\section{Introduction}
\label{intro}
Several searches for supersymmetric (SUSY) particles were performed
with data taken by the ATLAS experiment~\cite{Aad:2008zzm}.
For illustration, an example of a SUSY candidate event display is shown
in Fig.~\ref{fig:candidate}~\cite{Aaboud:2016ejt}.
The event display shows two muons, jets and large missing energy.
\begin{figure}[h]

\centering
\includegraphics[width=0.44\textwidth]{plots/candidate.jpg}
\includegraphics[width=0.5\textwidth]{plots/unification1.jpg}
\includegraphics[width=0.38\textwidth]{plots/dm1.jpg}
\includegraphics[width=0.54\textwidth]{plots/xsec.jpg}\\
\includegraphics[width=0.6\textwidth]{plots/naturalness1.jpg}
\caption{ATLAS event display of a SUSY candidate with two muons,
         jets and large missing energy~\cite{Aaboud:2016ejt}.
Illustration of gauge coupling unification~\cite{deBoer:2003xm},
dark matter candidates~\cite{Liu:2017drf} and
naturalness (light Higgs boson mass)~\cite{naturalness}.
Typical production cross-sections for SUSY reactions~\cite{Borschensky:2014cia}.
}
\label{fig:candidate}
\vspace*{-5mm}
\end{figure}

The search for SUSY particles is highly motivated in three aspects
1) gauge coupling unification~\cite{deBoer:2003xm},
2) dark matter candidates~\cite{Liu:2017drf} and
3) naturalness (light Higgs boson mass)~\cite{naturalness}.
These aspects are illustrated in Fig.~\ref{fig:candidate}.
An overview of all ATLAS SUSY searches is given in Ref.~\cite{atlassusy}.
The major expected production cross-sections for SUSY reactions are illustrated
in Fig.~\ref{fig:candidate}~\cite{Borschensky:2014cia}.

This article is structured in the following sections.
\ref{sec:inclusive}: inclusive production (0, 1, 2 leptons),
\ref{sec:stop}: direct stop production,
\ref{sec:bottom}: direct sbottom production,
\ref{sec:small}: small mass differences,
\ref{sec:ew}: electroweak direct production,
\ref{sec:photpnic}: photonic signatures,
\ref{sec:long}: long-lived particles,
\ref{sec:parity}: R-parity violation,
\ref{sec:displaced}: displaced vertices,
\ref{sec:pmssm}: pMSSM,
\ref{sec:prospects}: prospects
\ref{sec:conclusions}: conclusions and outlook.

\section{Inclusive production (0, 1, 2 leptons)}
\label{sec:inclusive}
Inclusive 2-6 jets searches for the third generation gluino mediation
were performed.
Feynman diagrams, an overview and results are summarized in Fig.~\ref{fig:inclusive}~\cite{ATLAS:2017cjl,ATLAS:2017vjw}.

\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/incl_fig_01a.png}
\includegraphics[width=0.24\textwidth]{plots/incl_fig_01b.png}
\includegraphics[width=0.24\textwidth]{plots/incl_fig_01c.png}
\includegraphics[width=0.24\textwidth]{plots/incl_fig_01d.png}
\includegraphics[width=0.24\textwidth]{plots/incl_fig_01e.png}
\includegraphics[width=0.24\textwidth]{plots/incl_fig_01f.png}
\includegraphics[width=0.24\textwidth]{plots/incl_fig_01g.png}\\
\includegraphics[width=0.24\textwidth]{plots/incl2_fig_01a.png}
\includegraphics[width=0.24\textwidth]{plots/incl2_fig_01b.png}
\includegraphics[width=0.24\textwidth]{plots/incl2_fig_02a.png}
\includegraphics[width=0.24\textwidth]{plots/incl2_fig_02b.png}
\includegraphics[width=0.24\textwidth]{plots/incl2_fig_02c.png}
\includegraphics[width=0.24\textwidth]{plots/incl2_fig_02d.png}\\
\includegraphics[width=0.27\textwidth]{plots/inclusive_overview.jpg}
\includegraphics[width=0.38\textwidth]{plots/incl_fig_12a.png}
\includegraphics[width=0.32\textwidth]{plots/incl2_fig_03b.png}
\includegraphics[width=0.38\textwidth]{plots/incl2_fig_10a.png}
\includegraphics[width=0.38\textwidth]{plots/incl2_fig_10b.png}
\caption{Inclusive production decay topologies,
        overview of inclusive searches,
        comparison of the observed and expected event yields
        as a function of signal region,
        number of selected b-jets and exclusion contours.
         }
\label{fig:inclusive}
\end{figure}

\section{Direct stop production}
\label{sec:stop}

Searches for direct stop production were performed.
Feynman diagrams and results are summarized
in Fig.~\ref{fig:stop}~\cite{Aaboud:2017nfd}
for final states with two leptons,
in Fig.~\ref{fig:stop2}~\cite{Aaboud:2017ayj}
for final states with jets plus missing transverse momentum,
and in Fig.~\ref{fig:stop3}~\cite{ATLAS-CONF-2017-037}
for final states with one isolated lepton, jets, and missing transverse momentum.

\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/stop_fig_01a.png}
\includegraphics[width=0.24\textwidth]{plots/stop_fig_01b.png}
\includegraphics[width=0.24\textwidth]{plots/stop_fig_01c.png}
\includegraphics[width=0.24\textwidth]{plots/stop_fig_01d.png}
\includegraphics[width=0.32\textwidth]{plots/stop_fig_09.png}
\includegraphics[width=0.32\textwidth]{plots/stop_fig_10.png}
\includegraphics[width=0.32\textwidth]{plots/stop_fig_11.png}
\caption{Scalar top decay topologies for the final states with two leptons
         and exclusion contours.
         }
\label{fig:stop}
\end{figure}

\begin{figure}[bp]
\centering
\includegraphics[width=0.24\textwidth]{plots/stop2_fig_01a.png}
\includegraphics[width=0.24\textwidth]{plots/stop2_fig_01b.png}
\includegraphics[width=0.24\textwidth]{plots/stop2_fig_01c.png}
\includegraphics[width=0.24\textwidth]{plots/stop2_fig_01d.png}
\includegraphics[width=0.32\textwidth]{plots/stop2_fig_08.png}
\includegraphics[width=0.32\textwidth]{plots/stop2_fig_09.png}
\includegraphics[width=0.32\textwidth]{plots/stop2_fig_13.png}
\includegraphics[width=0.32\textwidth]{plots/stop2_fig_10.png}
\includegraphics[width=0.32\textwidth]{plots/stop2_fig_11.png}
\includegraphics[width=0.32\textwidth]{plots/stop2_fig_12.png}
\caption{Scalar top decay topologies for the final state with jets plus missing transverse momentum and exclusion contours.
         }
\label{fig:stop2}
\vspace*{-10mm}
\end{figure}

\begin{figure}[tp]
\centering
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_01a.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_01b.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_21.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_22.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_23.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_24.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_25.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_26.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_28a.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_28b.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_28c.png}
\includegraphics[width=0.24\textwidth]{plots/stop3_fig_28d.png}
\caption{Scalar top decay topologies for the final states with one isolated lepton, jets
         and missing transverse momentum and exclusion contours.
         }
\label{fig:stop3}
\vspace*{-5mm}
\end{figure}

\section{Direct sbottom production}
\label{sec:bottom}

Searches for SUSY direct sbottom production were performed.
Feynman diagrams and results are summarized in Fig.~\ref{fig:sbottom}~\cite{Aaboud:2017wqg}.

\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/sbottom_fig_01a.png}
\includegraphics[width=0.24\textwidth]{plots/sbottom_fig_01b.png}
\includegraphics[width=0.24\textwidth]{plots/sbottom_fig_07a.png}
\includegraphics[width=0.24\textwidth]{plots/sbottom_fig_07b.png}
\caption{Scalar bottom decay topologies and exclusion contours.
         }
\label{fig:sbottom}
\vspace*{-5mm}
\end{figure}

\vspace*{-3mm}
\section{Small mass differences}
\label{sec:small}

Searches for SUSY with a small mass difference of squarks and neutralinos were performed.
A Feynman diagram and results are summarized in Fig.~\ref{fig:small}~\cite{ATLAS-CONF-2017-060}.

\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/small_fig_01b.png}
\includegraphics[width=0.3\textwidth]{plots/monojet_candidate.png}\\
\includegraphics[width=0.24\textwidth]{plots/small_fig_07a.png}
\includegraphics[width=0.24\textwidth]{plots/small_fig_07b.png}
\includegraphics[width=0.24\textwidth]{plots/small_fig_08a.png}
\includegraphics[width=0.24\textwidth]{plots/small_fig_08b.png}
\caption{Small mass difference of squarks and neutralinos decay topologies,
         an event display of a typical monojet with missing energy candidate
         and exclusion contours.
         }
\label{fig:small}
\vspace*{-10mm}
\end{figure}

\vspace*{-3mm}
\section{Electroweak direct production}
\label{sec:ew}

Searches for SUSY electroweak direct production with 2-leptons+0-jets, 2-leptons+jets,
and 3-leptons were performed.
Feynman diagrams, the 2-lepton invariant mass distribution for data, and the estimated SM backgrounds, an overview
and results are summarized in Fig.~\ref{fig:ew}~\cite{ATLAS-CONF-2017-039}.

\vspace*{-3mm}
\section{Photonic signatures}
\label{sec:photpnic}

Searches for SUSY with di-photon and missing energy were performed.
Feynman diagrams, missing transverse energy distribution,
and results are summarized in Fig.~\ref{fig:photon}
for the final states with two photons~\cite{ATLASCollaboration:2016wl},
and for the final states with one photon~\cite{ATLAS-CONF-2016-066}.

\clearpage
\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/ew_fig_01a.png}
\includegraphics[width=0.24\textwidth]{plots/ew_fig_01b.png}
\includegraphics[width=0.24\textwidth]{plots/ew_fig_01c.png}
\includegraphics[width=0.24\textwidth]{plots/ew_fig_01d.png}
\includegraphics[width=0.24\textwidth]{plots/ew_fig_01e.png}
\includegraphics[width=0.24\textwidth]{plots/ew_fig_02a.png}
\includegraphics[width=0.23\textwidth]{plots/ew_overview.png}
\includegraphics[width=0.26\textwidth]{plots/ew_fig_06a.png}
\includegraphics[width=0.23\textwidth]{plots/ew_fig_06b.png}
\includegraphics[width=0.24\textwidth]{plots/ew_fig_07a.png}
\includegraphics[width=0.24\textwidth]{plots/ew_fig_07b.png}
\includegraphics[width=0.24\textwidth]{plots/ew_fig_08.png}
\caption{Electroweak direct production with 2-leptons+0-jets, 2-leptons+jets, 3-leptons decay topologies,
         2-lepton invariant mass distribution,
         overview of inclusive searches
         and exclusion contours.
         }
\label{fig:ew}
\vspace{-10mm}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/photon_fig_01.png}
\includegraphics[width=0.38\textwidth]{plots/photon_fig_03.png}
\includegraphics[width=0.24\textwidth]{plots/photon2_fig_01.png}
\includegraphics[width=0.28\textwidth]{plots/photon2_fig_06a.png}
\includegraphics[width=0.32\textwidth]{plots/photon2_fig_07.png}
\includegraphics[width=0.32\textwidth]{plots/photon2_fig_08.png}
\caption{Two photons and one photon decay topologies,
         missing transverse energy distribution and exclusion contours.
         }
\label{fig:photon}
\vspace{-10mm}
\end{figure}

\section{Long-lived particles}
\label{sec:long}

Searches for SUSY particles with a long life time were performed.
Feynman diagrams, the reconstruction efficiency for such particles as a function of the decay radius
and results are summarized in Fig.~\ref{fig:long}~\cite{ATLAS-CONF-2017-017}.

\begin{figure}[h]
\centering
\includegraphics[width=0.38\textwidth]{plots/long_vertex.jpg}
\includegraphics[width=0.45\textwidth]{plots/long_fig_01.png}\\
\includegraphics[width=0.24\textwidth]{plots/long_fig_02a.png}
\includegraphics[width=0.24\textwidth]{plots/long_fig_02b.png}
\includegraphics[width=0.32\textwidth]{plots/long_fig_03.png}\\
\includegraphics[width=0.14\textwidth]{plots/long_fig_04a.png}
\includegraphics[width=0.14\textwidth]{plots/long_fig_04b.png}
\includegraphics[width=0.14\textwidth]{plots/long_fig_04c.png}\\
\includegraphics[width=0.29\textwidth]{plots/long_fig_08.png}
\includegraphics[width=0.29\textwidth]{plots/long_fig_09a.png}
\includegraphics[width=0.29\textwidth]{plots/long_fig_09b.png}
\caption{Views of the ATLAS vertex detector,
         long-lived SUSY particle decay topologies,
         reconstruction efficiency as a function of the decay radius,
         sketches of the different background components to the search for pixel tracklets
         and exclusion contours.
         }
\label{fig:long}
\vspace{-5mm}
\end{figure}

\section{R-parity violation}
\label{sec:parity}

Searches for SUSY particles with R-parity violation were performed.
An example is presented for an R-parity-violating stop quark coupling
to a final state with two leptons and two jets, at least one of which
is identified as a b-jet.
Feynman diagrams and results are summarized in Fig.~\ref{fig:rpv}~\cite{Aaboud:2017opj}.
Figure~\ref{fig:rpv2}~\cite{Aaboud:2017faq} shows the corresponding results for searches with a lepton
plus high (up to 12) jet multiplicity in the final states.

\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/rpv_fig_01.png}
\includegraphics[width=0.30\textwidth]{plots/rpv_fig_06.png}
\includegraphics[width=0.38\textwidth]{plots/rpv_fig_07.png}
\vspace*{-2mm}
\caption{R-parity-violating stop quark coupling to a final state with two leptons and two jets decay topologies
         and exclusion contours.
         }
\label{fig:rpv}
\vspace{-5mm}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/rpv2_fig_01a.png}
\includegraphics[width=0.24\textwidth]{plots/rpv2_fig_01b.png}
\includegraphics[width=0.24\textwidth]{plots/rpv2_fig_01c.png}
\includegraphics[width=0.24\textwidth]{plots/rpv2_fig_01d.png}\\
\includegraphics[width=0.24\textwidth]{plots/rpv2_fig_06f.png}
\includegraphics[width=0.24\textwidth]{plots/rpv2_fig_09a.png}
\includegraphics[width=0.24\textwidth]{plots/rpv2_fig_09b.png}
\includegraphics[width=0.24\textwidth]{plots/rpv2_fig_09c.png}
\vspace*{-2mm}
\caption{R-parity-violating SUSY particle decay topologies with high jet multiplicities decay topologies
         and exclusion contours.
         }
\label{fig:rpv2}
\end{figure}

\section{Displaced vertices}
\label{sec:displaced}

In split SUSY scenarios, the gluinos are long-lived enough to hadronise to R-hadrons
that can give rise to displaced vertices when they decay.
Searches for such massive particles were performed in events with displaced vertices
and large missing transverse momentum.
A Feynman diagram, densities of vertices in the inner ATLAS detector
and results are summarized in Fig.~\ref{fig:displaced}~\cite{Aaboud:2017iio}.

\begin{figure}[h]
\centering
\includegraphics[width=0.24\textwidth]{plots/displaced_fig_01.png}
\includegraphics[width=0.24\textwidth]{plots/displaced_fig_03a.jpg}
\includegraphics[width=0.24\textwidth]{plots/displaced_fig_08a.png}
\includegraphics[width=0.24\textwidth]{plots/displaced_fig_08b.png}
\includegraphics[width=0.24\textwidth]{plots/displaced_fig_09a.png}
\includegraphics[width=0.24\textwidth]{plots/displaced_fig_09b.png}
\includegraphics[width=0.24\textwidth]{plots/displaced_fig_10a.png}
\includegraphics[width=0.24\textwidth]{plots/displaced_fig_10b.png}
\vspace*{-2mm}
\caption{Massive particles in events with displaced vertices
         and large missing transverse momentum decay topologies,
         densities of observed vertices in the inner ATLAS detector
         and exclusion contours.
         }
\label{fig:displaced}
\vspace*{-5mm}
\end{figure}

\section{pMSSM}
\label{sec:pmssm}

The search result for supersymmetry after LHC Run-1 are interpreted in the phenomenological MSSM
(pMSSM)~\cite{Aaboud:2016wna}. A general parameter scan was performed with 19 parameters.
Feynman diagrams and results are summarized in Fig.~\ref{fig:pMSSM}~\cite{Aaboud:2016wna}.

\begin{figure}[h]
\centering
\includegraphics[width=0.3\textwidth]{plots/pMSSM_1.png}
\includegraphics[width=0.3\textwidth]{plots/pMSSM_2.png}
\includegraphics[width=0.38\textwidth]{plots/pMSSM_3.png}
\vspace*{-2mm}
\caption{pMSSM interpretation of the search results from LHC Run-1 data taking.
         Density of pMSSM points on the plane of relic density versus LSP mass.
         Fraction of model points excluded, with colour coding representing the
         fraction of model points excluded for each sparticle.
        }
\label{fig:pMSSM}
\vspace*{-5mm}
\end{figure}

\section{Prospects}
\label{sec:prospects}

An overview of the LHC operation for the coming years is given in Fig.~\ref{fig:prospects}.
The prospects for SUSY searches were studied for $\sqrt{s}= 14$\,TeV for average proton-proton interactions per
bunch crossing $\mu=200$, and a total integrated luminosity of 3000\,fb$^{-1}$.
Smearing functions for the upgraded ATLAS detector simulation and truth level particle corrections for detector
effects were implemented,
assuming 30\% systematic uncertainties on the background.
As an example, Fig.~\ref{fig:prospects} also shows the prospects for a search
with the ATLAS detector for direct pair production of a chargino and a
neutralino decaying via a W boson and the lightest Higgs boson in final states
with one lepton, two b-jets and missing transverse momentum at the high luminosity
LHC~\cite{ATL-PHYS-PUB-2015-032}.

\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{plots/HL-LHC-plan-2017-Plan-1.jpg}
\includegraphics[width=0.24\textwidth]{plots/prospects_fig_02.png}
\includegraphics[width=0.45\textwidth]{plots/prospects_fig_08.png}
\includegraphics[width=0.45\textwidth]{plots/prospects_fig_09.png}
\vspace{-2mm}
\caption{Overview of the LHC operation for the coming years.
         Prospects for a search for direct pair production
         of a chargino and a neutralino decaying via a W boson and
         the lightest Higgs boson in final states with one lepton,
         two b-jets and missing transverse momentum at the high luminosity LHC.
        }
\label{fig:prospects}
\vspace{-3mm}
\end{figure}

\section{Conclusions and outlook}
\label{sec:conclusions}

SUSY searches in ATLAS cover a large range of scenarios.
Inclusive squark and gluino searches resulted in mass limits up to above 2 TeV.
Direct stop limits increased from about 700\,GeV\,(Run-1)\,to about 950 GeV,
and direct sbottom limits increased from about 650\,GeV (Run-1) to about 950\,GeV.
Chargino mass limits range between 600 and 1100\,GeV.
Gluino limits reach up to 2\,TeV with photon signatures.
Searches of long-lived charginos, gluinos, R-parity violation (RPV) and pMSSM
resulted in stringent limits.
There are excellent prospects to advance SUSY searches with LHC Run-2
(completing 2015-2018 operation),
LHC Run-3 (2021-2023) and the high-luminosity HL-LHC (2026-).

\section*{Acknowledgment}
The project is supported by the Ministry of Education,
Youth and Sports of the Czech Republic
under projects LG15052 and LM2015058.


\section{Introduction}

Among the fields that the theory of dynamical systems finds its applications one can count the  models of swarm behaviour and consensus processes. Problems of this kind appear in physics of many natural phenomena and engineering applications: Immigrating groups of animals (flocks of birds and fish), robots that are in communication with each other on a bounded region, motions of groups of stars or galaxies, diffusive dynamics of microorganisms. The basic rule in the dynamics of the group consisting of $N$  individuals is that each agent moves in a spacetime path such that they are far enough from each other so they do not collide and keep a distance close enough so as to keep in communication with the neighboring agents, which corresponds to an evolution under an attractive-repulsive potential. In this setting, one of the simplest models is the ODE system
\begin{equation}
\frac{dx_i}{dt}=\sum_{j=1}^N a_{ij}(x_j-x_i), \qquad  i=1,2,...,N.
\end{equation}
In case of collective motion, $x_i$ stands for the position of an agent in 1-dimension. In opinion dynamics, which is the  evolution of opinions of individuals in a society through mutual interaction with other individuals, $x_i$ is the value of the "opinion" that the agent $i$ produces.

One of the questions to be answered in consensus problems is whether the agents in communication will arrive at a common decision
in the limit $t\rightarrow \infty$ or not. Or, are there subgroups with different opinion values in the long run? There are many variants of this model in literature \cite{Saber2007,Motsch2014,Bertozzi2012}. For a substantial review on applications of the theory of dynamical systems to decision dynamics, models of cultural change, language evolution, biological evolution and swarm motion one can see \cite{Castellano2009}.

The model in which the opinions of $N$ individuals are affected by their initial self-judgements and the weighted average of the group's opinion in discrete time is known as Friedkin- Johnsen model \cite{Friedkin2011}:
\begin{equation}\label{FJ}
x_i (k+1)=a_i \sum_{j=1}^N w_{ij} x_j (k)+(1-a_i ) x_i (0);   \quad     i=1,...,N,   \quad   k\geq 0.
\end{equation}
Here $x_i (k)$ is the opinion of the $i$th individual at time $k$,  $x_i (0)$ is its opinion at the beginning, $a_i$ is its sensitivity and $w_{ij}$ is a measure of the influence of agent $j$ on the decision of agent $i$. If one subtracts the term  $x_i (k)=(1+a_i-a_i)x_i (k)$ from both sides of \eqref{FJ} and organize the terms,
\begin{equation}\label{discrete}
 x_i (k+1)-x_i (k)=a_i \sum_{j=1}^N w_{ij} (x_j (k)-x_i (k))-(1-a_i )(x_i (k)-x_i (0))
\end{equation}
is  obtained \cite{Gabbay2014}.

In \cite{Gabbay2014}, instead of the discrete model \eqref{discrete}, the continuous counterpart
\begin{equation}\label{gab}
\frac{dx_i}{dt}=-\gamma_i (x_i-\mu_i )+\sum_{j=1}^N \kappa_{ij} h(x_j-x_i ), \quad         i=1,...,N
\end{equation}
is analyzed for long-time behavior according to the parameters included. Here the opinions of the agents in a group change under the influence of two main forces: One is the self-bias force, the other is the group influence force.
Self-bias force represents the agent's dependency on its initial opinion and is taken into account by the first term on the righthand side of \eqref{gab}. With $\gamma_i>0$, this force has a negative impact on the change of opinion. $\gamma_i$ is the strength of the $i$th individual's dependency on its initial opinion $\mu_i$. Group influence force is a result of the effect of other individuals on a specific agent, being represented by the second term on the righthand side of \eqref{gab}. $\kappa_{ij}>0$ is the coupling strength and is proportional to the frequency in which agent $j$ communicates with agent $i$.

$h(x_j-x_i )$  is the coupling function and depending on the opinion difference $x_j-x_i$ of the agents $i$ and  $j$, represents the measure of the strength of influence that agent $j$ has on agent   $i$. Due to the term $h(x_j-x_i )$  the model is a nonlinear one, taken in the form
\begin{equation}
h(x_j-x_i )= (x_j-x_i )\exp\Big[-\frac{1}{2}  \frac{(x_j-x_i )^2}{\lambda_i^2 }\Big].
\end{equation}
Therefore, the effects of individuals on each other, of which opinions are far away from each other,  are taken to be as less and less.

\cite{Gabbay2014} considers  the system \eqref{gab}  for a group of $N=3$ agents, explicitly in the form
\begin{align}
&\frac{dx_1}{dt}=-(x_1-\mu_1)+(\kappa+\nu)h(x_2-x_1),\nonumber\\
&\frac{dx_2}{dt}=-x_2+(\kappa-\nu)(h(x_1-x_2)+h(x_3-x_2)), \label{21}\\
&\frac{dx_3}{dt}=-(x_3-\mu_3)+(\kappa+\nu)h(x_2-x_3).\nonumber
\end{align}

\begin{figure}[!ht] \centering
\subfigure[$\kappa=1$]{\label{sek11}
\includegraphics[scale=0.50]{Fig1a.eps}} \hspace{2cm}
\subfigure[$\kappa=1.5$]{
\label{sek12}
\includegraphics[scale=0.5]{Fig1b.eps}}
\subfigure[$\kappa=3$]{\label{sek13}
\includegraphics[scale=0.5]{Fig1c.eps}}\hspace{2cm}
\caption{{Equilibrium outcomes in the dynamical system \eqref{21}  with  initial conditions: $x_1(0)=-2.5$, $x_2(0)=10^{-6}$, $x_3(0)=2.5$ and  initial disagreement: $\Delta\mu=\mu_3-\mu_1=5$. }}
\label{gabb}
\end{figure}

Figure \ref{gabb}, a reproduction of the findings in \cite{Gabbay2014} on numerical solution of \eqref{21},  indicates  the change that occurs on the opinion of agents in a chain network for three different values of the  coupling strength $\kappa$. They observe three distinct states which correspond to these values of the coupling strength. For $\kappa=1$, Figure \ref{sek11} demonstrates a state of Symmetric High Discord (SHD). In this state, the center node stays at zero and the natural biases of end nodes change only a small amount. For $\kappa=1.5$, Figure \ref{sek12} indicates Majority Rule state. In this state, center node approaches one of the end nodes generating a majority rule pair.
For $\kappa=3$, Figure \ref{sek13} shows the state of Symmetric Low Discord. In this state, the end nodes approach to the decision of the center node. Let us note that for  \cite{Gabbay2014} employs an initial condition of $x_2(0)=10^{-6}$ instead of a zero initial condition so as to create a small perturbation for $x_2$ in the direction of $x_3$.

In SHD state, the discord of members in the group is apparent. It is impossible that all group members agree to the same decision. In  MR state, nodes that form a majority pair accept a mutual decision. Although majority rule is adequate to reach a decision, it is not the decision of the group. If majority rule is sufficient for arriving at a decision, it will be the \emph{policy of the group} \cite{Gabbay2014}.  In SLD state, all group members agree on a common decision. This decision is very close to the opinion of center node, albeit we do not give any precise definition of this "closeness".

The long-time behaviour (the stable picture) of the system is analyzed in the parameter space $(\Delta \mu,\kappa)$; where $\Delta \mu=\mu_3-\mu_1$ is the maximum initial opinion difference between agents and $\kappa$ is a uniform coupling strength, to distinguish the behaviours (a) symmetric high discord-SHD,  (b) majority rule-MR, (c) symmetric low discord-SLD. The information given by the analysis according to the parameters included in the system, without solving the system, is the final consensus level of the group: Disagreement (or \emph{deadlock}  \cite{Gabbay2014}), majority, or a level acceptable as an agreement.

Another occasion of interest to decision dynamics is the situation in which the group under consideration is including a leader. In such systems, the leader is defined as the power that controls the opinion dynamics of the group. Hegselmann- Krause (HK) model is an example to these systems \cite{HK2014}:
\begin{subequations}\label{HK}
\begin{align}
&\frac{dx_0}{dt}=u(t),\\
&\frac{dx_i}{dt}=\sum_{j=1}^N a_{ij}(x_j-x_i)+C_i(x_0-x_i), \quad i=1,2,...,N.
\end{align}
\end{subequations}
Here the leader is a "stubborn" agent which is not influenced by anyone but has an influence on everyone. As it is seen, the individual $x_0$ has a dynamics determined by the function $u(t)$, not affected by any other agent, and has an effect in each agent's steering equation and represents the leader of the group \cite{HK2014}.

With the motivation provided by the literature outlined above, this article is focused on the problem framed by the following questions:
\begin{itemize}
\item How can the opinion evolution given in \eqref{gab} be modeled in the presence of a leader? This is an open decision dynamics problem not analyzed in the existing literature, to the best of our knowledge, and the answer starts by considering systems \eqref{gab} and \eqref{HK} together.
\item Through construction of such a model, and by a similar machinery as in \cite{Gabbay2014} (via bifurcation analysis in parameter space), can we foresee and identify the final stable situation of this system as disagreement, majority or agreement?
\end{itemize}
When the systems \eqref{gab} and \eqref{HK} are considered together, the model that will be the base of our problem is of the form
\begin{subequations}\label{sysfull}
\begin{align}
&\frac{dx_0}{dt}=u(t),\\
&\frac{dx_i}{dt}=-\gamma_i(x_i-\mu_i)+\sum_{j=1}^N\kappa_{ij}h(x_j-x_i)+C_i(x_0-x_i), \quad i=1,2,...,N;
\end{align}
\end{subequations}
which is a combination of the dynamical systems considered in \cite{Gabbay2014} and \cite{HK2014}. Just to mention, $x_0(t)$ is the opinion function of a stubborn agent, called the  leader, as it is not affected by the opinion of anyone in the group while he/she influences the opinion of everyone in the group. The other individuals have the same dynamics in the system \eqref{gab} besides  being influenced from the evolution of $x_0$.  $C_i$ represents the strength of this influence on each agent \cite{HK2014}. This constant is utilized to tune the free decision maker's, the leader's effect on the individuals of the society.
\section{Analysis}
For simplicity we will consider only the case $u(t)\equiv 0$, which gives a leader at a constant decision $x_0$ at any time. Therefore, we are led to the dynamical system
\begin{equation}\label{sys}
\frac{dx_i}{dt}=-\gamma_i(x_i-\mu_i)+\sum_{j=1}^N\kappa_{ij}h(x_j-x_i)+C_i(x_0-x_i),
\end{equation}
for $i=1,2,...,N$. The last term in \eqref{sys} represents the leadership effect. See that if $x_0>x_i$ this term supports $x_i$ to increase to arrive at $x_0$, and vice versa if $x_0<x_i$, when  $C_i>0$. We see that this term works for making $x_i$ come and stay around the opinion $x_0$. In case $C_i<0$, the effect is to push $x_i$  away from the opinion $x_0$. The term with $C_i$ can also be viewed as a control term and $C_i$ as the control parameter. In what follows we will always consider $C_i$ as a  positive parameter to represent the dominance of the leader on the group and impose it on the system as $\pm C_i$ for an attractive or impulsive leader effect.

Following \cite{Gabbay2014}, we analyze the evolution of opinion on a network which has three nodes. The natural biases in this network are symmetric around zero: $\mu_1=-\Delta\mu/2$, $\mu_2=0$ and $\mu_3=\Delta\mu/2$. The network has a chain topology; that is, node 1 and 3 are not conjoined while both node 1, node 2 and node 2, node 3 are connected between each other. Therefore, node 1 and node 3 do not directly influence each other. The binary adjacency matrix is then
\begin{equation}A=
\begin{bmatrix}
0&1&0\\1&0&1\\0&1&0
\end{bmatrix}.
\end{equation}
Based on this topology, \eqref{sys} takes the form
\begin{align}\label{sys1}
&\frac{dx_1}{dt}=-(x_1-\mu_1)+(\kappa+\nu)h(x_2-x_1)+C_1(x_0-x_1),\nonumber\\
&\frac{dx_2}{dt}=-x_2+(\kappa-\nu)(h(x_1-x_2)+h(x_3-x_2))+C_2(x_0-x_2)\\
&\frac{dx_3}{dt}=-(x_3-\mu_3)+(\kappa+\nu)h(x_2-x_3)+C_3(x_0-x_3).\nonumber
\end{align}
In this setting, asymmetric coupling between the center node 2 and the end nodes is possible by virtue of the parameter $\nu$. If $\nu$ is a positive parameter then  node 2 has a bigger effect on the end nodes. If $\nu$ is negative the effect of the  end nodes is higher.

We define three new parameters $r$, $s$ and $\bar{x}$: $r$ is described as discord and expressed as $r=x_3-x_1$. $s$ is asymmetry of the opinions, defined as  $s=x_3-x_2-(x_2-x_1)=x_3-2x_2+x_1$. $\bar{x}$ is the average of opinions (mean node opinion): $\bar{x}=\frac{1}{3}(x_1+x_2+x_3)$. The  coupling function $h(x)$ is an odd function. In terms of these variables, \eqref{sys1} takes the form
\begin{align}
\frac{dr}{dt}&=-r+\mu_3-\mu_1+(C_3-C_1)x_0+C_1x_1-C_3x_3-(\kappa+\nu)(h(\frac{r+s}{2})+h(\frac{r-s}{2})),\nonumber\\
\frac{ds}{dt}&=-s+(C_3-2C_2+C_1)x_0-C_1x_1+2C_2x_2-C_3x_3+\mu_3+\mu_1\nonumber\\
              &-(3\kappa-\nu)(h(\frac{r+s}{2})-h(\frac{r-s}{2})), \label{rsx}\\
\frac{d\bar{x}}{dt}&=-\bar{x}+\frac{1}{3}(C_1+C_2+C_3)x_0+\frac{1}{3}(\mu_3+\mu_1)
                    -\frac{1}{3}(C_1x_1+C_2x_2+C_3x_3)\nonumber\\
                    &-\frac{2\nu}{3}(h(\frac{r+s}{2})-h(\frac{r-s}{2})).\nonumber
\end{align}

We shall assume that the leader applies a pulling force on node 1 while it applies a repelling force on node 3. That is, we assume that $C_1=C$, $C_2=0$ and $C_3=-C$ where $C>0$. The equations of motion in the variables $x_1$, $x_2$, $x_3$, just to stress, are
\begin{equation}\label{main}
\begin{split}
\frac{dx_1}{dt}&=-(x_1-\mu_1)+(\kappa+\nu)h(x_2-x_1)+C(x_0-x_1) \\
\frac{dx_2}{dt}&=-x_2+(\kappa-\nu)(h(x_1-x_2)+h(x_3-x_2))\\
\frac{dx_3}{dt}&=-(x_3-\mu_3)+(\kappa+\nu)h(x_2-x_3)-C(x_0-x_3)
\end{split}
\end{equation}
Figure \ref{imper} shows  the change of opinion by using three distinct values of the coupling strength $\kappa$. SHD, MR and SLD states correspond to $\kappa=0.5$, $\kappa=1.5$ and $\kappa=14$, respectively.

\begin{figure}[!ht] \centering
\subfigure[$\kappa=0.5$]{\label{sek21}
\includegraphics[scale=0.50]{Fig2a.eps}} \hspace{2cm}
\subfigure[$\kappa=1.5$]{\label{sek22}
\includegraphics[scale=0.50]{Fig2b.eps}}
\subfigure[$\kappa=14$]{\label{sek23}
\includegraphics[scale=0.56]{Fig2c.eps}}\hspace{2cm}
 \caption{{Equilibrium outcomes for the system \eqref{main} with control/leadership effect. $x_0=4$, $C=0.05$. Initial disagreement: $\Delta\mu=5$ and initial conditions: $x_1(0)=-2.5$, $x_2(0)=0$, $x_3(0)=2.5$.  }}
\label{imper}
\end{figure}

From these simulations we see that it is also possible to observe the the three different states of SHD, MR and SLD of \cite{Gabbay2014} for the modified system \eqref{main} with the specific leadership effect considered in our work.
In SHD state, the initial opinions of nodes in the group do not change almost at all, since the value of interaction forces, coupling strength and leadership effect, are very small. In MR state, the center node generates majority pair by approaching to one of end nodes. It selects node 1 to generate the majority pair because of the leadership effect. In SLD state, the distance  between the center node $x_2$ and end nodes $x_1$ and $x_3$ decreases a considerable amount.

When we rearrange \eqref{rsx}  for $C_1=C$, $C_2=0$ and $C_3=-C$ we obtain
 \begin{subequations}
\begin{align}
\label{33a}\frac{dr}{dt}&=-r+2C\bar{x}+\frac{C}{3}s-2C x_0+\mu_3-\mu_1-(\kappa+\nu)(h(\frac{r+s}{2})+h(\frac{r-s}{2})), \\
\label{33b}\frac{ds}{dt}&=-s+Cr+\mu_3+\mu_1-(3\kappa-\nu)(h(\frac{r+s}{2})-h(\frac{r-s}{2})),\\
\label{33c}\frac{d\bar{x}}{dt}&=-\bar{x}+\frac{C}{3}r+\frac{1}{3}(\mu_3+\mu_1)-\frac{2\nu}{3}(h(\frac{r+s}{2})-h(\frac{r-s}{2})).
\end{align}
\end{subequations}
where we have used the fact that $\displaystyle x_1+x_3=2\bar{x}+s/3$.

In the sequel, bifurcation analysis will be made in the chain network. The system will be investigated in regard to states identified above and the passage from one  state to another will analyzed. The most interesting of all, the existence of an imperfect pitchfork bifurcation will be shown and several bifurcation boundaries will be identified. While the leaderless system exhibits a pitchfork bifurcation, the perturbation of the model by a leader causes the qualitative behaviour to be perturbed to the imperfect one.

\eqref{33a} is examined in the equilibrium of SHD state ($s=0$). It means that the system does not have asymmetric coupling in this state. Furthermore, mean node opinion is zero ($\bar{x}=0$) in this state and the value of discord is accepted as $r\approx\Delta\mu+\theta$ for large $\Delta\mu$. \eqref{33a} is stated for $\bar{x}=0$ and $s=0$ as below
\begin{equation}
\label{34}\frac{dr}{dt}=-r-2C x_0+\mu_3-\mu_1-2(\kappa+\nu)h(\frac{r}{2}).
\end{equation}
We rearrange this equation for $r\approx\Delta\mu+\theta$ as
\begin{equation}
\label{35}0=\theta+2C x_0+2(\kappa+\nu)h(\frac{\Delta\mu+\theta}{2}).
\end{equation}
The Taylor expansion of the coupling function is found around $\theta=0$ in order to calculate the value of $\theta$ as follows
\begin{equation}
h(\frac{\Delta\mu+\theta}{2})\approx h(\frac{\Delta\mu}{2})+h'(\frac{\Delta\mu}{2})\frac{\theta}{2}.
\end{equation}
\eqref{35} is rearranged by utilizing this expansion to find
$$\theta=-\frac{2Cx_0+2(\kappa+\nu)h(\frac{\Delta\mu}{2})}{1+(\kappa+\nu)h'(\frac{\Delta\mu}{2})}$$
where $\displaystyle h(\frac{\Delta\mu}{2})=\frac{\Delta\mu}{2}e^{-\frac{\Delta\mu^{2}}{8}}$ and
$\displaystyle h'(\frac{\Delta\mu}{2})=\frac{1}{2}(1-\frac{\Delta\mu^{2}}{4})\ e^{-\frac{\Delta\mu^{2}}{8}}$.
In order to exhibit the existence of bifurcation, we consider small perturbations of s around $s=0$ in \eqref{33b}. The Taylor approximation of $h(\frac{r+s}{2})-h(\frac{r-s}{2})$ is
\begin{equation}h(\frac{r+s}{2})-h(\frac{r-s}{2})\approx h'(\frac{r}{2})s+\frac{1}{24}h'''(\frac{r}{2})s^{3}.\end{equation}
We rewrite \eqref{33b} by using this expansion as below
\begin{equation}
\label{38}\frac{ds}{dt}\approx Cr-(1+(3\kappa-\nu)h'(\frac{r}{2}))s-\frac{1}{24}(3\kappa-\nu)h'''(\frac{r}{2})s^{3}.
\end{equation}
\eqref{38} is rescaled with the parameters $\tau$ and $R$ as
\begin{subequations}
\begin{align}
\tau=[\frac{1}{24}(3\kappa-\nu)&h'''(\frac{r}{2})]t, \quad R=-\frac{1+(3\kappa-\nu)h'(\frac{r}{2})}{\frac{1}{24}(3\kappa-\nu)h'''(\frac{r}{2})},\\
\label{39b}&\frac{ds}{d\tau}=A+Rs-s^{3}
\end{align}
\end{subequations}
where $\displaystyle A=\frac{Cr}{\frac{1}{24}(3\kappa-\nu)h'''(\frac{r}{2})}$. \eqref{39b} is nothing but the general form of an imperfect pitchfork bifurcation.  The critical point of \eqref{39b} will be obtained for $\Delta=0$, where $\Delta$ is the discriminant of the cubic polynomial on the righthandside of \eqref{39b}. This condition is going to give us one of the boundaries separating the different regions in the stability diagram Figure 3.
\begin{rmk}
The main outcome that makes this paper a contribution to the literature is the emergence of  Eq. \eqref{39b}. In the leaderless case, it had been reported in \cite{Gabbay2014} that the dynamical model exhibits a pitchfork bifurcation with the prototype equation for $s$ being \eqref{39b} with $A=0$. In our case, the prototype equation for $s$ becomes the more general one \eqref{39b} and the system undergoes an imperfect pitchfork bifurcation in the existence of a leader.
\end{rmk}
\subsection{Boundary $\kappa_1$}
$\kappa_1$, depicted in Figure 3,  is the upper boundary of SHD in $\Delta\mu-\kappa$ parameter space. This boundary is found by using the critical point of the imperfect pitchfork bifurcation. This critical case is the point  where the righthand side of \eqref{39b} has a unique real root before and has three real roots afterwards, or vice versa. To mention,   for the general cubic equation  $ax^{3}+bx^{2}+c x+d=0$ the discriminant $\Delta$ is expressed as follows
\begin{equation}
\label{310}\Delta=b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd.
\end{equation}
It is clear that the critical point of the imperfect bifurcation is obtained when $\Delta=0$. To obtain $a, b, c, d$, \eqref{38} is rearranged for $r\approx\Delta\mu+\theta$ as
\begin{equation}
\frac{ds}{dt}\approx C(\Delta\mu+\theta)-(1+(3\kappa-\nu)h'(\frac{\Delta\mu+\theta}{2}))s-\frac{1}{24}(3\kappa-\nu)h'''(\frac{\Delta\mu+\theta}{2})s^{3}.
\end{equation}
By using the coefficients of above equation, $a, b, c, d$ are obtained as below
\begin{equation}
\begin{split}
&a=-\frac{1}{24}(3\kappa-\nu)h'''(\frac{\Delta\mu+\theta}{2}),\,\,\,\,\,\,\,\,\,\,b=0,\\
&c=-(1+(3\kappa-\nu)h'(\frac{\Delta\mu+\theta}{2})),\,\,\,\,\,d=C(\Delta\mu+\theta)
\end{split}
\end{equation}
These parameters are substituted in $\Delta = 0 $ to get
\begin{equation}\label{313c}
-32(1+(3\kappa-\nu)h'(\frac{\Delta\mu+\theta}{2}))^{3}=9C^{2}(\Delta\mu+\theta)^{2}(3\kappa-\nu)h'''(\frac{\Delta\mu+\theta}{2}).
\end{equation}
The Taylor approximations of $h'(\frac{\Delta\mu+\theta}{2})$ and $h'''(\frac{\Delta\mu+\theta}{2})$ are calculated around $\theta=0$ as follows
\begin{subequations}
\begin{align}
&h'(\frac{\Delta\mu+\theta}{2})\approx h'(\frac{\Delta\mu}{2})+h''(\frac{\Delta\mu}{2})\frac{\theta}{2},\\
&h'''(\frac{\Delta\mu+\theta}{2})\approx h'''(\frac{\Delta\mu}{2})+h^{(4)}(\frac{\Delta\mu}{2})\frac{\theta}{2}.
\end{align}
\end{subequations}
These expansions are substituted in \eqref{313c} as
\begin{align}
\label{315}-32(1+(3\kappa-\nu)\Big[ h'(\frac{\Delta\mu}{2})+h''(\frac{\Delta\mu}{2})\frac{\theta}{2}\Big]^{3}=9C^{2}(\Delta\mu+\theta)^{2}(3\kappa-\nu)\Big[ h'''(\frac{\Delta\mu}{2})+h^{(4)}(\frac{\Delta\mu}{2})\frac{\theta}{2}\Big]
\end{align}
where
\begin{subequations}\label{219}
\begin{align}
h'''(\frac{\Delta\mu}{2})   &=\frac{1}{16}(-6+3\Delta\mu^{2}-\frac{\Delta\mu^{4}}{8})e^{-\frac{\Delta\mu^{2}}{8}},\\
h^{(4)}(\frac{\Delta\mu}{2})&=\frac{1}{64}(30\Delta\mu-5\Delta\mu^{3}+\frac{\Delta\mu^{5}}{8})\ e^{-\frac{\Delta\mu^{2}}{8}}.
\end{align}
\end{subequations}
$\kappa_1$ is given in Figure 3 of the next Section, by the numerical solution of  \eqref{315} for $\kappa$ with varying values of $\Delta \mu$, making use of \eqref{219}.
\subsection{Boundary $\kappa_2$}
Now, following \cite{Gabbay2014}, we are going to analyze the situation where the system passes from MR state to SHD state, as the coupling constant is decreased. In the final picture Figure 3, this boundary $\kappa_2$ will be the curve above which MR and SHD both occur and below which the system is in SHD. The transition mechanism is a  saddle-node bifurcation, and $\kappa_2$ is a perturbed form of  that found in \cite{Gabbay2014}.
Stable and unstable equilibrium points collide with one another along this curve. To find this boundary, \eqref{33b} is analyzed around MR equilibrium point. In MR state, nodes $x_1$ and $x_2$  create a majority pair and $x_3$ is the minority node. $\displaystyle \varepsilon=\frac{x_1+x_2}{2}+\frac{\Delta\mu}{4}$ is defined as the change in the mean opinion of $x_1$ and $x_2$, and for large $\Delta\mu$, the term $\displaystyle h(\frac{r+s}{2})$ is negligible. The approximate values of $x_1$ and $x_3$  are assumed to be
\begin{equation}
x_1\approx-\frac{\Delta\mu}{2}+2\varepsilon-x_2, \qquad  x_3\approx\frac{\Delta\mu}{2}.
\end{equation}
The asymmetry is
\begin{equation}
s=x_3-2x_2+x_1=\frac{\Delta\mu}{2}-2x_2-\frac{\Delta\mu}{2}+2\varepsilon-x_2=-3x_2+2\varepsilon,
\end{equation}
which also gives
\begin{equation}
x_2=-\frac{s}{3}+\frac{2\varepsilon}{3}, \qquad  x_1=-\frac{\Delta\mu}{2}+\frac{s}{3}+\frac{4\varepsilon}{3}.
\end{equation}
The discord is
\begin{equation}\label{rr}
r=x_3-x_1=\frac{\Delta\mu}{2}+\frac{\Delta\mu}{2}-2\varepsilon-\frac{s}{3}+\frac{2\varepsilon}{3}=\Delta\mu-\frac{s}{3}-\frac{4\varepsilon}{3},\\
\end{equation}
yielding
\begin{equation}
\frac{r-s}{2}=\frac{1}{2}(\Delta\mu-\frac{s}{3}-\frac{4\varepsilon}{3}-s)=\frac{2}{3}(\frac{3}{4}\Delta\mu-s-\varepsilon).
\end{equation}
We define a new parameter as $\tilde{s}=\frac{3}{4}\Delta\mu-s-\varepsilon$ and when \eqref{33b} is rewritten for $\tilde{s}$ we get
\begin{equation}
\label{320}\frac{d\tilde{s}}{dt}=-(\tilde{s}-\frac{3}{4}\Delta\mu+\varepsilon)+Cr-(3\kappa-\nu)h(\frac{2}{3}\tilde{s}).
\end{equation}
\eqref{33c} in the equilibrium case (r.h.s. equal to zero) gives  the value of $\varepsilon$ as
\begin{equation}
\varepsilon=\frac{Cr}{2}+\nu h(\frac{2}{3}\tilde{s}).
\end{equation}
\eqref{320} is rearranged as below
\begin{equation}\label{322c}
\frac{d\tilde{s}}{dt}=-(\tilde{s}-\frac{3}{4}\Delta\mu)+\frac{C}{2}r-3\kappa h(\frac{2}{3}\tilde{s}).
\end{equation}
Considering  \eqref{33a} in the equilibrium case, we obtain
\begin{equation}\label{323d}
r(1-\frac{C}{2})=(C-1)\varepsilon+(1+\frac{C}{4})\Delta\mu-\frac{C}{3}\tilde{s}-2C x_0-\kappa h(\frac{2}{3}\tilde{s}).
\end{equation}
The value of $r$ in \eqref{rr} is reexpressed using $\tilde s$ as
\begin{equation}
\label{324}r(1-\frac{C}{2})=\frac{3}{4}(1-\frac{C}{2})\Delta\mu+\frac{1}{3}(1-\frac{C}{2}){\tilde{s}}-(1-\frac{C}{2})\varepsilon.
\end{equation}
\eqref{323d} and \eqref{324} together gives
\begin{equation}
\varepsilon=-(\frac{1}{2C}+\frac{5}{4})\Delta\mu+\frac{1}{3}(1+\frac{2}{C})\tilde{s}+4 x_0+\frac{2}{C}\kappa h(\frac{2}{3}\tilde{s})].
\end{equation}
\\
\eqref{322c} is rewritten for $\displaystyle r=\frac{\tilde{s}}{3}+\frac{3}{4}\Delta\mu-\varepsilon$ to get
\begin{equation}\label{326b}
\frac{d\tilde{s}}{dt}=-(1-\frac{C}{6})\tilde{s}+\frac{3}{4}(1+C)\Delta\mu-\frac{C}{2}\varepsilon-3\kappa h(\frac{2}{3}\tilde{s}).
\end{equation}
The value of $\varepsilon$ is substituted in \eqref{326b} to obtain
\begin{equation}
\label{327}\frac{d\tilde{s}}{dt}=-\frac{4}{3}\tilde{s}+(1+\frac{11C}{3})\Delta\mu-2Cx_0-4\kappa h(\frac{2}{3}\tilde{s}).
\end{equation}
To reveal the saddle-node bifurcation in the equilibrium point, the right side of \eqref{327} and its derivative are set equal to zero. Thus,
\begin{subequations}
\begin{align}
\label{328a}&\frac{4}{3}\tilde{s}-(1+\frac{11C}{3})\Delta\mu+2Cx_0=-4\kappa_2\tilde{s}e^{-\frac{2}{9}\tilde{s}^{2}},\\
\label{328b}&\frac{4}{3}=-4\kappa_2(1-\frac{4}{9}\tilde{s}^2)e^{-\frac{2}{9}\tilde{s}^{2}}.
\end{align}
\end{subequations}
In order to obtain $\tilde{s}$, \eqref{328a} and \eqref{328b} are solved together and we get
\begin{equation}
\tilde{s}^{3}-\frac{3}{4}(\Delta\mu+\frac{11C}{8}\Delta\mu-2Cx_0)\tilde{s}^{2}+(\frac{27}{16}+\frac{297C}{128})\Delta\mu-\frac{54C}{16}x_0=0.
\end{equation}
$\tilde{s}=\frac{3}{2}+\frac{12}{(8+11C)\Delta\mu}$ is a solution of above equation. \eqref{328a} is solved with respect to $\kappa_2$ as follows
\begin{equation}
\label{330}\kappa_2=\frac{1}{4\tilde{s}}(\Delta\mu+\frac{11C}{8}\Delta\mu-\frac{4}{3}\tilde{s}-2Cx_0)e^{\frac{2}{9}\tilde{s}^{2}}
\end{equation}
The value of $\tilde{s}$ is substituted in \eqref{330} to obtain
\begin{align}
\kappa_2&=\frac{(8+11C)\Delta\mu}{6[(8+11C)\Delta\mu+8]}\Big(\Delta\mu+\frac{11C}{8}\Delta\mu-\frac{16}{(8+11C)\Delta\mu}-2Cx_0-2\Big)e^{\frac{2}{9}(\frac{3}{2}+\frac{12}{(8+11C)\Delta\mu})^{2}}. \end{align}

\subsection{Boundary $\kappa_3$}
Let $\kappa_3$ be the lower boundary of SLD state. To find this boundary, we will consider \eqref{34} in the equilibrium case and the coefficient of linear term in the \eqref{38}. A system which consists of these equations indicates the existence of a pitchfork bifurcation between SLD and MR states. This system is constructed as follows.
\begin{subequations}
\begin{align}
\label{332a}&r+2C x_0-\Delta\mu=-(\kappa_3+\nu)re^{-\frac{r^{2}}{8}}, \\
\label{332b}&1=-\frac{1}{2}(3\kappa_3-\nu)(1-\frac{r^{2}}{4})e^{-\frac{r^{2}}{8}}.
\end{align}
\end{subequations}
To calculate $r$, \eqref{332a} and \eqref{332b} are divided to each other to have
\begin{equation}\label{333d}
r^{3}-\Delta\mu r^{2}+2Cx_0r^{2}-\frac{4}{3}r-8Cx_0+4\Delta\mu=0.
\end{equation}
$r\approx2+\frac{4}{3\Delta\mu}$ is the solution of \eqref{333d}, of order $O(\frac{1}{\Delta\mu})$. We solve \eqref{332a} for $\kappa_3$ as
\begin{equation}
\kappa_3+\nu=-\frac{r+2Cx_0-\Delta\mu}{r}e^{\frac{r^{2}}{8}}.
\end{equation}
For the value of $r$, this equation is rewritten to find
\begin{equation}
\kappa_3=\frac{1}{2(2+3\Delta\mu)}
         \Big\{-4\nu-6\nu\Delta\mu+\big[3\Delta\mu^{2}-6(1+Cx_0)\Delta\mu-4\big]e^{\frac{1}{8}(2+\frac{4}{3\Delta\mu})^{2}}\Big\}
\end{equation}
\begin{figure}[!ht] \centering
{\label{sek1a}
\includegraphics[scale=1]{Fig3.eps}} \hspace{1.5cm}
\caption{{Stability diagram of the triad \eqref{main} with symmetric coupling ($\nu=0$) for chain network}.}
\label{sek1}
\end{figure}
In Figure 4, the boundary $\kappa_4$ is the threshold beyond which Majority Rule does not occur. $\kappa_4$ was determined by direct simulation of the dynamical system and detecting the critical values of $\kappa$ versus $\Delta\mu$ that we note in the following table.

\begin{table}[h]
\centering
\caption{Boundary $\kappa_4$}
\label{my-label}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\Delta\mu$&$5$&$5.1$&$5.2$&$5.3$&$5.4$&$5.5$&$5.6$&$5.7$&$5.8$&$5.9$ \\ \hline
$\kappa_4$&$7.19$&$9.08$&$11.51$&$14.63$&$18.65$&$23.84$&$30.58$&$39.33$&$50.76$&$65.72$ \\ \hline
\end{tabular}
\end{table}

\section{Discussions and Future Work}

We want to conclude by pointing out new directions that the analysis in this work can be further taken to.

The system \eqref{sys} is indeed a modification to \eqref{gab}, including the leadership effect. See that \eqref{sys} can be written in the form
\begin{equation}\label{syss}
\frac{dx_i}{dt}=-(\gamma_i+C_i)\Big(x_i-\frac{\gamma_i\mu_i+x_0C_i}{\gamma_i+C_i}\Big)+\sum_{j=1}^N\kappa_{ij}h(x_j-x_i).
\end{equation}
Expressed this way, the leadership effect modifies \eqref{gab} in a way that each agent is under the stress of getting away from the modified bias $\displaystyle \frac{\gamma_i\mu_i+x_0C_i}{\gamma_i+C_i}$, including the opinion of himself and that of the leader. In this picture, we considered small values of $C_i$s so that the profiles SHD, MR and SLD of \cite{Gabbay2014} arise as behavioral patterns of the system and determined how the main results of \cite{Gabbay2014} change in the new case. The bifurcation analysis is done according to the parameters $\Delta \mu$ and $\kappa$. A natural problem would be a bifurcation analysis including the parameter $C_i$. This is a more fundamental question to answer: How the power  $C_i$ of the leader on the agents affect the agreement level of the system? Thus, bifurcation analysis in the parameter spaces $(\Delta \mu,C_i)$, $(\kappa,C_i)$, $(\Delta \mu, \kappa,C_i)$ are possible open problems in this direction.

What are possible behaviours of the system in that case? When the values of $C_i$ and the coupling constants are taken as different then we did here, it is possible to obtain the profiles given in Figures 4 and 5. In Figure 4, all the agents change their opinions a considerable amount compared to their initial opinions; with the final profile SHD in (a), MR in (b) and SLD in (c). The effect of the leader on each agent in the upward direction is quite apparent. This complete control case of \eqref{sys1} is going to be the subject of another paper.
\begin{figure}[!ht] \centering
\subfigure[$\kappa=1$]{\label{sek1a}
\includegraphics[scale=0.50]{Fig4a.eps}} \hspace{2cm}
\subfigure[$\kappa=1.5$]{\label{sek1b}
\includegraphics[scale=0.50]{Fig4b.eps}}
\subfigure[$\kappa=4$]{\label{sek1c}
\includegraphics[scale=0.50]{Fig4c.eps}}\hspace{2cm}
 \caption{{Equilibrium outcomes in the dynamical system \eqref{sys1} with complete control case: $C_1=C_2=C_3=C=0.2$, $x_0=8$. Initial disagreement: $\Delta\mu=5$ and initial conditions: $x_1(0)=-2.5$, $x_2(0)=0$, $x_3(0)=2.5$.}}
\label{sek1}
\end{figure}

In Figure 5(a), the strength of the leader on the individuals is quick and more dominant, making them reach a common decision very close to the leader's.  Figure 5(b) has been our favorite, as it exhibits the very interesting feature MR first, and an SLD afterwards.
\begin{figure}[!ht] \centering
\subfigure[$\kappa=14$]{\label{sek1a}
\includegraphics[scale=0.45]{Fig5a.eps}} \hspace{1cm}
\subfigure[$\kappa=1.5$]{\label{sek1b}
\includegraphics[scale=0.45]{Fig5b.eps}}
 \caption{{Numerical simulation of \eqref{sys1} with end nodes controlled: (a) $C_1=C_3=4$, $C_2=0$ (b) $C_1=-C_3=0.19$, $C_2=0$. Initial disagreement: $\Delta\mu=5$ and initial conditions: $x_1(0)=-2.5$, $x_2(0)=0$, $x_3(0)=2.5$ with $x_0=4$. }}
\label{sek1}
\end{figure}

The leadership effect we considered is in the form of a simple harmonic force $C_i(x_0-x_i)$. In literature there are forces of general power types; i.e., we could also consider forces of the type  $C_i|x-x_0|^p$, where the power $p$ can also assume negative values. If $p>0$, the effect of the leader on the agent becomes smaller when the agent's opinion is around the leader's opinion. If $p<0$, the effect of the leader on agents far away are not much; however, if an agent comes around the leader's opinion, the effect is rapidly increasing, due to the reciprocal dependence on the distance. As an example to the literature with this type of effect, we can mention \cite{Kolokolnikov2014}.


\section{Introduction}
Let $(A,\mathfrak{m},\Bbbk)$ denote a local ring. Let $\mathfrak{q} \subset A$ be an $\mathfrak{m}$-primary ideal
and $\underline a = a_1,\ldots,a_d$ be a system of parameters in $A$ such that $a_i \in \mathfrak{q}^{c_i},
i = 1,\ldots,d,$ with $c_i > 0$. The main interest in the present report is a comparison
of the multiplicities $e_0(\underline a;A)$ and $e_0(\mathfrak{q};A)$.

Let $M$ be a finitely generated
$A$-module. Note that for an ideal $\mathfrak{q} \subset A$ such that the length $\ell_A(M/\mathfrak{q}M)$ is finite, the multiplicity $e_0(\mathfrak{q};M)$ is defined as the leading term of the Hilbert-Samuel polynomial
\[
\ell_A(M/\mathfrak{q}^{n+1}M) = \sum_{i=0}^{d} e_i(\mathfrak{q};M) \binom{n+d-i}{d-i} \, \text{ for } n \gg 0, \text{ with } d = \dim_A M
\]
(see for instance \cite{ZS}, \cite{Ma} for all the details or more general \cite{RV} for
generalizations to filtered modules). With the previous assumption, clearly $e_0(\underline a;A) \geq
e_0(\mathfrak{q};A)$. We will discuss this relation in more detail. First let us recall some known results:
\begin{itemize}
	\item[(a)] For $\mathfrak{q} = \underline a A$ we get $e_0(\underline a^{\underline{n}};A) = n \cdot e_0(\underline a;A)$, where
	$\underline a^{\underline{n}} = a_1^{n_1}, \ldots, a_d^{n_d}$ and $n = n_1 \cdot \ldots \cdot n_d$ for some
	$(n_1, \ldots,n_d) \in \mathbb{N}^d$.
	\item[(b)] If $\underline a$ is a minimal reduction of $\mathfrak{q}$, then $e_0(\underline a;A) = e_0(\mathfrak{q};A)$.
	The converse is true (see \cite{Re1}) provided $A$ is formally equidimensional.
	\item[(c)]  It follows that $e_0(\underline a;A) \geq c \cdot e_0(\mathfrak{q};A)$, where $c = c_1\cdot \ldots \cdot c_d$
	with the above notation.
\end{itemize}

For the proof of (a) we refer to \cite{AB}. The first part of (b) is well-known, while the
converse is an outstanding result of Rees (see \cite{Re1}). The claim in (c) is easy to prove (see
\cite{BS} for details). The main goal of the present report is a discussion of the
difference $\chi := e_0(\underline a;A) - c \cdot e_0(\mathfrak{q};A) \geq 0$ of (c) in various
situations, its vanishing  resp. a simplified proof of some known results.

The importance of the understanding of $\chi$ has to do with Bezout's Theorem in the plane.
Let $C = V(F), D = V(G) \subset \mathbb{P}^2_{\Bbbk}, \Bbbk$ an algebraically closed field, be two curves in the projective plane without a common component. Then
\[
\sum_{P \in C\cap D} \mu(P;C,D) = \deg C \cdot \deg D,
\]
where $\mu(P;C,D)$ denotes the local intersection multiplicity of $P$ in $C \cap D$. In the case
of $P$ is the origin, it follows that $\mu(P;C,D) = e_0(f,g;A)$, where $A = \Bbbk[x,y]_{(x,y)}$ and $f,g$ denote
the equations in $A$. Since $C,D$ have no component in common, $\{f,g\}$ forms as system of parameters in $A$. Then
\[
e_0(f,g;A) \geq c\cdot d \cdot e_0(\mathfrak{m};A) = c\cdot d,
\]
since $e_0(\mathfrak{m};A) = 1$. Here $c, d$ denote the initial degree of $f,g$ respectively. This
estimate is well-known (see for instance \cite{BK} or \cite{Fi}) proved by resultants or Puiseux expansions.
Moreover equality holds if and only if $C,D$ intersect transversally in the origin. In other words, $f^{\star},g^{\star}$,
the initial forms of $f,g$ in the form ring $G_A(\mathfrak{m}) \cong \Bbbk[X,Y]$
are a homogeneous system of parameters.

Here we shall provide another argument with extensions to arbitrary local rings.  Let $A$ denote a local
ring with $\underline a = a_1,\ldots,a_d \in A$ a system of parameters. Put $c = c_1\cdot \ldots \cdot c_d$
for $a_i \in \mathfrak{m}^{c_i} \setminus \mathfrak{m}^{c_i +1}$.  In
his paper (see \cite{Pr}) the author claimed that $e_0(\underline a;A) = c \cdot  e_0(\mathfrak{m};A)$
if and only if the sequence of initial elements  $a_1^{\star}, \ldots ,a_d^{\star} \in G_A(\mathfrak{m})$
forms a regular sequence. This is not true as the following example shows.

\begin{example} \label{exam-1}
	Let $\Bbbk$ denote a field and $A = \Bbbk[|t^4,t^5,t^{11}|]
	\subset \Bbbk[|t|],$ where $t$ is an indeterminate over $\Bbbk.$ Then $A$ is a one-dimensional
	domain and therefore a Cohen-Macaulay ring with $A \simeq \Bbbk[|X,Y,Z|]/(X^4-YZ,Y^3-XZ,Z^2-X^3Y^2).$
	Clearly, the residue class $a = x$ of $X$ is a parameter with $a \in \mathfrak{m} \setminus
	\mathfrak{m}^2,$ so that $c = 1.$

	Furthermore, by easy calculations it follows that $e_0(a,A) = \ell_A(A/aA) = 4$ and
	$e_0(\mathfrak{m},A) = 4.$ So, the equation $e_0(a,A) = c \cdot e_0(\mathfrak{m},A)$ holds,
	while $G_A(\mathfrak{m}) = \Bbbk[X,Y,Z]/(XZ,YZ,Y^4,Z^2)$ is not a Cohen-Macaulay ring
	(see \cite[Section 3]{BS} for the details).
\end{example}

In Section 2 we start with some preliminaries and Koszul complexes. In Section 3 we derive some
new complexes from certain Koszul complexes important for the study of multiplicities. Euler
characteristics are the feature of Section 4. As an application we derive a short argument
for computing certain multiplicities as Euler characteristics of Koszul complexes (originally done
by Auslander and Buchsbaum (see \cite{AB}) and Serre (see \cite{jpS}) by spectral sequence arguments).
In Section 5 we study the equality $e_0(\underline a;M) = c_1 \cdot \ldots \cdot c_d \cdot e_0(\mathfrak{q};M)$. Under
some additional regularity condition on the sequence of initial forms $a_1^{\star}, \ldots, a_{d-1}^{\star}$
in $G_A(\mathfrak{q})$ we estimate the difference $\ell_A(M/\underline a M) - c_1 \cdot \ldots \cdot c_d \cdot e_0(\mathfrak{q};M)$.
As an application we get a bound of the local Bezout intersection numbers of two curves in the
projective plane without common component.

\section{Preliminaries}
First let us fix the notations we will use in the following. For the basics on $\mathbb{N}$-graded
structures we refer e.g. to \cite{GW}.

\begin{notation} \label{not-1}
	(A) We denote by $A$ a commutative Noetherian ring with $0 \not= 1$.
	For an ideal we write $\mathfrak{q} \subset A$. An $A$-module is denoted by $M$.
	Mostly we consider  $M$ as finitely generated. \\
	(B) We consider the Rees and form rings of $A$ with respect
	to $\mathfrak{q}$ by
	\[
	R_A(\mathfrak{q}) = \oplus_{n \geq 0} \mathfrak{q}^n \,T^n \subseteq A[T] \,
	\text{ and }\, G_A(\mathfrak{q}) = \oplus_{n \geq 0} \mathfrak{q}^n/\mathfrak{q}^{n+1}.
	\]
	Here $T$ denotes an indeterminate over $A$. Both rings are naturally $\mathbb{N}$-graded. For an $A$-module $M$ we define
	the Rees and form modules in the corresponding way by
	\[
	R_M(\mathfrak{q}) = \oplus_{n \geq 0} \mathfrak{q}^n M \,T^n \subseteq M[T] \,
	\text{ and } \, G_M(\mathfrak{q}) = \oplus_{n \geq 0} \mathfrak{q}^nM/\mathfrak{q}^{n+1}M.
	\]
	Note that $R_M(\mathfrak{q})$ is a graded $R_A(\mathfrak{q})$-module and
	$G_M(\mathfrak{q})$ is a graded $G_A(\mathfrak{q})$-module. Note that $R_A(\mathfrak{q})$
	and $G_A(\mathfrak{q})$ are both Noetherian rings. In case $M$ is a finitely generated
	$A$-module then $R_M(\mathfrak{q})$ resp. $G_M(\mathfrak{q})$ is finitely generated
	over $R_A(\mathfrak{q})$ resp. $G_A(\mathfrak{q})$. \\
	(C) There are the following two short exact sequences of graded modules
	\begin{gather*}
	0 \to R_M(\mathfrak{q})_{+}[1] \to R_M(\mathfrak{q}) \to G_M(\mathfrak{q}) \to 0 \text{ and }\\
	0 \to R_M(\mathfrak{q})_{+} \to R_M(\mathfrak{q}) \to M \to 0,
	\end{gather*}
	where $R_M(\mathfrak{q})_{+} = \oplus_{n > 0} \mathfrak{q}^n M \,T^n$. \\
	(D) Let $m \in M$ and $m \in \mathfrak{q}^c M \setminus \mathfrak{q}^{c+1} M$. Then we define
	$m^{\star} := m + \mathfrak{q}^{c+1} M \in [G_M(\mathfrak{q})]_c$. If $m \in \cap_{n \geq 1} \mathfrak{q}^n M$,
	then we write $m^{\star} = 0$. $m^{\star}$ is called the initial element of $m$ in $G_M(\mathfrak{q})$ and $c$ is called the initial degree of $m$.
	Here $[X]_n, n \in \mathbb{Z},$ denotes the $n$-th graded component of an $\mathbb{N}$-graded module $X$.
\end{notation}

For these and related results we refer to \cite{GW} and \cite{SH}. Another feature for the
investigations will be the use of Koszul complexes.

\begin{remark} \label{kos-1} ({\sl Koszul complex.})
	(A) Let $\underline{a} = a_1,\ldots,a_t$ denote a system of elements of the ring $A$. The Koszul complex
	$K_{\bullet}(\underline{a};A)$ is defined as follows: Let $F$ denote a free $A$-module with basis $e_1,\ldots,e_t$.
	Then $K_i(\underline{a};A) = \bigwedge^i F$ for $i = 1,\ldots,t$. A basis of $K_i(\underline{a};A)$ is given by the wedge
	products $e_{j_1} \wedge \ldots \wedge e_{j_i}$ for $1 \leq j_1 < \ldots < j_i \leq t$. The boundary
	homomorphism $K_i(\underline{a};A) \to K_{i-1}(\underline a;A)$ is defined by
	\[
	d_{j_1 \ldots j_i} :
	e_{j_1} \wedge \ldots \wedge e_{j_i} \mapsto \sum_{k=1}^{i} (-1)^{k+1} a_{j_k} e_{j_1}\wedge \ldots \wedge \widehat{e_{j_k}}
	\wedge \ldots \wedge e_{j_i}
	\]
	on the free generators $e_{j_1} \wedge \ldots \wedge e_{j_i}$. \\
	(B) Another way of the construction of $K_{\bullet}(\underline a;A)$ is inductively by the mapping cone.
	To this end let $X$ denote a complex of $A$-modules. Let $a \in A$ denote an element of $A$. The
	multiplication by $a$ on each $A$-module $X_i, i \in \mathbb{Z},$ induces a morphism of complexes
	$m_a : X \to X$. We define $K_{\bullet}(a;X)$ as the mapping cone $\operatorname{Mc} (m_a)$. Then we define inductively
	\[
	K_{\bullet}(a_1,\ldots,a_t;A) = K_{\bullet}(a_t,K_{\bullet}(a_1,\ldots,a_{t-1};A)).
	\]
	It is easily seen that
	\[
	K_{\bullet}(\underline a;A) \cong K_{\bullet}(a_1;A) \otimes_A \cdots \otimes_A K_{\bullet}(a_t;A).
	\]
	Therefore it follows that $K_{\bullet}(\underline a;A) \cong K_{\bullet}(\underline a_{\sigma};A)$, where
	$\underline a_{\sigma} = a_{\sigma(1)}, \ldots, a_{\sigma(t)}$ with a permutation $\sigma$ on $t$ letters.
	For an $A$-complex $X$ we define $K_{\bullet}(\underline a;X) = K_{\bullet}(\underline a;A)\otimes_A X$. We write
	$H_i(\underline a;X), i \in \mathbb{Z},$ for the $i$-th homology of $K_{\bullet}(\underline a;X)$. A short exact
	sequence of $A$-complexes $0 \to X' \to X \to X'' \to 0$ induces a long exact homology sequence
	for the Koszul homology
	\[
	\ldots \to H_i(\underline a;X') \to H_i(\underline a;X) \to H_i(\underline a;X'') \to H_{i-1}(\underline a;X') \to \ldots.
	\]
	Let $\underline a$ as above a system of $t$ elements in $A$ and $b \in A$. Then the mapping cone construction
	provides a long exact homology sequence
	\[
	\ldots \to H_i(\underline a;X) \to H_i(\underline a;X) \to H_i(\underline a,b;X) \to H_{i-1}(\underline a;X) \to H_{i-1}(\underline a;X) \to \ldots,
	\]
	where the homomorphism $H_i(\underline a;X) \to H_i(\underline a;X)$ is multiplication by $(-1)^i b$. Moreover, $\underline a H_i(\underline a;X) = 0$
	for all $i \in \mathbb{Z}$.

\end{remark}

For the proof of the last statement, we recall the following well-known argument.

\begin{lemma} \label{cone-1}
	Let $X$ denote a complex of $A$-modules. Let $a \in A$ denote an element.  Then $a H_i(a,X) = 0$ for
	all $i \in \mathbb{Z}.$
\end{lemma}

\section{The construction of complexes}
First we fix notations for this section. As above let $A$ denote a commutative Noetherian ring
and $\mathfrak{q} \subseteq A$. Let $\underline a = a_1,\ldots,a_t$ denote a system of elements of $A$.
Suppose that $a_i \in \mathfrak{q}^{c_i}$ for some integers $c_i \in \mathbb{N}$ for $i = 1,\ldots,t$.
Let $M$ denote a finitely generated $A$-module. We define two complexes here, see also \cite{KSch} for more detail.

\begin{notation} \label{kos-2}
	Let $n$ denote an integer. We define a complex $K_{\bullet}(\underline a,\mathfrak{q},M;n)$ in the following way:
	\begin{itemize}
		\item[(a)] For $0 \leq i \leq t$ put $K_i(\underline a,\mathfrak{q},M;n) = \oplus_{1\leq j_1 < \ldots < j_i \leq t}
		\mathfrak{q}^{n-c_{j_1}- \ldots- c_{j_i}}M$ and $K_i(\underline a,\mathfrak{q},M;n) = 0$ for $i >t$ or $i <0$.
		\item[(b)] The boundary map $K_i(\underline a,\mathfrak{q},M;n) \to K_{i-1}(\underline a,\mathfrak{q},M;n)$ is defined by maps on each of
		the direct summands $\mathfrak{q}^{n-c_{j_1}- \ldots- c_{j_i}}M$. On $\mathfrak{q}^{n-c_{j_1}- \ldots- c_{j_i}}M$
		it is the map given by $d_{j_1 \ldots j_i} \otimes 1_M$ restricted to $\mathfrak{q}^{n-c_{j_1}- \ldots- c_{j_i}}M$, where $d_{j_1 \ldots j_i}$ denotes the homomorphism as defined in \ref{kos-1}.
	\end{itemize}
	It is clear that the image of the map is contained in $\oplus_{1\leq j_1 < \ldots < j_{i-1} \leq t}
	\mathfrak{q}^{n-c_{j_1}- \ldots- c_{j_{i-1}}}M$. Clearly it is a boundary homomorphism.
	By the construction it follows that $K_{\bullet}(\underline a,\mathfrak{q},M;n)$ is a sub complex of the
	Koszul complex $K_{\bullet}(\underline a;M)$ for each $n \in \mathbb{N}$.
\end{notation}

Another way for the construction is the following.

\begin{remark} \label{rem-kos}
	Let $R_A(\mathfrak{q})$ and $R_M(\mathfrak{q})$ denote the Rees ring and the Rees module.
	For $a_i, i = 1,\ldots,t,$ we consider $a_i T^{c_i} \in [R_A(\mathfrak{q})]_{c_i}$. Then we have
	the system $\underline{aT^c} = a_1T^{c_1}, \ldots,a_tT^{c_t}$ of elements of $R_A(\mathfrak{q})$. Note
	that $\deg a_iT^{c_i} = c_i, i = 1,\ldots,t$. Then we may consider the Koszul complex
	$K_{\bullet}(\underline{aT^c};R_M(\mathfrak{q}))$. This is a complex of graded $R_A(\mathfrak{q})$-modules.
	It is easily seen that the degree $n$-component $[K_{\bullet}(\underline{aT^c};R_M(\mathfrak{q}))]_n$
	of $K_{\bullet}(\underline{aT^c};R_M(\mathfrak{q}))$ is the complex $K_{\bullet}(\underline a,\mathfrak{q},M;n)$ as introduced
	in \ref{kos-2}. We write $H_i(\underline a,\mathfrak{q},M;n)$ for the $i$-th
	homology of $K_{\bullet}(\underline a,\mathfrak{q},M;n)$ for $i \in \mathbb{Z}$.
\end{remark}

We come now to the definition of second complex.

\begin{definition} \label{def-1}
	With the previous notation we define $\mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n)$ the quotient of the
	embedding $K_{\bullet}(\underline a,\mathfrak{q},M;n) \to K_{\bullet}(\underline a;M)$. That is there is a short exact sequence
	of complexes
	\[
	0 \to K_{\bullet}(\underline a,\mathfrak{q},M;n) \to K_{\bullet}(\underline a;M) \to \mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n)
	\to 0.
	\]
	Note that $\mathcal{L}_i(\underline a,\mathfrak{q},M;n) \cong \oplus_{1 \leq j_1 < \ldots < j_i \leq t}M/\mathfrak{q}^{n-c_{j_1}- \ldots- c_{j_i}}M$.
	The boundary maps are those induced by the Koszul complex. We write $L_i(\underline a,\mathfrak{q},M;n)$
	for the $i$-th homology of $\mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n)$ and any $i \in \mathbb{Z}$.
\end{definition}

For a construction by mapping cones we need the following technical result. For a morphism $f: X \to Y$ we write
$C(f)$ for the mapping cone of $f$.

\begin{lemma} \label{cone-2}
	With the previous notation let $b \in \mathfrak{q}^d$ an element. The multiplication map by $b$ induces
	the following morphisms
	\[
	m_b(K) : K_{\bullet}(\underline a,\mathfrak{q},M;n-d) \to K_{\bullet}(\underline a,\mathfrak{q},M;n) \text{ and }
	m_b(\mathcal{L}) : \mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n-d) \to \mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n)
	\]
	of complexes. They induce isomorphism of complexes
	\[
	C(m_b(K)) \cong  K_{\bullet}(\underline a,b,\mathfrak{q},M;n) \text{ and } C(m_b(\mathcal{L})) \cong \mathcal{L}_{\bullet}(\underline a,b,\mathfrak{q},M;n).
	\]
\end{lemma}

\begin{proof}
	The proof follows easily by the structure of the complexes and the mapping cone construction.
\end{proof}

We begin with a few properties of the previous complexes.

\begin{theorem} \label{kos-3}
	Let $\underline a = a_1,\ldots,a_t$ denote a system of elements of $A$, $\mathfrak{q} \subset A$ an ideal and $M$ a
	finitely generated $A$-module. Let $n \in \mathbb{N}$ denote an integer.
	\begin{itemize}
		\item[(a)] $H_i(\underline a,\mathfrak{q},M;n) \cong H_i(\underline a_{\sigma},\mathfrak{q},M;n)$ and
		$L_i(\underline a,\mathfrak{q},M;n) \cong L_i(\underline a_{\sigma},\mathfrak{q},M;n)$ for all $i \in \mathbb{Z}$ and any $\sigma$,
		a permutation on $t$ letters.
		\item[(b)] $\underline a H_i(\underline a,\mathfrak{q},M;n) = 0$ and $\underline a L_i(\underline a,\mathfrak{q},M;n) = 0$ for all $i \in \mathbb{Z}$.
		\item[(c)] $H_i(\underline a,\mathfrak{q},M;n)$ and $L_i(\underline a,\mathfrak{q},M;n)$ are finitely generated $A/\underline a A$-modules
		for all $i \in \mathbb{Z}$.
	\end{itemize}
\end{theorem}

\begin{proof}
	The statement in (a) follows by virtue of the short exact sequence of complexes in \ref{def-1} and the long exact
	homology sequence. Note that the homology of Koszul complexes is isomorphic under permutations.

	The claim in (c) is a consequence of (b) since the homology modules $H_i(\underline a,\mathfrak{q},M;n)$ and $L_i(\underline a,\mathfrak{q},M;n)$
	are finitely generated $A$-modules.

	For the proof of (b) we follow the mapping cone construction of \ref{cone-2} with the arguments of \ref{cone-1}. To this end
	let $K_n = K_{\bullet}(\underline a,\mathfrak{q},M;n)$ and $C = C(m_b(K)_n) = K_{\bullet}(\underline a,b,\mathfrak{q},M;n)$. Then there is a short exact sequence
	of complexes
	\[
	0 \to K_n \to C \to K_{n-d}[-1] \to 0.
	\]
	The differential $\partial_i$ on $(x,y) \in C_i = (K_{n-d})_{i-1} \oplus (K_n)_i$ is given by
	\[
	\partial_i(x,y) = (d_{i-1}(x),d_i(y) +(-1)^{i-1}b x).
	\]
	Suppose that $\partial_i(x,y) = 0$, i.e., $d_{i-1}(x) =0$ and $d_i(y) = (-1)^i bx$. Then
	$$(y,0) \in (K_{n-d})_i\oplus (K_n)_{i+1} =C_{i+1} \text{ because } (K_n)_i \subseteq (K_{n-d})_i$$
	and therefore $\partial_{i+1}((-1)^iy,0) = b(x,y)$. That is $b H_i(C) = 0$ for all $i \in \mathbb{Z}$.

	In order to show the claim in (b) we use the previous argument. So let us consider
	$K_{\bullet}(\underline a,\mathfrak{q},M;n) = C(m_{a_t}(K_{\bullet}(\underline a',\mathfrak{q},M;n)))$, where
	$\underline a' = a_1,\ldots,a_{t-1}$. The previous argument shows $a_t H_i(\underline a,\mathfrak{q},M;n) = 0$ for all
	$i \in \mathbb{Z}.$ By view of (a) this finishes the proof in the case of $H_i(\underline a,\mathfrak{q},M;n)$.

	For the proof of $\underline a L_i(\underline a,\mathfrak{q},M;n) = 0$ we follow the same arguments. Instead
	of the injection
	$(K_n)_i \subseteq (K_{n-d})_i$ we use the surjection $(\mathcal{L}_n)_i \twoheadrightarrow (\mathcal{L}_{n-d})_i$, where
	$\mathcal{L}_n = \mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n)$. We skip the details here.
\end{proof}

\section{Euler characteristics}
Let $A$ denote a commutative ring. Let $X$ denote a complex of $A$-modules.

\begin{definition} \label{def-2}
	Let $X : 0 \to X_n \to \ldots \to X_1 \to X_0 \to 0$ denote a bounded complex of $A$-modules.
	Suppose that $H_i(X), i = 0, 1, \ldots, n,$ is an $A$-module of finite length. Then
	\[
	\chi_A(X) = \sum_{i=0}^n (-1)^i \ell_A(H_i(X))
	\]
	is called the Euler characteristic of $X$.
\end{definition}

We collect a few well known facts about Euler characteristics.

\begin{lemma} \label{char}
	Let $A$ denote a Noetherian commutative ring.
	\begin{itemize}
		\item[(a)] Let $0 \to X' \to X \to X'' \to 0$ denote a short exact sequence of complexes such that all the homology
		modules are of finite length. Then
		$
		\chi_A(X) = \chi_A(X') + \chi_A(X'').
		$
		\item[(b)] Suppose $X: 0 \to X_n \to \ldots \to X_1 \to X_0 \to 0$ is a bounded complex such
		that $X_i, i = 0,\ldots,n,$ is of finite length. Then
		$
		\chi_A(X) = \sum_{i=0}^n (-1)^i \ell_A(X_i)
		$
	\end{itemize}
\end{lemma}

\begin{proof}
	The statement in (a) follows by the long exact cohomology sequence derived by $0 \to X' \to X \to X'' \to 0$. The
	second statement might be proved by induction on $n$, the length of the complex $X$.
\end{proof}

As an application we get the following result about multiplicities, originally shown by
\cite{AB} and \cite{jpS}.

\begin{proposition} \label{mult-1}
	Let $(A,\mathfrak{m})$ be a local ring and $a_1,\ldots,a_d \in \mathfrak{m}$ a system of parameters
	for $M$, a finitely generated $A$-module. Then
	\[
	\chi_A(\underline a;M) = e_0(\underline a;M),
	\]
	where $\chi_A(\underline a;M) = \chi_A(K_{\bullet}(\underline a;M))$ and $e_0(\underline a;M)$ denotes the Hilbert-Samuel
	multiplicity.
\end{proposition}

\begin{proof} Let $\underline a = a_1,\ldots,a_d$ be the system of parameters and $\underline a A = \mathfrak{q}$.
	We choose $c_i = 1, i = 1,\ldots,d$. Then the short
	exact sequence of \ref{def-1} has the following form
	\[
	0 \to K_{\bullet}(\underline a,\underline a,M;n) \to K_{\bullet}(\underline a;M) \to \mathcal{L}_{\bullet}(\underline a,\underline a,M;n) \to 0.
	\]
	All of the three complexes have homology modules of finite length and therefore
	$\chi_A(\underline a;M) = \chi_A(K_{\bullet}(\underline a,\underline a,M;n))+ \chi_A(\mathcal{L}_{\bullet}(\underline a,\underline a,M;n))$ for all $n \in \mathbb{N}$.

	First we show that $\chi_A(K_{\bullet}(\underline a,\underline a,M;n)) = 0$ for all $n \gg 0$. To this end recall that
	$H_i(\underline a,\underline a,M;n) = [H_i(\underline{aT};R_M(\underline a))]_n$. We know that $\underline{aT} (H_i(\underline{aT};R_M(\underline a))) = 0$ for all
	$i = 0, \ldots,d$. Therefore $H_i(\underline{aT};R_M(\underline a))$ is a finitely generated module over
	$R_A(\underline a)/\underline{aT} R_A(\underline a) = A/\underline a A$. This implies that $[H_i(\underline{aT};R_M(\underline a))]_n = H_i(\underline a,\underline a,M;n) = 0$
	for all $n \gg 0$. That is $\chi_A(K_{\bullet}(\underline a,\underline a,M;n)) = 0$ for all $n \gg 0$.

	By view of Lemma \ref{char} (b) we get $\chi_A(\mathcal{L}_{\bullet}(\underline a,\underline a,M;n)) = \sum_{i=0}^d (-1)^i\binom{d}{i}
	\ell_A(M/\underline a^{n-i}M)$. For $n \gg 0$ the length $\ell_A(M/\underline a^nM)$ is given by the Hilbert polynomial
	$e_0(\underline a;M) \binom{d+n}{d} + \ldots + e_d(\underline a;M)$. Therefore, it follows that $\chi_A(\mathcal{L}_{\bullet}(\underline a,\underline a,M;n)) =
	e_0(\underline a;M)$. This completes the argument.
\end{proof}

The more general situation of a system of parameters $\underline a = a_1,\ldots, a_d$ of a finitely generated $A$-module
$M$ of a local ring $(A,\mathfrak{m})$ and an ideal $\mathfrak{q} \supset \underline a$ with $a_i \in \mathfrak{q}^{c_i},
i = 1,\ldots,d$ is investigated in the following.

\begin{proposition} \label{mult-2}
	With the previous notation we have the equality
	\[
	e_0(\underline a;M) = c_1 \cdot \ldots \cdot c_d \cdot e_0(\mathfrak{q};M) + \chi_A(K_{\bullet}(\underline a,\mathfrak{q},M;n))
	\]
	for all $n \gg 0$. In particular, for all $n \gg 0$ the Euler characteristic $\chi_A(K_{\bullet}(\underline a,\mathfrak{q},M;n))$
	is a constant.
\end{proposition}

\begin{proof}
	The proof follows by the inspection of the short exact sequence of complexes
	\[
	0 \to K_{\bullet}(\underline a,\mathfrak{q},M;n) \to K_{\bullet}(\underline a;M) \to \mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n) \to 0
	\]
	of \ref{def-1}. By view of Proposition \ref{mult-1} we have $e_0(\underline a;M)$ for the Euler characteristic
	of the complex in the middle. For the Euler characteristic on the right we get (see \ref{char} (b))
	\[
	\chi_A(\mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n)) = \sum_{i=0}^{d} (-1)^i \sum_{1 \leq j_1 < \ldots < j_i \leq d}
	\ell_A(M/\mathfrak{q}^{n-c_{j_1}-\ldots- c_{j_i}}M),
	\]
	which gives the first summand in the above formula (see also \cite{BS} for the details in the case of $M = A$).
	This finally proves the claim.
\end{proof}

For several reasons it would be interesting to have an answer to the following problem.

\begin{problem} \label{prob}
	With the notation of Proposition \ref{mult-2} it would be of some interest to give
	an interpretation of $\chi_A(\underline a,\mathfrak{q},M) := \chi_A(K_{\bullet}(\underline a,\mathfrak{q},M;n))$ for
	large $n \gg 0$ independently of $n$. By a slight
	modification of an argument given in \cite{BS} it follows that $\chi_A(\underline a,\mathfrak{q},M) \geq 0$.
\end{problem}

First we shall use the previous results in order to prove a few formulas for the multiplicity.
The following Theorem provides a simplified proof of some of the main results of Auslander and Buchsbaum (see \cite{AB}), originally proved by the use of spectral sequences.

\begin{theorem} \label{mult-3}
	Let $(A,\mathfrak{m})$ denote a local ring with $\underline a = a_1,\ldots,a_d, d = \dim_A M,$ a system
	of parameters for a finitely generated $A$-module $M$.
	\begin{itemize}
		\item[(a)] $e_0(\underline a;M) = e_0(\underline a';M/aM) - e_0(\underline a'; 0:_M a)$, where $a = a_1$ and $\underline a' = a_2,\ldots,a_d$.
		\item[(b)] $e_0(\underline a;M) = e_0(a,\underline a';M) + e_0(b,\underline a';M)$, where $a_1 = a\cdot b$ and
		$\underline a' = a_2,\ldots,a_d$.
		\item[(c)] $e_0(\underline a^{\underline{n}};M) = n \cdot e_0(\underline a;M)$, where $\underline a^{\underline{n}} = a_1^{n_1}, \ldots,a_d^{n_d}$ for $\underline{n} = (n_1,\ldots,n_d) \in \mathbb{N}^d$ and $n = n_1 \cdot \ldots \cdot n_d$.
	\end{itemize}
\end{theorem}

\begin{proof}
	We start with the comparison of the Koszul complexes $K_{\bullet}(b;M)$ and $K_{\bullet}(ab;M)$. This leads to the following commutative diagram with exact rows
	\[
	\xymatrix{
		&  &    & 0 \ar[d]   & 0 \ar[d] &  &  &  &  \\
		&  & 0  \ar[r]  & M \ar[d]_{b}  \ar@{=}[r] & M  \ar[r] \ar[d]^{ab} & 0 &  &  \\
		& 0  \ar[r] & 0:_M a \ar[r]  & M \ar[d]  \ar[r]^{a} & M \ar[d]  \ar[r] & M/aM \ar[r] & 0 \\
		&     &    & 0 & 0 &  &  &
		}
	\]
	where the columns are the Koszul complexes. That is, we have an exact sequence of complexes
		\[
		0 \to 0:_M a \to K_{\bullet}(b;M) \to K_{\bullet}(ab;M) \to M/aM \to 0.
		\]
	We tensor now by $K_{\bullet}(\underline a';A)$, a bounded
	complex of free $A$-modules. By the definition it induces an exact sequence of complexes
	\[
	0 \to K_{\bullet}(\underline a';0:_Ma) \to K_{\bullet}(b,\underline a';M) \to K_{\bullet}(ab,\underline a';M)
	\to K_{\bullet}(\underline a';M/aM) \to 0.
	\]
	Now we inspect the previous sequence in the case $b =1$. Then it follows that $ K_{\bullet}(1,\underline a';M)$
	is exact. By virtue of the Euler characteristic and by \ref{mult-1}, the statement in (a) follows.

	Next we look at the case of a general $b$. With the previous result (a) it implies that
	\[
	e_0(ab,\underline a';M) = e_0(a,\underline a';M) +e_0(b,\underline a';M)
	\]
	which proves (b).

	By induction and permutability of the sequence this yields the statement in (c).
\end{proof}

It is noteworthy to say that $e_0(\underline a';0:_Ma) \not= 0$ if and only if $\dim_A 0:_Ma = \dim_A M -1$.

\section{On an equality}
As before let $(A,\mathfrak{m})$ denote a local ring with $M$ a finitely generated $A$-module.
Let $\mathfrak{q}$ denote an ideal of $A$ such that $\ell_A(M/\mathfrak{q}M) < \infty$.
For a system of parameters $\underline a = a_1,\ldots,a_d, d = \dim_A M,$ of $M$ suppose that
$a_i \in \mathfrak{q}^{c_i} \setminus \mathfrak{q}^{c_i+1}, i = 1,\ldots,d$. Besides of the
Rees module $R_M(\mathfrak{q})$ we need the following.

\begin{definition} \label{eq-1}
	With the previous notation we investigate the ring $\mathfrak{R} = A[a_1T^{c_1},\ldots,a_dT^{c_d}] \subset A[T]$.
	Note that $\mathfrak{R}$ is a graded $A$-algebra. Because of $a_i \mathfrak{q}^n \subseteq \mathfrak{q}^{n+c_i},
	i = 1,\ldots,d$ it follows that the Rees module $R_M(\mathfrak{q})$ is an $\mathfrak{R}$-module.
\end{definition}

In the following we shall explore when $R_M(\mathfrak{q})$ is a finitely generated $\mathfrak{R}$-module.

\begin{lemma} \label{eq-2}
	With the previous notation the following conditions are equivalent:
	\begin{itemize}
		\item[(i)] $R_M(\mathfrak{q})$ is a finitely generated $\mathfrak{R}$-module.
		\item[(ii)] There is a positive integer $k$ such that $\mathfrak{q}^n M = \sum_{i=1}^{d} a_i\mathfrak{q}^{n-c_i}M$ for all $n > k$.
		\item[(iii)] The initial forms $a_1^{\star}, \ldots, a_d^{\star}$ are a system of parameters
		of $G_M(\mathfrak{q})$.
	\end{itemize}
\end{lemma}

\begin{proof}
	First we prove (i) $\Longrightarrow$ (ii). If $R_M(\mathfrak{q})$ is finitely generated over $\mathfrak{R}$,
	then $$R_M(\mathfrak{q}) = \mathfrak{R}(m_1T^{d_1},\ldots , m_rT^{d_r}) \text{ for } m_iT^{d_i} \in R_M(\mathfrak{q})_{d_i}.$$
	Let $k = \max \{d_1,\ldots , d_r\}$ and $n > k$. Let $m \in \mathfrak{q}^nM$ and therefore
	\[
	m T^n = \sum_{j=1}^{r} (\sum_{\underline{\alpha}_j} r_{\underline{\alpha}}^{(j)}(a_1T^{c_1})^{\alpha_{j,1}}
	\cdots (a_dT^{c_d})^{\alpha_{j,d}})\,  m_j T^{d_j} \text{ with } r_{\underline{\alpha}}^{(j)} \in A
	\]
	and $d_j + \sum c_i \alpha_{j,i} \geq n$. The last inequality implies
	\[
	\sum c_i \alpha_{j,i} \geq n - d_j \geq n-k > 0.
	\]
	Therefore for each $j$ there is an $i$ such that $\alpha_{j,i} > 0$ and
	$mT^n \in \sum_{i=1}^{d} a_i \mathfrak{q}^{n-c_i}MT^n$,
	which proves the statement in (ii).

	For the converse we claim $R_M(\mathfrak{q}) = \mathfrak{R}(M,\mathfrak{q}MT, \ldots, \mathfrak{q}^kM T^k).$
	Let $m T^n$ with $m \in \mathfrak{q}^nM$. If $n \leq k$ then clearly $mT^n \in \mathfrak{R}(M,\mathfrak{q}MT, \ldots, \mathfrak{q}^kM T^k)$. If $n > k$, then $m = \sum_{i=1}^{d} a_i m_i, m_i \in \mathfrak{q}^{n-c_i}M$.
	By induction $m_i T^{n-c_i} \in \mathfrak{R}(M,\mathfrak{q}MT, \ldots, \mathfrak{q}^kM T^k)$ and therefore
	$mT^n = \sum_{i=1}^{d} a_iT^{c_i}m_i T^{n-c_i} \in\mathfrak{R}(M,\mathfrak{q}MT, \ldots, \mathfrak{q}^kM T^k)$,
	which finishes the argument.

	For the equivalence of (ii) and (iii) recall that $a_1^{\star}, \ldots, a_d^{\star}$ is a homogeneous
	system of parameters if and only if $[G_M(\mathfrak{q})/(a_1^{\star}, \ldots, a_d^{\star})G_M(\mathfrak{q})]_n = 0$ for all $n > k$. This is equivalent to
	\[
	\mathfrak{q}^nM = \sum_{i=1}^{d} a_i \mathfrak{q}^{n-c_i}M + \mathfrak{q}^{n+1}M \text{ for all } n > k.
	\]
	By Nakayama Lemma this is equivalent to the condition in (ii).
\end{proof}

For the following we define $c = c_1 \cdot \ldots \cdot c_d$ and $e_i = c/c_i, i = 1,\ldots,d$. Then
$\underline a^{\underline{e}} = a_1^{e_1}, \ldots,a_d^{e_d}$ is a system of parameters of $M$ and
$a_i^{e_i} \in \mathfrak{q}^c$. With these notation we get the following commutative diagram
\[
\begin{array}{ccc}
R_M(\mathfrak{q}^c) & \subset & R_M(\mathfrak{q})\\
 \cup & & \cup \\
 \mathfrak{S} & \subset & \mathfrak{R},
\end{array}
\]
where $\mathfrak{S} = A[a_1^{e_1}T^c, \ldots, a_d^{e_d}T^c]$. It is easily seen that $\mathfrak{S} \subset  \mathfrak{R}$ is a finitely generated extension since it is integral. Note that $a_i^{e_i}T^c = (a_iT^{c_i})^{e_i},
i = 1,\ldots,d$.

\begin{corollary} \label{eq-3}
	With the previous notation the following conditions are equivalent:
	\begin{itemize}
		\item[(i)] The initial forms $a_1^{\star}, \ldots, a_d^{\star}$ are a system of parameters
		of $G_M(\mathfrak{q})$.
		\item[(ii)] There is an integer $k > 0$ such that $\mathfrak{q}^{nc} M = \sum_{i=1}^{d} a_i^{e_i} \mathfrak{q}^{nc-c}M$ for all $n > k$.
		\item[(iii)] The initial forms $(a_1^{e_1})^\star,\ldots,(a_d^{e_d})^\star$ in $G_A(\mathfrak{q}^c)$ are
		a system of parameters in $G_M(\mathfrak{q}^c)$.
	\end{itemize}
\end{corollary}

\begin{proof}
	We have the isomorphism $R_M(\mathfrak{q}^c) \cong M[\mathfrak{q}^cT^c] =: \mathfrak{M}$ and
	$$
	R_M(\mathfrak{q}) \cong \mathfrak{M} \oplus \mathfrak{q}T \mathfrak{M} \oplus \ldots
	\oplus  \mathfrak{q}^{c-1}T^{c-1} \mathfrak{M}.
	$$
	Whence $\mathfrak{S} \subset R_M(\mathfrak{q}^c)$ is a finitely generated extension if and only if
	$\mathfrak{S} \subset R_M(\mathfrak{q})$ is a finitely generated extension. Then Artin-Rees
	lemma yields the equivalence of (i) and (ii) by view of \ref{eq-2}.

	The equivalence of (ii) and (iii) follows by Lemma \ref{eq-2}.
\end{proof}

\begin{corollary} \label{eq-4}
	With the previous notation suppose that $a_1^{\star}, \ldots, a_d^{\star}$ forms a system of parameters
	in $G_M(\mathfrak{q})$. Then $e_0(\underline a;M) = c_1 \cdot \ldots \cdot c_d \cdot e_0(\mathfrak{q};M)$ and therefore   $\chi_A(K_{\bullet}(\underline a,\mathfrak{q},M;n)) = 0$ for $n \gg 0$.
\end{corollary}

\begin{proof}
	By view of \ref{eq-3} we get $(\mathfrak{q}^c)^n M = \sum_{i=1}^{d} a_i^{e_i} (\mathfrak{q}^c)^{n-1} M$ and
	$({\mathfrak{q}^c})^l M \subset \underline a^{\underline{e}}M \subset \mathfrak{q}^cM$ for some $l \in \mathbb{N}$.
	Moreover it follows that $(\mathfrak{q}^c)^{n+k}M \subseteq (\underline a^{\underline{e}})^k M \subseteq (\mathfrak{q}^c)^k M$ for all $k \geq l$. Then
	\[
	\ell_A(M/(\mathfrak{q}^{c})^{n+k}M) \geq \ell_A(M/(\underline a^{\underline{e}})^k M) \geq
	\ell_A(M/(\mathfrak{q}^c)^k M),
	\]
	which implies that $e_0(\mathfrak{q}^c;M) = e_0(\underline a^{\underline{e}};M)$. Because of $e_0(\mathfrak{q}^c;M) =
	c^d\cdot e_0(\mathfrak{q};M)$ as easily seen and $e_0(\underline a^{\underline{e}};M) = e_1 \cdot \ldots \cdot e_d\cdot e_0(\underline a;M)$ (see \ref{mult-3}). This finishes the proof.
\end{proof}

The previous result is a generalization of \cite[Theorem 5.1]{BS} to the situation of finitely
generated $A$-modules. In the case of a formally equidimensional ring the converse is also true
(see \cite[Theorem 5.2]{BS}).

\section{The subregular case}
As a consequence of the definition of $\mathcal{L}_{\bullet}(\underline a,\mathfrak{q},M;n)$ we get the following
equality
\[
c_1\cdot \ldots \cdot c_d \cdot e_0(\mathfrak{q};M) = \sum_{i=1}^{d} (-1)^i \ell_A(L_i(\underline a,\mathfrak{q},M;n)) \text{ for all } n \gg 0.
\]
We have $L_0(\underline a,\mathfrak{q},M;n) \cong M/(\underline a,\mathfrak{q}^n)M$. The other homology modules are
difficult to describe. For the vanishing of some of them in relation to the existence of $G_M(\mathfrak{q})$-regular
sequences we refer to \cite{K}.

In the previous section we have shown that $e_0(\underline a;M) = c \cdot e_0(\mathfrak{q};M)$ provided
$a_1^{\star}, \ldots, a_d^{\star}$ forms a system of parameters of $G_M(\mathfrak{q})$. In the
next, we consider the case that $(a_1^{\star}, \ldots, a_d^{\star})G_A(\mathfrak{q})$ contains
a $G_M(\mathfrak{q})$-regular sequence of length $d-1$.

We start with the behavior of $\chi(\underline a,\mathfrak{q},M;n)$ by passing to a certain element.
To this end, let $\underline a' = a_2,\ldots,a_d, a = a_1$ and $c_1 =f$.

\begin{lemma} \label{reg-1}
	Suppose that $a^{\star}$ is $G_M(\mathfrak{q})$ regular. Then  $\chi(\underline a,\mathfrak{q},M;n) = \chi(\underline a',\mathfrak{q},M/aM;n)$ for all $n \in \mathbb{Z}$.

\end{lemma}

\begin{proof}
	If $a^{\star}$ is $G_M(\mathfrak{q})$-regular, then $0:_M a = 0$ and
	$\mathfrak{q}^{n}M :_M a = \mathfrak{q}^{n-f}M$ for all $n \geq 0$. So it follows that
	$0:_{R_M(\mathfrak{q})} aT^f = 0$ and $R_M(\mathfrak{q})/(aT^f)R_M(\mathfrak{q}) \cong R_{M/aM}(\mathfrak{q})$.

	Moreover, there is the following diagram with exact rows
	\[
	\xymatrix{
		&  & 0 \ar[d]   & 0 \ar[d] &  &  &    \\
		& 0  \ar[r]  & R_M(\mathfrak{q})(-f) \ar@{=}[d]  \ar@{=}[r] & R_M(\mathfrak{q})(-f)  \ar[r] \ar[d]^{aT^f} & 0 &  &  \\
		& 0 \ar[r]  & R_M(\mathfrak{q})(-f) \ar[d]  \ar[r]^{aT^f} & R_M(\mathfrak{q}) \ar[d]  \ar[r] & R_{M/aM}(\mathfrak{q}) \ar[r] & 0 & \\
		&    & 0 & 0 &  &  &
	}
	\]
	where the columns are the Koszul complexes $K_{\bullet}(1;R_M(\mathfrak{q}))(-f)$ and
	$K_{\bullet}(aT^f;R_M(\mathfrak{q}))$ resp. That is, we have a short exact sequence of complexes
	\[
	0 \to K_{\bullet}(1;R_M(\mathfrak{q}))(-f) \stackrel{aT^f}{\longrightarrow}
	K_{\bullet}(aT^f;R_M(\mathfrak{q})) \to R_{M/aM}(\mathfrak{q}) \to 0.
	\]
	By tensoring with $K_{\bullet}(\underline{a'T^{c'}}, R_A(\mathfrak{q}))$ it provides a
	short exact sequence of complexes
	\[
	0 \to K_{\bullet}(1,\underline{a'T^{c'}},R_M(\mathfrak{q}))(-f) \stackrel{aT^f}{\longrightarrow} K_{\bullet}(\underline{a T^c};R_M(\mathfrak{q})) \to K_{\bullet}(\underline{a'T^{c'}};R_{M/aM}(\mathfrak{q}))\to 0.
	\]
	Since the first Koszul complex in the previous sequence is exact (see \ref{cone-1}),
	the claim follows by the definition.
\end{proof}

The following result is a particular case of \cite[VIII, Lemma 3]{ZS}
resp. to \cite[Lemma 1.6]{RV},  a generalization to filtered modules.

\begin{lemma} \label{reg-2}
	Let $\mathfrak{q} \subset A$ denote an ideal such that $\ell_A(M/\mathfrak{q}M) < \infty$.
	\begin{itemize}
		\item[(a)] Suppose that $a^{\star}$ is $G_M(\mathfrak{q})$ regular. Then
		$f \cdot e_0(\mathfrak{q};M) = e_0(\mathfrak{q};M/aM)$.
		\item[(b)] Let $\dim_A M = 1$ and $a \in \mathfrak{q}^f$ be a parameter of $M$. Then
		$$
		f\cdot e_0(\mathfrak{q};M) = \ell_A(M/aM) - \ell_A(\mathfrak{q}^nM :_M a/\mathfrak{q}^{n-f}M)
		$$
		for all $n \gg 0$. In particular $\ell_A(\mathfrak{q}^nM :_M a/\mathfrak{q}^{n-f}M)$ is a constant
		for all $n \gg 0$.
	\end{itemize}
\end{lemma}

\begin{proof}
	The statements follow by counting the length in the exact sequence
	\[
	0 \to \mathfrak{q}^n M :_M a/\mathfrak{q}^{n-f} M \to M/\mathfrak{q}^{n-f} M \stackrel{a}{\to}
	M/\mathfrak{q}^n M \to M/(a,\mathfrak{q}^n) M \to 0
	\]
	for all $n \in \mathbb{N}$. If $a^{\star}$ is $G_M(\mathfrak{q})$ regular, then
	$\mathfrak{q}^n M :_M a = \mathfrak{q}^{n-f} M$ for all $n \in \mathbb{N}$. If
	$\dim_A M = 1$ and $a \in \mathfrak{q}^f$ is a parameter of $M$, then
	$\ell_A(M/\mathfrak{q}^nM) - \ell_A(M/\mathfrak{q}^{n-f}M) = f \cdot e_0(\mathfrak{q};M)$
	for all $ n \gg 0$. Moreover $\mathfrak{q}^n M \subset aM$ for all $n \gg 0$. For details see \cite[VIII, Lemma 3]{ZS} resp. \cite[Lemma 1.6]{RV}.
\end{proof}

Now we are prepared for the main situation of this section.

\begin{corollary} \label{reg-3}
	With the previous notation assume that $a_1^{\star},\ldots,a_{d-1}^{\star}$ is a $G_M(\mathfrak{q})$-regular
	sequence. Then
	\[
	c_1\cdot \ldots \cdot c_d \cdot e_0(\mathfrak{q};M) = \ell_A(M/\underline a M) - \ell_A((\underline a',\mathfrak{q}^n)M:_M a_d/
	(\underline a',\mathfrak{q}^{n-c_d})M)
	\]
	for all $n \gg 0$ where $\underline a' = a_1,\ldots,a_{d-1}$.
\end{corollary}

\begin{proof}
	By an iterative application of \ref{reg-2} (a) it follows $c_1\cdot \ldots\cdot c_{d-1} \cdot e_0(\mathfrak{q};M)
	= e_0(\mathfrak{q};M/\underline a' M)$. By view of \ref{reg-2} (b) it implies $c_d \cdot e_0(\mathfrak{q};M/\underline a' M)
	= \ell_A(M/\underline a M) - \ell_A((\underline a',\mathfrak{q}^n)M:_M a_d/
	(\underline a',\mathfrak{q}^{n-c_d})M)$. Putting both together yields the claim.
\end{proof}

It is a problem to give an interpretation of the constant $\ell_A((\underline a',\mathfrak{q}^n)M:_M a_d/
(\underline a',\mathfrak{q}^{n-c_d})M)$ in intrinsic data of the module $M$. As a partial result in this direction we get the
following.

\begin{lemma} \label{reg-4}
	With the notation and assumption of \ref{reg-3} we have the following inequality
	\[
	\ell_A(M/\underline a M) \geq c \cdot e_0(\mathfrak{q};M) + \mathfrak{x},
	\]
	where $\mathfrak{x}=  \ell_A([\Ext^{d-1}_{G_A(\mathfrak{q})}(G_A(\mathfrak{q})/(\underline{a}^{\star}),G_M(\mathfrak{q}))]_{n-s-1})$
	is a constant for $n \gg 0$ and  $s$ denotes $s = c_1+\ldots +c_d$.
\end{lemma}

\begin{proof}
	There is an injection of
	\[
	\mathfrak{X} := [(\underline a'^{\star})G_M(\mathfrak{q}) : a_d^{\star}/(\underline a'^{\star})G_M(\mathfrak{q})]_{n-c_d-1} \cong
	((\underline a',\mathfrak{q}^n)M:_M a_d) \cap (\underline a',\mathfrak{q}^{n-c_d-1})M)/ (\underline a',\mathfrak{q}^{n-c_d})M
	\]
	into $(\underline a',\mathfrak{q}^n)M:_M a_d/ (\underline a',\mathfrak{q}^{n-c_d})M$. Moreover, there is an isomorphism
	$$
	\mathfrak{X} \cong [\Hom_{G_A(\mathfrak{q})}(G_A(\mathfrak{q})/(\underline{a}^{\star}), G_M(\mathfrak{q})/
	(\underline a'^{\star})G_M(\mathfrak{q}))]_{n-c_d-1}.
	$$
	Since $\underline a'^{\star}$ is a $G_M(\mathfrak{q})$-regular sequence of length $d-1$, it follows that
	\[
	\mathfrak{X} \cong [\Ext^{d-1}_{G_A(\mathfrak{q})}(G_A(\mathfrak{q})/(\underline{a}^{\star}),G_M(\mathfrak{q}))]_{n-s-1},
	\]
	which proves the claim.
\end{proof}

We conclude with a geometric application on the local Bezout numbers.

\begin{example} \label{reg-5}
	Let $C = V(F), D = V(G) \subset \mathbb{P}^2_{\Bbbk}$ be two curves in the projective plane without a common
	component. Let $\mu(P;C,D)$ denote the local intersection multiplicity of $P$ in $C \cap D$. In the case
	of $P$ is the origin, it follows that $\mu(P;C,D) = \ell_A(A/(f,g))$, where $A = \Bbbk[x,y]_{(x,y)}$ and $f,g$ denote
	the equations in $A$. Since $C,D$ have no component in common, $\{f,g\}$ forms as system of parameters in $A$.
	Let $\mathfrak{m}$ denote the maximal ideal of $A$. Then $B := \Bbbk[X,Y] = G_A(\mathfrak{m})$ and $1 = e_0(\mathfrak{m};A)$.
	We distinguish two cases:
	\begin{itemize}
		\item[1.] $C$ and $D$ intersect transversally in the origin. Then $f^{\star},g^{\star}$ form a
		homogeneous system of parameters in $B$ and therefore
		\[
		\ell_A(A/(f,g)) = c \cdot d,
		\]
		where $c,d$ denote the initial degree of $f,g$ resp.
		\item[2.] Suppose that $C$ and $D$ do not intersect transversally. Then
		\[
		\ell_A(A/(f,g)) \geq c\cdot d +t,
		\]
		where $t$ denotes the number of common tangents of $f$ and $g$ at the origin counted with multiplicities.
	\end{itemize}
	\begin{proof}
		First note that $A$ is a Cohen-Macaulay ring and therefore $e_0(f,g;A) = \ell_A(A/(f,g))$. Then the
		equality in the first case is a consequence of \ref{eq-4}. To this end note that $f^{\star},g^{\star}$
		forms a system of parameters in $B$ provided $C$ and $D$ intersect transversally in the origin.

		For the second case we use \ref{reg-4}. To this end we have to show that $\mathfrak{x} =t$. We
		put $\mathfrak{Y} = f^{\star}B:_B g^{\star}/f^{\star}B$. Since $f^{\star},g^{\star}$ are not relatively
		prime, we write $f^{\star} = h\cdot f', g^{\star} = h \cdot g'$ with homogeneous polynomials $f',g',h \in B$,
		where $f',g'$ are relatively prime. Then
		\[
		\mathfrak{Y} = f'hB:_B g'h/f'hB \cong f'B/f'hB \cong B/hB[-\deg f']
		\]
		and $\dim_{\Bbbk} \mathfrak{Y}_n = \deg h$ for all $n \gg 0$. Since $\deg h$ is the number of common tangents
		counted with multiplicities, this confirms the second case.
	\end{proof}
\end{example}

The second case was also proved by Byd\u{z}ovsk\'y (see \cite{By}). Note that this is an
improvement of the corresponding result in \cite{Fi} where it is shown that $\ell_A(A/(f,g)) \geq c\cdot d +1$
in case there is a common tangent.

A further discussion of the difference $\ell_A(A/(f,g)) - c\cdot d - t \geq 0$ is given in \cite{BS}.
There is also another approach by blowing-ups.

{\bf Acknowledgement:} The authors are grateful to the reviewer for bibliographical comments
and suggestions.