you may conclude that for any $a,b\in \mathbb{R}$ the inequality
$$
|f(b)-f(a)|\leq C|b-a|
$$
holds.


text:
~\\
\section{\sc\bf Theorem on covalent Carter diagrams}
  \label{sec_transition}

  \subsection{The size of linkage system $\mathscr{L}(\Gamma)$}
  The following proposition together with corollaries constitute the main result of this section:
  \begin{proposition}
     \label{prop_Estim}
     For the Carter diagram $\Gamma$, we have $\mid \mathscr{L}(\Gamma) \mid \leq \mathscr{E}$, where
     $\mathscr{E}$ is given by Table \ref{tab_val_Estim}.
  \end{proposition}
   In sections \S\ref{sec_stratif_D}-\S\ref{sec_stratif_A}, we check that for each Carter diagram $\Gamma$,
  there exists at least $\mathscr{E}$ linkage diagrams in the linkage system $\mathscr{L}(\Gamma)$.
  Together with Proposition \ref{prop_Estim}, we get the following
  \begin{corollary}
    \label{corol_Estim}
     For the Carter diagram $\Gamma$, we have $\mid \mathscr{L}(\Gamma) \mid = \mathscr{E}$
     (where $\mathscr{E}$ is given by Table \ref{tab_val_Estim}).
  \end{corollary}
 ~\\
  \begin{table}[H]
  \Small
  \centering
  \renewcommand{\arraystretch}{1.5}
  \begin{tabular} {||c|c||c|c||}
  \hline \hline
      The Carter          &   $\mathscr{E}$ & The Carter          & $\mathscr{E}$ \\
        diagram $\Gamma$  &                 & diagram $\Gamma$    & \\
    \hline \hline
       $E_6$, ~$E_6(a_1)$, ~$E_6(a_2)$ &  $54$  & $A_3$    & $14$  \\
    \hline
       $E_7$, $E_7(a_1)$, $E_7(a_2)$, $E_7(a_3)$, $E_7(a_4)$ & $56$ & $A_4$    & $30$  \\
   \hline
       $D_4$, ~$D_4(a_1)$ &  $24$ & $A_5$    & $62$ \\
    \hline
       $D_5$, ~$D_5(a_1)$ &  $42$ & $A_6$    & $126$ \\
    \hline
       $D_6$, ~$D_6(a_1)$, ~$D_6(a_2)$  &  $76$ & $A_7$    & $184$\\
    \hline
       $D_7$, ~$D_7(a_1)$, ~$D_7(a_2)$ & $142$ & $A_8$    & $90$ \\
    \hline
       $D_l$, ~$D_l(a_k)$, $l > 7$ & $2l$ & $A_l$, $l > 8$  & $(l+1)(l+2)$\\
   \hline  \hline
  \end{tabular}
  \vspace{2mm}
  \caption{\hspace{3mm}$\mid \mathscr{L}(\Gamma) \mid \leq \mathscr{E}$}
  \label{tab_val_Estim}
  \end{table}
 ~\\

   By Proposition \ref{prop_Estim} and calculations of this section we get
  \begin{corollary}
    \label{corol_Estim_2}
     For the Carter diagram $E_i(a_k)$ and $D_i(a_k)$,  we have
   \begin{equation*}
     \begin{split}
      & \mid \mathscr{L}(E_6(a_k)) \mid ~=~ \mid \mathscr{L}(E_6) \mid ~=~ 54, \text{ where } k = 1,2, \\
      & \mid \mathscr{L}(E_7(a_k)) \mid ~=~ \mid \mathscr{L}(E_7) \mid ~=~ 56, \text{ where } k = 1,2,3,4, \\
      & \mid \mathscr{L}(D_l(a_k)) \mid ~=~ \mid \mathscr{L}(D_l) \mid,
        \text{ where } l \geq 4 \text{ and } k = 1 \leq k \leq \big [ \tfrac{l-2}{2} \big ].
     \end{split}
   \end{equation*}
  \end{corollary}

   For each Carter diagram $\Gamma \in  \mathsf{C4} \coprod \mathsf{DE4} \coprod \mathsf{A}$,
   the values $\mathscr{B}^{\vee}_{\Gamma}(\gamma^{\nabla})$, the number of components, and number of
   linkage diagrams are presented in Table \ref{tab_val_Bmin1}, see Theorem \ref{th_full_descr}.

  \subsection{Transition between a partial root system $\mathcal{P}$ and a root system $\mathcal{R}$}
    \label{sec_part_root_and_comp}

 \begin{theorem}[On bijection of covalent root systems]
  \label{prop_bijection_root_syst}
  {\rm(i)}
  {\it Consider a covalent pair of Carter diagrams $\{\widetilde\Gamma, \Gamma\}$ out of the following list:
   \begin{equation}
    \label{eq_covalent_pairs}

     \begin{split}
      & \{D_l(a_k), D_l\} \text{ for } l \geq 4 \text{ and } 1 \leq k \leq \big [\tfrac{l-2}{2} \big ], \\
      & \{E_6(a_k), E_6\} \text{ for }  k = 1,2, \\
      & \{E_7(a_k), E_7\} \text{ for }  k = 1,2,3,4.
     \end{split}
   \end{equation}
  Let $\mathcal{R}$ (resp. $\mathcal{P}$) be the root system (resp. partial root system) corresponding to
  $\Gamma$ (resp. $\widetilde\Gamma$).
   There exists the map $M : \mathcal{P} \longmapsto \mathcal{R}$
   given by Table \ref{tab_part_root_syst} is the invertible linear map.}

 {\rm(ii)} Let $\widetilde\tau$ be a root in $\varPhi$, where $\varPhi$ is the primary root system.
  Then $\widetilde\tau$ is a root of the partial root system $\mathcal{P}$
  if and only if $\widetilde\tau$ is a root of the  root system $\mathcal{R}$.
 \end{theorem}

  \PerfProof
 (i)  For all pairs \{$\mathcal{P}$, $\mathcal{R}$\} from Table \ref{tab_part_root_syst},
 it is easy to check by \eqref{eq_Kac} that vectors images under the map $M$ are roots.
 For example, for case (2), the value $\mathscr{B}(\beta_{k+1})$ is as follows:
 \begin{equation*}
   \Small
   \begin{split}
    \mathscr{B}(\beta_{k+1}) =
             & \mathscr{B}(\tau_1)  + 4\sum\limits_{i=2}^k\mathscr{B}(\tau_i) + \mathscr{B}(\tau_{k+1}) +
                \mathscr{B}(\overline\tau_{k+1}) + \\
             & 4(\tau_1, \tau_2) + 8\sum\limits_{i=2}^{k-1}(\tau_{k-1}, \tau_k) + 4(\tau_k, \tau_{k+1}) + 4(\tau_k, \overline\tau_{k+1}) = \\
             & 2 + 8(k-1) + 2 + 2 - 4  - 8(k-2) - 4 - 4 = 2 + 8(k-1) - 8(k-1) = 2.
   \end{split}
 \end{equation*}
  Thus, $\beta_{k+1}$ is the root.  It only remains to check some inner products.
~\\

  \begin{table}[H]

  \centering
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular} {|c|c|c|c|}
    \hline
    \hline
      & Partial root system & Root system  & Linear maps  \\
      &     $\mathcal{P}$    &  $\mathcal{R}$  &
            $M : \mathcal{P} \longmapsto \mathcal{R}$ and  $M^{-1} : \mathcal{R} \longmapsto \mathcal{P}$ \\
     &  &  & {\Small ($M$ (resp. $M^{-1}$) acts as the identity on all} \\
     &  &  & {\Small  not mentioned $\alpha_i$ and $\beta_j$)} \\
    \hline
    \hline
     1 &
     $\begin{array}{c} \addlinespace
      \includegraphics[scale=0.9]{D4a1_p_root_syst.eps}  \end{array}$  &
     $\begin{array}{c}  \addlinespace
       \includegraphics[scale=0.9]{D4_root_syst.eps}  \end{array}$
           & \footnotesize
         $\begin{array}{cl}
         \addlinespace
         M & \hspace{-3mm}:=
         \begin{cases}
           \alpha_3  = -(\alpha_1 + \beta_1  + \beta_2)
         \end{cases}
         \\  \addlinespace
         M^{-1} &  \hspace{-3mm}:=
          \begin{cases}
           \beta_2 = -(\alpha_1 + \alpha_3 + \beta_1)
          \end{cases}
       \end{array}$ \\
    \hline
    2 &
     $\begin{array}{c} \\ \addlinespace
      \includegraphics[scale=0.9]{Dlak_p_root_syst.eps}  \end{array}$  &
     $\begin{array}{c} \\ \addlinespace
       \includegraphics[scale=0.9]{Dl_root_syst.eps}  \end{array}$
        & \footnotesize
     $\begin{array}{cl}
       \addlinespace
        M & \hspace{-3mm}:=
          \begin{cases}
             \beta_{k+1} =
              -(\tau_1 + 2\sum\limits_{i=2}^k\tau_i +  \tau_{k+1} + \overline{\tau}_{k+1}), \\
               \qquad\qquad\qquad\qquad\qquad\qquad\qquad k \geq 2, \\
              \beta_2 = -(\tau_1 + \tau_2 + \overline{\tau}_2).
          \end{cases} \\ \addlinespace
        M^{-1} & \hspace{-3mm}:=
          \begin{cases}
             \overline{\tau}_{k+1} =
              -(\tau_1 + 2\sum\limits_{i=2}^k\tau_i +  \tau_{k+1} + \beta_{k+1}), \\
               \qquad\qquad\qquad\qquad\qquad\qquad\qquad k \geq 2, \\
              \overline{\tau}_2 = -(\tau_1 + \tau_2 + \beta_2).
          \end{cases} \\ \addlinespace
     \end{array}$ \\
    \hline
    \hline
     3 &  $\begin{array}{c} \addlinespace
      \includegraphics[scale=0.8]{E6a1_p_root_syst.eps}  \end{array}$  &
     $\begin{array}{c} \addlinespace
       \includegraphics[scale=0.8]{E6_root_syst.eps}  \end{array}$
            & \footnotesize
         $\begin{array}{cl}
           \addlinespace
         M & \hspace{-3mm}:=
         \begin{cases}
           \beta_3 = -(\alpha_1 + \beta_1 + \alpha_3 + \widetilde\beta_3)
         \end{cases}
         \\  \addlinespace
         M^{-1} &  \hspace{-3mm}:=
          \begin{cases}
            \widetilde\beta_3 = -(\beta_3 + \alpha_1 + \beta_1 + \alpha_3)
          \end{cases}
       \end{array}$ \\
    \hline
     4 &  $\begin{array}{c} \\ \addlinespace
      \includegraphics[scale=0.8]{E6a2_p_root_syst.eps}  \end{array}$  &
     $\begin{array}{c} \\ \addlinespace
       \includegraphics[scale=0.8]{E6_root_syst.eps}  \end{array}$
        & \footnotesize
         $\begin{array}{cl}
         \addlinespace
         M & \hspace{-3mm}:=
         \begin{cases}
           \beta_2 = -(\alpha_2 + \beta_1 + \alpha_3 + \widetilde\beta_2) \\
           \beta_3 = -(\alpha_1 + \beta_1 + \alpha_3 + \widetilde\beta_3)
         \end{cases}
         \\  \addlinespace
         M^{-1} &  \hspace{-3mm}:=
          \begin{cases}
            \widetilde\beta_2 = -(\beta_2 + \alpha_2 + \beta_1 + \alpha_3) \\
            \widetilde\beta_3 = -(\beta_3 + \alpha_1 + \beta_1 + \alpha_3)
          \end{cases}
         \end{array}$ \\
    \hline
    \hline
     5 &  $\begin{array}{c}
      \includegraphics[scale=0.8]{E7a1_p_root_syst.eps}  \end{array}$  &
     $\begin{array}{c}
       \includegraphics[scale=1.0]{E7_root_syst_3.eps}  \end{array}$
        & \footnotesize
         $\begin{array}{cl}
         M & \hspace{-3mm}:=
         \begin{cases}
           \alpha_3 = -(\beta_2 + \alpha_2 + \beta_3 + \widetilde\alpha_3)
         \end{cases}
         \\  \addlinespace
         M^{-1} &  \hspace{-3mm}:=
          \begin{cases}
            \widetilde\alpha_3 = -(\alpha_3 + \beta_2 + \alpha_2 + \beta_3)
          \end{cases}
         \end{array}$
         \\
      \hline
     6 &   $\begin{array}{c}
      \includegraphics[scale=0.9]{E7a2_p_root_syst.eps}  \end{array}$ &
     $\begin{array}{c}
       \includegraphics[scale=1.0]{E7_root_syst_3.eps}  \end{array}$
        & \footnotesize
       $\begin{array}{cl}
       \addlinespace
        M & \hspace{-3mm}:=
         \begin{cases}
         & \hspace{-2mm}\beta_2 = -( \alpha_2 + \beta_1 + \widetilde\alpha_3 + \widetilde\beta_2), \\
         & \hspace{-2mm}\beta_3 = 2\widetilde\alpha_3 + \beta_1 + \widetilde\beta_3 + \widetilde\beta_2,  \\
         & \hspace{-2mm} \alpha_3 = -\widetilde\beta_3.
          \end{cases}
          \\  \addlinespace
         M^{-1} & \hspace{-3mm}:=
         \begin{cases}
         & \hspace{-2mm}\widetilde\alpha_3 = \alpha_3 + \beta_2 + \alpha_2 + \beta_3, \\
         & \hspace{-2mm}\widetilde\beta_2 =  \alpha_3 + 2\beta_2 + 2\alpha_2 + \beta_1 + \beta_3, \\
         & \hspace{-2mm}\widetilde\beta_3 = -\alpha_3.
          \end{cases}
         \end{array}$
          \\
      \hline
     7 &  $\begin{array}{c} \\
      \includegraphics[scale=0.9]{E7a3_p_root_syst.eps}  \end{array}$  &
     $\begin{array}{c} \\

       \includegraphics[scale=1.0]{E7_root_syst_3.eps}  \end{array}$
         & \footnotesize
        $\begin{array}{cl}
        \addlinespace
        M& \hspace{-3mm}:=
         \begin{cases}
         & \hspace{-2mm}\beta_4 = \alpha_4 + \widetilde\beta_3 - \alpha_2 - \beta_1 - \alpha_1, \\
         & \hspace{-2mm}\alpha_3 = \widetilde\alpha_3 + \widetilde\beta_3 - \alpha_2 - \beta_2, \\
         & \hspace{-2mm}\beta_3 = -\widetilde\beta_3.
          \end{cases}
          \\ \addlinespace
         M^{-1} & \hspace{-3mm}:=
         \begin{cases}
         & \hspace{-2mm}\alpha_4 = \beta_4 + \alpha_1  + \beta_1 + \alpha_2 + \beta_3, \\
         & \hspace{-2mm}\widetilde\alpha_3 = \alpha_3 + \beta_2 + \alpha_2 + \beta_3,  \\
         & \hspace{-2mm}\widetilde\beta_3 = -\beta_3.
          \end{cases}
         \end{array}$
        \\
      \hline
     8 &  $\begin{array}{c} \\ \addlinespace
      \includegraphics[scale=0.9]{E7a4_p_root_syst.eps} \end{array}$   &
     $\begin{array}{c} \\ \addlinespace
       \includegraphics[scale=1.0]{E7_root_syst_3.eps}   \end{array}$
         & \footnotesize
        $\begin{array}{cl} \addlinespace
        M & \hspace{-3mm}:=
         \begin{cases}
         & \hspace{-2mm}\beta_4 = -(2\alpha_4 + \beta_3 + \widetilde\beta_4 - \beta_2), \\
         & \hspace{-2mm}\alpha_3 = -(\widetilde\alpha_1 + \beta_3 + \alpha_2 + \beta_2), \\
         & \hspace{-2mm}\alpha_1 = -(\beta_1 + \alpha_2 + \beta_2 - \alpha_4 - \widetilde\beta_4).
          \end{cases}
          \\ \addlinespace
         M^{-1} & \hspace{-3mm}:=
         \begin{cases}
         & \hspace{-2mm}\widetilde\alpha_1 =  -(\alpha_3 + \beta_2 + \alpha_2 + \beta_3), \\
         & \hspace{-2mm}\alpha_4 =  -(\beta_4 + \alpha_1 + \beta_1 + \alpha_2 + \beta_3), \\
         & \hspace{-2mm}\widetilde\beta_4 =  \beta_4 + 2\alpha_1 + 2\beta_1 + 2\alpha_2 + \beta_2 + \beta_3.
          \end{cases}
          \\ \addlinespace
         \end{array}$
        \\
      \hline
      \hline
\end{tabular}
  \vspace{2mm}
  \caption{\hspace{3mm}The bijective linear maps between root systems and partial root systems.}

  \label{tab_part_root_syst}
  \end{table}

  \underline{$(1)$ {\it Pair } \{$D_4(a_1)$, $D_4$\}.} Let us check relations for $\alpha_3$:
 \begin{equation*}
  \Small
   \begin{array}{cl}
    (\alpha_3, \alpha_1) ~= & -(\mathscr{B}(\alpha_1) + (\beta_1, \alpha_1) + (\beta_2, \alpha_1)) = -(2 - 2) = 0, \\
    (\alpha_3, \alpha_2) ~= & -(\alpha_1, \alpha_2) + (\beta_1, \alpha_2) + (\beta_2, \alpha_2) = -1 + 1 = 0, \\
    (\alpha_3, \beta_1) ~= & -(\alpha_1, \beta_1) + (\beta_1, \beta_1) + (\beta_2, \beta_1) = 1 - 2 = -1. \\
   \end{array}
 \end{equation*}

  \underline{$(2)$ {\it Pair } \{$D_l(a_k)$, $D_l$\}.}
  The root $\beta_{k+1}$ is connected with $S = \{\tau_1, \dots, \tau_l \}$ at point $\tau_2$:

  \begin{equation*}
   \Small
   \begin{split}
   \text{ For } & k \geq 2, \\
      & \beta_{k+1} \perp \tau_1, \text{ since } (\tau_1, \tau_1 + 2\tau_2) = 0, \\
      & (\beta_{k+1}, \tau_2) = -((\tau_2, \tau_1) + 2\mathscr{B}(\tau_2) + 2(\tau_2,\tau_3)) = -(-1 + 4 -2) = -1, \\
      & \beta_{k+1} \perp \tau_i \text{ for } 3 \leq i \leq k-1, \text{ since } (\tau_i, \tau_{i-1} + \tau_i + \tau_{i+1}) = 0, \\
      & \beta_{k+1} \perp \tau_k, \text{ since } (\tau_k, 2\tau_{k-1} + 2\tau_k + \tau_{k+1} +  \overline\tau_{k+1}) = 0, \\
      & \beta_{k+1} \perp \tau_{k+1}, \text{ since } (\tau_{k+1}, 2\tau_k + \tau_{k+1}) = 0, \\
      & \beta_{k+1} \perp \tau_{k+2}, \text{ since } (\tau_{k+2}, \tau_{k+1} +  \overline\tau_{k+1}) = 0, \\
      & \beta_{k+1} \perp \tau_i \text{ for }  i > k+2. \\
   \text{ For } & k = 1, \\
      & \beta_2 \perp \tau_1, \text{ since } (\tau_1, \tau_1 + \tau_2 + \overline\tau_2) = 2 - 1 - 1 = 0, \\
      & (\beta_2, \tau_2) = -((\tau_2, \tau_1) + \mathscr{B}(\tau_2) + (\tau_2,\overline\tau_2)) = -(-1 + 2)  = -1, \\
      & \beta_2 \perp \tau_3, \text{ since } (\tau_2, \tau_3) + (\overline\tau_2, \tau_3) = -1 + 1  = 0, \\
      & \beta_2 \perp \tau_i \text{ for }  i > 3. \\
   \end{split}
 \end{equation*}
 By Remark \ref{rem_max_root} the root $\beta_{k+1}$ is uniquely determined, so the root
 $\overline\tau_{k+1}$ is also uniquely determined by $\beta_{k+1}$.

 \underline{$(3)$ {\it Pair } \{$E_6(a_1)$, $E_6$\}.} For the root $\beta_3 = -(\alpha_1 + \beta_1 + \alpha_3 + \widetilde\beta_3)$, we have
 $\beta_3 \perp  \beta_1, \alpha_2, \alpha_3, \beta_2$ and
 \begin{equation*}
     \Small \begin{array}{c}
   (\beta_3, \alpha_1) = -(\mathscr{B}(\alpha_1) +(\alpha_1, \beta_1)) = -1.
    \end{array}
 \end{equation*}

 \underline{$(4)$ {\it Pair } \{$E_6(a_2)$, $E_6$\}.} For roots $\beta_2 = -(\alpha_2 + \beta_1 + \alpha_3 + \widetilde\beta_2)$ and
 $\beta_3 = -(\alpha_1 + \beta_1 + \alpha_3 + \widetilde\beta_3)$, the following relations hold:
  \begin{equation*}
    \Small
     \begin{split}
     & \beta_2 \perp \alpha_1, \alpha_3, \beta_1, \quad \beta_3 \perp \alpha_2, \alpha_3, \beta_1, \quad
       (\beta_2, \alpha_2) = -1, \quad (\beta_3, \alpha_1) = -1,  \\
     & (\beta_2, \beta_3) = \mathscr{B}(\beta_1) + \mathscr{B}(\alpha_3) +
        (\beta_1, \alpha_1 + 2\alpha_3 + \alpha_2) +
        (\widetilde\beta_3, \alpha_2 + \alpha_3) + (\widetilde\beta_2, \alpha_1 + \alpha_3) =  2 + 2 - 4 + 0 + 0 = 0.
     \end{split}
 \end{equation*}

 \underline{$(5)$ {\it Pair } \{$E_7(a_1)$, $E_7$\}.} Here, $\alpha_3 = -(\beta_2 + \alpha_2 + \beta_3 + \widetilde\alpha_3)$.
 We have $\alpha_3 \perp \alpha_1, \beta_3, \beta_1, \alpha_2, \beta_4$ and
 \begin{equation*}
    \Small
  \begin{array}{c}
     (\alpha_3, \beta_2) = -((\mathscr{B}(\beta_2) + (\beta_2, \alpha_2)) = -1.
         \end{array}
 \end{equation*}

\underline{$(6)$ {\it Pair } \{$E_7(a_2)$, $E_7$\}.}
 For roots $\beta_2 = -( \alpha_2 + \beta_1 + \widetilde\alpha_3 + \widetilde\beta_2)$ and
 $\beta_3 = 2\widetilde\alpha_3 + \beta_1 + \widetilde\beta_3 + \widetilde\beta_2$, we have
 $\beta_2 \perp \beta_1, \alpha_1, \beta_4$ and $\beta_3 \perp \beta_1,\alpha_1, \beta_4$.
 Additionally, the following relations hold:
 \begin{equation*}
  \Small
   \begin{split}
      (\beta_2, \alpha_3) = &(\widetilde\beta_3, \widetilde\alpha_3) = -1, \\
      (\beta_2, \alpha_2) = &-((\alpha_2, \alpha_2) + (\alpha_2, \beta_1)) = -(2 - 1) = -1, \\
      (\beta_2, \beta_3) = &-(\alpha_2 + \beta_1 + \widetilde\alpha_3 + \widetilde\beta_2,
                   2\widetilde\alpha_3 + \beta_1 + \widetilde\beta_3 + \widetilde\beta_2) = \\
                      &-\big ((\alpha_2, \beta_1) + (\beta_1, \beta_1) + 2(\beta_1, \widetilde\alpha_3) +
                       2(\widetilde\alpha_3, \widetilde\alpha_3) + (\alpha_3, \beta_1) + (\alpha_3, \widetilde\beta_3)  \\
                      & + (\alpha_3, \widetilde\beta_2) + (\widetilde\beta_2, \widetilde\beta_2) + 2(\alpha_3, \widetilde\beta_2) \big ) =
                     1 - 2 + 2 - 4 + 1 + 1 + 1  - 2 + 2 = 0. \\
      (\beta_3, \alpha_3) = &-(\widetilde\beta_3,\widetilde\beta_3) - 2(\widetilde\alpha_3, \widetilde\beta_3) = 0. \\
   \end{split}
 \end{equation*}

 \underline{$(7)$ {\it Pair } \{$E_7(a_3)$, $E_7$\}.}
 Here, $\beta_4 \perp \beta_3, \alpha_2, \alpha_3, \beta_1, \beta_2$ and $(\beta_4, \alpha_1) = -1$,
 For example,
 \begin{equation*}
  \Small
   \begin{split}
     & (\beta_4, \beta_1) = (\alpha_4 + \widetilde\beta_3 - \alpha_2 - \beta_1 - \alpha_1, \beta_1) =
        -(\alpha_2 + \beta_1 + \alpha_1, \beta_1) =  0, \\
     & (\beta_4, \beta_2) = (\alpha_4 + \widetilde\beta_3 - \alpha_2 - \beta_1 - \alpha_1, \beta_2) =
        (\alpha_4 - \alpha_2, \beta_2) =  0, \\
     & (\beta_4, \alpha_2) = (\alpha_4 + \widetilde\beta_3 - \alpha_2 - \beta_1 - \alpha_1, \alpha_2) =
        (-\beta_1 -\alpha_2 + \widetilde\beta_3, \alpha_2) =  0. \\
   \end{split}
 \end{equation*}
 Further, $\alpha_3 \perp \beta_3, \alpha_2, \beta_1, \alpha_1$ and $(\alpha_3, \beta_2) = -1$.
 For example,
 \begin{equation*}
  \Small
   \begin{split}
     & (\alpha_3, \alpha_2) = (\widetilde\alpha_3 + \widetilde\beta_3 - \alpha_2 - \beta_2, \alpha_2) =
           -(\alpha_2, \alpha_2) -(\beta_2, \alpha_2) + (\widetilde\beta_3, \alpha_2) = 0, \\
     & (\alpha_3, \beta_1) = (\widetilde\alpha_3 + \widetilde\beta_3 - \alpha_2 - \beta_2, \beta_1) =
          (\beta_1, \widetilde\alpha_3) - (\beta_1, \alpha_2) = 0.
   \end{split}
 \end{equation*}
 Similarly, $\beta_3 \perp \alpha_2, \alpha_1, \beta_1$ and $(\beta_3, \alpha_2) = -1$.

 \underline{$(8)$ {\it Pair } \{$E_7(a_4)$, $E_7$\}.}
 We have $\beta_4 \perp  ~\beta_1, \alpha_2, \beta_3, \beta_2$.
 The remaining cases for $\beta_4$:
\begin{equation*}
  \Small
   \begin{split}
     \qquad
      (\beta_4, \alpha_1) = & ~(2\alpha_4, -\alpha_4 + \beta_2 - \widetilde\beta_4) + (\beta_3, \alpha_2 - \alpha_4) +
        (\widetilde\beta_4, -\widetilde\beta_4 -\alpha_4) -(\beta_2, \beta_2 + \alpha_2 - \alpha_4) = \\
         & 0 + 0 -1 + 0 = -1,  \\
      (\beta_4, \alpha_3) = & ~(2\alpha_4, \beta_3 + \beta_2) + (\beta_3, \beta_3 + \widetilde\alpha_1 + \alpha_2) +
          (\widetilde\beta_4, \widetilde\alpha_1) + (-\beta_2, \beta_2 + \alpha_2) = 0 + 0 + 1 -1 = 0. \\
  \end{split}
 \end{equation*}
 Further, $\alpha_3 \perp ~\alpha_2, \beta_1, \beta_3$ and remaining cases for $\alpha_3$ are
 as follows:
 \begin{equation*}
  \Small
   \begin{split}
      (\alpha_3, \beta_2) = & ~-(\beta_2, \beta_2) - (\beta_2,  \widetilde\alpha_1) = -2 +1 = -1, \\
      (\alpha_3, \alpha_1) = & ~(\widetilde\alpha_1, \beta_1 - \widetilde\beta_4) + (\beta_3, \alpha_2 - \alpha_4)
         + (\alpha_2, \beta_1 + \beta_2 + \alpha_2) + (\beta_2, \beta_2 + \alpha_2 - \alpha_4) = 0. \\
   \end{split}
 \end{equation*}
~\\
 Finally, $\alpha_1 \perp ~\alpha_2, \beta_2, \beta_3$ and
       $(\alpha_1, \beta_1) = ~-(\beta_1, \beta_1) -(\beta_1, \alpha_2) = -1.$
~\\

  \index{$S = \{\tau_1,\dots,\tau_l\}$, $\Gamma$-associated root subset}
  (ii) Let $\widetilde{S} = \{\widetilde\tau_1,\dots,\widetilde\tau_l\}$ (resp. $S = \{\tau_1,\dots,\tau_l\}$)
  be the $\widetilde\Gamma$-associated (resp. $\Gamma$-associated) root subset
  generating the partial root system $\mathcal{P}$ (resp. root system $\mathcal{R}$).
  Let $M$ be matrix given by Table \ref{tab_part_root_syst}, $M$ acts as follows:
\begin{equation}
  \label{eq_matr_N_1}
    M\widetilde\tau_i = \tau_i.
\end{equation}
   Let $\widetilde\tau$ be a certain root in $\mathcal{P}$.  Then
 \begin{equation*}
     \widetilde\tau = \sum\limits_{i = 0}^l t_i\widetilde\tau_i,
 \end{equation*}
  where $t_i$ are some integer coefficients.
  Consider the vector $\tau := M\widetilde\tau$. We have
 \begin{equation*}
     \tau = M\widetilde\tau = \sum\limits_{i = 0}^l t_i{M}\widetilde\tau_i = \sum\limits_{i = 0}^l t_i\tau_i.
 \end{equation*}
  Thus, $\tau$ belongs to the integer lattice spanned by $S$.  Since $M^{-1}$ is the integer matrix then
  the vector $\widetilde\tau = M^{-1}\tau$ also belongs to the integer lattice spanned by $S$.
  In addition, $\widetilde\tau \in \mathcal{P} \subset \varPhi$, i.e., $\widetilde\tau$ is the root in $\varPhi$.
  Therefore, $\widetilde\tau$ is the root in the root system $\mathcal{R}$.
  The converse is similar.

  Thus, linear transformations $M : \mathcal{P} \longmapsto \mathcal{R}$ and $M^{-1} : \mathcal{R} \longmapsto \mathcal{P}$
  map roots of $\mathcal{P}$ onto roots of $\mathcal{R}$ and realize the bijection of $\mathcal{P}$ and $\mathcal{R}$.
 \qed
~\\

The matrix $M$ maps is called the {\it transition matrix} from $\widetilde{S}$ onto $S$.

 \begin{corollary}
  \label{corol_biject}
 {\rm (i)} For any pair \eqref{eq_covalent_pairs},
 the root system $\varPhi(\Gamma')$ contains the root subsystem $\mathcal{R}$ if and only if
 $\varPhi(\Gamma')$ contains the partial root subsystem $\mathcal{P}$.

 {\rm (ii)} For any pair \eqref{eq_covalent_pairs},
 \begin{equation}
    \label{eq_number_of_roots}
     \mid \mathcal{R} \mid = \mid \mathcal{P} \mid.
 \end{equation}
 \qed
 \end{corollary}

  \begin{corollary}
     \label{cor_exist_Dynkin_ext}
      For any Carter diagram out of the list \eqref{eq_covalent_pairs},
      there exists the Dynkin extension.
  \end{corollary}
  \PerfProof For Dynkin diagrams $\Gamma = A_l, D_l, E_l$, the Dynkin extensions exist since
  $\varPhi(E_l) \subset \varPhi(E_{l+1})$ for $l < 8$, $\varPhi(A_l) \subset \varPhi(A_{l+1})$
  and $\varPhi(D_l) \subset \varPhi(D_{l+1})$.
  For non-Dynkin Carter diagrams $\widetilde\Gamma = E_l(a_k), D_l(a_k)$,
  the Dynkin extensions exist by Corollary \ref{corol_biject}(i):
  Since $\varPhi(\Gamma) \subset \varPhi(\Gamma')$ then $\varPhi(\widetilde\Gamma) \subset \varPhi(\Gamma')$.
  \qed

 \begin{corollary}
  {\rm (i)}
   Let $S = \{\tau_1, \dots, \tau_{l-1}\}$ be an $A_{l-1}$-associated subset of roots in $\varPhi(D_l)$.
   There is the only root $\overline\tau_k$ such $S \cup \overline\tau_k$ is the $D_l(a_{k-1})$-associated subset,
   see Fig. \ref{Dlak_tk_tk1}$(a)$.

  {\rm (ii)} If $\overline\tau_k$ (resp. $\overline\tau_{k+1}$) is a root such that the subset
  $S \cup \overline\tau_k$ (resp. $S \cup \overline\tau_{k+1}$) constitutes the $D_l(a_{k-1})$-associated
  (resp. $D_l(a_k)$-associated) subset, then
 \begin{equation}
    \label{eq_rel_tk_tk1}
         \tau_k + \tau_{k+1} + \overline\tau_{k+1} - \overline\tau_k = 0,
 \end{equation}
  see Fig. \ref{Dlak_tk_tk1}$(d)$.
 \end{corollary}
\begin{figure}[H]
\centering
\includegraphics[scale=0.7]{Dlak_tk_tk1}
\caption{ \hspace{3mm}Relation between $\overline\tau_k$ and
$\overline\tau_{k+1}$}

\label{Dlak_tk_tk1}
\end{figure}
\PerfProof
 \rm (i) By Remark \ref{rem_max_root}, the maximal root $\mu_{max}$ is uniquely determined.
 The vector $\beta_k$ defined in Table \ref{tab_part_root_syst}(case $(2)$) coincides with
 $\mu_{max}$, i.e., there exists only one root $\beta_k$ connected with $\tau_2$ and orthogonal
 to roots $\{\tau_1,\tau_3, \dots, \tau_l\}$. Thus, the vector
 \begin{equation}
  \label{eq_iniq_def_2}
     \overline\tau_k = -(\tau_1 + 2\sum\limits_{i=2}^{k-1}\tau_i + \tau_k + \beta_k)
 \end{equation}
  is also uniquely determined:
 \begin{equation}
  \label{eq_iniq_def_2a}
     \beta_k = \beta_{k+1}.
 \end{equation}

 \index{dotted ! - edge}
 \rm (ii)  By \eqref{eq_iniq_def_2a}, we have
 \begin{equation*}

    -(\tau_1 + 2\sum\limits_{i=2}^{k-1}\tau_i + \tau_k + \overline\tau_k) =
    -(\tau_1 + 2\sum\limits_{i=2}^k\tau_i + \tau_{k+1} + \overline\tau_{k+1}).
 \end{equation*}
 In other words,
 \begin{equation*}
    \overline\tau_k = \tau_k + \tau_{k+1} + \overline\tau_{k+1},
 \end{equation*}
 i.e., \eqref{eq_rel_tk_tk1} holds. Further,
 \begin{equation*}
    (\overline\tau_k, \overline\tau_{k+1})  = (\tau_k, \overline\tau_{k+1}) +  (\overline\tau_{k+1},\overline\tau_{k+1}) =
        -1 + 2 = 1.
 \end{equation*}
 Therefore, $\overline\tau_k$ and $\overline\tau_{k+1}$ are connected by a dotted edge,
 see Fig. \ref{Dlak_tk_tk1}$(d)$. \qed
~\\

\subsection{Relation of linkage diagrams lying in $\mathscr{L}(\widetilde\Gamma)$ and in $\mathscr{L}(\Gamma)$}
  \label{sec_rel_coval_2}

Denote by $\gamma^{\nabla}_{\Gamma}$ and $\gamma^{\nabla}_{\widetilde\Gamma}$ the following vectors:
\begin{equation}
  \label{eq_def_gamma_theta_nabla}
   \gamma^{\nabla}_{\Gamma} =
   \left (
   \begin{array}{c}
       (\gamma, \tau_1) \\
       \dots \\
       (\gamma, \tau_l)
   \end{array}
     \right ), \qquad
      \gamma^{\nabla}_{\widetilde\Gamma} =
   \left (
   \begin{array}{c}
       (\gamma, \widetilde\tau_1) \\
       \dots \\
       (\gamma, \widetilde\tau_l)
   \end{array}
     \right ).
\end{equation}

\begin{lemma}
  \label{lem_contragred_map}
 Let $\{\widetilde\Gamma,  \Gamma\}$ be the covalent pair of Carter diagrams from \eqref{eq_covalent_pairs}.

{\rm(i)} The vector $\gamma^{\nabla}_{\widetilde\Gamma}$ is the
linkage label vector in the linkage system
$\mathscr{L}(\widetilde\Gamma)$ if and only if
$\gamma^{\nabla}_{\Gamma}$ is the linkage label vector in the
linkage system $\mathscr{L}(\Gamma)$.

{\rm(ii)} Let $\gamma^{\nabla}_{\widetilde\Gamma}$,
$\theta^{\nabla}_{\widetilde\Gamma} \in
\mathscr{L}(\widetilde\Gamma)$ and
 $\gamma^{\nabla}_{\Gamma}$, $\theta^{\nabla}_{\Gamma} \in \mathscr{L}(\Gamma)$.
 Then $\gamma^{\nabla}_{\widetilde\Gamma}$, $\theta^{\nabla}_{\widetilde\Gamma}$ coincide if and only if
 $\gamma^{\nabla}_{\Gamma}$, $\theta^{\nabla}_{\Gamma}$ coincide.
\end{lemma}

\PerfProof
  Let $L$ be the subspace spanned by roots of $\widetilde{S} = \{\widetilde\tau_1, \dots \widetilde\tau_l\}$,
  we will write $L = [\widetilde\tau_1, \dots \widetilde\tau_l]$. By Theorem \ref{prop_bijection_root_syst}
  we have
\begin{equation}
 \label{eq_L_two_forms}
   L = [\widetilde\tau_1, \dots \widetilde\tau_l] = [\tau_1, \dots \tau_l].
\end{equation}

  (i) The fact that $\gamma^{\nabla}_{\widetilde\Gamma}$
  is the linkage label vector in the linkage system $\mathscr{L}(\widetilde\Gamma)$
  means that $\gamma$ is the linkage root linearly independent of roots of $\widetilde{S}$, see \S\ref{sec_linkage}.
  In turn, by \eqref{eq_L_two_forms}, this means,
  that $\gamma$ is also linearly independent of roots of $S$ and $\gamma^{\nabla}_{\Gamma}$
  is the linkage label vector in the linkage system $\mathscr{L}(\Gamma)$.

  (ii) By \eqref{eq_labels_proj} and \eqref{eq_def_gamma_theta_nabla}
 \begin{equation}
   \label{eq_gamma_theta}
   \begin{split}
    & \gamma^{\nabla}_{\widetilde\Gamma} = \mathscr{B}_{\widetilde\Gamma}\gamma_L, \qquad
      \theta^{\nabla}_{\widetilde\Gamma} = \mathscr{B}_{\widetilde\Gamma}\theta_L, \\

    & \gamma^{\nabla}_{\Gamma} = \mathscr{B}_{\Gamma}\gamma_L, \qquad
      \theta^{\nabla}_{\Gamma} = \mathscr{B}_{\Gamma}\theta_L. \\
   \end{split}
 \end{equation}
  where $\gamma_L$ (resp. $\theta_L$) is the projection of $\gamma$ (resp. $\theta$) on $L$.
  Note that projections of $\gamma$ and $\theta$ on $L$ do not depend on choice of the diagram $\widetilde\Gamma$.
  By \eqref{eq_gamma_theta} from $\gamma^{\nabla}_{\widetilde\Gamma} = \theta^{\nabla}_{\widetilde\Gamma}$ we get
  $\mathscr{B}_{\widetilde\Gamma}\gamma_L = \mathscr{B}_{\widetilde\Gamma}\theta_L$.
  Since $\mathscr{B}_{\widetilde\Gamma}$ is positive definite (Proposition \ref{restr_forms_coincide}),
  we have $\gamma _L = \theta_L$. Again, by \eqref{eq_gamma_theta} we have $\gamma^{\nabla}_{\Gamma} = \theta^{\nabla}_{\Gamma}$.
  \qed

 \begin{corollary}
    \label{corol_Estim_2}
     For any covalent pair of Carter diagrams $\{\widetilde\Gamma, \Gamma\}$ out of \eqref{eq_covalent_pairs} we have
   \begin{equation*}
      \mid \mathscr{L}(\widetilde\Gamma) \mid ~=~ \mid \mathscr{L}(\Gamma) \mid.
   \end{equation*}
  \end{corollary}

   This follows directly from Lemma \ref{lem_contragred_map}.
  \qed

  \subsection{Linkage system components}
    \label{sec_linkage_syst_comp}
  \index{Dynkin extension ! - $\Gamma <_D \Gamma'$}
  \index{$\Gamma <_D \Gamma'$, Dynkin extension}
  \index{linkage system component}
  \index{stratum size}
  \index{ $\varPhi(S)$, partial root system}
  \index{root system ! - partial}
  \index{partial root system}
  \index{root system}

   Consider a Carter diagram  $\Gamma$.
   Let $\Gamma <_D \Gamma'$ be a Dynkin extension, $S$ a certain $\Gamma$-associated root subset,
   $S \subset \varPhi(\Gamma')$, where  $\varPhi(\Gamma')$ is the root system associated with $\Gamma'$.
 ~\\

   Recall that the root subset $\varPhi(S)$ consisting of roots of $\varPhi$
   linearly dependent on roots of $S$ is said to be a partial root system.
   The subset of roots $\varPhi(\Gamma')\backslash\varPhi(S)$ is said to be the root stratum, see \S\ref{sec_Dynkin_ext}.
   The number of roots in the root stratum $\varPhi(\Gamma')\backslash\varPhi(S)$
   is said to be the {\it stratum size}:
 \begin{equation}
  \label{eq_regext2}
 \mid \varPhi(\Gamma')\backslash\varPhi(S) \mid \quad = \quad  \mid\varPhi(\Gamma')\mid - \mid\varPhi(S)\mid,
 \end{equation}
  see \S\ref{sec_link_syst_comp}. We define the set of linkage diagrams $\mathscr{L}_{\Gamma'}(\Gamma)$ as follows:
  \index{linkage diagrams ! - in $\mathscr{L}_{\Gamma'}(\Gamma)$}
 \begin{equation}
  \label{eq_regext3}
   \mathscr{L}_{\Gamma'}(\Gamma) = \{\gamma^{\nabla} \mid \gamma \in \varPhi(\Gamma')\backslash\varPhi(S)\}.
 \end{equation}
 The set $\mathscr{L}_{\Gamma'}(\Gamma)$ is said to be the {\it linkage system component}
 (or, {\it $\Gamma'$-component} of $\mathscr{L}(\Gamma)$).
 There are several Dynkin extensions and, accordingly,
 several linkage system components for the given Carter diagram $\Gamma$:
 \begin{equation*}
     \Gamma <_D \Gamma_1, \dots, \Gamma <_D \Gamma_n.
 \end{equation*}
  The linkage system $\mathscr{L}(\Gamma)$ is the union of the sets $\mathscr{L}_{\Gamma_i}(\Gamma)$
  taken for all Dynkin extensions of $\Gamma$:
 \begin{equation*}
     \mathscr{L}(\Gamma) \quad = \quad  \bigcup_{\Gamma <_D \Gamma_i} \mathscr{L}_{\Gamma_i}(\Gamma).
 \end{equation*}
 For example, the Dynkin extensions for $\Gamma = A_7$ are as follows:
 \begin{equation*}
     A_7 <_D A_8, \quad A_7 <_D  D_8, \quad A_7 <_D E_8,
 \end{equation*}
 the corresponding linkage system components are
 $\mathscr{L}_{A_8}(A_7)$, $\mathscr{L}_{D_8}(A_7)$, $\mathscr{L}_{E_8}(A_7)$ and
 the linkage system $\mathscr{L}(A_7)$ is the union of linkage system components of
 $A$-, $D$-, and $E$-type:
 \begin{equation*}
     \mathscr{L}(A_7) ~=~  \mathscr{L}_{A_8}(A_7) \cup
                                      \mathscr{L}_{D_8}(A_7) \cup  \mathscr{L}_{E_8}(A_7).
 \end{equation*}

 For every root $\gamma \in \varPhi(\Gamma')\backslash\varPhi(S)$,
 there exist the linkage diagram $\gamma^{\nabla} \in \mathscr{L}_{\Gamma'}(\Gamma)$.
 There exist pairs of roots $\gamma_1, \gamma_2 \in \varPhi(\Gamma')\backslash\varPhi(S)$
 such that $\gamma_1^{\nabla} = \gamma_2^{\nabla}$, see Lemma \ref{lem_coinc_linkage}.
 Thus, from \eqref{eq_regext2} we deduce that:
 \begin{equation}
   \label{eq_regext4}
     \mid \mathscr{L}_{\Gamma'}(\Gamma) \mid  \quad \leq \quad  \mid\varPhi(\Gamma')\mid - \mid\varPhi(S)\mid.
 \end{equation}

config: arxiv


text:
\begin{document}

\allowdisplaybreaks

\date{\today}

\begin{abstract}
In this paper, we study the local well-posedness of two types of generalized Cucker-Smale (in short C-S) flocking models. We consider two different communication weights, singular and regular ones, with nonlinear coupling velocities $v|v|^{\beta-2}$ for $\beta > \frac{3-d}{2}$. For the singular communication weight, we choose $\psi^1(x) = 1/|x|^{\alpha}$ with $\alpha \in (0,d-1)$ and $\beta \geq 2$ in dimension $d > 1$. For the regular case, we select $\psi^2(x) \geq 0$ belonging to $(L_{loc}^\infty \cap \mbox{Lip}_{loc})(\R^d)$ and $\beta \in (\frac{3-d}{2},2)$. We also remark the various dynamics of C-S particle system for these communication weights when $\beta \in (0,3)$.
\end{abstract}

\maketitle \centerline{\date}

\tableofcontents

\section{Introduction}
In the last years, collective problems as a dynamic feature of autonomous agents are active recent subjects in many different disciplines such as statistical physics, mathematics, biology, and control theory, etc., due to its engineering, physical, and biological applications \cite{Bir,CDF,CKFL,DM,LP,PE,PEG,TT}. Among various mathematical models describing the interactions between the individuals, our interest in the paper lies the flocking model which was introduced by Cucker-Smale \cite{CS1,CS2}. Cucker-Smale(in short C-S) model is a type of Newton particle model motivated by the work of Vicsek et. al. \cite{Vic}, and rigorous asymptotic flocking estimates depending on the decay rate of regular communication weight were provided in \cite{CS2}. Later, these estimates were improved and refined in the literature \cite{CFRT,HL,HT}.

Despite of its novelty of C-S  model for flocking dynamics, there are several drawbacks in real applications. Among them, our study is dealing with two issues; collision avoidance between individuals and general coupling for velocities of them. For the real applications of C-S model, e.g., unmanned aerial vehicle, it is important to avoid any collision. However the original C-S model \cite{CS1,CS2} does not take into account this, and as a consequence, there are some studies to prevent the collisions by adding new forcing terms to control the distance between the individuals \cite{CD,PKH} or considering a modified communication weight \cite{ACHL}. Concerning the general coupling for velocities, the original C-S model has a linear coupling for velocities, but there is no specific physical reason for this. Thus we will consider the nonlinear velocity coupling which is averaged over the strength of the relative speed with a certain exponent.

More precisely, let $f = f(x,v,t)$ be the one-particle distribution function at a spatial domain
$x, v \in \bbr^d$ at time $t$ in dimension $d >1$.
In this situation the density function $f$ is determined by
\begin{equation}\label{k-CS-si}
\partial_t f + v \cdot \nabla_x f
 + \nabla_v \cdot \big[F_i(f)f\big] = 0,
  \quad (x, v) \in \bbr^d \times \bbr^d,~~t > 0,
\end{equation}
subject to initial data:
\begin{equation}\label{ini-k-CS}
f(x,v,0) =: f^0(x,v), \quad (x, v) \in \bbr^d \times \bbr^d,
\end{equation}
where $F_i$ denotes the alignment force between particles:
\begin{equation*}
F_i(f)(x,v,t) := -\int_{\bbr^d \times \bbr^d}
\psi^i(x-y)\nabla_v \phi(v-w)f(y,w,t)dydw, \quad i=1,2.
\end{equation*}
Here the potential function $\phi(v)$ for the coupling of velocity is given by
\begin{equation*}
\phi(v) = \frac1\beta|v|^\beta, \quad \beta > \frac{3-d}{2}
\end{equation*}
Note that if $\beta=2$, then $\nabla\phi(v) = v$, and the velocity coupling is the same with Cucker-Smale's one. For the communication weight $\psi^i$, we take the following two different forms:
\begin{equation}\label{condi:psi}
\psi^1(x) := \frac{1}{|x|^\alpha}, \quad \alpha \in (0,d-1) \quad \mbox{and} \quad 0 \leq \psi^2(x) = \psi^2(-x) \in \left(L_{loc}^\infty \cap \mbox{Lip}_{loc}\right)(\R^d).
\end{equation}
Here $\psi^2 \in \mbox{Lip}_{loc}(\R^d)$ implies that for any compact set $K \subset \R^d$, there is some constant $L_K > 0$ such that
\[
|\psi^2(x) - \psi^2(y)| \leq L_K|x-y|, \quad x,y \in K.
\]
In order to emphasize on the fact that $\beta$ depends on the two different communication weights $\psi^i$, we denote the exponent $\beta_i$ of the coupling of velocity corresponding to $\psi^i$ for $i=1,2$.

The purpose of the paper is to establish a local existence of unique weak solutions to the kinetic C-S equations \eqref{k-CS-si}-\eqref{ini-k-CS}. For the singular communication weight $F_1$, the singularity with respect to the position is allowed up to the one of Newtonian interactions, and the exponent $\beta_1$ of the potential function is greater than or equals to original Cucker-Smale's one, i.e., $\alpha \in (0,d-1)$ and $\beta_1 \geq 2$. In the other case, the communication weight $F_2$ is sufficiently regular, but the coupling for velocity has a singularity of the order of $\beta_2 \in \left( \frac{3-d}{2}, 2\right)$. For both cases, the general framework for the well-posedness of solutions can not be applied due to the singularity either in position or velocity, and it has not been addressed in the literature to the best of our knowledge. We adapted the idea from our recent work \cite{CCH} to overcome these difficulties. On the other hand, in case of enough regularity of the communication weight and coupling for velocity in the forcing term, the well-posedness for these kinetic equations describing the collective behaviour is included in \cite{CCR}. We also refer to \cite{HJ1,HJ2} for the issue of the mean-field derivation of Vlasov-type equations with singular kernel.

The rest of this paper is organized as follows. In Section \ref{sec2}, we briefly provide definition and properties of Wasserstein distances, and state our main results on well-posedness. Section 3 is devoted to give the details of the proof of a unique weak solution to the system \eqref{k-CS-si} for each case. Our strategy is first to construct approximate solutions, and obtain the uniform bounds of approximate solutions with respect to the regularization. Then we finally show that the approximate solutions are Cauchy sequence, and let the parameter of regularization tend to zero to have the existence of the weak solutions. Finally, in the last section we remark that the dynamics of the generalized C-S particles system can lead to collisions in finite time and we discussed the rate and the conditions under which this can happen.\newline

\noindent {\bf Notations:} $|\cdot|$ denotes the Euclidean distance, and $\mathcal{P}_p(\R^d)$ stands for the set of probability measures with bounded moments of order $p \in [1,\infty)$. For notational simplicity, we also use the following notations throughout the paper: For $1 \leq p \leq \infty$,
\[
\|g\|_{L^p} := \|g\|_{L^p(U)} \quad \mbox{where $U$ can be either $\R^d$ or $\R^d \times \R^d$},
\]
\[
\|g\|_{L^1 \cap L^p} := \|g\|_{L^1} + \|g\|_{L^p}\,, \quad \mbox{and} \quad \|g\| := \|g\|_{L^{\infty}(0,T;L^{1} \cap
L^{p})}.
\]

\section{Preliminaries and main results}\label{sec2}

\subsection{Mathematical tools}
In this
part, we present several definition and properties of Wasserstein
distances that will be mainly used in our arguments for the well-posedness.

\begin{definition}(Wasserstein p-distance) \label{defdp}
Let $\rho_1,~ \rho_2$ be two Borel probability measures on
$\bbr^d$. Then the Euclidean Wasserstein distance of order $1\leq
p<\infty$ between $\rho_1$ and $\rho_2$ is defined as
\begin{equation*}\label{wasser-p-dis}
d_p(\rho_1,\rho_2) := \inf_{\gamma} \left( \int_{\bbr^d \times
\bbr^d} |x-y|^p \, d\gamma(x,y) \right)^{1/p},
\end{equation*}
and, for $p=\infty$ (this is the limiting case, as $p \to
\infty$),
\begin{equation*}\label{wasser-inf-dis}
d_{\infty}(\rho_1,\rho_2) := \inf_{\gamma} \left(\sup_{(x,y) \in \text{supp}(\gamma)} | x- y | \right),
\end{equation*}
where the infimum runs over all transference plans, i.e., all
probability measures $\gamma$ on $\bbr^d \times \bbr^d$ with
marginals $\rho_1$ and $\rho_2$ respectively,
\[
\int_{\bbr^d \times \bbr^d} \phi(x) d\gamma(x,y) = \int_{\bbr^d}
\phi(x) \rho_1(x) dx,
\]
and
\[
\int_{\bbr^d \times \bbr^d} \phi(y) d\gamma(x,y) = \int_{\bbr^d}
\phi(y) \rho_2(y) dy,
\]
for all $\phi \in \mathcal{C}_b(\bbr^d)$.
\end{definition}
Note that $\mathcal{P}_p(\R^d), 1\leq p < \infty$ is a
complete metric space endowed with the $p$-Wassertein distance
$d_p$, see \cite{Vil}. We refer to \cite{GS,MC} for more
details in the case of the $d_\infty$ distance.

In particular $p=1$, Wasserstein-1 distance $d_1$ is equivalent to the bounded Lipschitz distance which is also called {\it Monge-Kantorovich-Rubinstein distance}:
\[
d_1(\rho_1,\rho_2) = \sup\left\{ \int_{\R^d} \varphi(\xi)(\rho_1(\xi) - \rho_2(\xi))d\xi \Big| \varphi \in \mbox{Lip}(\R^d), \mbox{ Lip}(\varphi) \leq 1\right\},
\]
where Lip($\R^d$) denotes the set of Lipschitz functions on $\R^d$, and Lip($\varphi$) the Lipschitz constant of a function $\varphi$.
We also remind the definition of the push-forward of a measure by
a mapping in order to give the relation between Wasserstein
distances and optimal transportation.

\begin{definition}
Let $\rho_1$ be a Borel measure on $\bbr^d$ and $\mathcal{T} :
\bbr^d \to \bbr^d$ be a measurable mapping. Then the push-forward
of $\rho_1$ by $\mathcal{T}$ is the measure $\rho_2$ defined by
\[
\rho_2(B) = \rho_1(\mathcal{T}^{-1}(B)) \quad \mbox{for} \quad B
\subset \bbr^d,
\]
and denoted as $\rho_2 = \mathcal{T} \# \rho_1$.
\end{definition}

We recall in the next proposition some classical properties, which proofs may be found in \cite{Vil}.

\begin{proposition}\label{prop-proper}
(i) The definition of $\rho_2 = \mathcal{T} \# \rho_1$ is equivalent
to
$$
\int_{\bbr^d} \phi(x)\, d\rho_2(x) = \int_{\bbr^d} \phi(\mathcal
T(x))\, d\rho_1(x)
$$
for all $\phi\in \mathcal{C}_b(\R^d)$. Given a probability measure with
bounded $p$-th moment $\rho_0$, consider two measurable mappings
$X_1,X_2 : \bbr^d \to \bbr^d$, then the following inequality
holds.
\[
d_p^p(X_1 \# \rho_0, X_2 \# \rho_0) \leq \int_{\bbr^d \times
\bbr^d} |x-y|^p d\gamma(x,y) = \int_{\bbr^d} | X_1(x) - X_2(x)|^p
d\rho_0(x).
\]
Here, we used as transference plan $\gamma = (X_1 \times
X_2)\#\rho_0$ in Definition \ref{defdp}. \newline

\noindent (ii) Given $\{\rho_k\}_{k=1}^{N}$ and $\rho$ in $\pp_1(\R^d)$, the followings are equivalent:
\begin{itemize}
\item $d_1(\rho_k,\rho) \to 0$ as $k \to +\infty$.
\item $\rho_k$ converges to $\rho$ weakly-* as measures and
\[
\int_{\R^d}|\xi|\rho_k(\xi) d\xi \to \int_{\R^d} |\xi| \rho(\xi) d\xi, \quad \mbox{as} \quad n \to +\infty.
\]
\end{itemize}
\end{proposition}
Finally, we recall {\it a priori} energy estimates of kinetic Cucker-Smale model.
\begin{lemma}\label{k-energy-est}
Let $f$ be any smooth solutions to the system \eqref{k-CS-si}. Then we have
\begin{align*}
\begin{aligned}
&(i) \,\,\,\, \frac{d}{dt} \int_{\bbr^d \times \bbr^d} f dx dv = 0, \quad \frac{d}{dt} \int_{\bbr^d \times \bbr^d} v f dx dv = 0, \cr
&(ii) \,\, \frac{1}{2}\frac{d}{dt} \int_{\bbr^d \times \bbr^d} |v|^2 f dx dv = -\frac{1}{2}\int_{\bbr^{2d} \times \bbr^{2d}} \psi^i(x-y) | v - w|^{\beta_i} f(x,v,t) f(y,w,t) dxdydv dw.
\end{aligned}
\end{align*}
\end{lemma}

\subsection{Main results}
In this part, we introduce the notion of weak solution and stability in our frameworks, and state our main results on well-posedness.

We first present two frameworks depending on singularity of communication weight and strength of velocity coupling.
\begin{itemize}
\item {\it Framework A (Singular communication weight and high strength of velocity coupling)}: For the force term $F_1$, initial data $f^0$ has a compact support in velocity belongs to $L^p$ for some $p$. The exponents $\alpha$ and $\beta_1$, appearing in the definition of $F_1$, and $p$ should satisfy
\begin{equation*}
(\alpha + 1)p' < d \quad \mbox{and} \quad \beta_1 \geq 2, \quad \mbox{respectively}.
\end{equation*}
where $p'$ is a conjugate of $p$, i.e., $p' := p/(p - 1)$.
\item {\it Framework B (Low strength of velocity coupling)}: For the force term $F_2$, initial data $f^0$ has a compact support in position and velocity, belongs to $L^p$ for some $p$, and the exponent $\beta_2$ of nonlinear velocity coupling satisfies $\beta_2 \in \left(\frac{3-d}{2}, 2\right)$. Here the exponent $\beta_2$ and $p$ should satisfy
\[
(3-2\beta_2)p' < d \,\, \mbox{ for } \,\, \beta_2 \in \left( \frac{3-d}{2}, 1\right) \quad \mbox{and} \quad (2-\beta_2)p' < d \,\, \mbox{ for } \,\, \beta_2 \in (1,2).
\]
\end{itemize}

\begin{definition}\label{def-ext} For a given $T \in (0,\infty)$, $f$ is a weak solution of \eqref{k-CS-si} on the time-interval $[0,T)$ if and only if the following condition are satisfied:
\begin{enumerate}
\item
$ f\in L^{\infty}(0,T; (L_+^1 \cap L^p)(\R^d \times\R^d)) \cap \mathcal{C}([0,T],\pp_1(\R^d \times \R^d)), $
\item For all $\Psi \in \mathcal{C}_c^{\infty}(\R^d \times \R^d \times [0,T])$,
\begin{align*}
\begin{aligned}
&\int_{\R^d \times \R^d}f(x,v,T)\Psi(x,v,T) dx dv - \int_0^T \int_{\R^d \times \R^d}
f\big(\partial_t\Psi + \nabla_x \Psi \cdot v
+ \nabla_v \Psi \cdot F_i(f) \big)dxdv dt
 \cr
&\qquad \qquad \qquad  = \int_{\R^d\times\R^d}
 f^0(x,v) \Psi^0(x,v)dxdv.
\end{aligned}
\end{align*}
\end{enumerate}
\end{definition}

We now state our main result on the local existence of a unique weak solution.

\begin{theorem}\label{ma-thm}
Suppose that either Framework A or Framework B hold, and the initial data $f^0$ satisfies
\begin{equation}\label{ini-condi}
f^0 \in (L_+^1 \cap L^p)(\R^d \times \R^d) \cap \pp_1(\R^d \times \R^d).
\end{equation}
Then there exist $T>0$ and a unique weak solution $f$ in the sense of Definition {\rm\ref{def-ext}} on the time interval $[0,T]$. Furthermore, if $f_i,i=1,2$ are two such solutions to \eqref{k-CS-si}, then we have the following $d_1$-stability estimate.
    \[
    \frac{d}{dt}d_1(f_1(t),f_2(t)) \leq Cd_1(f_1(t),f_2(t)), \quad \mbox{for} \quad t \in [0,T],
    \]
where $C$ is a positive constant.
\end{theorem}

\section{Local well-posedness of the generalized Cucker-Smale model}\label{sec-local-well}
In this section, we provide a detailed proof of Theorem \ref{ma-thm} in the Framework $A$. Since the arguments for the Framework $B$ are similar to this, we will give a sketch proof for it in the last part of this section. We also notice that it is enough to show Theorem \ref{ma-thm} in the Framework $A$ when $\beta_1=2$ due to the estimate of compact support of $f$ in velocity (see Lemma \ref{lem-gro-vel}).

\subsection{A regularized model}
In this part, we will consider a regularized model.
For this, we first introduce a standard mollifier $\theta$:
\[
\theta(x) = \theta(-x) \geq 0, \quad \theta \in \mathcal{C}_0^{\infty}(\R^d), \quad \mbox{supp }\theta \subset B(0,1), \quad \int_{\R^d} \theta(x) dx = 1,
\]
and we set a sequence of smooth mollifiers:
\[
\theta_{\e}(x) := \frac{1}{\e^d}\theta \left( \frac{x}{\e}\right).
\]
Here $B(0,1):=\{x \in \R^d : |x| \leq 1\}$. Then we regularize $\psi^1$ such as $\psi^1_\e := \psi^1 * \theta_\e$. Since $\psi^1_\e \in \mathcal{C}^{\infty}(\R^d)$, we deduce from well-posedness theories in \cite{CCR,HL,HT} that there exists a unique global solution $f_\e$ which has compact support in kinetic velocity to the following equations.
\begin{equation}\label{reg-k-CS-si}
\left\{ \begin{array}{ll}
\partial_t f_\e + v \cdot \nabla_x f_\e
 + \nabla_v \cdot \big[F_1(f_\e)f_\e\big] = 0, & \qquad (x, v) \in \bbr^d \times \bbr^d,~~t > 0,\\[2mm]
\displaystyle F(f_\e)(x,v,t) := \int_{\R^d \times \R^d} \psi^1_\e(x-y)(w - v)f_\e(y,w,t)dydw, & \qquad (x, v) \in \bbr^d \times \bbr^d,~~t > 0,\\[4mm]
f_\e(x,v,0) =: f^0(x,v), & \qquad (x,v) \in \R^d \times \R^d.
\end{array} \right.
\end{equation}
For the solution $f_\e$ to the system \eqref{reg-k-CS-si}, we will show the uniform $L^p$-bound of $f_\e$ in $\e$. For this, we first need to estimate the growth of the kinetic velocity. Consider the forward bi-characteristics $Z_\e(s) := \left( X_\e(s;0,x,v), V_\e(s;0,x,v)\right)$ satisfying the following ODE system:
\begin{align}\label{traj-vel-est}
\begin{aligned}
\frac{dX_\e(s)}{ds} &= V_\e(s),\cr
\frac{dV_\e(s)}{ds} &= \int_{\R^d \times \R^d} \psi^1_\e\left( X_\e(s) - y\right)(w - V_\e(s))f_\e(y,w,s) dy dw.
\end{aligned}
\end{align}
Set $\Omega_\e(t)$ and $R_\e^v(t)$ the $v$-projection of compact supp$f(\cdot,t)$ and maximum value of $v$ in $\Omega_\e(t)$, respectively:
\begin{equation*}
\Omega_\e(t) := \overline{\{ v \in \R^d : \exists (x,v) \in \R^d \times \R^d \mbox{ such that } f_\e(x,v,t) \neq 0 \}}, \quad R_\e^v(t):= \max_{v \in \Omega_\e(t)}|v|.
\end{equation*}
Then we have the following growth estimate for support of $f_\e$ in velocity.
\begin{lemma}\label{lem-gro-vel} Let $Z_\e(t)$ be the solution to the particle trajectory \eqref{traj-vel-est} issued from the compact supp$_{(x,v)}f^0$ at time $0$. Then we have
\[
R_\e^v(t) \leq R_\e^v(0) = R_0^v:= \max_{v\in \Omega(0)}|v|,
\]
i.e., the support of $f(x,v,t)$ in velocity is uniformly bounded by the one of $f^0(x,v)$.
\end{lemma}
\begin{proof} For the proof, we employ the same idea in \cite[Section 4]{CFRT}. We choose $V_\e(t)$ that make the value of $R_\e^v(t)$ such that $\frac{dR_\e^v(t)}{dt}$ is well-defined to obtain
\begin{align*}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}\left( R_\e^v(t) \right)^2 &= \frac{1}{2}\frac{d}{dt}\left|V_\e(t)\right|^2 = V_\e(t) \cdot \frac{d}{dt}V_\e(t) \cr
&= \int_{\R^d \times \R^d} \psi^1_\e\left(X_\e(t) - y\right)\left(w - V_\e(t)\right)\cdot V_\e(t)f_\e(y,w,t)dydw \cr
&\leq 0.
\end{aligned}
\end{align*}
Here we used the fact that for any $w \in \Omega_\e(t)$, $\left( w - V_\e(t) \right) \cdot V_\e(t) \leq 0.$
This completes the proof.
\end{proof}
\begin{remark}Set $\tilde{\Omega}_0 := B(0,R_0^v)$. Then it follows from Lemma \ref{lem-gro-vel} that $\Omega_\e(t) \subset \tilde{\Omega}_0$ for $t \geq 0$.
\end{remark}
We now show the $L^p$-estimate of $f_\e$ with the help of the estimate in Lemma \ref{lem-gro-vel}.
\begin{proposition}\label{prop-lp} Let $f_\e$ be the solution to the system \eqref{reg-k-CS-si}. Then there exists $T>0$ we have the uniform $L^1\cap L^p$-estimate of $f_\e$:
\[
\sup_{t \in [0,T]}\|f_\e\|_{L^1 \cap L^p} \leq C,
\]
where $C$ is a positive constant independent of $\e$.
\end{proposition}
\begin{proof} We first notice that the conservation of mass to the system \eqref{reg-k-CS-si}:
\[
\frac{d}{dt}\int_{\R^d \times \R^d} f_\e dx dv = 0.
\]
We next turn to $L^p$-estimate of $f_\e$. It is a straightforward to get
\[
\frac{d}{dt}\int_{\R^d \times \R^d} f_\e^p dx dv = -(p-1)\int_{\R^d \times \R^d} \left( \nabla_v \cdot (F_1(f_\e))\right) f_\e^p dx dv.
\]
For the estimate of $\|\nabla_v \cdot (F_1(f_\e))\|_{L^\infty}$, we use a cut-off function $\chi_1 \in \mathcal{C}_c^{\infty}(\R^d)$ defined by
\begin{equation*}
\chi_1(x) := \left\{
\begin{array}{ll}
1 & |x| \leq 1,\\[1mm]
0 & |x| >1.
\end{array} \right.
\end{equation*}
Then it follows from the assumption on the exponent $\alpha$  that
\[
\psi^1_\e(x) = \psi^1 * \theta_\e = (\psi^1(\chi_1 + (1 - \chi_1)))*\theta_\e = (\psi^1\chi_1)*\theta_\e + (\psi^1(1 - \chi_1))*\theta_\e,
\]
and
\[
\|(\psi^1 \chi_1)*\theta_\e\|_{L^{p'}} \leq \|\psi^1 \chi_1\|_{L^{p'}} \leq C, \quad \|(\psi^1(1 - \chi_1))*\theta_\e\|_{L^\infty} \leq \|\psi^1(1 - \chi_1)\|_{L^\infty} \leq 1.
\]
Thus we obtain
\begin{align*}
\begin{aligned}
|\nabla_v \cdot (F_1(f_\e))| &\leq d\int_{\R^d \times \R^d} |(\psi^1 \chi_1) * \theta_\e||f_\e| dy dw + d\int_{\R^d \times \R^d} |(\psi^1(1 - \chi_1))*\theta_\e||f_\e|dy dw \cr
&\leq C(R_0^v)^{\frac{1}{p'}}\|\psi^1 \chi_1\|_{L^{p'}}\|f_\e\|_{L^p} + \|\psi^1(1 - \chi_1)\|_{L^\infty}\|f_\e\|_{L^1}\cr
&\leq C\|f_\e\|_{L^1 \cap L^p},
\end{aligned}
\end{align*}
where $C$ is a positive constant independent of $\e$. Hence we have
\[
\frac{d}{dt}\|f_\e\|_{L^1 \cap L^p} \leq Cd\left(1 - \frac{1}{p}\right) \|f_\e\|^2_{L^1 \cap L^p},
\]
and this yields that there exists $T > 0$,
\[
\sup_{t \in [0,T]}\|f_\e\|_{L^1 \cap L^p} \leq C,
\]
where $C$ is a positive constant independent of $\e$.
\end{proof}

\begin{remark}\label{rmk-1-mom} 1. It is easy to find the estimate of first moments of $f_\e$. In fact, it directly follows from \eqref{reg-k-CS-si} that
\begin{align*}
\begin{aligned}
\frac{d}{dt}\int_{\R^d \times \R^d} |v|f_\e dx dv &= \int_{\R^d \times \R^d} \frac{v}{|v|} \cdot F_1(f_\e)f_\e dx dv \cr
& \leq \int_{\R^{2d} \times \R^{2d}} \psi^1_\e(x-y)|w|f_\e(x,v)f_\e
(y,w) dx dv dy dw \cr
&\qquad- \int_{\R^{2d} \times \R^{2d}} \psi^1_\e(x-y)|v|f_\e(x,v)f_\e
(y,w) dx dv dy dw \cr
&= 0,
\end{aligned}
\end{align*}
where we used $\psi^1_\e(x) = \psi^1_\e(-x)$, and interchange of variables $(x,v) \leftrightarrow (y,w)$. This yields
\begin{equation}\label{fir-mom-est}
\| vf_\e \|_{L^{\infty}(0,T;L^1)} \leq \| vf^0 \|_{L^1}.
\end{equation}
Since
\[
\frac{d}{dt}\int_{\R^d \times \R^d}|x| f_\e dx dv \leq \int_{\R^d \times \R^d}|v| f_\e dx dv,
\]
we deduce from \eqref{fir-mom-est} that
\[
\| xf_\e \|_{L^{\infty}(0,T;L^1)} \leq \| xf^0 \|_{L^1} + T\| vf^0 \|_{L^1}.
\]
2. It follows from the definition of $\psi^1_\e$ that
\begin{align}\label{j-1-est-1}
\begin{aligned}
\psi^1_\e(x) &= \int_{\R^d} \frac{1}{|x-y|^{\alpha}}\theta_\e(y)dy\cr
&\leq \int_{\left\{y: |y| < \frac{|x|}{2}\right\}}
\frac{\theta_{\e}(y)}{|x-y|^{\alpha}}  dy +
\int_{\left\{y: |y| \geq \frac{|x|}{2}\right\}} \frac{
\theta_{\e}(y)}{|x-y|^{\alpha}}  dy \cr &\leq
\frac{2^{\alpha}\e}{|x|^{\alpha}}\int_{\R^d}\theta_{\e}(y)dy
+ \mathbf{1}_{\{|x| \leq 2\e\}}\int_{\left\{y:~ \e \geq |y|
\right\}}\frac{\theta_{\e}(y)}{|x-y|^{\alpha}}dy \cr &\leq
\frac{C}{|x|^{\alpha}} +
\frac{C\e^{\alpha}}{|x|^{\alpha}}\int_{\left\{ y:~ \e \geq
|y| \right\}}\frac{\theta_{\e}(y)}{|x-y|^{\alpha}} dy \leq
\frac{C}{|x|^{\alpha}}.
\end{aligned}
\end{align}
Thus we obtain
\begin{equation}\label{key-ineq}
|\psi^1_\e(x) - \psi^1_\e(y)| \leq \frac{C|x-y|}{\min(|x|,|y|)^{1+\alpha}},
\end{equation}
where $C$ is independent of $\e$.
\end{remark}
We now show the growth estimate of $d_1(f_\e(t),f_{\e'}(t))$.
\begin{proposition}\label{prop-grow} Let $f_\e$ and $f_{e'}$ be two solutions to the system \eqref{reg-k-CS-si}. Then we have
\[
\frac{d}{dt}d_1(f_\e(t),f_{\e'}(t)) \leq C(d_1(f_\e(t),f_{\e'}(t)) + \e + \e'),
\]
for $\e, \e' > 0$. Here $C$ is independent of $\e$ and $\e'$.
\end{proposition}
\begin{proof}We first define flows $Z_\e := (X_\e,V_\e), Z_{\e'}:= (X_{\e'},V_{\e'}) : \R_+ \times \R_+ \times \R^d \times \R^d \to \R^d \times \R^d$ generated from \eqref{reg-k-CS-si} satisfying
\begin{equation}\label{traj}
\left\{ \begin{array}{ll}
\displaystyle \frac{d}{dt} X_\e(t;s,x,v) = V_\e(t;s,x,v), &\\[4mm]
\displaystyle \frac{d}{dt} V_\e(t;s,x,v) = F_1(f_\e)(Z_\e(t;s,x,v),t), &
\\[4mm]
(X_\e(s;s,x,v),V_\e(s;s,x,v)) = (x,v), &
\end{array} \right.
\end{equation}
and
\begin{equation}\label{traj2}
\left\{ \begin{array}{ll}
\displaystyle \frac{d}{dt}X_{\e'}(t;s,x,v) = V_{\e'}(t;s,x,v), &\\[4mm]
\displaystyle \frac{d}{dt} V_{\e'}(t;s,x,v) = F_1(f_{\e'})(Z_{\e'}(t;s,x,v),t), &
\\[4mm]
(X_{\e'}(s;s,x,v),V_{\e'}(s;s,x,v)) = (x,v), &
\end{array} \right.
\end{equation}
for all $s,t \in [0,T]$. Since $\psi^1_{\e}, \psi^1_{\e'} \in \mathcal{C}^{\infty}$, \eqref{traj} and \eqref{traj2} are well-defined for $s,t \in [0,T]$. We now choose an optimal transport map $\mt^0 = (\mt_1^0(x), \mt_2^0(v))$ between $f_\e(t_0)$ and $f_{\e'}(t_0)$ for fixed $t_0 \in [0,T)$, i.e., $f_{\e'}(t_0) = \mt^0 \# f_\e(t_0)$. It is known from \cite{CPJ} that such an optimal transport map exists when $f_\e(t_0)$ is absolutely continuous with respect to the Lebesgue measure. Then we apply the similar argument in \cite[Lemma 5.5]{HL} to obtain $f_\e(t) = Z_\e(t;t_0,\cdot,\cdot)\#f_\e(t_0)$ and $f_{\e'}(t) = Z_\e(t;t_0,\cdot,\cdot)\#f_{\e'}(t_0)$ using the mass transportation (not necessarily optimal) notation of push-forward. More precisely, we obtain that for any $g \in \mathcal{C}_c^1(\R^d \times \R^d \times [0,T])$,
\begin{align}\label{push-f}
\begin{aligned}
&\int_{\R^d \times \R^d} g(x,v,t)f_\e(x,v,t) dx dv - \int_{\R^d \times \R^d} g(x,v,t_0)f_\e(x,v,t_0) dx dv \cr
& \qquad = \int_{t_0}^t \int_{\R^d \times \R^d} \left( \partial_s g + v\cdot\nabla_x g + F_1(f_\e) \cdot \nabla_v g\right)f_\e(x,v,s)dxdvds.
\end{aligned}
\end{align}
We now choose
\[
g(x,v,t) := h(X_\e(s;t,x,v),V_\e(s;t,x,v)), \quad \mbox{for fixed } t,
\]
where $h \in \mathcal{C}_c^1(\R^d \times \R^d)$. This makes the r.h.s of \eqref{push-f} vanished and
\begin{equation}\label{push-f-2}
\int_{\R^d \times \R^d} h(x,v)f_\e(x,v,t) dx dv = \int_{\R^d \times \R^d}h(X_\e(t_0;t,x,v),V_\e(t_0;t,x,v))f_\e(x,v,t_0)dxdv.
\end{equation}
Thus we conclude $f_\e(t) = Z_\e(t;t_0,\cdot,\cdot) \#f_\e(t_0)$. Same argument can be applied to get $f_{\e'}(t) = Z_\e(t;t_0,\cdot,\cdot)\#f_{\e'}(t_0)$. We also notice that
\[
\mt^t \# f_\e(t) = f_{\e'}(t), \quad \mbox{where} \quad \mt^t = Z_{\e'}(t;t_0,\cdot,\cdot) \circ \mt^0 \circ Z_\e(t_0;t,\cdot,\cdot).
\]
By Definition \ref{defdp}, when $p=1$, we obtain
\[
d_1(f_\e(t),f_{\e'}(t)) \leq \int_{\R^d \times \R^d} |Z_\e(t;t_0,x,v) - Z_{\e'}(t;t_0,\mt^0(x,v))|f_\e(x,v,t_0) dx dv.
\]
Set
\[
Q_{\e,\e'}(t) := \int_{\R^d \times \R^d} |Z_\e(t;t_0,x,v) - Z_{\e'}(t;t_0,\mt^0(x,v))|f_\e(x,v,t_0) dx dv.
\]
Then straightforward computations yield
\begin{align*}
\begin{aligned}
&\frac{d}{dt}Q_{\e,\e'}(t) \Big|_{t = t_0+} \cr
&\leq \int_{\R^d \times \R^d} |V_\e(t;t_0,x,v) - V_{\e'}(t;t_0,\mt^0(x,v))|f_\e(x,v,t_0)dxdv \Big|_{t = t_0+} \cr
& + \int_{\R^d \times \R^d} \left| F_1(f_\e)(Z_\e(t;t_0,x,v),t) -F_1(f_{\e'})(Z_{\e'}(t;t_0,\mt^0(x,v)),t) \right|f_\e(x,v,t_0)dx dv \bigg|_{t = t_0+} \cr
&=: \mathcal{I} + \mathcal{J}.
\end{aligned}
\end{align*}
For the estimate of $\mathcal{I}$, it is easy to find
\begin{equation}\label{i-est}
\mathcal{I} = \int_{\R^d \times \R^d} |v - \mt^0_2(v)|f_\e(x,v,t_0) dx dv \leq C d_1(f_\e(t_0),f_{\e'}(t_0)).
\end{equation}
For the estimate of $\mathcal{J}$, we notice that
\begin{align*}
\begin{aligned}
\mathcal{J} &= \int_{\R^d \times \R^d} \bigg| \int_{\R^d \times \R^d} \psi^1_\e(x - y)(w - v)f_\e(y,w,t_0)dy dw \cr
&\quad -\int_{\R^d \times \R^d} \psi^1_{\e'}(\mt^0_1(x)- y)(w - \mt^0_2(v))f_{\e'}
(y,w,t_0)dy dw \bigg|f_\e(x,v,t_0)dx dv \cr
&= \int_{\R^d \times \R^d} \bigg| \int_{\R^d \times \R^d} \psi^1_\e(x - y)(w - v)f_\e(y,w,t_0)dy dw \cr
&\quad -\int_{\R^d \times \R^d} \psi^1_{\e'}(\mt^0_1(x)- \mt^0_1(y))(\mt^0_2(w) - \mt^0_2(v))f_{\e}
(y,w,t_0)dy dw \bigg|f_\e(x,v,t_0)dx dv.
\end{aligned}
\end{align*}
For notational simplicity, we omit the time dependency on $t_0$ in the rest of computations. We decompose $\mathcal{J}$ into two parts:
\[
\mathcal{J} = \int_{\R^d \times \R^d} | \mathcal{J}_1 + \mathcal{J}_2 | f_\e(x,v) dx dv,
\]
where
\begin{align*}
\begin{aligned}
\mathcal{J}_1 &:= \int_{\R^d \times \R^d} \left(\psi^1_\e(x-y) - \psi^1_{\e'}\left(\mt^0_1(x)-\mt^0_1(y)\right)\right)(w-v)f_\e(y,w)dydw,\cr
\mathcal{J}_2 &:= \int_{\R^d \times \R^d} \psi^1_{\e'}\left(\mt^0_1(x) - \mt^0_1(y)\right)\left( (w-v) - \left(\mt^0_2(w) - \mt^0_2(v)\right)\right)f_\e(y,w)dydw.
\end{aligned}
\end{align*}
For the estimates of $\mathcal{J}$, we divide it into two steps to make the reading easier.

\begin{itemize}
\item In Step A, we show
\begin{equation}\label{claim}
\int_{\R^d \times \R^d}|\mathcal{J}_1|f_\e dxdv \leq C\max(\|f_\e\|, \|f_{\e'}\|) d_1(f_\e(t_0), f_{\e'}(t_0)) + C \|f_\e\|^2(\e + \e'),
\end{equation}
where $C$ is independent of $\e$. \newline
\item In Step B, we show
\[
\int_{\R^d \times \R^d}|\mathcal{J}_2|f_\e dxdv \leq C\|f_{\e}\|\|f_{\e'}\|d_1(f_\e(t_0),f_{\e'}(t_0)).
\]
\end{itemize}

\textbf{Step A}: By adding and subtracting, we find that
\begin{align}\label{j-1-est-2}
\begin{aligned}
\mathcal{J}_1 &\leq \int_{\R^d \times \R^d} \left|(\psi^1_\e - \psi^1_{\e'})(x-y)\right||w-v|f_\e(y,w)dydw \cr
&\quad + \int_{\R^d \times \R^d} \left|\psi^1_{\e'}(x-y) - \psi^1_{\e'}\left(\mt^0_1(x)-\mt^0_1(y)\right)\right||w-v|f_\e(y,w)dydw.
\end{aligned}
\end{align}
It follows from a similar estimate to \eqref{j-1-est-1} that
\begin{align}\label{j-1-est-3}
\begin{aligned}
|\psi^1_\e(x) - \psi^1(x)| &\leq \int_{\R^d} |\psi^1(x-y) - \psi^1(x)|\theta_\e(y) dt\cr
&\leq 2\int_{\R^d} \left( \frac{1}{|x|^{1+\alpha}} + \frac{1}{|x-y|^{1+\alpha}}\right) |y|\theta_\e(y)dy \cr
&\leq 2\e\int_{\left\{ y:~ \e \geq
|y| \right\}} \left( \frac{1}{|x|^{1+\alpha}} + \frac{1}{|x-y|^{1+\alpha}}\right)\theta_\e(y)dy \cr
&\leq \frac{C\e}{|x|^{1+\alpha}}.
\end{aligned}
\end{align}
Then we use \eqref{j-1-est-3} to obtain
\begin{align}
&\int_{\R^{2d} \times \R^{2d}} \left|(\psi^1_\e - \psi^1)(x-y)\right||w-v|f_\e(y,w)f_\e(x,v)dxdvdydw \cr
& \quad \leq 2R_0^v\int_{\R^{2d} \times \tilde{\Omega}_0^2} \left|(\psi^1_\e - \psi^1)(x-y)\right|f_\e(y,w)f_\e(x,v)dxdvdydw\cr
& \quad \leq C\e\int_{\R^{2d} \times \tilde{\Omega}_0^2} \frac{1}{|x-y|^{1+\alpha}}f_\e(y,w)f_\e(x,v)dxdvdydw \cr
&\quad \leq C\e\int_{\R^d \times \R^d} \left( \int_{\{ y : |x-y| < 1\} \times \tilde{\Omega}_0} + \int_{\{ y : |x-y| \geq 1\} \times \tilde{\Omega}_0} \frac{1}{|x-y|^{1+\alpha}}f_\e(y,w) dydw\right)f_\e(x,v) dxdv\cr
&\quad \leq C\e\int_{\R^d \times \R^d} \left(\left(\int_{\{ y : |x-y|\leq 1\}}\frac{1}{|x-y|^{(1+\alpha)p'}}dy\right)^{\frac{1}{p'}}\|f_\e\|_{L^p} +\|f_\e\|_{L^1}\right)f_\e(x,v) dxdv\cr
&\quad\leq C\e\|f_\e\|^2 \leq C\e.\label{j-1-est-4}
\end{align}
Similarly, we get
\[
\int_{\R^{2d} \times \R^{2d}} \left|(\psi^1_{\e'} - \psi^1)(x-y)\right||w-v|f_\e(y,w)f_\e(x,v)dxdvdydw \leq C\e'.
\]
Thus we have
\begin{equation}\label{j-1-est-5}
\int_{\R^{2d} \times \R^{2d}} \left|(\psi^1_\e - \psi^1_{\e'})(x-y)\right||w-v|f_\e(y,w)f_\e(x,v)dxdvdydw \leq C(\e+\e').
\end{equation}
For a related to second term in the rhs of \eqref{j-1-est-2}, we employ \eqref{key-ineq} and interchange the variables $(x,v) \leftrightarrow (y,w)$ to find
\begin{align*}
\begin{aligned}
&\int_{\R^{2d} \times \tilde{\Omega}_0^2} \left|\psi^1_{\e'}(x-y) - \psi^1_{\e'}\left(\mt^0_1(x)-\mt^0_1(y)\right)\right||w-v|f_\e(x,v)f_\e(y,w)dxdvdydw\cr
&\quad \leq 2R_0^v\int_{\R^{2d} \times \tilde{\Omega}_0^2} \left( \frac{|\mt^0_1(x) - x|}{|\mt^0_1(x) - \mt^0_1(y)|^{1+\alpha}} + \frac{|\mt^0_1(x) - x|}{|x-y|^{1+\alpha}} \right)f_\e(x,v)f_\e(y,w)dxdvdydw \cr
&=:\mathcal{K}_1 + \mathcal{K}_2.
\end{aligned}
\end{align*}
By direct computations, we get
\begin{align*}
\begin{aligned}
\mathcal{K}_1 &= 2R_0^v \int_{\R^d \times \R^d} |\mt^0_1(x) - x|f_\e(x,v) \left( \int_{\R^d \times \Omega_0} \frac{1}{|\mt^0_1(x) - y|^{1+\alpha}}f_{\e'}(y,w) dydw\right) dxdv \cr
&\leq C\|f_{\e'}\|\int_{\R^d \times \R^d} |\mt^0_1(x) - x|f_\e(x,v) dxdv \leq C\|f_{\e'}\|d_1(f_\e(t_0),f_{\e'}(t_0)),
\end{aligned}
\end{align*}
where we used the same estimates in \eqref{j-1-est-4} to obtain
\[
\int_{\R^d \times \tilde{\Omega}_0} \frac{1}{|\mt^0_1(x) - y|^{1+\alpha}}f_{\e'}(y,w) dydw \leq C\|f_{\e'}\|.
\]
Similarly, we also obtain $\mathcal{K}_2 \leq C\|f_{\e}\|d_1(f_\e(t_0),f_{\e'}(t_0))$. This yields
\begin{align}\label{j-1-est-6}
\begin{aligned}
&\int_{\R^{2d} \times \R^{2d}} \left|\psi^1_{\e'}(x-y) - \psi^1_{\e'}\left(\mt^0_1(x)-\mt^0_1(y)\right)\right||w-v|f_\e(x,v)f_\e(y,w)dxdvdydw\cr
& \quad \qquad \leq C\max(\|f_\e\|, \|f_{\e'}\|) d_1(f_\e(t_0),f_{\e'}(t_0)).
\end{aligned}
\end{align}
We now combine \eqref{j-1-est-5} and \eqref{j-1-est-6} to conclude our desired claim. \newline

\textbf{Step B}: For the estimate of $\mathcal{J}_2$, we obtain
\begin{align*}
\begin{aligned}
\mathcal{J}_2 &\leq \int_{\R^{2d} \times \R^{2d}} \left|\psi^1_{\e'}\left( \mt^0_1(x) - \mt^0_1(y)\right)\right| \left| w - \mt^0_2(w)\right|f_\e(x,v) f_\e(y,w) dx dy dv dw \cr
&  \quad + \int_{\R^{2d} \times \R^{2d}} \left|\psi^1_{\e'}\left( \mt^0_1(x) - \mt^0_1(y)\right)\right| \left| v - \mt^0_2(v)\right| f_\e(x,v) f_\e(y,w) dx dy dv dw \cr
&=: \mathcal{J}_2^1 + \mathcal{J}_2^2.
\end{aligned}
\end{align*}
On the other hand, we can easily find that
\begin{align*}
\begin{aligned}
\mathcal{J}_2^1 &= \int_{\R^d \times \R^d} \left( \int_{\R^d \times \Omega_0 }\left|\psi^1_{\e'}\left( \mt^0_1(x) - \mt^0_1(y)\right)\right| f_\e(x,v) dx dv \right) \left|w - \mt^0_2(w)\right| f_\e(y,w)dydw \cr
&\leq C\|f_{\e'}\|\int_{\R^d \times \R^d}|w - \mt^0_2(w)|f_\e(y,w)dydw \leq C\|f_{\e'}\|d_1(f_\e(t_0),f_{\e'}(t_0)),
\end{aligned}
\end{align*}
where we used the estimates in \eqref{j-1-est-4} again. Similarly, we get
\[
\mathcal{J}_2^2 \leq C\|f_{\e'}\|d_1(f_\e(t_0),f_{\e'}(t_0)),
\]
and this deduces
\begin{equation}\label{j-2-est-1}
\mathcal{J}_2 \leq C\|f_{\e'}\|d_1(f_\e(t_0),f_{\e'}(t_0)).
\end{equation}
Thus we have
\[
\int_{\R^d \times \R^d}|\mathcal{J}_2|f_\e dxdv \leq C\|f_{\e}\|\|f_{\e'}\|d_1(f_\e(t_0),f_{\e'}(t_0)).
\]
We now combine \eqref{i-est}, \eqref{claim} and \eqref{j-2-est-1} to find
\[
\frac{d}{dt}Q_{\e,\e'}(t) \Big|_{t = t_0+} \leq  C(d_1(f_\e(t_0), f_{\e'}(t_0)) + \e + \e').
\]
We finally write the integral form, dividing $t - t_0$, and taking the limit $t \to t_0^+$ to conclude
\[
\frac{d}{dt}d_1(f_\e(t),f_{\e'}(t))\Big|_{t = t_0^+} \leq C(d_1(f_\e(t_0), f_{\e'}(t_0) + \e + \e').
\]
Since $t_0$ is arbitrary in $[0,T)$, this yields
\begin{equation*}
\frac{d}{dt}d_1(f_\e(t),f_{\e'}(t)) \leq C(d_1(f_\e(t_0), f_{\e'}(t_0) + \e + \e'),
\end{equation*}
where $C$ is independent of $\e$ and $\e'$.
\end{proof}
\subsection{Existence and uniqueness of weak solutions (limit as $\e \to 0$)}\label{sec3.2} It follows from Proposition \ref{prop-grow} that $\{f_\e\}_{\e >0}$ is a Cauchy sequence in $\mathcal{C}([0,T];\pp_1(\R^d \times \R^d))$, and this implies that there exists a limit curve of measure $f \in \mathcal{C}([0,T];\pp_1(\R^d \times \R^d))$, and $f \in L^{\infty}(0,T;(L_+^1 \cap L^p)(\R^d \times \R^d))$. Thus it only remains to show that $f$ is a solution of the Cucker-Smale model \eqref{k-CS-si}. Choose a test function $\Psi(x,v,t) \in \mathcal{C}_c^{\infty}(\R^d \times \R^d \times [0,T])$, then $f_\e$ satisfies
\begin{align}\label{weak-var}
\begin{aligned}
&\int_{\R^d \times \R^d} \Psi^0(x,v)f^0(x,v) dx dv \cr
& \quad = \int_{\R^d \times \R^d} \Psi(x,v,T)f_\e(x,v,T)dx dv + \int_0^T\int_{\R^d \times \R^d} f_\e(x,v,t)\partial_t\Psi(x,v,t)dxdvdt \cr
& \qquad -\int_0^T\int_{\R^d \times \R^d} (\nabla_x \Psi) \cdot v f_\e dx dvdt - \int_0^T\int_{\R^d \times \R^d} (\nabla_v \Psi) \cdot F_1(f_\e)f_\e dx dv dt.
\end{aligned}
\end{align}
We can easily show that the first, second, and third terms in the rhs of \eqref{weak-var} converge to
\begin{align*}
\begin{aligned}
&\int_{\R^d \times \R^d} \Psi(x,v,T)f(x,v,T)dx dv + \int_0^T\int_{\R^d \times \R^d} f(x,v,t)\partial_t\Psi(x,v,t)dxdvdt \cr
& \qquad -\int_0^T\int_{\R^d \times \R^d} (\nabla_x \Psi) \cdot v f dx dvdt \quad \mbox{as} \quad \e \to 0,
\end{aligned}
\end{align*}
since $f_\e \to f$ in $\mathcal{C}([0,T],\pp_1(\R^d \times \R^d))$. We also notice that
\begin{align*}
\begin{aligned}
&\left| \int_0^T \int_{\R^{2d} \times \R^{2d}} (\psi^1_\e - \psi^1)(x-y)(\nabla_v \Psi) \cdot (w-v)f_\e(x,v)f_\e(y,w) dxdvdydwdt \right| \cr
&\leq C\e R_0^v\|\nabla_v \Psi\|_{L^{\infty}(0,T;L^{\infty}(\R^d \times \R^d))}\int_0^T\int_{\R^{2d} \times \tilde{\Omega}_0^2} \frac{1}{|x-y|^{1+\alpha}}f_\e(x,v)f_\e(y,w) dxdvdydwdt \cr
& \leq C\e\|f_\e\|^2 \leq C\e \to 0 \quad \mbox{as} \quad \e \to 0\,,
\end{aligned}
\end{align*}
where we used the decomposition in local and far fields as in \eqref{j-1-est-4}.
Thus in order to obtain
\[
\int_0^T\int_{\R^d \times \R^d} (\nabla_v \Psi) \cdot F_1(f_\e)f_\e dx dv dt \to \int_0^T\int_{\R^d \times \R^d} (\nabla_v \Psi) \cdot F_1(f)f dx dv dt,
\]
it only remains to show
\begin{align}\label{ext-est-1}
\begin{aligned}
&\int_0^T \int_{\R^{2d} \times \R^{2d}} \psi^1(x-y) (\nabla_v \Psi) \cdot (w-v)f_\e(x,v)f_\e(y,w) dxdvdydwdt \cr
& \qquad \quad \to \int_0^T \int_{\R^{2d} \times \R^{2d}} \psi^1(x-y) (\nabla_v \Psi) \cdot (w-v)f(x,v)f(y,w) dxdvdydwdt,
\end{aligned}
\end{align}
as $\e \to 0$. For this, we introduce a cut-off function
$\chi_{\delta} \in \mathcal{C}_c^{\infty}(\R^d)$ such that
\[
\chi_{\delta}(x) = \left\{  \begin{array}{ll}
1 & \mbox{if} \quad |x|\leq \delta \\
0 & \mbox{if} \quad |x|\geq 2\delta
\end{array} \right.\,.
\]
Then since $(1 - \chi_{\delta}(x-y))\psi^1(x-y)(w-v)\cdot \nabla_v \Psi$ is a Lipschitz function and $f_\e \to f$ in $\mathcal{C}([0,T],\pp_1(\R^d \times \R^d))$, we find
\begin{align}\label{ext-est-2}
\begin{aligned}
&\int_0^T \int_{\R^{2d} \times \R^{2d}} (1 - \chi_{\delta})\psi^1(x-y) (\nabla_v \Psi) \cdot (w-v)f_\e(x,v)f_\e(y,w) dxdvdydwdt \cr
& \qquad \quad \to \int_0^T \int_{\R^{2d} \times \R^{2d}}(1 - \chi_{\delta}) \psi^1(x-y) (\nabla_v \Psi) \cdot (w-v)f(x,v)f(y,w) dxdvdydwdt,
\end{aligned}
\end{align}
as $\e \to 0$ for any $\delta > 0$.
On the other hand, the remaining term is estimated as follows:
\begin{align}\label{ext-est-3}
\begin{aligned}
&\int_0^T \int_{\R^{2d} \times \R^{2d}} \chi_{\delta}(x-y)\psi^1(x-y) (\nabla_v \Psi) \cdot (w-v)f_\e(x,v)f_\e(y,w) dxdvdydwdt \cr
&\quad \leq C\delta\int_0^T\int_{\{(x,y)\in \R^d \times \R^d :~|x-y|\leq
2\delta\} \times \tilde{\Omega}_0^2} \frac{1}{|x-y|^{1+\alpha}}f_\e(x,v)f_\e(y,w) dxdvdydwdt \cr
&\quad \leq C\delta \to 0 \quad \mbox{as} \quad \delta \to 0,
\end{aligned}
\end{align}
and similarly, we also have
\begin{equation}\label{ext-est-4}
\int_0^T \int_{\R^{2d} \times \R^{2d}}\chi_{\delta}(x-y) \psi^1(x-y) (\nabla_v \Psi) \cdot (w-v)f(x,v)f(y,w) dxdvdydwdt \leq C\delta \to 0,
\end{equation}
as $\delta \to 0$ due to the fact that $f$ has a compact support in velocity. Hence we conclude the convergence \eqref{ext-est-1} combining \eqref{ext-est-2}, \eqref{ext-est-3}, and \eqref{ext-est-4}. Uniqueness of the weak solutions $f_\e$ is just followed from Proposition \ref{prop-grow}. More specifically, let $f_1, f_2 \in L^{\infty}(0,T;(L_+^1\cap L^p)(\R^d \times \R^d)) \cap \mathcal{C}([0,T],\pp_1(\R^d \times \R^d))$ be the weak solutions to the system \eqref{k-CS-si} with same initial data $f^0 \in (L_+^1 \cap L^p)(\R^d \times \R^d) \cap \pp_1(\R^d \times \R^d)$ satisfying the framework $A$. Then Proposition \ref{prop-grow} yields that
\[
\frac{d}{dt}d_1(f_1(t),f_2(t)) \leq C\max( \|f_1\|, \|f_2\|)d_1(f_1(t),f_2(t)), \quad \mbox{for} \quad t \in [0,T].
\]
This completes the proof in the framework A.
\begin{proposition} Let $f$ be a weak solutions to \eqref{k-CS-si} on the time-interval $[0,T)$ in the sense of Definition {\rm\ref{def-ext}}. Then $f$ is determined as the push-forward of the initial density through the flow map generated by $(v,F_i(f))$.
\end{proposition}
\begin{proof}
Cconsider the following flow map:
\begin{equation}\label{flow}
\left\{ \begin{array}{ll}
\displaystyle \frac{d}{dt}X(t;s,x,v) = V(t;s,x,v), & \\[2mm]
\displaystyle \frac{d}{dt}V(t;s,x,v) = F_1(f)(X(t;s,x,v),V(t;s,x,v),t), &\\[4mm]
(X(s;s,x,v),V(s;s,x,v)) = (x,v),&
\end{array} \right.
\end{equation}
for all $s,t\in[0,T]$. Then since $f$ has compact support in $v$, the flow map \eqref{flow} is well-defined using the same argument in the proof of Proposition \ref{prop-grow}. Moreover we can use the similar argument to \eqref{push-f-2} to have
\[
\int_{\R^d \times \R^d} h(x,v)f(x,v,t) dx dv = \int_{\R^d \times \R^d}h(X(0;t,x,v),V(0;t,x,v))f^0(x,v)dxdv,
\]
for $t \in [0,T]$. This yields that $f$ is determined as the push-forward of the initial density through the flow map \eqref{flow}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{ma-thm} in the framework $B$] Similarly, we first regularize the nonlinear velocity coupling as $\nabla \phi_\e(v) := \frac{v}{|v|^{2-\beta_2} + \e}$, and define the $f_\e$ by this regularized system. Then we easily find that the estimates of support of $f$ in position and velocity, and first momentums using the same arguments in Lemma \ref{lem-gro-vel} and Remark \ref{rmk-1-mom}. The remaining parts are obtained by similar arguments as in Section \ref{sec-local-well}.
\end{proof}

\section{Rich dynamics of the generalized Cucker-Smale particle system}
In this part, we investigate the dynamics of the generalized Cucker-Smale particle system.
We consider Cucker-Smale particle system:
\begin{align}\label{par-gene}
\begin{aligned}
\frac{dx_i(t)}{dt} &= v_i(t),\cr
\frac{dv_i(t)}{dt} &= \frac1N \sum_{j=1}^N \psi^k(|x_j(t) - x_i(t)|)\frac{v_j - v_i}{|v_j - v_i|^{2-\beta}}, \quad k = 1,2,
\end{aligned}
\end{align}
subject to $(x_i(0),v_i(0)) =: (x_i^0,v_i^0)$. Here $\psi^k,k=1,2$ are given in \eqref{condi:psi}. We assume further that $\psi^k(s),k=1,2$ satisfy $\alpha \geq 1$ and
\[
0 < \psi^2(s_1) \leq \psi^2(s_2) \quad \mbox{for} \quad 0 \leq s_2 \leq s_1 < \infty, \quad \mbox{and} \quad \psi^2(s) \to 0 \quad \mbox{as} \quad s \to \infty,
\]
respectively.

For the system \eqref{par-gene} with $\beta=2$, there are only two papers \cite{ACHL,Pes} on the existence theory and large time behaviour to our best knowledge. In \cite{ACHL}, the authors identified the initial configurations that prevent the pairwise collisions in a finite-time when the singularity of communication weight is strong enough such as $\alpha \geq 1$. One-dimensional discrete C-S model \eqref{par-gene} was treated in \cite{Pes}, and showed existence  of piecewise weak solutions when the singularity of communication weight is sufficiently weak, $\alpha < 1$.

Note that the original alignment force  of Cucker-Smale model satisfies the conditions of $\psi^2$. Without loss of generality, we may assume that
\[
\sum_{j=1}^Nv_i(0) = 0.
\]
We set
\[
\|x\|_\infty := \max_{1\leq i \leq N}|x_i| \quad \mbox{and} \quad \|v\|_\infty := \max_{1\leq i \leq N}|v_i|.
\]
We notice that we can choose an index $i$ such that $\|v(t)\|_\infty = |v_i(t)|$ at any time $t$. Then it follows from \cite{CCR2,HHK} that
\[
\frac{d}{dt}\|v(t)\|_\infty^2 \leq -C_0\psi^k(2\|x(t)\|_\infty)\|v(t)\|_\infty^\beta, \quad \mbox{for} \quad \beta \in (0,3).
\]
It is also clear to obtain $\left| \frac{d\|x\|_\infty}{dt}\right| \leq \|v\|_\infty$. We now take the similar argument in \cite{ACHL,HL}, and define two Lyanpunov type functionals $\mathcal{E}_\pm(x,v)$:
\[
\mathcal{E}_\pm(x(t),v(t)) := \frac{1}{3-\beta}\|v(t)\|_\infty^{3-\beta} \pm \frac{C_0}{2}\Psi^k(2\|x(t)\|_\infty),
\]
where $\Psi^k(\cdot)$ is a primitive of $\psi^k$.

We next present two lemmas that can be obtained using the similar argument in \cite{ACHL,HL}.
\begin{lemma} Let $(x,v)$ be any smooth solutions to the system \eqref{par-gene}.
Then we have
\begin{align*}
\begin{aligned}
&(i) \,\,\,\, \mathcal{E}_\pm(x(t),v(t)) \leq \mathcal{E}_\pm(x_0,v_0).\cr
&(ii) \,\, \|v(t)\|_\infty^{3-\beta} + \frac{(3-\beta)C_0}{2}\left|\int_{2\|x_0\|_\infty}^{2\|x(t)\|_\infty} \psi^k(s) ds\right| \leq \|v_0\|_\infty^{3-\beta}.
\end{aligned}
\end{align*}
\end{lemma}
\begin{lemma}\label{lem:key2} Let $(x,v)$ be any smooth solutions to the system \eqref{par-gene}. If the initial data $(x_0,v_0)$ satisfies
\begin{equation}\label{basic-ini}
\|x_0\|_\infty > 0, \quad \|v_0\|_\infty^{3-\beta} < \frac{(3-\beta)C_0}{2}\min \left\{ \int_0^{2\|x_0\|_\infty} \psi^k(s) ds, \int_{2\|x_0\|_\infty}^\infty \psi^k(s)ds\right\},
\end{equation}
then there exist positive constants $x_m,x_M > 0$ such that
\[
\|x(t)\|_\infty \in [x_m, x_M], \quad \frac{d}{dt}\|v(t)\|_\infty^2 \leq -C_0\psi^k(2x_M)\|v(t)\|_\infty^\beta,
\]
where $x_m$ and $x_M$ are defined by
\[
\|v_0\|_\infty^{3-\beta} = \frac{(3-\beta)C_0}{2}\int_{2x_m}^{\|2x_0\|_\infty} \psi^k(s) ds \quad \mbox{and} \quad \quad \|v_0\|_\infty^{3-\beta} = \frac{(3-\beta)C_0}{2}\int_{2\|x_0\|_\infty}^{2x_M} \psi^k(s) ds,
\]
respectively.
\end{lemma}
\begin{remark}Note that if $\alpha \in [1,d-1)$ for $d>2$, then $\int_0^{2\|x_0\|_\infty}\psi^1(s) ds = \infty$ and this yields that we only need the following condition for $v_0$ in Lemma \ref{lem:key2}:
\[
\|v_0\|_\infty^{3-\beta} < \frac{(3-\beta)C_0}{2}\int_{2\|x_0\|_\infty}^\infty \psi^1(s) ds.
\]
\end{remark}
\begin{theorem} Let $(x,v)$ be any smooth solutions to the system \eqref{par-gene} with initial data $(x_0,v_0)$ satisfying \eqref{basic-ini}. Then the followings hold:\newline

$\bullet$ If $\beta = 2$, we have an exponential alignment between velocities:
\[
\|v(t)\|_\infty \leq \|v_0\|_\infty \exp\left\{ -\frac{C_0\psi^k(2x_M)t}{2}\right\}.
\]
Furthermore, if $\eta^0_{m,X} > \frac{\|v_0\|_\infty}{C_0\psi^k(2x_M)}$, then we have no finite-time collision between particles and
\[
\|v(t)\|_\infty \geq \|v_0\|_\infty \exp\left\{ - \psi^k(\eta_{m,X}^* )t\right\},
\]
where $\eta_{m,X}^*:= \eta^0_{m,X} - \frac{\|v_0\|_\infty}{C_0\psi^k(2x_M)} > 0$.\newline

$\bullet$ If $\beta \in (0,2)$, we have a finite-time alignment between velocities:
\[
\|v(t)\|_\infty \leq \left( \|v_0\|_\infty^{2-\beta} - \frac{(2-\beta)C_0\psi^k(2x_M)t}{2}\right)^{\frac{1}{2-\beta}}.
\]
Furthermore if $\eta^0_{m,X} > T^*\|v_0\|_\infty$, then we have no collision between particles, where
\[
T^* := \frac{4\|v_0\|^{2-\beta}}{(2-\beta)C_0\psi^k(2x_M)}.
\]

$\bullet$ If $\beta \in (2,3)$, we have an polynomial alignment between velocities:
\[
\|v(t)\|_\infty \leq \left( \|v_0\|_\infty^{2-\beta} + \frac{(\beta-2)C_0\psi^k(2x_M)t}{2}\right)^{-\frac{1}{\beta-2}}.
\]
\end{theorem}
\begin{proof} The inequalities for $\|v(t)\|_\infty$ are clearly obtained from the results in Lemma \ref{lem:key2}. Concerning the initial configuration for avoiding collisions between particles, a straightforward computation yields that for $\beta = 2$
\begin{align*}
\begin{aligned}
|\eta_{m,X}(t) - \eta_{m,X}^0| &\leq \left| \int_0^t \frac{d\eta_{m,X}(s)}{ds} ds\right|\cr
&\leq 2\|v_0\|_\infty\int_0^t e^{-\frac{C_0\psi^k(2x_M)s}{2}}ds\cr
&\leq \frac{\|v_0\|_\infty}{C_0\psi^k(2x_M)}.
\end{aligned}
\end{align*}
Thus we conclude that
\[
\eta_{m,X}(t) \geq \eta_{m,X}^0 - |\eta_{m,X}(t) - \eta_{m,X}^0| \geq \eta_{m,X}^0 - \frac{\|v_0\|_\infty}{C_0\psi^k(2x_M)} > 0.
\]
Similarly, for $\beta \in (0,2)$, we have
\[
|\eta_{m,X}(t) - \eta_{m,X}^0| \leq \int_0^t \|v(s)\|_\infty ds \leq T^*\|v_0\|_\infty,
\]
and this deduces
\[
\eta_{m,X}(t) \geq \eta_{m,X}^0 - T^*\|v_0\|_\infty > 0.
\]
This completes the proof.
\end{proof}
\begin{remark} In the case of $\beta = 2$, if we choose the initial data for position $x_0$ such that $\eta^0_{m,X} > \frac{\|v_0\|_\infty}{C_0\psi^k(2x_M)}$, then there is no collision between particles and alignment for velocities in a finite time. Similarly, if we select the initial data $x_0$ satisfying $\eta^0_{m,X} > T^*\|v_0\|_\infty$ when $\beta \in (0,2)$, then the particles do not collide each other until $T^*$.
\end{remark}

\section*{Acknowledgments}
JAC was partially supported by the project MTM2011-27739-C04-02
DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya.
JAC acknowledges support from the Royal Society by a Wolfson
Research Merit Award. YPC was partially supported by Basic Science Research Program through the National Research Foundation of Korea funded
by the Ministry of Education, Science and Technology (ref.
2012R1A6A3A03039496). JAC and YPC were supported by Engineering
and Physical Sciences Research Council grants with references
EP/K008404/1 (individual grant) and EP/I019111/1 (platform grant).


TITLE: limit of a sequence
QUESTION [3 upvotes]: I know from my intuition that the sequence
$$x_n=\left(1-\cfrac{1}{3}\right)^2 \left(1-\cfrac{1}{6}\right)^2 \left(1-\cfrac{1}{10}\right)^2\cdots \cdots\left(1-\cfrac{1}{\cfrac{n\left(n+1\right)}{2}}\right)^2,\quad n\geq2$$
is convergent.
But i don't know how to prove it.I almost try to apply every theorem I know (for eg ratio  test ,monotone convergence theorem,...). Help me to prove this.
Proof or idea is needed.Where does the sequence converge to?

REPLY [11 votes]: $$
\begin{align}
\prod_{k=2}^\infty\left(1-\frac2{k^2+k}\right)
&=\lim_{n\to\infty}\prod_{k=2}^n\frac{\color{#C00}{(k-1)}\color{#090}{(k+2)}}{\color{#00F}{k}\color{#C90}{(k+1)}}\\
&=\lim_{n\to\infty}\frac{\color{#C00}{(n-1)!}\,\color{#090}{(n+2)!/3!}}{\color{#00F}{n!/1!}\,\color{#C90}{(n+1)!/2!}}\\[3pt]
&=\frac13\lim_{n\to\infty}\frac{n+2}{n}\\[3pt]
&=\frac13
\end{align}
$$
Your product is just the square of this one.


TITLE: Baby Rudin - Theorem 1.35 Cauchy Schwartz
QUESTION [5 upvotes]: I'm stumped on the following difficulty while reading baby Rudin (p.15).
Let $A=\sum|a_j|^2, B=\sum|b_j|^2, C=\sum a_j \overline{b}_j$:
$$\begin{align}
\sum|Ba_j-Cb_j|^2 &= \sum(Ba_j-Cb_j)(B\overline{a}_j-\overline{Cb_j}) \\
&= B^2\sum|a_j|^2{\bf -B\overline{C}\sum a_j\overline{b}_j-BC\sum \overline{a}_j b_j}+|C|^2\sum|b_j|^2 \\
&=B^2A-B|C|^2=B(AB-|C|^2)
\end{align}$$
Note that the second line of the equation, the book indicates that the middle 2 term together to be zero. Why is that?

REPLY [3 votes]: Not quite. It's the last three terms in the second line that reduce to $-B|C|^{2}$. To see this, note that $\sum \overline{a_{j}}b_{j}=\sum\overline{a_{j}\overline{b_{j}}}=\overline{\sum a_{j}\overline{b_{j}}}=\overline{C}$. Hence,
$$
\begin{aligned}
&-B\overline{C}\sum a_{j}\overline{b_{j}}-BC\sum\overline{a_{j}}b_{j}+|C|^{2}\sum|b_{j}|^{2}\\
=&-B\overline{C}C-BC\overline{C}+C\overline{C}B\\
=&-BC\overline{C}\\
=&-B|C|^{2}
\end{aligned}
$$


TITLE: Get wrong answer on $\frac{\partial \mathbf{x}^{\top} \mathbf{A} \mathbf{x}}{\partial \mathbf{x}}$ when using graph
QUESTION [3 upvotes]: I can use the product rule to obtain $\frac{\partial \mathbf{x}^{\top} \mathbf{A} \mathbf{x}}{\partial \mathbf{x}} = \mathbf{x}^{\top} \frac{\partial \mathbf{A} \mathbf{x}}{\partial \mathbf{x}}+(\mathbf{A} \mathbf{x})^{\top} \frac{\partial \mathbf{x}}{\partial \mathbf{x}}$, and get the right result $\mathbf{x}^{\top}\left(\mathbf{A}+\mathbf{A}^{\top}\right)$. Here $x$ is a vector, $A$ is a matrix that is not a function of $x$.
However, when I try to calculate by using a graph, the result is different. I am not sure at which step I made the mistake. Please point out where I made it wrong. I used the graph attached (red numbers are the derivatives) and the equations as follows.

Let $
q_{2}=q_{1}x$, $q_{1}=(A^{T} x)^{T}$,  and $k=A^Tx$. I want to find $\frac{d q_{2}}{d x}$
$\frac{d q_{2}}{d x}=\frac{\partial q_{2}}{\partial x}+\frac{\partial q_{2}}{\partial q_{1}} \frac{\partial q_{1}}{\partial x}= q_{1}+x \frac{\partial q_{1}}{\partial x}= q_{1}+x \frac{\partial q_{1}}{\partial k} \frac{\partial k}{\partial x}  =x^TA+xIA^T = x^TA+xA^T \ne x^{\top}\left({A}+{A}^{\top}\right)$, $I$ is the NxN identity matrix.
The result I got here is different. I know it is not correct, but I couldn't figure out where I lost it.
Thank you!

REPLY [3 votes]: In the line
$$\frac{d q_{2}}{d x}=\frac{\partial q_{2}}{\partial x}+\frac{\partial q_{2}}{\partial q_{1}} \frac{\partial q_{1}}{\partial x}= q_{1}+x \frac{\partial q_{1}}{\partial x}=\dots$$
In terms of dimensions of matrices, it looks to me like you have written
$$ [1\times n] = [1\times n] + [1\times n][n\times n] = [1\times n] + [n\times 1][n\times n] = \dots $$
because  $x$ is a column vector. The product of matrices $[n\times 1][n\times n]$ is not defined, so there is an error here.
Let me try to rewrite in more explicit notation (I identify $\mathbb R^n$ with column vectors $\mathbb R^{n\times 1}$)
\begin{align} q&:\mathbb R^n \to \mathbb R,&q(x) &= x^TAx,\\
q_1&: \mathbb R^n \to \mathbb R^{1\times n}, & q_1(x) &= (A^Tx)^T = x^TA,\\
q_2&:\mathbb R^{1\times n} \times \mathbb R^n\to\mathbb R, & q_2(a,b) &= ab
\end{align}
Thus
$$q(x) = q_2(q_1(x),x)$$
Since there are subtly different spaces involved here, I'll elect to use a notation that works for arbitrary spaces, namely  the Frechet derivative $d$ which always satisfies the chain rule
$$ d(f\circ g)(x)[h] = df(g(x))[dg(x)[h]],$$
and then the only game is to figure out what spaces each differential belongs to.
Letting $\partial$ denote a partial Frechet derivative, we have
$$ dq(x)[h]= \partial_aq_2(q_1(x),x)[dq_1(x)[h]] + \partial_b q_2(q_1(x),x)[h].$$
Now, $q_1$ is linear, and $q_2$ is separately linear in each variable. For any linear map $L$, the Frechet derivative $dL(x)[h]=L(h)$. (The nice thing is that this covers multiplication of matrices on the left and on the right.) Thus for $h\in\mathbb R^n$ and $v\in\mathbb R^{1\times n}$,
\begin{align}
dq_1(x)[h]&=q_1(h)=h^TA,\\
\partial_a q_2(a,b)[v]&= vb,\\
\partial_b q_2(a,b)[h]&= ah.
\end{align}
therefore \begin{align}
 dq(x)h&=dq_1(x)[h]x + q_1(x)h\\
&=h^T Ax + x^TA h \\
&= x^TA^Th +x^TAh \\
&= x^T(A^T+A)h = ((A^T+A)x)\cdot h
\end{align}
and finally we can read the Jacobian from the final line:
$$ \frac {dq}{dx}(x) = x^T(A^T+A).$$
(And, the gradient vector is its transpose $\nabla q(x) = (A^T+A)x$.)
Equivalently, one has to understand that the Jacobian of a map $\mathbb R^n \to \mathbb R^{1\times n}$ acts on the right: if $g$ is $\mathbb R^{1\times n}$-valued, then
$ f(x+h) \approx f(x) + \frac{df}{dx}(x)h $ cannot make sense by dimension counting (remember $h$ is a column vector). So we are forced to accept that the correct approximation is $f(x+h)\approx f(x) + h^T \frac{df}{dx}(x)$. What this means is that the order of terms in chain rule is reversed! Indeed, the standard chain rule formula for the composition of a $\mathbb R^{n\times 1}$-vector valued function $f=f(y)$ and $g=g(x)$ is
$$ \frac{d(f\circ g)}{dx}(x) = \frac{df}{dy}(g(x))\frac{dg}{dx}(x)$$
If we take the matrix transpose of this equation, we obtain the chain rule for row-vector-valued functions $h=f^T$:
$$\frac{d(h\circ g)}{dx}(x) = \left(\frac{dg}{dx}(x)\right)^T\frac{dh}{dy}(g(x)).$$
A similar issue occurs in the above where the input is the row vector, and the approximation for $q_2$ reads
$ q_2(a+v,b)\approx q_2(a,b) + v \frac{\partial q_2}{\partial a}(a,b)$. Hence (and I think deriving from first principles is easiest)
\begin{align} q_2(q_1(x+h),b)&\approx q_2(q_1(x),b) + (q_1(x+h)-q_1(x)) \frac{\partial q_2}{\partial a}(q_1(x),b)
\\&\approx q_2(q_1(x),b) + h^T\frac{dq_1}{dx}(x) \frac{\partial q_2}{\partial a}(q_1(x),b)
\\& = q_2(a,b) +  \frac{\partial q_2^T}{\partial a}(q_1(x),b)\frac{dq_1^T}{dx}(x)h \end{align}  The correct chain rule is therefore, as painful as it seems,
$$\frac{dq}{dx}(x)= \frac{\partial q_2^T}{\partial a}(q_1(x),x)\frac{dq_1^T}{dx}(x) + \frac{\partial q_2}{\partial b}(q_1(x),x)$$
It would be easier IMO to avoid this mess by not using a $\mathbb R^{1\times n}$-valued function.


TITLE: $\mathbb{R}^2$ diagonally cut in half through point of origin, vector space?
QUESTION [0 upvotes]: I'm starting my math course at uni in a few days and I have a relatively simple question for most of you. I'd like to know whether $\mathbb{R}^2$ diagonally cut in half through the point of origin is a vector space (the space below the cut I mean). I'd say it's not a v-space because scalar multiplication leads out of the set if you take $-1$ as a scalar for the vector $(0,-x)$ so you end up with $(0,x)$ above the cut. I'm new to this type of task so I'd appreciate if someone can verify this for me. Also, how do you show this in a mathematically correct fashion?
Thanks in advance!

REPLY [2 votes]: You are right. We say $U$ is a vector subspace (or linear subspace) of some vector space $V$ if:
(1) (contains the zero vector) $0 \in U$
(2) (Closure under addition) If $u_1, u_2 \in U$ implies $u_1 + u_2 \in U$
(3) (Closure under scalar mult.) if $u \in U$, then $au \in U$ for some constant $a$
In your case $V = \mathbb{R}^2$. To show that your set is not a vector subspace you only need to show one of the above conditions does not hold.


TITLE: What's the intuition that helps to understand factorial of real number?
QUESTION [2 upvotes]: I just learn that gamma function can be used for real number factorial.
How to intuitively interpret (1/2)! ?

REPLY [4 votes]: When we talk about it as a "real factorial", we are not really saying that it should have an easy way of finding it for every real number, but instead talking about a thing called extensions.
It means "okay, we have this function in the natural numbers, and it would be awesome to find a real function that is both nice and follows things we would expect"
If you ask the real functions for one such that $f(x+1)=x*f(x)$ (which works as a nice way of generalising the factorial, as $(n+1)!=n*(n)!$), that $f(1)=1$, that it is logarithmic-convex (roughly meaning it keeps growing faster, something you expect seeing the factorial) and that it is analytic (a technical but very important condition for Calculus), a theorem called the Bohr-Mollerup Theorem tells you there is only one function following all those things
So we called it the Gamma function and say it is the real factorial, and now the "factorial" of 1/2 is just the value of this function. It's not like it has a reason to take the value it takes, it just happens to be the value that the only function nice enough for our standards takes, and it works amazingly well when using it this way