Our nation is anxious and torn by the tragic shooting of Michael Brown and the subsequent conflicting reports from the police officer Darren Wilson and bystanders. These events of course are painful and personal in their own right. They do not stand alone but are part of a larger conflictual tragic and frequently misunderstood story of race relations in our country. These sins are not one sided. Perfection in thought, emotion, or action is not necessary in order to carry valid points of view.
My call to all is to not let our perspectives start with the portrayal of events in news reports, commentary, or live footage. Rather our perspective starts with who you are as a daughter or son of God redeemed through the death of Jesus who suffered unjustly at the hands and hard hearts of the favored elite of the day. He did this with a heart of love and compassion for all even those who put him to death asking His Father to forgive them. Jesus did not take a side with the normal points of view in the issue divides of the generation but called people from every side and view point to realign with the truth and grace of his Father in Heaven.
The role of the church--and each one of us as part of the church called disciples-- is to align with the One who says " I am the Way, the Truth, and the Life. And from this solid foundation to minister grace and truth in love into conflicts and loss of hope.
Several weeks ago as we studied Ephesians chapter 2 the dominant theme was that where the racial divides in the past is no longer the decisive dividing line. The dividing line is who is following Jesus and who is not following Jesus. This new group of Jesus followers are a family that is colorful and diverse. This transformed people then carry a message in action, attitude, and conversation that listens and shares the heart of Christ.
At CCF we have a unique opportunity to listen to one another and hear the pain and hopes. Will you join me in the following?
Can we ask God to not let us speak unless we speak grace and truth spoken together with a heart of love?
Can we in the words of Francis of Assisi seek to understand more then be understood?
Can we plead with Jesus to activate his role as the Good Shepherd, Prince of Peace, and Lord of this world?
Can we walk with one another in bonds of peace and the unity of the Spirit in the name of Christ Jesus and reach across divides that usually alienate and separate?
Can we not be content with a pretense of peace because we avoid topics but rather cultivate a culture that seeks peace and pursues it by listening, loving, caring, and facing Jesus, and taking actions together that we know please the Spirit.
Can we express that "Jesus is Lord" is the dominant identification and reality statement in our lives. That there's no other identity that defines me more completely: not my race, education profession not even my name.
The Bible says: Human anger does not produce the righteousness God desires. (James 1:20) and also: Don't sin by letting anger control you. (Ephesians 4:26) The thought is emotional passion is not sin but of itself it does not bring about the reconciliation that furthers the plan and purpose of God. The passion is useful to press us into Jesus and his body the church so our responses and sustained with the power of the Spirit and produce enduring transformation.


text:
\section*{Introduction}

The local Euler obstruction is an invariant defined by MacPherson in \cite{MacPherson} as one of the main ingredients in his proof of the Deligne-Grothendieck conjecture about the existence of Chern classes for singular varieties. The local Euler obstruction of $X$ at the origin, where $X$ is a sufficiently small representative of the equidimensional analytic germ $(X,0)$, is denoted by ${Eu}_{X}(0)$.

After MacPherson's pioneer work, the local Euler obstruction has been studied by many authors. In \cite{LT}, L\^e and Teissier showed that the Euler obstruction is equal to an alternated sum of multiplicities of generic polar varieties of $X$ at $0$. In \cite{BLS}, Brasselet, L\^{e} and Seade proved a Lefschetz type formula for the local Euler obstruction: they related this invariant with the topology of the Milnor fibre on $X$ of a generic linear function.

In \cite{BMPS}, Brasselet, Massey, Parameswaran and Seade defined a new invariant associated to a function $f$ defined on the variety $X$, called the Euler obstruction of the function and denoted by ${Eu}_{f,X}(0)$. In \cite{STV}, Seade, Tib\u{a}r and Verjovsky showed that, up to sign, the Euler obstruction of a function is equal to the number of critical points of a Morsefication of $f$ lying on the regular part of $X$ which shows that this invariant can be seen as a generalization of the Milnor number (\cite{BMPS,STV}).

Based on the multiplicity formula of L\^e and Teissier, J.-P. Brasselet conjectured that a relative version of the Euler obstruction should satisfy a similar multiplicity formula. In  \cite{DG}, the authors define an invariant called the Brasselet number, denoted by ${B}_{f,X}(0)$, and proved that it satisfies a Lê-Greuel type formula, giving an answer to Brasselet's conjecture.

When $f$ has isolated singularity, ${B}_{f,X}(0)= {Eu}_{X}(0)-{Eu}_{f,X}(0)$ and in the general case, ${B}_{f,X}(0)= {Eu}_{X}(0)-{D}_{f,X}(0)$, where ${D}_{f,X}(0)$ is the defect defined and studied in Section 6 of \cite{BMPS}. Such defect is related to the L\^e numbers (Proposition 6.1 in \cite{BMPS}), when $X$ is smooth, and to the L\^e-Vogel numbers (Theorem 6.2 in \cite{BMPS}) in the general case. Therefore, considering these relations and the result of Seade, Tib\u{a}r and Verjovsky mentioned above, the defect ${D}_{f,X}(0)$ appears to be a good singular version of the Milnor number, whereas the Brasselet number appears to be a good relative version of the Euler obstruction.

The Euler obstruction of a map, defined by Grulha \cite{Grulha2008}, and the Chern number of collections of forms, defined by Ebeling and Gusein-Zade \cite{Ebeling2007a}, were defined in the first decade of the 21st century and, in \cite{brasselet2010euler}, Brasselet, Grulha and Ruas related these two invariants. The Euler obstruction of a map is a generalization of the Euler obstruction of a function, defined in \cite{BMPS}.

The  Chern number of collections of forms has a foudantion similar to the Euler obstruction of maps; however it has traveled a different trajectory ``avoiding'' cellular decompotitions, which are, in some sense, covered by the study of the loci of collections of forms.  That is another way to present a generalization of the local Euler obstruction and, in \cite{GaffneyGrulhaJofSing}, Gaffney and Grulha present an algebraic treatment to study the Chern number.

Since the Euler obstruction of a function-germ $g$ at the origin gives important topological information about $g$, more precisely, it counts the number of Morse points on the regular part of $X$ of a generic perturbation of $g$, a natural question has risen: which kind of topological information could be encoded inside the Euler obstruction of a map and how this invariant is related with the Euler obstruction of the coordinate functions of $f$.

 In this paper we relate all these invariants and present geometrical  information that are captured by the Euler obstruction of a map, and by the Chern number, in the context of an application $f: (X,0) \to (\mathbb{C}^{2},0).$

\begin{center}\textbf{Acknowledgements}\end{center}

The authors are grateful to professor Brasselet, to whom they dedicate this article on the occasion of his 75th birthday. The first author started his research on the local Euler obstruction and its generalizations during his Ph.D. in the Universit\'{e} de la M\'{e}diterran\'{e}e (currently called Aix-Marseille Universit\'{e})  under the advisory of professor Jean-Paul Brasselet. Some questions answered in this work are strong related and raised from inicial questions proposed by Professor Brasselet to the first author during his Ph.D or elaborated by the doctoral jury during his thesis defense in 2007.

The authors also thank professors Dutertre, Gaffney, Ruas and Seade for all productive conversations about Euler obstruction, GSV-index and related topics for their important remarks and suggestions that were crucial to the progress of this work. We also thank professors Ebeling and Zach for the fruitful conversations during the Thematic Program on Singularity Theory, at IMPA, Rio de Janeiro, Brazil, 2020, which were essential in the development of this paper.

The first author was supported by FAPESP, under grant 2019/21181-02, and by CNPq, grant 303046/2016-3. The second author was partially supported by DMA-UFTM. The third author was supported by FAPESP, grant 2015/25191-9. The authors also thank PROBAL (CAPES-DAAD), grant 88881.198862/2018- 01.

\section{Local Euler obstruction and and generalizations}

\hspace{0,5cm} In this section, we recall  the definition of the local Euler obstruction, the Euler obstruction of a function and the Brasselet number, both generalizations of the local Euler obstruction. The main references for this section are \cite{Ms1,MacPherson,DG, santana2019brasselet}.

Let $(X,0)\subset(\mathbb{C}^n,0)$ be an equidimensional reduced complex analytic germ of dimension $d$ in a open set $U\subset\mathbb{C}^n.$ Consider a complex analytic Whitney stratification $\{V_{\lambda}\}$ of $U$ adapted to $X$ such that $\{0\}$ is a stratum. We choose a small representative of $(X,0),$ denoted by $X,$ such that $0$ belongs to the closure of all strata. We write $X=\cup_{i=0}^{q} V_i,$ where $V_0=\{0\}$ and $V_q$ denotes the regular part $X_{reg}$ of $X.$ We suppose that $V_0,V_1,\ldots,V_{q-1}$ are connected and that the analytic sets $\overline{V_0},\overline{V_1},\ldots,\overline{V_q}$ are reduced. We write $d_i=dim(V_i), \ i\in\{1,\ldots,q\}.$ Note that $d_q=d.$

Let $G(d,n)$ be the Grassmannian manifold, $x\in X_{reg}$ and consider the Gauss map $\phi: X_{reg}\rightarrow U\times G(d,n)$ given by $x\mapsto(x,T_x(X_{reg})).$

\begin{definition}
The closure of the image of the Gauss map $\phi$ in $U\times G(d,n)$, denoted by $\tilde{X}$, is called \textbf{Nash modification} of $X$. It is a complex analytic space endowed with a holomorphic projection map $\nu:\tilde{X}\rightarrow X.$
\end{definition}

Consider the extension of the tautological bundle $\mathcal{T}$ over $U\times G(d,n).$ Since \linebreak$\tilde{X}\subset U\times G(d,n)$, we consider $\tilde{T}$ the restriction of $\mathcal{T}$ to $\tilde{X},$ called the \textbf{Nash bundle}, and $\pi:\tilde{T}\rightarrow\tilde{X}$ the projection of this bundle.

In this context, denoting by $\varphi$ the natural projection of $U\times G(d,n)$ at $U,$ we have the following diagram:

$$\xymatrix{
\tilde{T} \ar[d]_{\pi}\ar[r] & \mathcal{T}\ar[d] \\
\tilde{X}\ar[d]_{\nu}\ar[r] & U\times G(d,n)\ar[d]^{\varphi} \\
X\ar[r] & U\subseteq\mathbb{C}^n \\}  $$

Adding the stratum $U \setminus X$ we obtain a Whitney stratification of $U$. Let us denote the restriction to $X$ of the tangent bundle of $U$ by $TU|_{X}$. We know that a stratified vector field $v$ on $X$ means a continuous section of $TU|_{X}$ such that if $x \in V_{i} \cap
 X$ then $v(x) \in T_{x}(V_{i})$. From Whitney's condition (a), one has the following lemma.

\begin{lemma} \cite{brasselet1981classes}.
	Every stratified vector field $v$, non-null on a subset $A \subset X$, has a canonical lifting to a non-null section $\tilde{v}$ of the Nash bundle $\widetilde {\mathcal{T}}$ over $\nu^{-1}(A) \subset \widetilde{X}$.
\end{lemma}

Now consider a stratified radial vector field $v(x)$ in a neighborhood of $\left\{0\right\}$ in $X$, {\textit{i.e.}}, there is $\varepsilon_{0}$ such that for every $0<\varepsilon \leq \varepsilon_{0}$, $v(x)$ is pointing outwards the ball ${B}_{\varepsilon}$ over the boundary ${S}_{\varepsilon} := \partial{{B}_{\varepsilon}}$.

The following interpretation of the local Euler obstruction has been given by Brasselet and Schwartz in \cite{brasselet1981classes}. As said before, the original definition is presented in \cite{MacPherson}.

\begin{definition}
	Let $v$ be a radial vector field on $X \cap {S}_{\varepsilon}$ and $\tilde{v}$ the lifting of $v$ on $\nu^{-1}(X \cap {S}_{\varepsilon})$ to a section of the Nash bundle. The \textbf{local Euler obstruction} (or simply the \textbf{Euler obstruction}) ${\rm Eu}_{X}(0)$ is defined to be the obstruction to extending $\tilde{v}$ as a nowhere zero section of $\widetilde {\mathcal{T}}$ over $\nu^{-1}(X \cap {B}_{\varepsilon})$.
\end{definition}

More precisely, let $\mathcal{O}{(\tilde{v})} \in \mathbb{H}^{2d}(\nu^{-1}(X \cap
{B_{\varepsilon}}), \nu^{-1}(X \cap {S_{\varepsilon}}), \mathbb{Z})$ be the
obstruction cocycle to extending $\tilde{v}$ as a nowhere zero section of
$\widetilde {\mathcal{T}}$ inside $\nu^{-1}(X\cap {B_{\varepsilon}})$. The
local Euler obstruction ${\rm Eu}_{X}(0)$ is defined as the evaluation of the
cocycle $\mathcal{O}(\tilde{v})$ on the fundamental class of the pair $[\nu^{-1}(X
\cap {B_{\varepsilon}}), \nu^{-1}(X \cap {S_{\varepsilon}})]$ and therefore
it is an integer.

In \cite{BLS}, Brasselet, Lê and Seade proved a formula to make the calculation of the Euler obstruction easier.

\begin{theorem}(Theorem 3.1 of \cite{BLS})
Let $(X,0)$ and $\mathcal{V}$ be given as before, then for each generic linear form $l,$ there exists $\varepsilon_0$ such that for any $\varepsilon$ with $0<\varepsilon<\varepsilon_0$ and $\delta\neq0$ sufficiently small, the Euler obstruction of $(X,0)$ is equal to

$$Eu_X(0)=\sum^{q}_{i=1}\chi(V_i\cap B_{\varepsilon}\cap l^{-1}(\delta)).Eu_{X}(V_i),$$

\noindent where $\chi$ is the Euler characteristic, $Eu_{X}(V_i)$ is the Euler obstruction of $X$ at a point of $V_i, \ i=1,\ldots,q$ and $0<|\delta|\ll\varepsilon\ll1.$
\end{theorem}

Let us give the definition of another invariant introduced by Brasselet, Massey, Parameswaran and Seade in \cite{BMPS}. Let $f:X\rightarrow\mathbb{C}$ be a holomorphic function with isolated singularity at the origin given by the restriction of a holomorphic function $F:U\rightarrow\mathbb{C}$ and denote by $\overline{\nabla}F(x)$ the conjugate of the gradient vector field of $F$ at $x\in U,$ $$\overline{\nabla}F(x):=\left(\overline{\frac{\partial F}{\partial x_1}},\ldots, \overline{\frac{\partial F}{\partial x_n}}\right).$$

Since $f$ has an isolated singularity at the origin, for all $x\in X\setminus\{0\},$ the projection $\hat{\zeta}_i(x)$ of $\overline{\nabla}F(x)$ over $T_x(V_i(x))$ is nonzero, where $V_i(x)$ is a stratum containing $x.$ Using this projection, the authors constructed, in \cite{BMPS}, a stratified vector field on $X,$ denoted by $\overline{\nabla}f(x).$ Let $\tilde{\zeta}$ be the lifting of $\overline{\nabla}f(x)$ as a section of the Nash bundle $\tilde{T}$ over $\tilde{X}$, without singularity on $\nu^{-1}(X\cap S_{\varepsilon}).$

Let $\mathcal{O}(\tilde{\zeta})\in\mathbb{H}^{2n}(\nu^{-1}(X\cap B_{\varepsilon}),\nu^{-1}(X\cap S_{\varepsilon}))$ be the class of the obstruction cocycle for extending $\tilde{\zeta}$ as a non zero section of $\tilde{T}$ inside $\nu^{-1}(X\cap B_{\varepsilon}).$

\begin{definition}
The \textbf{local Euler obstruction of the function} $f, Eu_{f,X}(0)$ is the evaluation of $\mathcal{O}(\tilde{\zeta})$ on the fundamental class $[\nu^{-1}(X\cap B_{\varepsilon}),\nu^{-1}(X\cap S_{\varepsilon})].$
\end{definition}

For an equivalent definition of the Euler obstruction of a function using forms one can see  \cite{DutertreGrulhaJofSing}.

The next theorem compares the Euler obstruction of a space $X$ with the Euler obstruction of function defined on $X.$

\begin{theorem}\label{Euler obstruction of a function formula}(Theorem 3.1 of \cite{BMPS})
Let $(X,0)$ and $\mathcal{V}$ be given as before and let \linebreak$f:(X,0)\rightarrow(\mathbb{C},0)$ be a function with an isolated singularity at $0.$ For $0<|\delta|\ll\varepsilon\ll1,$ we have
 $$Eu_{f,X}(0)=Eu_X(0)-\sum_{i=1}^{q}\chi(V_i\cap B_{\varepsilon}\cap f^{-1}(\delta)).Eu_X(V_i).$$
\end{theorem}

Let us recall some important definitions. Let $\mathcal{V}=\{V_{\lambda}\}$ be a stratification of a reduced complex analytic space $X.$

\begin{definition}
Let $p$ be a point in a stratum $V_{\beta}$ of $\mathcal{V}.$ A \textbf{degenerate tangent plane of $\mathcal{V}$ at $p$} is an element $T$ of the Grassmanian manifold of $n_{\alpha}$-planes in $\mathbb{C}^n$ such that $T=\displaystyle\lim_{p_i\rightarrow p}T_{p_i}V_{\alpha},$ where $p_i\in V_{\alpha}$, $V_{\alpha}\neq V_{\beta}$ and $n_{\alpha}=\dim V_{\alpha}.$
\end{definition}

\begin{definition}
Let $(X,0)\subset(U,0)$ be a germ of complex analytic space in $\mathbb{C}^n$ equipped with a Whitney stratification and let $f:(X,0)\rightarrow(\mathbb{C},0)$ be a holomorphic function, given by the restriction of a holomorphic function $F:(U,0)\rightarrow(\mathbb{C},0).$ Then $0$ is said to be a \textbf{generic point}\index{holomorphic function germ!generic point of} of $f$ if the hyperplane $Ker(d_0F)$ is transverse in $\mathbb{C}^n$ to all degenerate tangent planes of the Whitney stratification at $0.$
\end{definition}

Now, let us see the definition of a Morsefication of a function.

\begin{definition}
Let $\{V_0,V_1,\ldots,V_q\},$ with $0\in V_0,$ a Whitney stratification of the complex analytic space $X.$ A function $f:(X,0)\rightarrow(\mathbb{C},0)$ is said to be \textbf{Morse stratified} if $\dim V_0\geq1, f|_{V_0}: V_0\rightarrow\mathbb{C}$ has a Morse point at $0$ and $0$ is a generic point of $f$ with respect to $V_{i},$ for all $ i\neq0.$
\end{definition}

A \textbf{stratified Morsefication}\index{holomorphic function germ!stratified Morsefication of} of a germ of holomorphic function $f:(X,0)\rightarrow(\mathbb{C},0)$ is a perturbation $\tilde{f}$ of $f$ such that $\tilde{f}$ is Morse stratified.

In \cite{STV}, Seade, Tib\u{a}r and Verjovsky proved that the Euler obstruction of a function $f$ is also related to the number of Morse critical points of a stratified Morsefication of $f.$

\begin{proposition}(Proposition 2.3 of \cite{STV})\label{Eu_f and Morse points}
Let $f:(X,0)\rightarrow(\mathbb{C},0)$ be a germ of holomorphic function with isolated singularity at the origin. Then, \begin{center}
$Eu_{f,X}(0)=(-1)^dn_{reg},$
\end{center}
where $n_{reg}$ is the number of Morse points in $X_{reg}$ in a stratified Morsefication of $f.$
\end{proposition}

Let $X$ be a reduced complex analytic space (not necessarily equidimensional) of dimension $d$ in an open set $U\subseteq\mathbb{C}^n$ and let $f:(X,0)\rightarrow(\mathbb{C},0)$ be a holomorphic map. We write $X^f=X\cap\{f=0\}.$

\begin{definition}\label{good stratification}
A \textbf{good stratification of $X$ relative to $f$} is a stratification $\mathcal{V}$ of $X$ which is adapted to $X^f$ such that $\{V_{\lambda}\in\mathcal{V},V_{\lambda}\nsubseteq X^f\}$ is a Whitney stratification of $X\setminus X^f$ and such that for any pair $(V_{\lambda},V_{\gamma})$ such that $V_{\lambda}\nsubseteq X^f$ and $V_{\gamma}\subseteq X^f,$ the $(a_f)$-Thom condition is satisfied, that is, if $p\in V_{\gamma}$ and $p_i\in V_{\lambda}$ are such that $p_i\rightarrow p$ and $T_{p_i} V(f|_{V_{\lambda}}-f|_{V_{\lambda}}(p_i))$ converges to some $\mathcal{T},$ then $T_p V_{\gamma}\subseteq\mathcal{T}.$
\end{definition}

Let $\mathcal{V}$ be a good stratification of $X$ relative to $f.$

\begin{definition}
The \textbf{critical locus of $f$ relative to $\mathcal{V}$}, $\Sigma_{\mathcal{V}}f,$ is given by the union \begin{center}$\Sigma_{\mathcal{V}}f=\displaystyle\bigcup_{V_{\lambda}\in\mathcal{V}}\Sigma(f|_{V_{\lambda}}).$\end{center}
\end{definition}

Let $g:(X,0)\rightarrow(\mathbb{C},0)$ be a holomorphic function-germ.

\begin{definition}
If $\mathcal{V}=\{V_{\lambda}\}$ is a stratification of $X,$ the \textbf{relative polar variety of $f$ and $g$ with respect to $\mathcal{V}$}, denoted by $\Gamma_{f,g}(\mathcal{V}),$ is the the union $\cup_{\lambda}\Gamma_{f,g}(V_{\lambda}),$ where $\Gamma_{f,g}(V_{\lambda})$ denotes the closure in $X$ of the critical locus of $(f,g)|_{V_{\lambda}\setminus X^f}.$
\end{definition}

\begin{definition}
If $\mathcal{V}=\{V_{\lambda}\}$ is a stratification of $X,$ the \textbf{symmetric relative polar variety of $f$ and $g$ with respect to $\mathcal{V}$}, $\tilde{\Gamma}_{f,g}(\mathcal{V}),$ is the union $\cup_{\lambda}\tilde{\Gamma}_{f,g}(V_{\lambda}),$ where $\tilde{\Gamma}_{f,g}(V_{\lambda})$ denotes the closure in $X$ of the critical locus of $(f,g)|_{V_{\lambda}\setminus (X^f\cup X^g)},$  $X^f=X\cap \{f=0\}$ and $X^g=X\cap \{g=0\}. $
\end{definition}

\begin{definition}\label{definition prepolar}
Let $\mathcal{V}$ be a good stratification of $X$ relative to a function
$f:(X,0)\rightarrow(\mathbb{C},0).$ A function $g :(X, 0)\rightarrow(\mathbb{C},0)$ is \textbf{prepolar with respect to $\mathcal{V}$ at the origin} if the origin is a stratified isolated critical point, that is, $0$ is an isolated point of $\Sigma_{\mathcal{V}}g.$
\end{definition}

\begin{definition}\label{definition tractable}
A function $g :(X, 0)\rightarrow(\mathbb{C},0)$ is \textbf{tractable at the origin with respect to a good stratification $\mathcal{V}$ of $X$ relative to $f :(X, 0)\rightarrow(\mathbb{C},0)$} if $dim_0 \ \tilde{\Gamma}^1_{f,g}(\mathcal{V})\leq1$ and, for all strata $V_{\alpha}\subseteq X^f$,
$g|_{V_{\alpha}}$ has no critical point in a neighbourhood of the origin except perhaps at the origin itself.

\end{definition}

We recall the definition of the Brasselet number introduced by Dutertre and Grulha \cite{DG}. Let $f: (X,0)\rightarrow(\mathbb{C},0)$ be a germ of holomorphic function, $X$ a suficiently small representative of the germ, and let $\mathcal{V}$ be a good stratification of $X$ relative to $f.$ We denote by $V_1,\ldots, V_q$ the strata of $\mathcal{V}$ that are not contained in $\{f=0\}$ and we assume that $V_1,\ldots, V_{q-1}$ are connected and that $V_{q}=\linebreak X_{reg}\setminus \{f=0\}.$ Note that $V_q$ could be not connected.

\begin{definition}
Suppose that $X$ is equidimensional. Let $\mathcal{V}$ be a good stratification of $X$ relative to $f.$ The \textbf{Brasselet number} of $f$ at the origin, $B_{f,X}(0),$ is defined by \begin{center}
$B_{f,X}(0)=\sum_{i=1}^{q}\chi(V_i\cap f^{-1}(\delta)\cap B_{\varepsilon})Eu_X(V_i),$
\end{center}
where $0<|\delta|\ll\varepsilon\ll1.$
\end{definition}

\noindent\textbf{Remark:} If $V_q^i$ is a connected component of $V_{q},$ $Eu_X(V_q^i)=1.$

Notice that if $f$ has a stratified isolated singularity at the origin, then \linebreak$B_{f,X}(0)=Eu_{X}(0)-Eu_{f,X}(0)$ (see Theorem \ref{Euler obstruction of a function formula}).

In \cite{DG}, Dutertre and Grulha proved formulas describing the topological relation between the Brasselet number and a number of certain critical points of a special type of perturbation of functions. Let us now present some of these results. First we need the definition of a special type of Morsefication, introduced by Dutertre and Grulha.

\begin{definition}
A \textbf{partial Morsefication} of $g:f^{-1}(\delta)\cap X\cap B_{\varepsilon}\rightarrow\mathbb{C}$ is a function $\tilde{g}: f^{-1}(\delta)\cap X\cap B_{\varepsilon}\rightarrow\mathbb{C}$ (not necessarily holomorphic) which is a local Morsefication of all isolated critical points of $g$ in $f^{-1}(\delta)\cap X\cap \{g\neq 0\}\cap B_{\varepsilon}$ and which coincides with $g$ outside a small neighborhood of these critical points.
\end{definition}

Let $g : (X, 0) \rightarrow (\mathbb{C},0)$ be a holomorphic function which is tractable at the origin with respect to $\mathcal{V}$ relative to $f .$ Then $\tilde{\Gamma}_{f,g}$ is a complex analytic curve and for $0<|\delta|\ll1$ the critical points of $g|_{f^{-1}(\delta)\cap X}$ in $B_{\varepsilon}$ lying outside $\{g=0\}$ are isolated.
Let $\tilde{g}$ be a partial Morsefication of $g:f^{-1}(\delta)\cap X\cap B_{\varepsilon}\rightarrow\mathbb{C}$ and, for each \linebreak$i\in\{1,\ldots, q\},$ let $n_i$ be the number of stratified Morse critical points of $\tilde{g}$ appearing on $V_i\cap f^{-1}(\delta)\cap \{g\neq 0\}\cap B_{\varepsilon}.$

\begin{theorem}\label{Corollary 4.3 of DG}(Corollary 4.3 of \cite{DG})
Suppose that $X$ is equidimensional and that $g$ is tractable at the origin with respect to $\mathcal{V}$ relative to $f.$ For $0<|\delta|\ll\varepsilon\ll 1,$ we have\begin{center}
 $\chi(X\cap f^{-1}(\delta)\cap B_{\varepsilon},Eu_X)-\chi(X\cap g^{-1}(0)\cap f^{-1}(\delta)\cap B_{\varepsilon},Eu_X)=(-1)^{d-1}n_q.$
 \end{center}
\end{theorem}

If one supposes, in addition, that $g$ is prepolar, a consequence of this result is a Lê-Greuel type formula for the Brasselet number.

\begin{theorem}\label{4.4 DG}(Theorem 4.4 of \cite{DG})
Suppose that $X$ is $d$-equidimensional and that $g$ is prepolar with respect to $\mathcal{V}$ at the origin. For $0<|\delta|\ll\varepsilon\ll 1,$ we have\begin{center}
 $B_{f,X}(0)-B_{f,X^g}(0)=(-1)^{d-1}n_q,$
 \end{center}
 where $n_q$ is the number of stratified Morse critical points on the top stratum $V_q\cap f^{-1}(\delta)\cap B_{\varepsilon}$ appearing in a Morsefication of $g:X\cap f^{-1}(\delta)\cap B_{\varepsilon}\rightarrow \mathbb{C}.$
\end{theorem}

\hspace{0,5cm} In \cite{santana2019brasselet}, the author generalized the last formula to the case where the critical locus of $g,$ $\Sigma_{\mathcal{V}} g,$ is one-dimensional and that $\Sigma_{\mathcal{V}} g\cap \{f=0\}=\{0\}.$ We can decompose $\Sigma_{\mathcal{V}}g$ into branches $b_j, \Sigma_{\mathcal{V}} g=\bigcup_{\alpha=1}^q\Sigma g|_{V_{\alpha}}\cup\{0\}=b_1\cup\ldots\cup b_r,$ where $b_j\subseteq V_{\alpha},$ for some $\alpha\in\{1,\ldots,q\}.$  Let $\delta$ be a regular value of $f, 0<|\delta|\ll1,$ and let us write, for each $j\in\{1,\ldots,r\}, f^{-1}(\delta)\cap b_j=\{x_{i_1},\ldots,x_{i_{k(j)}}\}$ and $m_{f,b_j}=k(j).$  Let $\varepsilon$ be sufficiently small such that the local Euler obstruction of $X$ and of $X^g$ are constant on $b_j\cap B_{\varepsilon}$. In this case, we denote by $Eu_{X}(b_j)$ (respectively, $Eu_{X^g}(b_j)$) the local Euler obstruction of $X$ (respectively, $X^g$) at a point of $b_j\cap B_{\varepsilon}.$

\begin{theorem}\label{Generalization 6.4 of DG}
Suppose that $g$ is tractable at the origin with respect to  $\mathcal{V}$ relative to $f.$ Then, for $0<|\delta|\ll\varepsilon\ll1,$
\begin{center}
$B_{f,X}(0)-B_{f,X^g}(0)-\sum_{j=1}^{r}m_{f,b_j}(Eu_X(b_j)-Eu_{X^g}(b_j))=(-1)^{d-1}n_q,$
\end{center}
where $n_q$ is the number of stratified Morse critical points of a partial Morsefication of \linebreak$g:X\cap f^{-1}(\delta)\cap B_{\varepsilon}\rightarrow\mathbb{C}$ appearing on $X_{reg}\cap f^{-1}(\delta)\cap \{g\neq 0\}\cap B_{\varepsilon}.$
\end{theorem}

\section{Relations between the Euler obstruction of a Map, of a Function and the Brasselet number}

The notion of local Chern obstruction extends the notion of local Euler obstruction in the case of collections of germs of 1-forms. More precisely, Ebeling and Gusein-Zade perform the following construction.

Let $(X, 0)\subset(\mathbb{C}^{n},0)$ be the germ of a $d$-equidimensional reduced complex analytic variety at the origin. Let $\{\omega_j^{(i)}\}$ be a collection of germs of 1-forms on $(\mathbb{C}^{n},0)$ such that $i=1,\ldots, s$; $j=1, \ldots, d-k_{i}+1$, where the $k_{i}$ are non-negative integers with $\sum_{i=1}^{s}k_{i}=d$. Let $\varepsilon>0$ be small enough so that there is a representative $X$ of the germ $(X, 0)$ and representatives $\{\omega_{j}^{(i)}\}$ of the germs of 1-forms inside the ball $B_{\varepsilon}(0)\subset\mathbb{C}^{n}.$

\begin{definition}
	For a fixed $i$, the locus of the subcollection $\{\omega_{j}^{(i)}\}$ is the set of points $x\in X$ such that there exists a sequence $x_{n}$ of points from the non-singular part $X_{{\rm{reg}}}$ of the variety $V$ such that the sequence $T_{x_{n}}X_{{\rm{reg}}}$ of the tangent spaces at the points $x_{n}$ has a limit $L$ (in $G(d,n)$) and the restrictions of the 1-forms $\omega_{1}^{(i)}, \ldots, \omega_{d-k_{i}+1}^{(i)}$ to the subspace $L\subset T_{x}\mathbb{C}^{n}$ are linearly dependent.
\end{definition}

\begin{definition}\label{specialpoint} A point $x\in X$ is called a {\it special point} of the collection $\{\omega_{j}^{(i)}\}$ if it is in the intersection of the loci of the subcollections $\{\omega_{j}^{(i)}\}$ for each $i=1,\ldots, s$. The collection $\{\omega_{j}^{(i)}\}$ of 1-forms has an isolated special point at $\{0\}$ if it has no special point on $X$ in a punctured neighbourhood of the origin.
\end{definition}

Note that in Definition \ref{specialpoint}, for each fixed $i$, we require each subcollection $\{\omega_{j}^{(i)}\}$ to be linearly dependent when restricted to the same limit plane. Also note that if an element of the collection has less than maximal rank at a point, then it is linearly dependent on all planes passing through the point.

Let $\{\omega_{j}^{(i)}\}$ be a collection of germs of 1-forms on $(X,0)$ with an isolated special point at the origin. Let $\nu:\tilde{X}\rightarrow X$ be the Nash transformation of the variety $X$ and $\tilde{T}$ be the Nash bundle. The collection of 1-forms $\{\omega_{j}^{(i)}\}$ gives rise to a section $\Gamma(\omega)$ of the bundle
$$\tilde{\mathbb{T}}=\bigoplus_{i=1}^{s}\bigoplus_{j=1}^{d-k_{i}+1}\tilde{T}_{i,j}^{*},$$ where $\tilde{T}_{i,j}^{*}$ are copies of the dual Nash bundle $\tilde{T}^{*}$ over the Nash transformation $\tilde{X}.$

Let $\mathbb{D}\subset\tilde{\mathbb{T}}$ be the set of pairs $(x,\{\alpha_{j}^{(i)}\})$ where $x\in\tilde{X}$ and the collection of 1-forms $\{\alpha_{j}^{(i)}\}$ is such that $\alpha_{1}^{(i)},\ldots, \alpha_{n-k_{i}+1}^{(i)}$ are linearly dependent for each $i=1, \ldots, s.$

\begin{definition}
	Let $0$ be a special point of the collection $\{\omega_{j}^{(i)}\}$. The local Chern obstruction ${\rm{Ch}}_{X,0}\{\omega_{j}^{(i)}\}$ of the collection of germs of 1-forms $\{\omega_{j}^{(i)}\}$ on $(X,0)$ at the origin is the obstruction to extend the section $\Gamma(\omega)$ of the fibre bundle $\tilde{\mathbb{T}}\backslash\mathbb{D}\rightarrow \tilde{X}$ from $\nu^{-1}(X\cap S_{\varepsilon})$ to $\nu^{-1}(X\cap B_{\varepsilon})$.
\end{definition}

We can see that we have the correct obstruction dimension as follows. For each $\tilde{\mathbb{T}}_i=\bigoplus_{j=1}^{d-k_{i}+1}\tilde{T}_{i,j}^{*}$ let $\mathbb{D}_{i}\subset\tilde{\mathbb{T}}_{i}$ be the set of pairs $(x,\{\alpha_{j}^{(i)}\})$, where $x\in\tilde{X}$ and the collection $\{\alpha_{j}^{(i)}\}$ is such that $\alpha_{1}^{(i)}, \ldots,\alpha_{d-k_{i}+1}^{(i)}$ are linearly dependent. Then, the set $\tilde{\mathbb{T}}_i\backslash\mathbb{D}_{i}$ is a Stiefel manifold, with associated obstruction dimension equal to $k_i$, and therefore the obstruction dimension for $\tilde{\mathbb{T}}$ is $\sum_{i=1}^{s} k_i=d$.

The following result is a consequence of \cite[Proposition 3.3]{Ebeling2007a}.

\begin{proposition}\rm{
		Let $(X,0)\subset(\mathbb{C}^{n},0)$ be the germ of a $d$-equidimensional reduced complex analytic variety at the origin. Let $\{\omega_{j}^{(i)}\}$ be a collection of germs of $1$-forms on $(\mathbb{C}^{n},0)$ such that $i=1, \ldots, s$; $j=1, \ldots, d-k_{i}+1,$ where the $k_i$ are non-negative integers with $\sum_{i=1}^{s}k_{i}=d$. Let $0$ be an isolated special point for the collection. If $\omega^{(i)}$, $i=2, \ldots, s$, are generic collections of linear forms, then the number ${\rm{Ch}}_{X,0}\{\omega_{j}^{(i)}\}$ does not depend on the choice of the subcollections $\omega^{(i)}$, $i=2, \ldots, s$.}
\end{proposition}

In in \cite{Grulha2008} Grulha defined a notion of the Euler obstruction of a map, but in his construction the obstruction depends on a certain cellular decomposition. Some years latter, in \cite{brasselet2010euler} Brasselet, Grulha and Ruas compared the notion of the Euler obstruction of a map and the Chern number. In that paper they also proved that the Euler obstruction does not depend on a generic choice in its constuction. Based on this, we define the Euler obstruction of a map in terms of collection of forms.

\begin{definition}
	Let $X$ be an equidimensional complex variety of dimension $d$, $f:(X,0)\to \mathbb{C}^{p}$, a holomorphic map, with $0\leq p \leq d$ and $\omega_{1}=\{df_{1}, df_{2},...,df_{p}\}$, with $df_{i}$ the differential of the coordinate functions of $f$, and $\omega_{2}$ a generic collection, in such way that $0$ is a special point of the collection of collections $\omega = \{\omega_{1},\omega_{2}\}$. We define the Euler obstruction of the map $f$ at the origin, denoted by $Eu
^{*}_{X,f}(0)={\rm{Ch}}_{X,0}\{\omega_{j}^{(i)}\}$.

\end{definition}

\section{The Euler obstruction of a map and Morse critical points}

Consider a map-germ $f:(X,0)\rightarrow(\mathbb{C}^2,0), f(x)=(f_1(x),f_2(x)),$ given as the restriction of a holomorphic map $F:(U,0)\rightarrow(\mathbb{C}^2,0), F(x)=(F_1(x),F_2(x)),$  where $X\subset U\subset\mathbb{C}^n$ has dimension $d>2.$ We aim to compare the local  Euler obstruction of the map $f$ and the Brasselet number of its coordinate functions.

Let $\mathcal{V}$ be a good stratification of $X$ relative to $f_2,$ which always exists (see \cite{Ms1}). Consider the collection $\{\omega^{(1)},\omega^{(2)}\}$ of $1$-forms,
where $\{\omega^{(1)}\}=\{df_1,df_2\}$ and $\{\omega^{(2)}\}=\{l_1,\ldots, l_{d}\}$ is a generic collection of linear forms \cite{ebeling2005radial}. Notice that, by \cite{brasselet2010euler}, $Eu^{*}_{f,X}(0)=Ch_{X,0}\{\omega^{(i)}_j\}.$ Hence,  the number $Ch_{X,0}\{\omega^{(i)}_j\}$ depends only on the map-germ $f.$

From now on, we assume that the collection $\{\omega^{(1)},\omega^{(2)}\}$ of $1$-forms has an isolated special point at the origin.

\begin{lemma}\label{correspondence between special and critical points}
	 Suppose that $f_1$ is tractable at the origin with respect to the good stratification $\mathcal{V}$ of $X$ relative to $f_2$. Then the Euler obstruction $Eu^{*}_{f, X\cap f_2^{-1}(\delta)}(x_0)$ of $f$ at a point $x_0\in X\cap f_2^{-1}(\delta)$ is equal to the number of Morse critical points of a Morsefication of $f_{1}|_{X_{reg}\cap f_{2}^{-1}(\delta)}$ appearing in $X_{reg}\cap f_2^{-1}(\delta),$ for $0<|\delta|\ll\varepsilon\ll1.$
\end{lemma}
\noindent\textbf{Proof.}
Let $p$ be a nondegenerate Morse critical point of a perturbation $f_1|_{X_{reg}\cap f_{2}^{-1}(\delta)}+tL$ of $f_1|_{X_{reg}\cap f_{2}^{-1}(\delta)}$, where $L$ is a generic linear form and $0<|t|\ll\varepsilon.$ Then $d_pf_1|_{T_pX_{reg}\cap f_{2}^{-1}(\delta)}+tL(p)=0$, which implies that $p$ is in the locus of a generic perturbation of $\{df_1|_{T_pX_{reg}\cap f_{2}^{-1}(\delta)},df_2|_{T_pX_{reg}\cap f_{2}^{-1}(\delta)}\}.$ Since $Ch_{X\cap f_{2}^{-1}(\delta),x_0}\{\omega^{(i)}_j\}$ does not depend on the (generic) choice of the subcollection $\omega^{(2)},$ $p$ is one of the points counted by this Chern obstruction.

Conversely, let $p\in X_{reg}\cap f_{2}^{-1}(\delta)$ be a nondegenerated special point of a small perturbation  $\{\omega^{(i)}_j+tl^{(i)}_j\}$ of $\{\omega^{(i)}_j\},$ where $0<|t|\ll\varepsilon$ and $l^{(i)}_j$ are generic linear forms.  Then $\{df_1|_{T_pX_{reg}\cap f_{2}^{-1}(\delta)}+tl^{(1)}_1,df_2|_{T_pX_{reg}\cap f_{2}^{-1}(\delta)}+tl^{(1)}_2\}$ is linearly dependent. Hence, there exist $\lambda_1,\lambda_2\in\mathbb{C}$ such that \begin{eqnarray*}
 df_2|_{T_pX_{reg}\cap f_{2}^{-1}(\delta)}&=&\frac{\lambda_1}{\lambda_2}df_1|_{T_pX_{reg}\cap f_{2}^{-1}(\delta)}+\frac{t\lambda_1}{\lambda_2}l^{(1)}_1-tl^{(1)}_2\\
 &=&\lambda df_1|_{T_pX_{reg}\cap f_{2}^{-1}(\delta)}+tl,
\end{eqnarray*}
\noindent where $l$ is the linear form $\frac{\lambda_1}{\lambda_2}l^{(1)}_1-l^{(1)}_2.$ This implies that $p$ is a critical point of $f_{1}|_{X_{reg}\cap f_2^{-1}(\delta)}+tl,$ where $ 0<|\delta|\ll\varepsilon$. Therefore, $p$ is a nondegenerated critical point of a perturbation of $f_{1}|_{X_{reg}\cap f_2^{-1}(\delta)}$ (see Theorem 3.2 of \cite{Ms1}).

Since the classes of perturbations of the collection of forms $\{\omega^{(i)}_j\}$ and of the function $f_{1}|_{X_{reg}\cap f_2^{-1}(\delta)}$ are the same, we conclude that $Ch_{X\cap f_2^{-1}(\delta),x_0}\{\omega^{(i)}_j\}$ is equal to the number $n_{reg}$ of Morse critical points of a Morsefication of $f_{1}|_{X_{reg}\cap f_{2}^{-1}(\delta)}$ appearing in $X_{reg}\cap f_2^{-1}(\delta).$\qed

\begin{proposition}\label{obstrucao de Euler da aplicacao e Numero de Brasselet}
	If $f_1$ is prepolar at the origin with respect to the good stratification $\mathcal{V}$ of $X$ relative to $f_2,$ then
	$Eu^{*}_{f,X\cap f_2^{-1}(\delta)}(x_0)=(-1)^{d-1}(B_{f_2,X}(0)-B_{f_2,X\cap f_1^{-1}(0)}(0)),$ where $x_0\in X\cap f_2^{-1}(\delta)$
\end{proposition}
\noindent\textbf{Proof.} If $f_1$ is prepolar at the origin with respect to $\mathcal{V},$ $f_1$ is tractable at the origin with respect to $\mathcal{V}.$ Hence, applying Lemma \ref{correspondence between special and critical points} and Theorem \ref{4.4 DG}, we obtain \begin{eqnarray}
Eu^{*}_{f,X\cap f_2^{-1}(\delta)}(x_0)=n_{reg}=(-1)^{d-1}(B_{f_2,X}(0)-B_{f_2,X\cap f_1^{-1}(0)}(0)),
\end{eqnarray}
\noindent where $n_{reg}$ is the number of Morse critical points of a Morsefication of $f_{1}|_{X_{reg}\cap f_{2}^{-1}(\delta)}$ appearing in $X_{reg}\cap f_2^{-1}(\delta).$\qed

A consequence of the last result is a relation between the Euler obstruction of a map and the Euler obstruction of a function.

\begin{corollary}
	Suppose that $f_2$ has an isolated singularity at the origin. If $f_1$ is prepolar at the origin with respect to the good stratification $\mathcal{V}$ of $X$ induced by $f_2$ and $x_0\in X\cap f_2^{-1}(\delta),$ then
	$Eu^{*}_{f,X\cap f_2^{-1}(\delta)}(x_0)=(-1)^{d-1}(Eu_{f_2,X\cap f_1^{-1}(0)}(0)-Eu_{f_2,X}(0)).$
\end{corollary}
\noindent\textbf{Proof.} If $f_2$ has an isolated singularity at the origin, then $B_{f_2,X}(0)=Eu_{X}(0)-Eu_{f_2,X}(0)$ and $B_{f_2,X\cap f_1^{-1}(0)}(0)=Eu_{X\cap f_1^{-1}(0)}(0)-Eu_{f_2,X\cap f_1^{-1}(0)}(0).$ Since $f_1$ is prepolar at the origin with respect to $\mathcal{V},$ then $ Eu_{X\cap f_1^{-1}(0)}(0)=Eu_{X}(0).$ Hence \begin{eqnarray*}
B_{f_2,X\cap f_1^{-1}(0)}(0)-B_{f_2,X}(0)&=&Eu_{X\cap f_1^{-1}(0)}(0)-Eu_{f_2,X\cap f_1^{-1}(0)}(0)\\
&-&Eu_{X}(0)+Eu_{f_2,X}(0)\\
&=&Eu_{f_2,X}(0)-Eu_{f_2,X\cap f_1^{-1}(0)}(0).
\end{eqnarray*}
\qed

\vspace{0,1cm}

In the last result of this section, we apply formulas of  \cite{santana2019brasselet} to compare the Brasselet number and the Euler obstruction of a map.

\begin{comment}

\begin{theorem}
	Let $v^{(2)}=(v_1,v_2)$ a $2$-frame without singularities on the boundary $\partial X^{\sigma}=X\cap\partial\sigma$ and $v_2$ a radial vector field. If $f_1$ is prepolar at the origin with respect to $\mathcal{V},$ then
	$Eu_{X}(0)=B_{f_2,X}(0).$.
\end{theorem}
\noindent\textbf{Proof.}

In one hand, from Theorem 10.2.1 of \cite{BrasseletSeadeSuwa2009} (the original reference \cite{brasselet1981classes})one has

$$Eu_{X}(v^{(2)},0)=Eu_{X}(0)\cdot Ind_{Sch}(v^{2},\sigma).$$

But as $v^{(2)}$ is generic we have $Ind_{Sch}(v^{2},\sigma)=1$, therefore $Eu_{X}(v^{(2)},0)=Eu_{X}(0)$.

In the other hand, let $z_0=(\alpha,\delta)\in\mathbb{C}^2, 0<|z_0|\ll\varepsilon\ll1.$ Since we are working on the cell $F_1^{-1}(0)\cap B_{\varepsilon}, \alpha=0.$ By Theorem 6.1 of \cite{Grulha2008}, \begin{eqnarray}\label{Th 6.1}
Eu(v^{(2)},X,0)=\sum_{\alpha}\chi(V_{\alpha}\cap f^{-1}(z_0)\cap B_{\varepsilon})Eu_X(V_{\alpha})+Eu_{f,X}(0).
\end{eqnarray}

Since $f_1$ is prepolar at the origin with respect to $\mathcal{V}, f_1^{-1}(0)\setminus\{0\}$ intersects each strata of $\mathcal{V}$ transversely.
Hence, \begin{eqnarray*}
	B_{f_2,X^{f_1}}(0)&=&\sum\chi(V_{\alpha}\cap f_1^{-1}(0)\cap f_2^{-1}(\delta) \cap B_{\varepsilon})Eu_{X\cap f_1^{-1}(0)}(V_{\alpha}\cap f_1^{-1}(0))\\
	&=&\sum\chi(V_{\alpha}\cap f_1^{-1}(0)\cap f_2^{-1}(\delta)\cap B_{\varepsilon})Eu_{X}(V_{\alpha}).
\end{eqnarray*}

Using Equation \ref{Th 6.1} and Proposition \ref{obstrucao de Euler da aplicacao e Numero de Brasselet}, we obtain
\begin{eqnarray*}
	Eu(v^{(2)},X,0)&=&\sum_{\alpha}\chi(V_{\alpha}\cap f^{-1}(z_0)\cap B_{\varepsilon})Eu_X(V_{\alpha})+Eu_{f,X}(0)\\
	&=&B_{f_2,X^{f_1}}(0)+Eu_{f,X}(0)\\
	&=&B_{f_2,X}(0)-(-1)^{d-1}n_{reg}+Eu_{f,X}(0)\\
	&=&B_{f_2,X}(0).

\end{eqnarray*}

Therefore $Eu_{X}(0)=B_{f_2,X}(0)$.

\qed
\end{comment}

\begin{proposition}
	 If $f_1$ is tractable at the origin with respect to the good stratification $\mathcal{V}$ of $X$ relative to $f_2,$ $\dim_0\Sigma_{\mathcal{V}}f_1=1$, $\Sigma_{\mathcal{V}}f_1\cap\{f_2=0\}=\{0\}$ and $x_0\in X\cap f_2^{-1}(\delta),$ then

	$$B_{f_2,X}(0)=\sum_{\alpha}\chi(V_{\alpha}\cap f^{-1}(z_0)\cap B_{\varepsilon})Eu_X(V_{\alpha})+(-1)^{d-1}Eu^{*}_{f,X\cap f_2^{-1}(\delta)}(x_0).$$

\end{proposition}

\noindent\textbf{Proof.} Let $z_0=(\alpha,\delta)\in\mathbb{C}^2, 0<|z_0|\ll\varepsilon\ll1,$ be a regular value of $f.$ Following \cite{santana2019brasselet}, since $f_1$ is tractable at the origin with respect to $\mathcal{V},$
\begin{eqnarray*}
	\sum_{\alpha}\chi(V_{\alpha}\cap f^{-1}(z_0)\cap B_{\varepsilon})Eu_X(V_{\alpha})&=&B_{f_2,X^{f_1}}(0)-\sum_{j=1}^rm_{f,b_j}(Eu_{X^{f_1}}(b_j)-Eu_X(b_j)).
\end{eqnarray*}

Also, by Theorem 3.2 of \cite{santana2019brasselet}, \begin{eqnarray*}
	B_{f_2,X^{f_1}}(0)=B_{f_2,X}(0)+\sum_{j=1}^rm_{f,b_j}(Eu_{X^{f_1}}(b_j)-Eu_{X}(b_j))-(-1)^{d-1}n_{reg},
\end{eqnarray*}
\noindent where $n_{reg}$ denotes the number of stratified Morse critical points of a partial Morsefication of $f_{1}|_{X\cap f_2^{-1}(\delta)}$ appearing in $X_{reg}\cap f_2^{-1}(\delta).$

Hence, \begin{eqnarray*}
	B_{f_2,X}(0)=\sum_{\alpha}\chi(V_{\alpha}\cap f^{-1}(z_0)\cap B_{\varepsilon})Eu_X(V_{\alpha})+(-1)^{d-1}n_{reg}.
\end{eqnarray*}
Using Lemma \ref{correspondence between special and critical points}, we obtain the formula.\qed

\section{Applications}

\subsection{Lê-Iomdine formula for the Euler obstruction of a map}

Let $f$ be a function with a one-dimensional critical set defined over an open subset of $\mathbb{C}^n$ and $l$ a generic linear form on $\mathbb{C}^n$. In \cite{iomdin1974complex}, Iomdin gave an algebraic relation between the Euler characteristic of the Milnor fibre of $f$ and the Euler characteristic of the Milnor fibre of $f+l^N,$ where $N$ is sufficiently large non-negative integer number. In \cite{le1980ensembles}, Lê proved this same relation in a more geometrical approach and with a way to obtain the Milnor fibre of $f$ by attaching a certain number of $n$-cells to the Milnor fibre of $f|_{\{l=0\}}.$

In \cite{Ms2}, Massey worked with a function $f$ with critical locus of higher dimension defined over a nonsingular space and defined the Lê numbers and cycles, which provides a way to numerically describe the Milnor fibre of this function with nonisolated singularity. In that work, Massey presented a relation between the Lê numbers of $f$ and $f+l^N,$ obtaining a Lê-Iomdin type formula between these numbers.

We keep the previous hypothesis about $f_1$ and we suppose, in addition, that  $f_2$ has an isolated singularity at the origin. Consider the perturbation $\tilde{f_1}=f_1+f_2^N$ of $f_1$, where $N>>1$ is an integer.  In \cite{santana2019local}, the author proved that, if $N>>1$ is  sufficiently large and $\delta$ is a regular value of $f_2$,  the number $n_{reg}$ of Morse critical points of a stratified Morsefication of $f_1|_{X\cap f_2^{-1}(\delta)}$ appearing on $X_{reg}\cap f_2^{-1}(\delta)\cap B_{\varepsilon}$ and the number $\tilde{n_{reg}}$ of Morse critical points of a stratified Morsefication of $\tilde{f_1}|_{X\cap f_2^{-1}(\delta)}$ appearing on $X_{reg}\cap f_2^{-1}(\delta)\cap B_{\varepsilon}$ can be compared  as follows \begin{eqnarray}\label{cor 3.12}
\tilde{n_{reg}}=n_{reg}+(-1)^{d-1}\sum_{i=1}^rm_{f_2,b_j}Eu_{f_1,X\cap f_2^{-1}(\delta)}(b_j).
\end{eqnarray}

In that paper, it was also proved that the number $m_{reg}$ of Morse critical points of a stratified Morsefication of $f_2|_{X\cap f_1^{-1}(\alpha)}$ appearing on $X_{reg}\cap f_1^{-1}(\alpha)\cap B_{\varepsilon}$ and the number $\tilde{m_{reg}}$ of Morse critical points of a stratified Morsefication of $f_2|_{X\cap \tilde{f_1}^{-1}(\alpha^{\prime})}$ appearing on $X_{reg}\cap \tilde{f_1}^{-1}(\alpha^{\prime})\cap B_{\varepsilon}$ can be compared as follows \begin{eqnarray}\label{lemma 4.3}
\tilde{m_{reg}}=m_{reg}+(-1)^{d-1}N\sum_{i=1}^rm_{f_2,b_j}Eu_{f_2,X\cap \tilde{f_1}^{-1}(\alpha^{\prime})}(b_j),
\end{eqnarray}
\noindent where $ 0<|\alpha|,|\alpha^{\prime}|\ll|\delta|\ll\varepsilon\ll1, \alpha$ is a regular value of $f_1$ and $\alpha^{\prime}$ is a regular value of $\tilde{f_1}.$
Applying these formulas and the results proved in the last section, we obtain a Lê-Iomdine type formula for the Euler obstruction of a map.   We aim to compare the Euler obstruction of the map $f=(f_1,f_2)$ with the Euler obstruction of a perturbation $\tilde{f}=(\tilde{f_1},f_2).$

\begin{corollary}
	For a sufficiently large integer number $N>>1,$ if $f_1$ is tractable at the origin with respect to $\mathcal{V}$ and  $x_0\in X\cap f_2^{-1}(\delta),$ then $$Eu^{*}_{\tilde{f},X\cap f_2^{-1}(\delta)}(x_0)=Eu^{*}_{f, X\cap f_2^{-1}(\delta)}(x_0)+(-1)^{d-1}\sum_{i=1}^rm_{f_2,b_j}Eu_{f_1,X\cap f_2^{-1}(\delta)}(b_j).$$
\end{corollary}
\noindent\textbf{Proof.}
Using Lemma \ref{correspondence between special and critical points}, $Eu^{*}_{\tilde{f},X\cap f_2^{-1}(\delta)}(x_0)=\tilde{n_{reg}}$ and $Eu^{*}_{f,X\cap f_2^{-1}(\delta)}(x_0)=n_{reg}.$ Applying these equalities to Equation \ref{cor 3.12}, we obtain the  equality claimed.\qed

\vspace{0,2cm}

We can also compare the Euler obstruction of the map $g=(f_2,f_1)$ with the Euler obstruction of a perturbation $\tilde{g}=(f_2, \tilde{f_1}).$

\begin{corollary}
	For a sufficiently large integer number $N>>1,$ if $f_1$ is tractable at the origin with respect to $\mathcal{V}$ and  $x_0\in X\cap f_2^{-1}(\delta),$ then $$Eu^{*}_{\tilde{g},X\cap \tilde{f_1}^{-1}(\alpha)}(x_0)=Eu^{*}_{g, X\cap f_1^{-1}(\alpha)}(x_0)+N(-1)^{d-1}\sum_{i=1}^rm_{f_2,b_j}Eu_{f_2,X\cap \tilde{f_1}^{-1}(\alpha)}(b_j),$$ where $ 0<|\alpha|\ll\varepsilon\ll1.$
\end{corollary}
\noindent\textbf{Proof.}
By Lemma \ref{correspondence between special and critical points}, $Eu^{*}_{\tilde{g},X\cap \tilde{f_1}^{-1}(\alpha^{\prime})}(x_0)=\tilde{m_{reg}}$ and $Eu^{*}_{g,X\cap {f_1}^{-1}(\alpha)}(x_0)=m_{reg}.$ Applying these equalities to Equation \ref{lemma 4.3}, we obtain the equality claimed.\qed

\subsection{Euler obstruction of a map, Chern obstruction and number of cusps}

Applying results of \cite{BOT}, we present in this section an application that generalizes Proposition 3.2 of \cite{brasselet2010euler}, computing the number of cusps of a stabilization $f_{s}:X_{s}\to \mathbb{C}^{2}$. We also present the relation between the multiplicity of the singular set of the application and the Euler obstruction of a map as follows.

\begin{proposition} Let $f:(X,0)\rightarrow(\mathbb{C}^2,0), f(x)=(f_1(x),f_2(x)),$ be a holomorphic map-germ. Consider the collection $\{\omega^{(1)},\omega^{(2)}\}$ of $1$-forms on $\mathbb{C}^n$,
where $\{\omega^{(1)}\}=\{df_1,df_2\}$ and $\{\omega^{(2)}\}=\{l_1,\ldots, l_{d}\}$ are generic. Assume that the collection $\{\omega^{(1)},\omega^{(2)}\}$ of $1$-forms has an isolated special point at the origin. Then, $$Eu^{*}_{f,X}(0)=\# \ \Sigma(\tilde{f})\cap H,$$ where $\Sigma(\tilde{f})$ is the singular set of a generic perturbation of $f$ and  $H$ is a $\ell$-plane in $\mathbb{C}^n$ in general position with $\Sigma(\tilde{f})$ and $\ell$ is the codimension of $\Sigma(\tilde{f})$.
\end{proposition}

\begin{proof} By Corollary 1 and Proposition 2 of \cite{Ebeling2007a} and by the relation between the Chern number and the Euler obstruction of a map \cite{brasselet2010euler}, we have
\begin{equation*}
  \begin{array}{ccl}
   Eu^{*}_{f,X}(0)  &=&  Ch_{X,0}\{\omega_j^{(i)}\}  \\
      &=&  \# \{\text{special points of } \{\tilde{\omega}_j^{(i)}\} \text{ on } X_{reg}\}  \\
       & = & \# \ \text{locus}\{d\tilde{f_1}, d\tilde{f_2}\}\cap\text{locus}\{l_1,\ldots, l_{d}\}\\
       & = & \Sigma(\tilde{f}|_{X_{reg}})\cap H\\
        & = & \Sigma(\tilde{f})\cap H.\\

  \end{array}
\end{equation*}
\end{proof}

In the next result, we consider $Ind_{X,0}\{\omega_j^{(i)}\}$ the GSV-index for collections of 1-forms defined by Ebeling and Gusein-Zade \cite{EGZLondon}, which is a generalization of the GSV-index defined in \cite{GSV}.

\begin{definition}
Let $f:(X,0)\rightarrow(\mathbb{C}^2,0)$ be an $A$-finite holomorphic map-germ, where $(X,0)\subset(\mathbb{C}^n,0)$ is a 2-dimensional isolated complete intersection singularity (ICIS). Given $\mathcal{F}:(\mathcal{X},0)\rightarrow(\mathbb{C}\times\mathbb{C}^2,0)$, $\mathcal{F}(s,x)=(s,f_s(x))$, a stabilisation of $f$, we consider a small representative $\mathcal{F}:\mathcal{X}\rightarrow D\times B_{\varepsilon}$, where $D$ is an open neighbourhood of $0$ in $\mathbb{C}$ and $B_{\varepsilon}$ is a ball of radius $\varepsilon$ in $\mathbb{C}^2$. We define by $c(f_s)$ the number of cusps of $f_s:X_s\rightarrow B_{\varepsilon}$, for $s\in D\setminus\{0\}$.
\end{definition}

In \cite{BOT} the authors show that the number of cusps $c(f_s)$ is independent of $s\in D\setminus\{0\}$ and of the stabilisation $\mathcal{F}$.

\begin{proposition}
Let $f:(X,0)\rightarrow(\mathbb{C}^2,0), f(x)=(f_1(x),f_2(x))$, be an $A$-finite holomorphic map-germ, where $(X,0)\subset(\mathbb{C}^n,0)$ is a 2-dimensional ICIS. Consider the collection $\{\omega_j^{(i)}\}=\{\omega^{(1)}, \omega^{(2)}\}$ of 1-forms on $(\mathbb{C}^n,0)$, with an isolated special point at the origin, given by $\omega^{(1)}=\{df_1,df_2\}$ and $\{\omega^{(2)}\}=\{df_1,d\Delta\}$, where $\Delta$ is the determinant of the jacobian matrix of $f$. Then $$Ch_{X,0}\{\omega_j^{(i)}\}=c(f_s)-Ind_{X,0}\{\ell_j^{(i)}\},$$ where $c(f_s)$ is the number of cusps of a stable deformation $f_s$ in a stabilisation of $f$ and $\{\ell_j^{(i)}\}$ is a generic collection  of linear functions on $\mathbb{C}^n$.
\end{proposition}

\begin{proof} Let us consider a stabilisation of $f$ as in Definition 3.2 of \cite{BOT}. We denote by $X_s$ a smoothing of $(X,0)$ and by $f_s:X_s\rightarrow \mathbb{C}^2$ a small representative of a stable map in the stabilisation. By Corollary 3 of \cite{Ebeling2007a}, for a generic collection $\{\ell_j^{(i)}\}$ of linear functions on $\mathbb{C}^n$, we have that $$Ch_{X,0}\{\omega_j^{(i)}\}= Ind_{X,0}\{\omega_j^{(i)}\}-Ind_{X,0}\{\ell_j^{(i)}\}.$$ Following \cite{EGZLondon}, $Ind_{X,0}\{\omega_j^{(i)}\}$ is equal to the sum of the indices of the singular points $q_1.\ldots, q_l$ of $\{\omega_j^{(i)}\}$ on the smoothing $X_s$ of $X$ in a neighbourhood of the origin. Since the local Chern obstruction coincides with the index on a smooth manifold and $f_s:X_s\rightarrow \mathbb{C}^2$ is stable, thus finitely determined, it follows from \cite[ Proposition 3.2]{brasselet2010euler} that
\begin{equation*}
  \begin{array}{ccl}
     Ind_{X,0}\{\omega_j^{(i)}\}  &=& \displaystyle\sum_{i=1}^{\l}Ind_{X_s,q_i}\{{\omega}_j^{(i)}\} \\
     &=&  \displaystyle\sum_{i=1}^{\l}Ch_{X_s,q_i}\{{\omega}_j^{(i)}\} \\
        & = & \displaystyle\sum_{i=1}^{\l}c_i({f_s})\\
        & = & c(f_s),
  \end{array}
\end{equation*} where $c_i$ is the number os cusps of $f_s$ close to $q_i$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Brian Smith Podcasts
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redpajama_set_name: RedPajamaCommonCrawl


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Marine Harvest slashes harvesting volume by 20,000 tonnes
Marine Harvest achieved an operational EBIT of EUR 175 million in the second quarter of 2018.
Supported by the good results, a strong market outlook, and a solid financial position, the Board has proposed a quarterly dividend of NOK 2.60 per share, according to a press release from the company.
"The global demand for salmon has been very good so far this year. This has led to high prices in all markets in the second quarter. The result has also been impacted by strong performances by our business units in Norway and Chile, which both had their best second quarter ever," says CEO Alf-Helge Aarskog.
Marine Harvest Group reported operational revenues of EUR 888 million in the second quarter of 2018. Total harvest volume was 78 346 tonnes in the quarter. Harvest guidance for 2017 has been reduced from 400 000 to 380 000 tonnes, including Northern Harvest, because of biological issues in Norway, Scotland and Canada.
Salmon of Norwegian origin achieved an operational EBIT per kilo of EUR 2.56 in the second quarter, while salmon of Scottish and Canadian origin reported operational EBIT per kilo of EUR 2.38 and EUR 0.98 respectively. Salmon of Chilean origin reported operational EBIT per kilo of EUR 1.86 in the quarter. MH Feed reported an operational EBIT of EUR 0.7 million. MH Consumer Products reported an operational EBIT of EUR 13.1 million.
After being given the approval by Canada's Competition Bureau to buy Northern Harvest in June, Marine Harvest completed the transaction in July. Marine Harvest has through this acquisition expanded its global footprint to include Atlantic Canada. Northern Harvest will be consolidated from the third quarter.
"This is an important investment for Marine Harvest and shows that we are committed to growing the industry in Canada. We believe the prospect for salmon farming in Atlantic Canada is great. Although this is a new area for us, we are working in close cooperation with authorities, local communities and other industry players to realize success," says Aarskog.
Eco Clean Team has been awarded a six-figure sum to develop detergents from salmon waste


The fine folks behind the indie sci-fi film Coherence have offered Reel Artsy readers a chance to watch a special screening of the film on June 19th at the East Village Cinemas in NYC. If you're in the New York area and you'd like to attend simply RSVP here, and you'll get added to the list. It's on a first-come-first-serve basis. The screening starts at 7:40pm.
The film co-stars Reel Artsy "fempire" favorite Lorene Scafaria (Seeking A Friend for the End of the World) and Nicholas Brendon (Buffy the Vampire Slayer).


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Q: Where are logs stored when connecting (or failing to do so) to a file server? I have a user running Ubuntu. When they are in the office, connected directly to the local network, they are unable to connect to a file server just down the hall from there office. To do so, he must connect first connect to our company's VPN.
I'd like to find the logs for this connection attempt to hopefully shine light on what's causing this issue.

A: If you are connecting to a Samba share, then you may find some information in the /var/log/syslog file. When connecting to a network share, something like this will be recorded in the log:
Feb  3 15:04:56 Carbon kernel: [ 2364.612766] CIFS: No dialect specified on mount. Default has changed to a more secure dialect, SMB2.1 or later (e.g. SMB3.1.1), from CIFS (SMB1). To use the less secure SMB1 dialect to access old servers which do not support SMB3.1.1 (or even SMB3 or SMB2.1) specify vers=1.0 on mount.
Feb  3 15:04:56 Carbon kernel: [ 2364.612770] CIFS: Attempting to mount \\192.168.0.241\Archives

If there is an error, it will appear shortly after this sort of line.

redpajama_set_name: RedPajamaStackExchange


Stir all ingredients together and store in a tightly covered jar for up to 2 months. TO USE: : Put 2 to 3 heaping tablespoons of the mix into a mug and fill with boiling water or, for an extra rich drink, hot milk. top with whipped cram and cinnamon for a special treat.


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Q: Commands API not working correctly in Macs - possible Chrome bug? I want to register a command to use with my extension. Following the documentation present here, I added this to the manifest:
       {
        "name": "My extension",
        ...
        "commands": {
          "toggle-feature-foo": {
            "suggested_key": {
              "default": "Ctrl+Shift+Y",
              "mac": "Command+Shift+Y"
            },
            "description": "Toggle feature foo"
          }
        },
        ...
      }

When I load the extension in Windows Chrome, everything works as it should, but when I load it on a Mac, the shortcut itself (Command+Shift+Y) is not registered but the description is. It will show the description - Toggle feature foo - with no shortcut.
While testing this, there were no other extensions installed. I tested it on 2 Mac with the same result. Am I missing something?
Thank you

A: I think the reason is that this key combination (Cmd+Shift+Y) is already used by OS X as a system-wide shortcut - and therefore cannot be overridden by Chrome.

A: This has been reported as a bug: https://bugs.chromium.org/p/chromium/issues/detail?id=657985&can=4&q=&colspec=ID%20Pri%20M%20Stars%20ReleaseBlock%20Component%20Status%20Owner%20Summary%20OS%20Modified


A crisp, clean, tightly bound softcover copy appearing unread, presenting light shelf wear only.
The Faces Of Science Fiction. Intiate portraits of the men and women who shape the way we look at the future.
Szabla Wschodnia I Jej Typy Narodowe (National Types of Eastern Sabres).
Four Portraits and One Subject: Bernard DeVoto. The Historian; The Writer; The Citizen; The Personality, With A Bibliography Of His Writings.
American Series Of Popular Biographies, New Hampshire Edition. This Volume Contains Biographical Sketches of Representative Citizens of The State Of New Hampshire.


Have you noticed you're seeing less patients than a few years ago? Or having to bulk bill more patient? A recent article by the ABC said Australians are putting off seeing a doctor because they are unable to afford the out of pocket expenses. However, during the 2017-2018 financial year, Medicare bulk billing rate hit an all-time high of 86%, and more than 133 million GP services were bulk billed.
With the decreasing costs paid to doctors, practices and clinics are looking for ways to cut costs. One solution is to decrease staff costs and limit the amount of staff, including receptionists. If efficiencies aren't created, this can than mean a lack of service to patients.
Market your practice. Create content or share content about when people should see a doctor.
Today for any business, not just the medical industry, having a good service and experience for patients is key.


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Home » Newpages Blog » Interview with Paulette Licitra
Interview with Paulette Licitra
I don't even think of Alimentum necessarily as "food writing." Alimentum is this funny thing in the middle because it's fiction and creative writing and it's poetry, but it's about food—so some people think of it as "food writing." The magazine started as a bridge between literary writing and food, since they were a big part of my life. Some lit mags do food-themed issues once in a while, but I wondered: why not a journal that was food all the time? We started in 2005 and we still get terrific submissions from writers who somehow incorporate food as a catalyst or character in their stories and poems.
The biannual Alimentum: The Literature of Food is a passionate, aesthetically pleasing print magazine filled with poetry, fiction and creative nonfiction. It has been lauded in such places as the New York Times, The Chicago Tribune, and The Times Literary Supplement. In 2010 it won Gourmand's International World Cookbook award for "Best Food Magazine in the World." In July 2012 it will become on-line only.
I recently spoke with Publisher Paulette Licitra about the decision to go online, electronic vs. print submissions and other nuts-and-bolts details involved in switching formats, as well as Alimentum's "Eat and Greet" tours and what we can expect next from this excellent publication.
I caught up with Licitra right before she headed out for a gig of "Duette," the acoustic band she plays in with partner Duane Spencer, Alimentum's Music Editor. Licitra recently moved from New York City to Nashville, Tennessee where, in addition to making music and publishing a magazine, the Institute of Culinary Education graduate teaches Italian cooking classes. Here are the highlights of our conversation:
How did you get into food writing?
I don't even think of Alimentum necessarily as "food writing." Alimentum is this funny thing in the middle because it's fiction and creative writing and it's poetry, but it's about food—so some people think of it as "food writing."
The magazine started as a bridge between literary writing and food, since they were a big part of my life. Some lit mags do food-themed issues once in a while, but I wondered: why not a journal that was food all the time? We started in 2005 and we still get terrific submissions from writers who somehow incorporate food as a catalyst or character in their stories and poems.
We have a diverse audience—including people who read Saveur and Bon Appetit and are interested in recipes and who's the trendiest chef and all that stuff—but then they get into Alimentum and it's a whole other thing. A lot of lit journals might have readers who love to read literature, but we also have people who just love food. That's what helps us—we've got a couple of audiences.
Alimentum is a very successful print magazine—it's been written up in the New York Times and gotten lots of other good press, and the layout is gorgeous. . . . Tell us why you decided to go online.
Partly for financial reasons and partly because we were looking to reach a bigger audience. We're not associated with a university budget. We have sponsors who help pay our costs, but in this economy it was getting harder. Then there was this pull to make the website bigger—so we made the decision to switch over.
It's almost as if Alimentum is busting out and can't be contained between those two small covers! We have a fabulous web editor, Eric LeMay, who is also a terrific writer, and he has a million great ideas for the website. The home page is a menagerie of pictures and places to jump into. It looks like a salon-style gallery of images, but each picture will bring you to a piece of content: food-related fiction, nonfiction, poetry, book reviews, recipe poems (poems you can cook from), art galleries, songs, "Eat and Greet" events, Alimentum news, and a store for cool food-related stuff. We'll also have a department called "featurettes" which will be mostly videos from the field on food-related experiences. And we'll feature favorite food blogs from around the country and the world.
Will it be free?
Totally free.
How will that affect the current subscribers?
We told our subscribers that we'll be starting an Alimentum Press, with a "Best of Alimentum" Anthology, and also single author books—like a collection of poetry or short stories of writers we've worked with. We're offering any subscribers who have subscriptions left these future books. They seem pretty excited about that.
Will the website content be updated frequently?
We're thinking definitely monthly for all of the features, and for some we may change more frequently.
Will this change anything about the type of work you look for? I know the fiction is already pretty short so should lend itself well, but sometimes, at least in journalism, people go for extra-snappy-sounding writing on the Internet. Will that affect the writing?
No—the only thing that will be affected is length. We've been wrestling over longer pieces—over 2,000 words—that we love, thinking of how we might make them episodic.
We still want our kind of literature. We want the sensibility that is Alimentum. That is very hard to describe but we know what we want when we see it. I think our readers get it and that's what they love too, and we want to keep with that tradition.
It's not only about the food because it is literature . . . so it's not just straight, like an ode-to-an-orange kind of thing; it's got to have some personal mix in there, and it's got to have a voice and a story. Food is such a deep part of everybody's life that if someone can express what that relationship is it can be pretty amazing.
We'll also continue our "Eat and Greet Tours." We've done a lot of them in Nashville, and one in New York, but more are coming. I also took a small group of avid cooks on a trip to Rome, Italy to food shop and cook there. That's another audience opening up.
The tour in New York was a two-day tour. We explored the great culinary neighborhoods in Queens, and when we'd take a break from eating and shopping, we'd sit at a café and do some writing. We do other tours where it's mostly about food and not about writing. We've visited cafes that do their own coffee roasting, artisanal chocolate makers, CSA farms, restaurant kitchens . . . we did an Indian Culture tour in Nashville, and visited a Hindu temple where the guide talked to us about Hinduism and gave us a tour, and then we had a wonderful Indian lunch. So it was a culture-plus-food experience.
We're making videos of the tours too, and they're becoming little stories on their own.
Generally when people hear a magazine is going online, they're like, "Oh no! That's the end of that publication!" I was going to ask you why Alimentum is going to be different, but it sounds like you're answering that.
That's exactly how and why it will be different.
I've gotten several letters from people expressing a lot of sadness—one bookstore, Kitchen Arts & Letters up in New York, wrote "We've been carrying you from day one . . . I'm so sorry to hear this but I know that wherever you're going, it's going to be great. . . ." Our readers really love the journal and are sad to see the print edition go, but they seem to be equally excited about the new format.
How are you going to archive the web issues?
That's something our web editors are trying to figure out, but everything is going to be there. You'll be able to access whatever's been online. All of the back print issues are still available (except for #1). They are for sale and many people collect them. We're also going to have a store with cool food-related items.
Up to now you've only been accepting submissions by snail mail. Is this going to continue?
It's so funny that this is an issue. The other day Eric, our Web Editor, and Ruth Polleys, our Assistant Web Editor, were trying to convince me to use a submission manager online. I still don't understand why that's important. Peter Selgin, our Fiction and Nonfiction Editor, and Cortney Davis, our Poetry Editor, and I seem fine with the snail mail (maybe we just have luddite brains or something), but I've been told: "Now that you're an online journal you should be accepting submissions digitally." I don't know. What do you think?
I think one reason it's supposed to be easier is that when you accept the work you can actually cut and paste it into the publication. . . .
Once we accept a piece of writing, we ask them to send us a word doc.
Well, I can think of a reason not to do it—it's pretty easy to submit online, and I wonder whether the added steps you have to take to submit it via snail mail might mean that you'd get better work.
I don't know—Eric was saying you might get more submissions too. . . .
But do you want them?
I remember in the beginning, when we accepted e-mail submissions, that Cortney noticed poems sometimes lost their formatting. But I'm sure with a submission manager program that probably doesn't happen.
But I just thought of a possible plus: now that we're an online journal, when you're reading a submission on your computer, rather than a piece of paper, it can help you better visualize how the piece works for a digital format . . . maybe that's a reason there.
Right. But maybe you should hold firm—especially now because you're doing so many new things. I assume the amount of submissions would go way up as people receive links from each other and are like "Oh, I should submit here. . . . "
Which is great—I'd love to get twice the amount of submissions.
Alimentum debuted online on July 4. To check out their great new features and find out whether they decide to begin using a submission manager, stay tuned to their website: www.alimentumjournal.com.
Tanya Angell Allen is a writer from New Haven, Connecticut. Her poems and essays have been published in such places as The Lyric, Expansive Poetry & Music, and The New York Times.
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The mission of the Foreign Investors Council is to actively promote and develop a predictable, competitive and sustainable business climate in Serbia through open dialogue with the state and other relevant stakeholders. The organisation within this process has changed significantly since the Council was founded 16 years ago, though it has remained consistent with the FIC's principles.
How has the growing number of foreign investors and competitors – and the rise of new business associations – influenced your offer of membership?
– They have strengthened those characteristics that are of crucial importance for us: independence and the dissemination of knowledge and best practices. We offer companies a safe environment in which they can freely exchange knowledge, experiences and best practises with other foreign investors that operate in Serbia. What is essential for business is that through FIC membership they reduce risks to their business, as they always have fresh information about upcoming regulatory changes and an ability to influence them. But perhaps the most important specificity of the FIC is that the voice of every member is essential because we represent only the common interests of companies around which a consensus exists.
Thanks to everything mentioned above, the FIC is recognised as an organisation with high quality based on the top regulatory expertise possessed by our members. As an association with more than 70 per cent of members coming from the EU, while the remaining members operate on the European market, we have a specific position as a powerful promoter of Serbia's economic integration into Europe.
When it comes to changes, I would also like to mention the expected Law on Lobbying, which is currently under parliamentary procedure. The first draft of this law threatened to significantly regress the quality of public-private dialogue and condition it through the use of lobbying services. It is good that the Ministry of Justice, through a deep public debate, improved the final text of the law significantly and created the space for a continuation of the dialogue between the economy and the state, either directly or via associations. However, the fact remains that most countries around the world, including in the EU, don't have laws on lobbying, but rather fight against corruption through a developed system of various legal solutions.
How effective have the new forms of dialogue with the Government, which you introduced in the last two years, proved to be?
– The novelties that we introduced have had varying results. The "Dialogue for Change" project has proven to be a very effective mechanism for overcoming specific barriers to doing business, but also a good format for exchanging information and strengthening mutual understanding between foreign investors and the state. Today the "Dialogue for Change" is a high-level event that's closed to the public, with a limited number of participants. Each event is dedicated to addressing a particular segment of the business climate and brings together groups of directors of member companies and the relevant minister.
The last such meeting was organised in May this year with the Minister of Labour, Employment, Veterans and Social Policy and the FIC member companies that are the largest employers. Such a format allows one to speak in a very open way about how to modernise labour regulations in line with contemporary international and European standards and in a way that enables greater economic growth while simultaneously guaranteeing workers' rights. We will undoubtedly continue this project, with taxes as the next topic planned.
On the other hand, we last year formed a Working Group for implementing recommendations contained within the White Book, which was the creative idea of then Serbian Prime Minister and current President Aleksandar Vučić. The purpose of this body is to enable more regular active dialogue at the highest level between the Government and the FIC and to encourage accelerated improvement of the business climate in Serbia. Since it was established in January last year, this Working Group has achieved only limited success and, unfortunately, hasn't realised its full potential. Nevertheless, through this or some other format, we are ready to continue participating in an active dialogue with the Government with the aim of strengthening the competitiveness of the Serbian market.