text:
\section{Corrected Lemma and its consequences}

We present here a corrected version of Lemma 5.5 in \cite{BiSp}.

\begin{lemma} \label{RFDAEs:lemma00}
Let $E\in\R^{n\X n}$ and $C\in C\big(\R, \R^{n\X n}\big)$ be respectively a matrix and a
matrix-valued function such that
\begin{equation}\label{RFDAEs:ex1a}
   \ker\, C^T(t) = \ker\, E^T,\;\forall \,t\in\R, \text{ and $\dim\ker\, E^T>0$},
  \end{equation}
Put $r=\mathop\mathrm{rank}\nolimits\, E$, and let $P, Q\in \R^{n\X n}$ be orthogonal matrices that realize a
singular value decomposition for $E$.  Then it follows that
  \begin{equation} \label{RFDAEs:continuousSVD}
P^TC(t) Q
    = \begin{pmatrix} \widetilde{C}_{11}(t) & \widetilde{C}_{12}(t)  \\
      0 & 0 \end{pmatrix},\quad \forall t\in \R ,
  \end{equation}
with $\widetilde C_{11}\in C\big(\R, \R^{r\X r} \big)$ and
$\widetilde C_{12}\in C\big(\R, \R^{r\X n} \big)$.

If, furthermore,
\begin{equation}\label{RFDAEs:ex1b}
   \ker\, C(t) = \ker\, E,\;\forall \,t\in\R,
  \end{equation}
then  $\widetilde{C}_{12}(t)\equiv 0$. Namely, in this case,
\begin{equation} \label{RFDAEs:continuousSVD2}
P^TC(t) Q
    = \begin{pmatrix} \widetilde{C}_{11}(t) & 0  \\
      0 & 0 \end{pmatrix},\quad \forall t\in \R ,
  \end{equation}
with $\widetilde{C}_{11}(t)$ nonsingular for all $t\in\R$.
\end{lemma}

\begin{proof}
Our proof is essentially a singular value decomposition (see, e.g., \cite{GoVL}) argument,
based on a technical result from \cite{dae}.

Observe that \eqref{RFDAEs:ex1a} imply $\mathop\mathrm{rank}\nolimits E =\mathop\mathrm{rank}\nolimits C(t)=r>0$ for all $t\in\R$.
In fact,
\begin{multline*}
\mathop\mathrm{rank}\nolimits E=\mathop\mathrm{rank}\nolimits E^T=n-\dim\ker E^T=\\
=n-\dim\ker C(t)^T =\mathop\mathrm{rank}\nolimits C(t)^T=\mathop\mathrm{rank}\nolimits C(t).
\end{multline*}
Since $\mathop\mathrm{rank}\nolimits C(t)$ is constantly equal to $r>0$, by inspection of the proof of Theorem 3.9
of \cite[Chapter 3, \S 1]{dae} we get the existence of orthogonal matrix-valued functions
$U, V\in C\big(\R, \R^{n\X n}\big)$ and $C_r\in C(\R, \R^{r\X r})$ such that, for all
$t\in\R$, $\det C_r(t)\neq 0$ and
\begin{equation}\label{svdcont}
    U^T(t)C(t)V(t) =
        \begin{pmatrix}
         C_r(t) & 0 \\
           0 & 0
        \end{pmatrix}.
\end{equation}

Let $U_r,V_r\in C\big(\R,\R^{n\X r}\big)$ and $U_0,V_0\in C(\R,\R^{n\X (n-r)})$ be
matrix-valued functions formed, respectively, by the first $r$ and $n-r$ columns of $U$ and
$V$. An argument involving Equation \eqref{svdcont} shows that, for all $t\in\R$, the
space $\mathop\mathrm{im}\nolimits C(t)$ is spanned by the columns of $U_r(t)$. Also, \eqref{svdcont} imply that the
columns of $V_0(t)$, $t\in\R$, belong to $\ker C(t)$ for all $t\in\R$. A dimensional argument
shows that they constitute a basis $\ker C(t)$. Analogously, transposing  \eqref{svdcont},
we see that the columns of $V_r(t)$ and $U_0(t)$ are bases of $\mathop\mathrm{im}\nolimits C(t)^T$ and $\ker C(t)^T$
respectively.\footnote{In fact, the orthogonality of the matrices $V(t)$ and $U(t)$ for all
$t\in\R$, imply that the columns of $U_r(t)$, $V_r(t)$, $U_0(t)$ and $V_0(t)$ are respective
orthogonal bases of the spaces $\mathop\mathrm{im}\nolimits C(t)$, $\mathop\mathrm{im}\nolimits C(t)^T$, $\ker C(t)^T$ and $\ker C(t)$.}

Let now $P_r,Q_r$ and $P_0,Q_0$ be the matrices formed taking the first $r$ and $n-r$
columns of $P$ and $Q$, respectively. Since $P$ and $Q$ realize a singular value
decomposition of $E$, proceeding as above one can check that the columns of $P_r$, $Q_r$,
$P_0$ and $Q_0$ span $\mathop\mathrm{im}\nolimits E$, $\mathop\mathrm{im}\nolimits E^T$, $\ker E^T$, and $\ker E$, respectively.

We claim that $P_0^TU_r(t)$ is constantly the null matrix in $\R^{(n-r)\X r}$. To prove this,
it is enough to show that for all $t\in\R$, the columns of $P_0$ are orthogonal to those of
$U_r(t)$. Let $v$ and $u(t)$, $t\in\R$, be any column of $P_0$ and of $U_r(t)$, respectively.
Since for all $t\in\R$ the columns of $U_r(t)$ are in $\mathop\mathrm{im}\nolimits C(t)$, there is a vector
$w(t)\in\R^n$ with the property that $u(t)=C(t)w(t)$, and
\begin{equation*}
    \langle v, u(t)\rangle= \langle v, C(t)w(t)\rangle=
    \langle C(t)^Tv, w(t)\rangle=0,\quad \forall t\in\R,
\end{equation*}
because $v\in\ker E^T=\ker C(t)^T$ for all $t\in\R$. This proves the claim. A similar argument
shows that $P_r^TU_0(t)$ is identically zero as well.

Since for all $t\in\R$
\[
 P^TU(t)
=\begin{pmatrix}
                        P_r^TU_r(t) & 0 \\
                             0 & P_0^TU_0(t)
                  \end{pmatrix}\quad
\]
is nonsingular, we deduce in particular that so is $P_r^TU_r(t)$.

Let us compute the matrix product $P^TC(t)Q$ for all $t\in\R$. We omit here, for the sake of
simplicity, the explicit dependence on $t$.
  \begin{equation*}\label{PTCQ}
    \begin{aligned}
      P^TCQ &= P^TUU^TCVV^T Q
            =\begin{pmatrix}
                        P_r^TU_r & 0 \\
                             0 & P_0^TU_0
                  \end{pmatrix}
        \begin{pmatrix}
                        C_r & 0 \\
                             0 & 0
                  \end{pmatrix}\begin{pmatrix}
                        V_r^TQ_r & V_r^TQ_0 \\
                             V_0^TQ_r & V_0^TQ_0
                  \end{pmatrix}\\
      &= \begin{pmatrix} P_r^TU_r C_r V_r^TQ_r & P_r^TU_rC_rV_r^TQ_0 \\
                                             0 & 0
         \end{pmatrix},
    \end{aligned}
  \end{equation*}
which proves \eqref{RFDAEs:continuousSVD}.
\smallskip

Let us now assume that also \eqref{RFDAEs:ex1b} holds. We claim that in this case
$V_0^TQ_r$ is identically zero. To see this we proceed as done above for the products
$P_0^TU_r$ and $P_r^TU_0$.
Let $v(t)$, $t\in\R$, be any column of $V_0(t)$, hence a vector of $\ker C(t)$ for
all $t\in\R$, and let $q$ be a column of $Q_r(t)$. Since the columns of $Q_r$ lie in
$\mathop\mathrm{im}\nolimits E^T$, there is a vector $\ell\in\R^n$ with the property  that $q=E^T\ell$, and
\begin{equation*}
    \langle v(t), q\rangle= \langle v(t), E^T\ell\rangle=
    \langle E v(t), \ell\rangle=0,\quad \forall t\in\R,
\end{equation*}
because $v(t)\in\ker C(t)=\ker E$ for all $t\in\R$. This proves the claim. A similar
argument shows that $V_r^TQ_0(t)$ is identically zero as well. Hence,
\[
 V(t)^TQ
=\begin{pmatrix}
                        V_r(t)^TQ_r & 0 \\
                             0 & V_0(t)^TQ_0
                  \end{pmatrix}
\]
thus $V_r^T(t)Q_0$, and $V_0^T(t)Q_r$ are nonsingular. Also, plugging $V_0^TQ_r=0$ in
the above expression for $P^TCQ$ one gets (we omit again the explicit dependence on $t$)
  \begin{equation}
    P^TCQ = \begin{pmatrix} P_r^TU_r C_r V_r^TQ_r & 0 \\
                                             0 & 0
            \end{pmatrix}.
  \end{equation}
Which proves the assertion because  $P_r^TU_r$, $C_r$, and $V_r^TQ_r$ are nonsingular.
\end{proof}

In view of the corrected version of the above lemma, the statement of Corollary 5.7 of \cite{BiSp}
can be rewritten as follows:

\begin{corollary}\label{corcorretto}
Consider Equation
\begin{equation} \label{RFDAEs:exx1}
E \dot{\boldsymbol{\mathrm{x}}}(t) = \mathcal{F}\big(\boldsymbol{\mathrm{x}}(t)\big)  + \lambda C(t) S(\boldsymbol{\mathrm{x}} _t),
\end{equation}
where the maps $C\colon\R\to\R^{n\X n}$ and $S\colonBU\big((-\infty, 0], \R^n\big)\to\R^n$ are continuous, $E$ is a
(constant) $n\X n$ matrix, $\mathcal{F}$ is locally Lipschitz
and $S$ verifies condition \textrm{(\textbf{K})}{} in \cite{BiSp}. Suppose also that $C$ and $E$ satisfy
\eqref{RFDAEs:ex1a} and \eqref{RFDAEs:ex1b}, and that $C$ is $T$-periodic. Let $r>0$ be
the rank of $E$ and assume that there exists an orthogonal basis of $\R^n\simeq\mathbb{R}^r\times\mathbb{R}^{n-r}$ such that
$E$ has the form
\begin{equation*}
  E\simeq\begin{pmatrix}
               E_{11} & E_{12} \\
               0      & 0
        \end{pmatrix},\text{ with $E_{11}\in\R^{r\X r}$ invertible and
         $E_{12}\in\R^{r\X (n-r)}$}.
\end{equation*}
Assume also that, relatively to this decomposition of $\R^n$,
$\partial_2{\mathcal{F}}_2(\xi,\eta)$ is invertible for all $x=(\xi,\eta)\in\mathbb{R}^r\times\mathbb{R}^{n-r}$.

Let $\Omega$ be an open subset of $[0,+\infty)\X C_T(\R^n)$ and suppose that
$\deg(\mathcal{F},\Omega\cap\R^n)$ is well-defined and nonzero. Then, there exists
a connected subset $\Gamma$ of nontrivial $T$-periodic pairs for \eqref{RFDAEs:exx1}
whose closure in $\Omega$ is noncompact and meets the set
$\big\{(0,\su{\boldsymbol{\mathrm{p}}})\in \Omega:\mathcal{F}(\boldsymbol{\mathrm{p}}) =0\big\}$.
\end{corollary}

This result follows as in \cite{BiSp} taking into account the modified version of the lemma.

\section{Associated vector Fields and RFDEs on $M$}\label{sec2}

In this section, following \cite[Chapter 4, \S 5]{dae} (compare also \cite{spaDAE}), we associate
to \eqref{RFDAEs:01} a RFDE on $M = g^{-1}(0)$.

We first discuss the notion of solution to a retarded functional DAE of the form \eqref{RFDAEs:01}.
Let $f\colon\mathbb{R}^k\times\mathbb{R}^s\to\R\sp{k}$ and $g\colon\mathbb{R}^k\times\mathbb{R}^s\to\R\sp{s}$ be given maps with $f$ continuous
and $g\in C^{\infty}(\mathbb{R}^k\times\mathbb{R}^s, \R\sp{s})$ with the property that $\partial_2 g (p,q)$ is invertible for
any $(p,q)\in\R^k\times\R^s$.  Given $T>0$, consider also a continuous and $T$-periodic in the
first variable  map $h\colon\R\times\burkrs\to\R\sp{k}$.  A solution of \eqref{RFDAEs:01}, for a
given $\lambda\geq 0$, consists of a pair of functions $(x,y)\in C(I,\mathbb{R}^k\times\mathbb{R}^s)$, $I\subseteq\R$ an
interval with $\inf I=-\infty$, such that $x$ and $y$ are bounded and uniformly continuous on any
half-line of the form $(-\infty, b]$ with $b\leq\sup I$, and
\begin{subequations}\label{daedef}
 \begin{equation}\label{daemanif}
  g\big(x(t), y(t)\big)=0,\quad\text{ for all $t\in I$},
 \end{equation}
and, eventually,
\begin{equation}\label{daeeq}
      \dot  x(t) = f\big(x(t), y(t)\big) + \lambda h(t, x_t, y_t).
\end{equation}
\end{subequations}
The latter assertion means that there exists a subinterval $J\subseteq I$ with $\sup J=\sup I$ on
which \eqref{daeeq} holds.
Observe that, by the Implicit Function Theorem, $y$ is $C^1$ on $J$. Therefore, a solution of
\eqref{RFDAEs:01} is a function $\zeta:=(x,y)$ which is bounded and uniformly continuous on any
half-line of the form $(-\infty, b]$ with $b\leq\sup I$, and is eventually a $C^1$ function, i.e.,
$\zeta\in C^1(J,\mathbb{R}^k\times\mathbb{R}^s)$.

Let us now associate tangent vector fields on $M$ to $f$ and $h$. Recall that given a differentiable
manifold $N\subseteq\R^n$, a continuous map $w\colon N\to\R^n$ with the property that for any $p\in N$,
$w(p)$ belongs to the tangent space $T_pN$ to $N$ at $p$ is called a \emph{tangent vector field on
$N$}. Similarly, a time-dependent \emph{functional (tangent vector) field on $N$} is a map
$W\colon\R\XBU\big((-\infty, 0], N\big)\to\mathbb{R}^k\times\mathbb{R}^s$, such that $W(t,\varphi,\psi)\in T_{(\varphi(0),\psi(0))}N$, for all
$(t,\varphi,\psi)\in\R\XBU\big((-\infty, 0], N\big)$.

Consider the maps
$\Psi\colon M\to\mathbb{R}^k\times\mathbb{R}^s$ and $\Upsilon \colon \R\X\bum \to \mathbb{R}^k\times\mathbb{R}^s$ defined as follows:
\begin{subequations}\label{campiv}
\begin{align}
  &\Psi (p, q)= \big( f(p, q), [\partial_2 g(p, q)]^{-1}
  \partial_1 g  (p, q) f(p, q) \big),\,\, \textrm{and} \label{RFDAEs:03}   \\
  &\Upsilon (t, \varphi, \psi)= \big( h(t, \varphi, \psi), -[\partial_2 g(
  \varphi(0), \psi(0))]^{-1} \partial_1 g (\varphi(0), \psi(0)) h(t,
  \varphi, \psi) \big). \label{RFDAEs:04}
\end{align}
\end{subequations}
Using the fact that, given a point $(p,q)\in M$, $T_{(p,q)}M$ is the kernel $\ker d_{(p,q)}g$ of
the differential $d_{(p,q)}g$ of $g$ at $(p, q)$, it can be easily proved that $\Psi$ is tangent to
$M$ in the sense that $\Psi(p,q)$ belongs to $T_{(p,q)}M$ for all $(p,q)\in M$ (compare, e.g.\
\cite{spaDAE}). Similarly, we have that $\Upsilon$ is tangent to $M$, in the sense that
$\Upsilon(t,\varphi,\psi)\in T_{(\varphi(0),\psi(0))}M$, for all $(t,\varphi,\psi)\in\R\X\bum$. In
other words, we see that $\Psi$ is a tangent vector field, whereas $\Upsilon$ is a time-dependent
functional field on $M$. Since $h$ is assumed $T$-periodic in the first variable, so is $\Upsilon$.
Notice that, for any $\lambda\geq0$, the map of $\R\times BU\big((-\infty, 0],M)$ in $\mathbb{R}^k\times\mathbb{R}^s$, defined
by
\begin{equation*}
  (t,\varphi,\psi)\mapsto \Psi\big(\varphi(0), \psi(0)\big)+\lambda
  \Upsilon(t,\varphi, \psi),
\end{equation*}
is a functional tangent vector field as well.

We claim that \eqref{RFDAEs:01} is equivalent to the following RFDE on $M$, which keeps implicitly
account of the algebraic condition $g(x,y)=0$:
\begin{equation} \label{RFDAEs:05}
\dot \zeta(t) = \Psi \big(\zeta(t)\big) + \lambda  \Upsilon (t, \zeta_t),
\end{equation}
where we have used the compact notation $\zeta_t = (x_t, y_t)$, in the sense that $\zeta=(x,y)$ is
a solution of \eqref{RFDAEs:05} in an interval $I\subseteq\R$ if and only if so is $(x,y)$ for
\eqref{RFDAEs:01}. To verify the claim, let $\zeta =(x, y)$ be a solution of \eqref{RFDAEs:01},
defined on $I\subseteq \R$. Let $J\subseteq I$ be a subinterval where \eqref{daeeq} holds. By
differentiation of the algebraic equation $g\big(x(t),y(t)\big)=0$, one gets
\begin{equation*}
  0=\partial_1 g(x(t),y(t))\dot x(t) + \partial_2 g(x(t),y(t))\dot y(t),
\end{equation*}
whence
\begin{equation} \label{RFDAEs:02}
    \dot y(t) = [\partial_2 g(x(t), y(t))]^{-1}
    \partial_1 g (x(t), y(t)) \left[ f(x(t), y(t)) + \lambda h(t, x_t, y_t)\right],
\end{equation}
when $t\in J$. Hence, the solutions of \eqref{RFDAEs:01} correspond to those of \eqref{RFDAEs:05}.
The converse correspondence is more straightforward and follows from the fact that a solution
$\zeta=(x,y)$ of \eqref{RFDAEs:05} defined on an interval $I$ with $\inf I=-\infty$  satisfies
identically $\big(x(t),y(t)\big)\in M$, which implies \eqref{daedef}, and eventually fulfills
\[
\dot\zeta(t)=\Psi\big(\zeta(t)\big)+\lambda \Upsilon (t,\zeta_t),
\]
whose first component is \eqref{daeeq}.

\medskip
We now introduce the important technical assumption \textrm{(\textbf{K})}{} below on the function $h$. This
hypothesis implies a similar property, called condition \textrm{(\textbf{H})}{} (discussed e.g.\ in \cite{lavesp}), for
the induced vector $\Upsilon$ on $M$ defined in \eqref{RFDAEs:04} that plays a central role in
\cite{fupespa2}. This fact allows us to apply the methods of \cite{fupespa2} to our situation.

Throughout this paper, we shall suppose that $f$ is locally Lipschitz and that $h$ satisfies the
following assumption \textrm{(\textbf{K})}{}:
\begin{definition}
We say that $\mathcal{K}\colon\R\times BU\big((-\infty,0],\R^n\big)\to\R^d$ satisfies \textrm{(\textbf{K})}{} if,
given any compact subset $C$ of $\R\times BU\big((-\infty,0],\R^n\big)$, there exists $\ell\geq 0$
such that
\begin{equation*}
| \mathcal{K}(t,\varphi) - \mathcal{K}(t,\psi)|_{d}
   \leq \ell\sup_{t\le 0} | \varphi(t)-\psi(t) |_n,
\end{equation*}
for all $(t,\varphi)\,,(t,\psi) \in C$. Here $|\cdot|_{n}$ and $|\cdot|_d$ represent the Euclidean
norm in $\R^{n}$ and $\R^d$, respectively. Furthermore, we say that condition \textrm{(\textbf{K})}{} holds locally
in $\R\X BU\big((-\infty,0],\R^n\big)$ if for any $(\tau,\eta)\in\R\X BU\big((-\infty,0],\R^n\big)$
there exists a neighborhood of $(\tau,\eta)$ in which \textrm{(\textbf{K})}{} holds.
\end{definition}

It can be proved (see e.g.\ \cite{lavesp}) that if a functional field on $M$ satisfies \textrm{(\textbf{H})}{} locally,
then any associated initial value problem admits a unique solution. This shows, given the
equivalence of \eqref{RFDAEs:01} and \eqref{RFDAEs:05}, that if $f$ and $h$ satisfy \textrm{(\textbf{K})}{} then
any initial value problem associated to \eqref{RFDAEs:01} has unique initial solution.

One could show that if \textrm{(\textbf{K})}{} is satisfied locally, then it is also satisfied globally. However, the
local condition is easier to check. It holds, for instance, when $\mathcal{K}$ is $C^1$ or, more
generally, locally Lipschitz in the second variable.

The assumption that $h$ satisfies \textrm{(\textbf{K})}{} means that for any compact subset $C$ of
$\R\X BU\big((-\infty,0],\mathbb{R}^k\times\mathbb{R}^s\big)$, there exists a constant $\ell\geq 0$ such that
\begin{equation*}
| h(t,\varphi_1,\psi_1) - h(t,\varphi_2,\psi_2)|_{k}
   \leq \ell\sup_{t\le 0} \Big(| \varphi_1 (t)-\varphi_2(t) |_k +|\psi_1(t)-\psi_2(t)|_s\Big)
\end{equation*}
for all $(t,\varphi_1,\psi_1)\,,(t, \varphi_2,\psi_2) \in C$. Here $|\cdot|_k$ and $|\cdot|_s$
represent the Euclidean norm in  $\R^k$ and $\R^s$, respectively. Observe that if $f\colon\mathbb{R}^k\times\mathbb{R}^s\to\R^k$
is a locally Lipschitz tangent vector field, and $h$ is a functional field satisfying \textrm{(\textbf{K})}{}, then for
any $\lambda\in[0,+\infty)$ the map of $\R\times BU\big((-\infty,0],\mathbb{R}^k\times\mathbb{R}^s)$ in $\R^k$, given by
\[
(t,\varphi,\psi)\mapsto f\big(\varphi(0),\psi(0)\big)+\lambda h(t,\varphi,\psi),
\]
verifies \textrm{(\textbf{K})}{} as well.

If $\Psi$ and $\Upsilon$ are the functional fields on $M$ defined in \eqref{campiv}, it is easy to
see that for any $\lambda\in[0,+\infty)$ the map of $\R\times BU\big((-\infty,0],M)$ in $\mathbb{R}^k\times\mathbb{R}^s$,
given by
\[
(t,\varphi,\psi)\mapsto\Psi\big(\varphi(0),\psi(0)\big)+\lambda\Upsilon(t,\varphi,\psi),
\]
verifies the condition \textrm{(\textbf{H})}{} discussed in \cite{lavesp,fupespa2}.

\section{The degree of the tangent vector field $\Psi$}

In this section we introduce some basic notions about the degree of tangent vector fields on
manifolds. Let $N\subseteq\R^n$ be a differentiable manifold. Recall that if $w\colon N\to\R^n$ is a
tangent vector field on $N$ which is (Fr\'echet) differentiable at $p\in N$ and $w(p) = 0$, then
the differential $d_p w \colon T_pN\to\R^n$ maps $T_pN$ into itself (see e.g.\ \cite{milnor}),
so that, the determinant $\det\, d_p w$ of $d_p w$ is defined.  In the case when $p$ is a
nondegenerate zero (i.e.\ $d_p w \colon T_p N\to \R^n$ is injective), $p$ is an isolated zero
and $\det\, d_p w \ne 0$.  Let $W$ be an open subset of $N$ in which we assume $w$ admissible for
the degree, that is we suppose the set $w^{-1}(0)\cap W$ is compact. Then, it is possible to
associate to the pair $(w, W)$ an integer, $\deg(w, W)$, called the degree (or characteristic) of
the vector field $w$ in $W$ (see e.g.\ \cite{fupespa,difftop}), which, roughly speaking, counts
(algebraically) the zeros of $w$ in $W$ in the sense that when the zeros of $w$ are all
nondegenerate, then the set $w^{-1}(0)\cap W$ is finite and
\begin{equation*} \label{RFDAEs:deg}
\deg(w, W) = \sum_{q \in w^{-1}(0)\cap W} \mathop\mathrm{sign}\nolimits\, \det\, d_q w.
\end{equation*}
The concept of degree of a tangent vector field is related to the classical one of Brouwer degree
(whence its name), but the former notion differs from the latter when dealing with manifolds.
In particular, the former does not need the orientation of the underlying manifolds. However,
when $N=\R^n$, the degree of a vector field $\deg(w, W)$ is essentially the well known Brouwer
degree of $w$ on $W$ with respect to $0$ (recall that in Euclidean spaces vector fields can be
regarded as maps). For the main properties of the degree we refer e.g.\ to
\cite{fupespa, difftop, milnor}.

Let now $g\colon\mathbb{R}^k\times\mathbb{R}^s\to\R\sp{s}$ and $f\colon\R\times\mathbb{R}^k\times\mathbb{R}^s\to\R\sp{k}$ be given maps such that $f$
is continuous and $g$ is $C^{\infty}$ with the property that $\partial_2 g (p, q)$ is invertible
for all $(p,q)\in\mathbb{R}^k\times\mathbb{R}^s$. Let $\Psi$ be the tangent vector field on $M= g^{-1}(0)$ given by
\eqref{RFDAEs:03}.

A crucial requisite for the remainder of the paper is that the degree of $\Psi$ is nonzero. The
following consequence of \cite[Th.\ 4.1]{spaDAE} (see also \cite{CaSp}) allows us to replace
this condition with a more manageable one, at least in principle.

\begin{proposition} \label{RFDAEs:teo1}  Let $F\colon\mathbb{R}^k\times\mathbb{R}^s\to\mathbb{R}^k\times\mathbb{R}^s$ be given by
\[
(p, q) \mapsto \big(f(p, q), g(p, q)\big)
\]
and let $V\subseteq\mathbb{R}^k\times\mathbb{R}^s$ be an open set.  Then, if either $\deg(\Psi, M\cap V)$ or $deg(F,V)$ is
well defined, so is the other, and
\begin{equation}\label{valasso}
   |\deg (\Psi, M\cap V)| = |\deg (F, V)|.
\end{equation}
\end{proposition}

\begin{proof}
Follows immediately from Theorem 4.1 in \cite{spaDAE} and the excision property.
\end{proof}

\section{Connected sets of $T$-periodic solutions}\label{sec4}

This section is concerned with the set of periodic solutions to
\eqref{RFDAEs:01}. As in Section \ref{sec2} we are given maps
$f\colon\mathbb{R}^k\times\mathbb{R}^s\to\R^k$, $g\colon\mathbb{R}^k\times\mathbb{R}^s\to\R^s$ and $h\colon \R \X
\burkrs \to \R\sp{k}$, and we assume that
\begin{enumerate}
\item $f$ is locally Lipschitz;
\item $g$ is $C^\infty$ and such that $\det \partial_2g(p, q) \ne 0$
  for all $(p, q)\in\mathbb{R}^k\times\mathbb{R}^s$;
\item $h$ satisfies \textrm{(\textbf{K})}{} and, given $T > 0$, is $T$-periodic with
  respect to its first variable.
\end{enumerate}

Denote by $C_T(\mathbb{R}^k\times\mathbb{R}^s)$ the Banach space of all the continuous $T$-periodic functions assuming
values in $\mathbb{R}^k\times\mathbb{R}^s$ with the usual supremum norm. We say that $(\mu, \xi)\in[0,+\infty)\XC_T (\R^k\times\R^s)$
is a \emph{$T$-periodic pair} for \eqref{RFDAEs:01} if $\xi=(x, y)$ satisfies \eqref{RFDAEs:01}
for $\lambda = \mu$. Here, as well as in what follows, the elements of $C_T (\R^k\times\R^s)$ will be written
as pairs whenever convenient. In this way, $T$-periodic pairs actually will be often written as
triples. Moreover, given $(p,q)\in\mathbb{R}^k\times\mathbb{R}^s$, denote by $\big(\su p,\su q\big)$ the element of
$C_T (\R^k\times\R^s)$ that is constantly equal to $(p, q)$. A $T$-periodic pair of the form $(0,\su p,\su q)$
is called \emph{trivial}.

Let $F\colon\mathbb{R}^k\times\mathbb{R}^s\to\mathbb{R}^k\times\mathbb{R}^s$ be the vector field given by
\begin{equation}\label{RFDAEs:F}
  F(p, q) = \big(f(p, q), g(p, q)\big).
\end{equation}
It can be easily verified that $(\su p,\su q)$ is a constant solution
of \eqref{RFDAEs:01} for $\lambda=0$ if and only if $F(p,q)=(0,
0)$. Thus, with the above notation, the set of trivial $T$-periodic
pairs can be written as
\begin{equation*}
  \big\{ (0,\su p,\su q) \in [0,+\infty) \X C_T (\R^k\times\R^s) : F(p, q) = (0, 0)\big\}.
\end{equation*}
The following convention is very handy. Given subsets $\Omega$ and $X$
of $[0,+\infty)\XC_T (\R^k\times\R^s)$ and of $\mathbb{R}^k\times\mathbb{R}^s$, respectively, with
$\Omega\cap X$ we denote the set of points of $X$ that, regarded as
constant functions, lie in $\Omega$. Namely,
\[
\Omega\cap X=\{(p,q)\in X: (0,\su p,\su q)\in\Omega\}.
\]

The next result provides an insight into the topological structure of
the set of $T$-periodic solutions of \eqref{RFDAEs:01}.

\begin{theorem}\label{RFDAEs:prop1}
  Let $f$, $h$ and $g$ be as above.  Let also $F\colon \mathbb{R}^k\times\mathbb{R}^s\to\mathbb{R}^k\times\mathbb{R}^s$ be
  defined as in \eqref{RFDAEs:F}.  Let $\Omega$ be an open subset of
  $[0,+\infty)\X C_T (\R^k\times\R^s)$ and Assume that $\deg\big(F,
  \Omega\cap(\mathbb{R}^k\times\mathbb{R}^s) \big)$ is well-defined and nonzero. Then, the set
  of nontrivial $T$-periodic pairs of \eqref{RFDAEs:01}, admits a
  connected subset whose closure in $\Omega$ is noncompact and meets
  the set of trivial $T$-periodic pairs in $\Omega$, i.e.\ the set
  $\big\{ (0,\su p,\su q)\in\Omega:F(p,q)=(0,0)\big\}$.
  In particular, the set of $T$-periodic pairs for \eqref{RFDAEs:01}
  contains a connected component that meets $\big\{(0,\su p,\su
  q)\in\Omega:F(p,q)=(0,0)\big\}$ and whose intersection with $\Omega$
  is not compact.
\end{theorem}

\begin{proof}
Let $\Psi$ and $\Upsilon$ be as in \eqref{campiv}. Then \eqref{RFDAEs:01} is equivalent
to \eqref{RFDAEs:05} on $M=g^{-1}(0)$.  Denote by $C_T(M)$ the metric subspace of the
Banach space $C_T(\mathbb{R}^k\times\mathbb{R}^s)$, of all the continuous $T$-periodic functions taking values in
$M$. Let also $\mathcal{O}$ be the open subset of $[0,+\infty)\X C_T(M)$ given by
\[
  \mathcal{O}=\Omega\cap \Big([0,+\infty)\X C_T(M)\Big).
\]
Given $Y\subseteq M$, by $\mathcal{O}\cap Y$ we mean the set of all those points of $Y$ that,
regarded as constant functions, lie in $\mathcal{O}$. With this convention one clearly has
$\Omega\cap Y=\mathcal{O}\cap Y$ and, in particular, $\Omega\cap M=\mathcal{O}\cap M$. This
identity and Proposition \ref{RFDAEs:teo1} imply that
\[
  \deg(\Psi,\mathcal{O}\cap M)=\deg(\Psi,\Omega\cap
         M)=\pm\deg\big(F,\Omega\cap(\mathbb{R}^k\times\mathbb{R}^s)\big)\neq 0.
\]
Thus, Theorem 4.1 in \cite{fupespa2} yields the existence of a connected subset $\Lambda$ of
\[
  \big\{ (\lambda,x,y)\in \mathcal{O}:\text{$(x,y)$ is a nonconstant
    solution of \eqref{RFDAEs:05} }\big\},
\]
  whose closure $\cl{\Lambda}$ in $\mathcal{O}$ is not compact and meets the set
\[
  \big\{(0,\su p,\su q)\in\mathcal{O}:\Psi(p,q)=(0,0)\big\},
\]
that coincides with $\big\{(0,\su p,\su q)\in\Omega:F(p,q)=(0,0)\big\}$.

Clearly, each $(\lambda,x,y)\in\Lambda$ is a nontrivial $T$-periodic pair of \eqref{RFDAEs:01}.
Since $M$ is closed in $\mathbb{R}^k\times\mathbb{R}^s$, it is not difficult to prove that any set which is closed in
$\mathcal{O}$ is closed in $\Omega$ too, and vice versa. Thus, $\cl\Lambda$ coincides with
the closure of $\Lambda$ in $\Omega$. The first part of the assertion follows.

Let us prove the last part of the assertion. Consider the connected component $\Gamma$ of the
set of $T$-periodic pairs that contains the connected set $\Lambda$. We shall now show that
$\Gamma$ has the required properties.  Clearly, $\Gamma$ meets the set
$\big\{(0,\su{p},\su{q})\in\Omega:g(p,q)=0\big\}$ because the closure of $\Lambda$ in $\Omega$
does. Moreover, $\Gamma\cap\Omega$ cannot be compact, since it contains the (noncompact)
closure of $\Lambda$ in $\Omega$.
\end{proof}

\begin{remark}\label{obs1}
  Let $\Omega$ be as in Theorem \ref{RFDAEs:teo1}, and assume that
  $\Gamma$ is a connected component of $T$-periodic pairs of
  \eqref{RFDAEs:01} that meets $\{(0,\su p,\su
  q)\in\Omega:F(p,q)=(0,0)\}$ and whose intersection with $\Omega$ is
  not compact. Ascoli's Theorem implies that any bounded set of
  $T$-periodic pairs is relatively compact.  Then, the closed set
  $\Gamma$ cannot be both bounded and contained in $\Omega$. In
  particular, if $\Omega$ is bounded, then $\Gamma$ necessarily meets
  the boundary of $\Omega$.
\end{remark}

The following corollary ensures the existence of a Rabinowitz-type
branch of $T$-periodic pairs.

\begin{corollary}\label{cor1}
  Let $f$, $h$ and $g$ be as in Theorem \ref{RFDAEs:teo1}. Let
  $V\subseteq\mathbb{R}^k\times\mathbb{R}^s$ be open and assume that $\deg(F,V)$ is well
  defined and nonzero. Then, there exists a connected component
  $\Gamma$ of $T$-periodic pairs of \eqref{RFDAEs:01} that meets the
  set
  \[
  \big\{(0,\su{p},\su{q})\in [0,+\infty)\times C_T(\mathbb{R}^k\times\mathbb{R}^s): (p,q)\in
  V\cap F^{-1}(0,0)\big\}
  \]
  and is either unbounded or meets
  \[
  \big\{(0,\su{p},\su{q})\in [0,+\infty)\times C_T(\mathbb{R}^k\times\mathbb{R}^s): (p,q)\in
  F^{-1}(0,0)\setminus V\big\}.
  \]
\end{corollary}

\begin{proof}
  Consider the open subset $\Omega$ of $[0,+\infty)\times C_T(\mathbb{R}^k\times\mathbb{R}^s)$
  given by
  \begin{multline*}
    \Omega=\big([0,+\infty)\times C_T(\mathbb{R}^k\times\mathbb{R}^s)\big)\setminus \\
    \setminus \big\{ (0,\su{p},\su{q})\in [0,+\infty)\times
    C_T(\mathbb{R}^k\times\mathbb{R}^s): (p,q)\in F^{-1}(0,0)\setminus V\big\}.
  \end{multline*}
Clearly, we have $\Omega\cap(\mathbb{R}^k\times\mathbb{R}^s)=V$. Hence $\deg\big(F,\Omega\cap(\mathbb{R}^k\times\mathbb{R}^s)\big)\neq 0$.
Theorem \ref{RFDAEs:teo1} implies the existence of a connected component $\Gamma$ of
$T$-periodic pairs of \eqref{RFDAEs:01} that meets $\{(0,\su p,\su q)\in\Omega:F(p,q)=0\}$
and whose intersection with $\Omega$ is not compact. Because of Remark \ref{obs1}, if
$\Gamma$ is bounded, then it meets the boundary of $\Omega$ which is given by
  \[
  \big\{ (0,\su{p},\su{q})\in [0,+\infty)\times C_T(\mathbb{R}^k\times\mathbb{R}^s): (p,q)\in
  F^{-1}(0,0)\setminus V\big\}.
  \]
And the assertion is proved.
\end{proof}

\medskip
\begin{example}
The well-known logistic equation (see, e.g. \cite{brauer-cast})
  \begin{equation}\label{RFDAEs:logisteq}
    \dot x=\alpha x -\beta x^2
  \end{equation}
is sometimes used as a model for a population $x$ with birth and mortality rate $\alpha x$ and
$\beta x^2$, respectively. Consider a generalization of \eqref{RFDAEs:logisteq} where the mortality
rate $y$ is related to the population by the implicit relation $g(x,y)=0$. This generalized model
is expressed by the following DAE:
  \begin{equation*}
    \left\{
      \begin{array}{l}
        \dot x= \alpha x - y ,\\
        g(x,y)=0.
      \end{array}
    \right.
  \end{equation*}
If we allow the population's fertility to undergo periodic oscillations ---say $\lambda h(t,x_t)$
with $\lambda\geq 0$ ---depending possibly on the history of the population, the above model
can be modified into the following RFDAE:
  \begin{equation}\label{RFDAEs:genlogeq}
    \left\{
      \begin{array}{l}
        \dot x(t)= \alpha x(t) - y(t)+\lambda h(t,x_t),\\
        g\big(x(t),y(t)\big)=0.
      \end{array}
    \right.
  \end{equation}
Examples of the perturbation $h(t, x_t)$ can obtained taking inspiration from models describing
the dynamics of animals populations (see, e.g.\ \cite{brauer-cast, burt-1}) in which the delay
depends on time.

Here, however, we are interested in Equation \eqref{RFDAEs:genlogeq} in itself regardless of its
biological meaning. In particular, we wish to look at how Theorem \ref{RFDAEs:prop1} can be applied
to it. Notice that it could be impossible to get a biologically relevant result merely
from such an application. In fact, \eqref{RFDAEs:genlogeq} makes sense as a population model only
as long as $x\geq 0$, but there is no guarantee that the $x$-component of all the solutions in the
branch of $T$-periodic pairs provided by this theorem are nonnegative.

Consider, for instance, the case when $\alpha>0$ and $g(x,y)=y^3+y-x^5$. Let
$\Omega=[0,+\infty)\X C_T(\mathbb{R}^k\times\mathbb{R}^s)$. The map $F$ defined in \eqref{RFDAEs:F} is given by
$F(x,y)=(\alpha x -y, y^3+y-x^5)$, and a simple direct computation shows that
$\Omega\cap F^{-1}(0,0)$ consists of the singleton $\{(0,0)\}$ and that
$\deg(F,\Omega\cap\R^2)=-1$.  Hence, Theorem \ref{RFDAEs:prop1} yields an unbounded connected
component of periodic $2\pi$-periodic pairs emanating from the trivial $2\pi$-periodic pair
$(0,\su 0,\su 0)$.
\end{example}

\section{An application}
This section is primarily intended as an illustration of our main result Theorem \ref{RFDAEs:prop1}.
For this reason we will not pursue maximal generality but restrict ourselves to simple situations.
Below, we consider retarded periodic perturbations of a particular class of implicit ordinary
differential equations. Namely, we study equations of the following form:
\begin{equation} \label{RFDAEs:strange1}
E \dot{\boldsymbol{\mathrm{x}}}(t) =  \mathcal{F}\big(\boldsymbol{\mathrm{x}}(t)\big) + \lambda \mathcal{H}(t, \boldsymbol{\mathrm{x}} _t),
\qquad \lambda\geq 0,
\end{equation}
where $E\colon \R^n\to\R^n$ is a linear endomorphism of $\R^n$, $\mathcal{F}\colon \R^n\to\R^n$
and $\mathcal{H}\colon\R\XBU\big((-\infty, 0], \R^n\big)\to\R^n$ are continuous maps with $\mathcal{F}$ locally Lipschitz
and $\mathcal{H}$ verifies condition \textrm{(\textbf{K})}{}.

Equation \eqref{RFDAEs:strange1}, when $\lambda=0$, is quite a particular case of semi-linear DAE
(see e.g.\ \cite{RR0} and references therein). Such equations, even in the further particular case
when $\mathcal{F}$ is linear, have some practical interest. In fact, they can be used to model
such things as electrical circuits or chemical reactions (see e.g.\ \cite{Sch}). Our approach here
is inspired to that of \cite{dae,RR0} for the linear, constant coefficients, case.

We will show how, in some circumstances, \eqref{RFDAEs:strange1} can be transformed into a
\mbox{RFDAE} of type \eqref{RFDAEs:01} by the means of relatively simple linear transformations. We
will apply the results of the previous section to the resulting RFDAE. A first example of the above
mentioned transformation is considered in the following remark:

\begin{remark}\label{RFDAEs:prop-nonlin-ex}
Consider Equation \eqref{RFDAEs:strange1} and let $r>0$ be the rank of $E$. Assume that there exists
a orthogonal basis of $\R^r\X\R^{n-r}$ with respect to which $E$ can be written in the following
block form:
\begin{subequations}\label{RFDAEs:ex0}
\begin{equation}\label{RFDAEs:ex0a}
  E\simeq\begin{pmatrix}
               E_{11} & E_{12} \\
               0      & 0
        \end{pmatrix},\text{ with $E_{11}\in\R^{r\X r}$ invertible and
         $E_{12}\in\R^{r\X (n-r)}$}.
\end{equation}
Assume also that in this basis $\mathcal{H}$ has, with a slight abuse of notation, the following
form:
\begin{equation}\label{RFDAEs:ex0b}
\mathcal{H}(t, \varphi)=\begin{pmatrix}
              \mathcal{H}_1(t, \varphi) \\
               0
                                \end{pmatrix},\;
         \textrm{ with }\, \mathcal{H}_1\colon \R
      \times BU\big((-\infty, 0], \R^n\big)\to \R^r.
\end{equation}
  \end{subequations}
In $\R^n\simeq\R^r\X\R^{n-r}$ put $\boldsymbol{\mathrm{x}} =(\xi,\eta)$ and let
$J_E\colon\R^r\X\R^{n-r}\to\R^r\X\R^{n-r}$ be the linear transformation represented by
the following block matrix:
\begin{equation*}
    \begin{pmatrix}
      E_{11}^{-1} & -E_{11}^{-1}E_{12} \\
      0           &   I
    \end{pmatrix}.
\end{equation*}
Let $(x,y)=J_E^{-1}(\xi,\eta)$, and let $\mathcal{F}_1(\xi,\eta)$ and $\mathcal{F}_2(\xi,\eta)$
denote the projection of $\mathcal{F}(\xi,\eta)$ onto the first and second factor, respectively,
of $\R^r\X\R^{n-r}$. Then, in the new variables $x$ and $y$ Equation \eqref{RFDAEs:strange1}
becomes, with a slight abuse of notation,
\[
EJ_E \begin{pmatrix}
      \dot x\\
      \dot y
     \end{pmatrix} =
     \begin{pmatrix}
      \mathcal{F}_1\big(J_E(x,y)\big) \\
      \mathcal{F}_2\big(J_E(x,y)\big)
     \end{pmatrix}+\lambda
     \begin{pmatrix}
      \mathcal{H}_1\big(t,J_E( x_t,y_t)\big) \\
      0
     \end{pmatrix},
\]
or, equivalently,
\begin{equation} \label{RFDAEs:test}
\left\{\begin{array}{l}
        \dot{{x}} = \widetilde{\mathcal{F}}_1({x}, {y}) + \lambda
        \widetilde{\mathcal{H}}_1 (t,{x}_t, {y}_t),\\
        \widetilde{\mathcal{F}}_2({x}, {y}) =0,
      \end{array}\right.
\end{equation}
where $\t{\mathcal{F}}_i(x,y)=\mathcal{F}_i\big(J_E(x,y)\big)$, $i=1,2$, and
$\t{\mathcal{H}}_1(t,\varphi) =\mathcal{H}_1\big(t,J_E\circ\varphi\big)$, for any
$(t,\varphi)\in\R\XBU\big((-\infty, 0], \R^n\big)$.  Furthermore, since $\mathcal{H}$ satisfies \textrm{(\textbf{K})}{}, it
is not difficult to prove that $\t{\mathcal{H}}_1$ satisfies \textrm{(\textbf{K})}{} as well.
\end{remark}

\begin{example}\label{ex1app}
 Consider the following DAE in $\R^2\X\R^3$:
\begin{equation}\label{daeex}
 \left\{
\begin{array}{l}
 \dot\xi_1+\dot\xi_2+\dot\eta =\xi_2,\\
 \dot\xi_1 = -\xi_1+\xi_2^2+\eta ,\\
 0=\eta ^3+\eta +\xi_1,
\end{array}
\right.
\end{equation}
which can be written as the implicit ODE below where $\boldsymbol{\mathrm{x}}=(\xi_1,\xi_2,\eta)$
\begin{equation}\label{impeqex}
 E\dot\boldsymbol{\mathrm{x}}=\mathcal{F}(\boldsymbol{\mathrm{x}}),
\end{equation}
where $E$ is the endomorphism of $\R^2\X\R$ represented by the block matrix
\[
\left( \begin{array}{cc|c}
          1 & 1 & 1 \\
          1 & 0 & 0 \\
          \hline
          0 & 0 & 0 \\
\end{array} \right)
\]
and $\mathcal{F}\colon\R^3\to\R^3$ is given by
$\mathcal{F}(\xi_1,\xi_2,\eta )=(\xi_2,-\xi_1+\xi_2^2+\eta ,\eta ^3+\eta +\xi_1)$. Let
$J_E$ be the linear transformation of $\R^3\simeq\R^2\X\R$ represented by the block matrix
\[
 \left( \begin{array}{cc|c}
          0 & 1 & 0 \\
          1 & -1 & -1 \\
          \hline
          0 & 0 & 1 \\
        \end{array} \right),
\]
and put $(x_1,x_2,y)=J_E^{-1}(\xi_1,\xi_2,\eta )$. One has that
\[
\mathcal{F}\big(J_E(x_1,x_2,y)\big)
      =\big( x_1 -x_2 -y, (x_1 -x_2- y)^2 +y-x_2, y^3  + y + x_2\big),
\]
As in Remark \ref{RFDAEs:prop-nonlin-ex}, for $\xi=(\xi_1,\xi_2)$, let $\mathcal{F}_1(\xi,\eta)$
and $\mathcal{F}_2(\xi,\eta)$ denote the projection of $\mathcal{F}(\xi,\eta)$ onto the first and
second factor, respectively, of $\R^2\X\R$. Put also
\[
\t{\mathcal{F}}_i(x,y)=\mathcal{F}_i\big(J_E(x,y)\big),\quad i=1,2,
\]
where $x=(x_1,x_2)$. Proceeding as in Remark \ref{RFDAEs:prop-nonlin-ex}, we transform Equation
\eqref{daeex} into
\[
 \left\{ \begin{array}{l}
        \dot x = \t{\mathcal{F}}_1(x, y), \\
        \t{\mathcal{F}}_2(x, y) =0,
      \end{array}\right.
\]
that can be written more explicitly as follows:
\begin{equation*}
    \left\{ \begin{array}{l}
        \dot x_1 = x_1 -x_2- y, \\
        \dot x_2 = (x_1 -x_2+ y)^2  +y -x_2, \\
        y^3  + y + x_2 =0.
      \end{array}\right.
  \end{equation*}
\end{example}

\medskip
Theorem \ref{RFDAEs:prop1} combined with the above Remark \ref{RFDAEs:prop-nonlin-ex},
yields Proposition \ref{RADEs:prop01-nonlinearcase} below concerning the set of $T$-periodic
solutions of \eqref{RFDAEs:strange1}. We use here the convention on the subsets of
$[0,+\infty)\XC_T (\R^r\times\R^{n-r})$ introduced in section 4. We also need to introduce some further
notation.

A pair $(\lambda,\boldsymbol{\mathrm{x}} )\in[0,+\infty)\X C_T(\R^n)$ is a \emph{$T$-periodic pair
for \eqref{RFDAEs:strange1}} if $\boldsymbol{\mathrm{x}} $ is a solution of \eqref{RFDAEs:strange1}
corresponding to $\lambda$. A $T$-periodic pair $(0,\boldsymbol{\mathrm{x}} )$ for
\eqref{RFDAEs:strange1} is \emph{trivial} if $\boldsymbol{\mathrm{x}} $ is constant.

\begin{proposition} \label{RADEs:prop01-nonlinearcase}
Consider Equation \eqref{RFDAEs:strange1} where $E\colon\R^n\to\R^n$ is linear,
$\mathcal{F}\colon\R^n\to\R^n$ and $\mathcal{H}\colon\R\XBU\big((-\infty, 0], \R^n\big)\to\R^n$ are continuous maps such
that $\mathcal{F}$ is locally Lipschitz and $\mathcal{H}$ verifies condition \textrm{(\textbf{K})}{} and is
$T$-periodic in the first variable. Assume, as in Remark \ref{RFDAEs:prop-nonlin-ex}, that $r>0$
is the rank of $E$ and that there exists an orthogonal basis of $\R^n\simeq\mathbb{R}^r\times\mathbb{R}^{n-r}$ such that $E$
and $\mathcal{H}$ can be represented as in \eqref{RFDAEs:ex0}. Relatively to this decomposition
of $\R^2$ suppose that $\partial_2{\mathcal{F}}_2(\xi,\eta)$ is invertible for all $(\xi,\eta)\in\mathbb{R}^r\times\mathbb{R}^{n-r}$.

Let $\Omega$ be an open subset of $[0,+\infty)\X C_T(\R^n)$ and suppose that
$\deg(\mathcal{F},\Omega\cap\R^n)$ is well-defined and nonzero. Then, there exists
a connected subset $\Gamma$ of nontrivial $T$-periodic pairs for \eqref{RFDAEs:strange1}
whose closure in $\Omega$ is noncompact and meets the set
$\big\{(0,\su{\boldsymbol{\mathrm{p}}})\in \Omega:\mathcal{F}(\boldsymbol{\mathrm{p}}) =0\big\}$.
\end{proposition}

\begin{proof}
Let $J_E$ be the linear transformation introduced in Remark \ref{RFDAEs:prop-nonlin-ex},
and consider the map $\h{J_E}\colon[0,+\infty)\X C_T(\R^n)\to [0,+\infty)\X C_T(\R^n)$
given by
\[
 \h{J_E}(\lambda,\psi)=(\lambda,J_E\circ\psi).
\]
Observe that since $J_E$ is invertible, $\h{J_E}$ is continuous and invertible, with $\h{J_E}^{-1}$
given by $\h{J_E}^{-1}(\lambda,\psi)=(\lambda,{J_E}^{-1}\circ\psi)$ and, hence, continuous. With the
convention on the subsets of $[0,+\infty)\X C_T(\R^n)$ introduced in section \ref{sec4}, we have
\[
 \h{J_E}^{-1}(\Omega)\cap\R^n={J_E}^{-1}(\Omega\cap\R^n).
\]

According to Remark \ref{RFDAEs:prop-nonlin-ex}, under our assumptions Equation
\eqref{RFDAEs:strange1} is equivalent to the RFDAE \eqref{RFDAEs:test}. We now show that
Theorem \ref{RFDAEs:prop1} can be applied to Equation \eqref{RFDAEs:test}. Define
$\t{\mathcal{F}}\colon\R^n\to\R^n$ by
$
\t{\mathcal{F}}(\boldsymbol{\mathrm{p}})=\big(\t{\mathcal{F}}_1(\boldsymbol{\mathrm{p}})\,,
                                 \,\t{\mathcal{F}}_2(\boldsymbol{\mathrm{p}})\big)
                           =\mathcal{F}\big(J_E(\boldsymbol{\mathrm{p}})\big)
$.
The property of invariance under diffeormorphism of the degree (also called topological
invariance, see e.g.\ \cite{fupespa}) yields
\begin{equation}
\begin{split}\label{RFDAEs:Fcorsiva}
   \deg\big(\mathcal{F},\Omega\cap\R^n\big) =&\,
    \deg\big({J_E}^{-1}\circ\mathcal{F}\circ J_E,J_{E}^{-1}(\Omega\cap\R^n)\big)\\
         &= \deg\big({J_E}^{-1}\circ\t{\mathcal{F}},\h{J_{E}}^{-1}(\Omega)\cap\R^n\big).
\end{split}
\end{equation}
Also, it is not difficult to show that
\begin{equation}\label{id2}
 \deg\big({J_E}^{-1}\circ\t{\mathcal{F}},\h{J_{E}}^{-1}(\Omega)\cap\R^n\big) =
  \mathop\mathrm{sign}\nolimits\det(J_E) \deg\big(\t{\mathcal{F}},\h{J_{E}}^{-1}(\Omega)\cap\R^n\big),
\end{equation}
so that, being $\deg\big(\mathcal{F},\Omega\cap\R^n\big)$ nonzero by assumption,
\eqref{RFDAEs:Fcorsiva}--\eqref{id2} yield
\[
 \deg\big(\t{\mathcal{F}},\h{J_{E}}^{-1}(\Omega)\cap\R^n\big)\neq 0.
\]
Hence, Theorem \ref{RFDAEs:prop1} yields a connected set $\Xi\subseteq \h{J_E}^{-1}(\Omega)$
of $T$-periodic pairs of \eqref{RFDAEs:test} whose closure in $\h{{J_E}}^{-1}(\Omega)$ is
noncompact and meets  the set
\[
\big\{(0,\su{\boldsymbol{\mathrm{p}}})\in\h{J_E}^{-1}(\Omega):\t{\mathcal{F}}(\boldsymbol{\mathrm{p}})=0\big\}.
\]
Since $\h{J_E}$ is a homeomorphism, it is not difficult to show that $\Gamma = \h{J_E}(\Xi)$
has the required properties.
\end{proof}

\begin{example}
Let $\mathcal{H}\colon\R\X BU\big((-\infty,0],\R^3\big)\to\R^3$ be as in \eqref{RFDAEs:ex0b}
with $r=2$. Assume also that $\mathcal{H}$ is $T$-periodic continuous in the first variable.
Consider the retarded perturbation $\lambda\mathcal{H}(t,\xi_t)$ of Equation \eqref{daeex} in
Example \ref{ex1app}. Namely,
\[
 E\dot\boldsymbol{\mathrm{x}}(t)=\mathcal{F}\big(\boldsymbol{\mathrm{x}}(t)\big)+\lambda \mathcal{H}(t,\boldsymbol{\mathrm{x}}_t),
\]
where, using the same notation of Example \ref{ex1app}, we put $\boldsymbol{\mathrm{x}}=(\xi_1,\xi_2,\eta)$.
Take $\Omega=[0,+\infty)\X C_T(\R^2\X\R)$. Observe that $\mathcal{F}^{-1}(0,0,0)=\{(0,0,0)\}$
so that the degree of $\mathcal{F}$ in $\Omega\cap\R^3$ is well-defined. A simple computation
shows that $\deg(\mathcal{F},\Omega\cap\R^3)=\deg(\mathcal{F},\R^3)=-1$. Thus, Proposition
\ref{RADEs:prop01-nonlinearcase} yields the existence of a connected subset of nontrivial
$T$-periodic pairs for the above equation whose closure in $\Omega$ is noncompact and meets
the set
\[
\big\{(0,\su{\boldsymbol{\mathrm{p}}})\in \Omega:\mathcal{F}(\boldsymbol{\mathrm{p}}) =0\big\}=
\{ (0,\su 0,\su 0,\su 0)\in \Omega\}.
\]
\end{example}
\bigskip

Observe that Proposition \ref{RADEs:prop01-nonlinearcase} seems to impose some rather
severe constraints on the form of $E$ and $\mathcal{H}$ in Equation \eqref{RFDAEs:strange1}.
In fact, with the help of some linear transformation, one can sometimes lift these
restrictions. This is the case when the perturbing term $\mathcal{H}$ has a particular
`separated variables' form that agrees with $E$ in the sense of Equation \eqref{RFDAEs:ex1a}
below. Namely, we consider the following equation:
\begin{equation} \label{RFDAEs:exx1}
E \dot{\boldsymbol{\mathrm{x}}}(t) = \mathcal{F}\big(\boldsymbol{\mathrm{x}}(t)\big)  + \lambda C(t) S(\boldsymbol{\mathrm{x}} _t),
\end{equation}
where $C\colon \R \to \R^{n\X n}$, $S \colon BU\big((-\infty, 0], \R^n\big)\to \R^{n}$ are continuous maps, $E$ is a
(constant) $n\X n$ matrix, and $\mathcal{F}$ is as in Equation \eqref{RFDAEs:strange1}.  We
also assume that $C$ and $E$ agree in the following sense:
  \begin{equation}\label{RFDAEs:ex1a}
   \ker\, C^T(t) = \ker\, E^T,\;\forall \,t\in\R, \text{ and $\dim\ker\, E^T>0$},
  \end{equation}
As a consequence of the well-known Rouch\'e-Capelli Theorem we get
\begin{multline*}
n-\mathop\mathrm{rank}\nolimits\, E=n-\mathop\mathrm{rank}\nolimits\, E^T=\dim\ker\, E^T=\\
=\dim\ker\, C(t)^T =n-\mathop\mathrm{rank}\nolimits\, C(t)^T=n-\mathop\mathrm{rank}\nolimits\, C(t).
\end{multline*}
Thus, we have that
  \begin{equation}\label{RFDAEs:ex1aa}
      \mathop\mathrm{rank}\nolimits\, E =\mathop\mathrm{rank}\nolimits\, C(t) \text{ is constant and greater than $0$ for all $t\in\R$}.
    \end{equation}
This is a singular value decomposition (see, e.g., \cite{GoVL}) argument based on the
following technical result from linear algebra:

\begin{lemma} \label{RFDAEs:lemma00}
Let $E\in\R^{n\X n}$ and $C\in C\big(\R, \R^{n\X n}\big)$ be as in \eqref{RFDAEs:ex1a}. Put
$r=\mathop\mathrm{rank}\nolimits\, E$, and let $P, Q\in \R^{n\X n}$ be orthogonal matrices that realize a  singular
value decomposition for $E$.  Then it follows that
  \begin{equation} \label{RFDAEs:continuousSVD} PC(t) Q^T
    = \begin{pmatrix} \widetilde{C}_{11}(t) & 0  \\
      0 & 0 \end{pmatrix},\quad \forall t\in \R ,
  \end{equation}
  with $\widetilde C_{11}\in C\big(\R, \R^{r\X r} \big)$ invertible
  for any $t\in\R$.
\end{lemma}

\medskip
We will provide a proof for Lemma \ref{RFDAEs:lemma00} for the sake of completeness but,
before doing that, we show how it can be used to convert Equation \eqref{RFDAEs:exx1} into
\eqref{RFDAEs:strange1}.  We begin with an example.

\begin{example}
  Consider Equation \eqref{RFDAEs:exx1} with
  \begin{equation*}
    \label{RFDAEs:commutingmatrices}
E= \begin{pmatrix}
      0 & 2 & 0 & 0 \\
      1 & 0 & 0 & 0 \\
      0 & 0 & 0 & 0 \\
      0 & 0 & 0 & 1 \\
    \end{pmatrix}
\quad\textrm{ and }\quad
C(t) = \begin{pmatrix}
      c(t) & 0 & 0 & 0 \\
      0 & c(t) & 0 & 0 \\
      0 & 0  & 0 & 0 \\
      0 & 0  & 0 & d(t) \\
    \end{pmatrix},\,\,
  \end{equation*}
where, for any $t\in \R$, $c(t)= \sin(t) + 2$ and $d(t)=\cos(t)+3$. It can be easily verified
that, with this choice of $E$ and $C$, \eqref{RFDAEs:ex1a} is satisfied. Here, clearly, $r=3$
and $n=4$. Consider the following orthogonal matrices
  \begin{equation*}
    P =\begin{pmatrix}
      0 & 1 & 0 & 0 \\
      1 & 0 & 0 & 0 \\
      0 & 0 & 0 & 1 \\
      0 & 0 & 1 & 0 \\
    \end{pmatrix}\,\, \textrm{and}\,\, Q =\begin{pmatrix}
      1 & 0 & 0 & 0 \\
      0 & 1 & 0 & 0 \\
      0 & 0   & 0 & 1 \\
      0 & 0   & 1 & 0 \\
    \end{pmatrix},
  \end{equation*}
that realize a singular value decomposition for $E$, that is, in block-matrix form in
$\R^4\simeq\R^3\X\R$,
  \begin{equation*}
    PEQ^T = \left(\begin{array}{ccc|c}
        1 & 0 & 0 & 0\\
        0 & 2 & 0& 0\\
        0 & 0 & 1 &  0 \\
        \hline
        0&  0 & 0  & 0 \\
      \end{array}\right)\,\,
    \textrm{and}\,\,
    PC(t)Q^T =\left(\begin{array}{ccc|c}
        0    & c(t) & 0 & 0 \\
        c(t) &   0  & 0 & 0\\
        0    &   0  & d(t) & 0  \\
        \hline
        0& 0  &0  & 0 \\
      \end{array} \right).
  \end{equation*}
Let us consider the orthogonal change of coordinates $\boldsymbol{\mathrm{x}} =Q^T x$. Multiplying
\eqref{RFDAEs:exx1} by $P$ on the left we get the following equivalent equation:
\begin{equation}\label{svdeq1}
 PEQ^T\dot x(t)=P\mathcal{F}\big(Q^Tx(t)\big)+\lambda PC(t)Q^TQS(Q^Tx_t) .
\end{equation}
Set $\t E= PEQ^T$, $\t{\mathcal{F}}(x)=P\mathcal{F}(Q^Tx)$ for all $x\in\R^4$, and finally,
put $\t{\mathcal{H}}(t,\varphi) = PC(t)Q^TQS(Q^T\varphi)$ for all
$(t,\varphi)\in\R\X BU\big((-\infty,0],\R^4\big)$. Thus \eqref{svdeq1} can be rewritten as
\[
 \t E\dot x(t)=\t{\mathcal{F}}\big(x(t)\big)+\lambda \t{\mathcal{H}}(t,x_t).
\]
It is easily verified that $\t E$ and $\t{\mathcal{H}}$ satisfy \eqref{RFDAEs:ex0}, so that
\eqref{svdeq1} is precisely of the form \eqref{RFDAEs:strange1}. In other words, we have
transformed \eqref{RFDAEs:exx1}, for $E$ and $C$ as above, into an equation of the form
considered in Proposition \ref{RADEs:prop01-nonlinearcase}.
\end{example}

Let us now consider Equation \eqref{RFDAEs:exx1} more in general. Let $r>0$ be the rank of
$E$, and assume that \eqref{RFDAEs:ex1a} is satisfied. Then Lemma \ref{RFDAEs:lemma00} yields
orthogonal matrices $P$ and $Q$ in $\R^{n\X n}$ such that, for every $t\in\R$, $PC(t)Q^T$ is
as in \eqref{RFDAEs:continuousSVD} and realize a singular value decomposition of $E$. That is
\begin{equation}\label{RFDAEs:peq}
  PEQ^T  = \begin{pmatrix} \widetilde E_{11}
    & 0 \\ 0 & 0 \end{pmatrix}
\end{equation}
where $\widetilde E_{11}\in \R^{r\X r}$ is a diagonal matrix with positive diagonal elements.
As in the above example, consider the orthogonal change of coordinates $\boldsymbol{\mathrm{x}} =Q^T x$ in
Equation \eqref{RFDAEs:exx1} and multiply by $P$ on the left. We get the equivalent equation
\begin{equation}\label{svdtransf}
 \t E\dot x(t)=\t{\mathcal{F}}\big(x(t)\big)+\lambda \t{\mathcal{H}}(t,x_t).
\end{equation}
where $\t E$, $\t{\mathcal{F}}$ and $\t{\mathcal{H}}$ are given by $\t E= PEQ^T$,
$\t{\mathcal{F}}(x)=P\mathcal{F}(Q^Tx)$ for all $x\in\R^n$, and
$\t{\mathcal{H}}(t,\varphi) = PC(t)Q^TQS(Q^T\varphi)$ for all $(t,\varphi)\in\R\XBU\big((-\infty, 0], \R^n\big)$. A
straightforward computation shows that $\t E$ and $\t{\mathcal{H}}$ satisfy conditions
\eqref{RFDAEs:ex0}. Therefore, \eqref{svdtransf} is of the form considered in Proposition
\ref{RADEs:prop01-nonlinearcase} from which we deduce the following consequence:

\begin{corollary}
Consider Equation \eqref{RFDAEs:exx1} where the maps $C\colon\R\to\R^{n\X n}$ and
$S\colonBU\big((-\infty, 0], \R^n\big)\to\R^n$ are continuous, $E$ is a (constant) $n\X n$ matrix, and $\mathcal{F}$
is such that $\mathcal{F}$ is locally Lipschitz and $S$ verifies condition \textrm{(\textbf{K})}{}. Suppose
also that $C$ and $E$ satisfy \eqref{RFDAEs:ex1a} and that $C$  is $T$-periodic. Let $r>0$ be
the rank of $E$ and assume that there exists an orthogonal basis of $\R^n\simeq\mathbb{R}^r\times\mathbb{R}^{n-r}$ such that
$E$ is as in \eqref{RFDAEs:ex0}. Assume also that, relatively to this decomposition,
$\partial_2{\mathcal{F}}_2(\xi,\eta)$ is invertible for all $x=(\xi,\eta)\in\mathbb{R}^r\times\mathbb{R}^{n-r}$.

Let $\Omega$ be an open subset of $[0,+\infty)\X C_T(\R^n)$ and suppose that
$\deg(\mathcal{F},\Omega\cap\R^n)$ is well-defined and nonzero. Then, there exists
a connected subset $\Gamma$ of nontrivial $T$-periodic pairs for \eqref{RFDAEs:exx1}
whose closure in $\Omega$ is noncompact and meets the set
$\big\{(0,\su{\boldsymbol{\mathrm{p}}})\in \Omega:\mathcal{F}(\boldsymbol{\mathrm{p}}) =0\big\}$.
\end{corollary}

\begin{proof}
Consider the map $\h Q\colon [0,+\infty)\X C_T(\R^n)\to [0,+\infty)\X C_T(\R^n)$ given
by $\h Q(\lambda,\psi)=(\lambda,Q\psi)$. Clearly, $T$-periodic pairs of \eqref{RFDAEs:exx1}
correspond to those of \eqref{svdtransf} under $\h Q$. The invariance under diffeomorphisms
of the degree (or topological invariance, compare e.g.\ \cite{fupespa}) implies
\[
\deg\big(\t{\mathcal{F}}, Q(\Omega)\cap\R^n\big)\neq 0.
\]
The assertion follows immediately applying Proposition \ref{RADEs:prop01-nonlinearcase} to
Equation \eqref{svdtransf}.
\end{proof}

\medskip
We conclude this section with a proof of our technical Lemma.

\begin{proof}[Proof of Lemma \ref{RFDAEs:lemma00}]
Since the dimension of $\ker C(t)$ is constantly equal to $r>0$, by inspection of the proof
of Theorem 3.9 of \cite[Chapter 3, \S 1]{dae} we get the existence of orthogonal matrix-valued
functions $U, V\in C\big(\R, \R^{n\X n}\big)$ and $C_r\in C(\R, \R^{r\X r})$ such that, for all
$t\in\R$, $\det C_r(t)\neq 0$ and
\begin{equation}\label{svdcont}
    U^T(t)C(t)V(t) =
        \begin{pmatrix}
         C_r(t) & 0 \\
           0 & 0
        \end{pmatrix}.
\end{equation}

Let $U_r,V_r\in C\big(\R,\R^{n\X r}\big)$ and $U_0,V_0\in C(\R,\R^{n\X (n-r)})$ be matrix-valued
functions formed, respectively, by the first $r$ and $n-r$ columns of $U$ and $V$.
A simple argument involving Equation \eqref{svdcont} shows that the columns of $V_0(t)$, $t\in\R$,
are in $\ker C(t)$ and, since there are $n-r=\dim\ker C(t)$ of them, we have that the columns of
$V_0(t)$ actually span $\ker C(t)$. In fact, the orthogonality of the matrix $V(t)$, $t\in\R$,
imply that the columns of $V_0(t)$ form an orthogonal basis of $\ker C(t)$. A similar argument
proves that the columns of $U_0(t)$ are vectors of $\R^n$ that constitute an orthogonal basis of
$\ker C(t)^T$ for all $t\in\R$. Observe also that since $\mathop\mathrm{im}\nolimits C(t)$ is orthogonal to $\ker C(t)^T$
for all $t\in\R$, it follows that the columns of $U_r(t)$ are an orthogonal basis for $\mathop\mathrm{im}\nolimits C(t)$
and that those of $V_r(t)$ so are for $\mathop\mathrm{im}\nolimits C(t)^T$.

Similarly, let $P_r,Q_r$ and $P_0,Q_0$ be the matrices formed taking the first $r$ and $n-r$
columns of $P$ and $Q$, respectively. Since $P$ and $Q$ realize a singular value decomposition
of $E$, one can check that the columns of $P_r$, $Q_r$, $P_0$ and $Q_0$ span $\mathop\mathrm{im}\nolimits E$, $\mathop\mathrm{im}\nolimits E^T$,
$\ker E^T$, and $\ker E$, respectively.

We claim that $P_0^TU_r(t)$ is constantly the null matrix. To prove this, it is enough to
show that for all $t\in\R$, the columns of $P_0$ are orthogonal to those of $U_r(t)$. Let $v$
and $u(t)$, $t\in\R$, be any column of $P_0$ and of $U_r(t)$, respectively. Since for all $t\in\R$
the columns of $U_r(t)$ are in $\mathop\mathrm{im}\nolimits C(t)$, there is a vector $w(t)\in\R^n$ with the property that
$u(t)=C(t)w(t)$, and
\begin{equation*}
    \langle v, u(t)\rangle= \langle v, C(t)w(t)\rangle=
    \langle C(t)^Tv, w(t)\rangle=0,\quad t\in\R,
\end{equation*}
because $v\in\ker E^T=\ker C(t)^T$ for all $t\in\R$. This proves the claim. A similar argument
shows that also $P_r^TU_0(t)$, $V_r^T(t)Q_0$, and $V_0^T(t)Q_r$ are also identically zero.

Since for all $t\in\R$
\[
 P^TQ
=\begin{pmatrix}
                        P_r^TU_r(t) & 0 \\
                             0 & P_0^TU_0(t)
                  \end{pmatrix}\quad
\text{ and }\quad
 V(t)^TQ
=\begin{pmatrix}
                        V_r(t)^TQ_r & 0 \\
                             0 & V_0(t)^TQ_0
                  \end{pmatrix}
\]
are nonsingular, we deduce in particular that so are $P_r^TU_r(t)$ and $V_r(t)^TQ_r$.

Let us compute the matrix product $P^TC(t)Q$ for all $t\in\R$. We omit here, for the sake of
simplicity, the explicit dependence on $t$.
  \begin{equation*}
    \begin{aligned}
      P^TCQ &= P^TUU^TCVV^T Q
            =\begin{pmatrix}
                        P_r^TU_r & 0 \\
                             0 & P_0^TU_0
                  \end{pmatrix}
        \begin{pmatrix}
                        C_r & 0 \\
                             0 & 0
                  \end{pmatrix}\begin{pmatrix}
                        V_r^TQ_r & 0 \\
                             0 & V_0^TQ_0
                  \end{pmatrix}\\
      &= \begin{pmatrix} P_r^TU_r C_r V_r^TQ_r & 0 \\
                                             0 & 0
         \end{pmatrix}.
    \end{aligned}
  \end{equation*}
Which proves the assertion because  $P_r^TU_r$, $C_r$, and $V_r^TQ_r$ are nonsingular.
\end{proof}

redpajama_set_name: RedPajamaArXiv


Gehan Homes has begun to pre-sale at Cadence at Gateway! This is the second builder to begin pre-sales at Cadence, following Pulte's June launch. The demand to secure a home in the intimate Cadence at Gateway community has been growing with Pulte selling 25 homes within a few weeks. Harvard Investments, developer of Cadence at Gateway and an Arizona-based real estate investment and development company, is finishing up the amenities in preparation for its planned October 20, 2018 grand opening.
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Arstanbek Abdylaev, scourge of the Kyrgnet, has struck again. Noted  on Global Voices before for predicting a planet shorn of seasonal transition, this ex-presidential candidate and current head of the "People's Academy" is back to tell the universe what he really meant, having been mocked and misunderstood by netizens back in November 2011.
This time round , Asakeev was limited to a bit-part performance, his coveted position as Abdylaev's number two apparently usurped by Pak. Pak, whose mastery of Russian exceeds that of both his colleagues, proceeded to explain that the original phrase "there will be no winter", had been taken too literally, and that actually it had a "big energetic and informational bloc" capable of creating a "moment of total quantum leap" for humanity.
"Four months ago we wrote to the Kazakh ambassador – we said, you are going to [suffer] terrorist attacks – mass upheavals  – they laughed at us. We wrote to the leader of the Russian Federation – Putin. We said that there would be war in Arabia, and they laughed at us, but there was [war]. Now my words are being proven, not by a historian, or an academic, or paper, but time," Abdylaev said.
Where could the press conference possibly go from there? Well, despite apparently being ridiculed by the Russian leadership, journalists heard, Moscow still has a place in Abdylaev's New World Order – chiefly as the brawn behind Kyrgyzstan's brain. Since Europe will soon be starved, disease-ridden and submerged under water, he reasoned, Russia would have no choice but to turn East.
"Why are we writing to Russia? Russia and Kyrgyzstan will conserve humanity. That's why I call Putin "a complex bio-robot"… We will give a program to the Russian leader, Putin, and he will do it, because he has the power," Abdylaev said.
With a bio robot, you give him a program and he does it, correct? But Putin is a complex bio robot – he himself does it.
Perhaps this difficult, semi-mechanic nature is the source of Putin's reputed marital troubles ….
This phrase [there will be no winter]… is also a code. Anyone who has heard the phrase "there will be no winter" has already received the code and they are using it, even if they are not fully conscious of this. That code contained here among the Kyrgyz should arrive in every person, and through it every person should come to a condition of over-standing; they will become more than human and they will be aware of their true capabilities and their real meanings as human beings… This is why Arstanbek [Abdylaev] called Putin a bio-robot. Barack Obama is also a bio-robot. Other bio-robots stand behind these people and behind these people are other people and today these people are all bio-robots. This code [there will be no winter] allows these bio-robots to become human and write constitutions for the future era, the era we have called the golden era.
So, what is this strange "People's Academy?" Aside from obtuse letters to world leaders, do they have any publications or academic accreditation? Increasingly they are beginning to sound like a cult, a fact that could get Abdylaev into trouble in a country where the state is often hostile towards obscure religious movements .
"Who are they kidding, it has been -20 here for two weeks," quipped [ru] alexxx8999  .
"Is there a doctor in the studio! Or aren't there any doctors in Kyrgyzstan? How can they allow abnormal people airtime?" asked [ru] breeedfrtf  .
"Who let them off the building site?" asked [ru] TheMacsander , a probable reference to the fact that many Kyrgyz work abroad in Russia as labour migrants.
"Judging by the commentaries, Russians differ from fascists only in their stupidity," raged BeksKazama  in response.
But other RuNet types saw reasons to envy their small, mountainous Central Asian neighbour, and took Abdylaev's public appearance as a cue to bash United Russia, the political machine that took victory in Russia's recent elections to the state Duma – a vote tainted  by allegations of massive fraud.
"Kyrgyz wake up, don't call the Russians to your aid! Otherwise you will soon have United Russia and corruption, in other words the whole bouquet 'from Russia with love!'" said [ru] marysimon79  .
"O Great Valley of Chuika [Chui-grown marijuana] Ototo – there will be no winter," said [ru] Mariyajuri .
N.B According to Bishkek-based citizen media portal Kloop, Abdylaev's second press conference has already surpassed his first in terms of online popularity. Once again, the catalyst  for a spike in viewing figures was a prime-time slot on Russian satirist Stas Davydov's internet show "ThisisHorosho ".


The association enables global members and Austrians to be actively involved in development cooperation. This can be through club membership, advocacy, by working for or visiting our events, or by means of support in kind or cash donations.
The annual fee for a club membership is currently 30 Euros. For students with low income, and those living in eastern European countries the annual fee costs 25 Euros. All other membership arrangements are found in the club-laws. The activities are documented in a weblog on the HIBA website as well as HIBA the Facebook site, giving the donors transparency about the usage of donations and resources.
Join us and become a member of the Hilfsverein Baileke (HIBA) association! So you too can give children a future.


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The Foundation receives a very high number of applications and around 3 in 100 are successful.
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Complete the eligibility form on the Foundation's website. Once you have successfully passed the eligibility stage, you will receive an e-mail with a link to the application form.
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A wedding album and slipcase for Molly and Archie Clark. On the front of the album and slipcase I inlaid two of their linoleum prints that they used as table cards for there wedding. The green and brown I used on the album and case were their wedding colors.


text:
Obec Lutopecny se nachází v okrese Kroměříž ve Zlínském kraji. Žije zde  obyvatel. Součástí obce je i vesnice Měrůtky.

Název
Na vesnici bylo přeneseno původní (posměšné) označení jejích obyvatel lutopecni – "skrblíci" ("kterým bylo líto chleba"). Od konce 14. století byla užívána i varianta Litopecny.

Historie
První písemná zmínka o obci pochází z roku 1290.

V Lutopecnách se narodil i generálmajor Alois Šiška, československý pilot, který ve druhé světové válce létal v 311. československé bombardovací peruti RAF. Známý je především pro nouzové přistání s bombardérem Vickers Wellington na hladině Severního moře, na kterém on a zbytek jeho posádky strávili v nafukovacím záchranném člunu 6 dní téměř bez vody a potravin, než byli vyplaveni na nizozemském pobřeží, kde byli následovně zajati německou pobřežní stráží a drženi v zajetí až do konce války. Alois Šiška je nositelem řady vyznamenání, mezi které patří i nejvyšší státní vyznamenání České republiky, Řád Bílého lva III. třídy, in memoriam.

V Lutopecnách žil známý katolický aktivista Augustin Navrátil, jedna z čelných postav českého katolického disentu za komunistického režimu, vydavatel katolického samizdatu, signatář Charty 77 a iniciátor petice Podněty katolíků k řešení situace věřících občanů – jedné z nejmasovějších akcí manifestujících požadavky na respektování práv věřících občanů a náboženské svobody (podepsalo ji 501 590 lidí).

Obyvatelstvo

Struktura
Vývoj počtu obyvatel za celou obec i za jeho jednotlivé části uvádí tabulka níže, ve které se zobrazuje i příslušnost jednotlivých částí k obci či následné odtržení.

Pamětihodnosti
 Kříž u silnice do Měrůtek
 Zvonice
 Pomník padlých
 Restaurace U Vojtěcha
 Rodiště gen. Aloise Šišky, pilota RAF, hrdiny bitvy o Británii

Galerie

Doprava
Územím obce prochází dálnice D1, silnice II/428 v úseku Morkovice-Slížany – Kroměříž a silnice III/36726 Hradisko – Měrůtky. Jihozápadním cípem katastrálního území prochází krátký úsek silnice III/43218 Popovice – Zlobice.

Slavní rodáci
 Augustin Navrátil
 Amálie Pouchlá
 Alois Šiška

Odkazy

Reference

Související články
 Římskokatolická farnost Hradisko u Kroměříže

Externí odkazy

Vesnice v okrese Kroměříž
Obce v okrese Kroměříž
Mikroregion Kroměřížsko
Spolek pro obnovu venkova Ječmínek
Sídla v Litenčické pahorkatině

redpajama_set_name: RedPajamaWikipedia


text:
package org.jboss.weld.tests.contexts.conversation.servlet;

import java.io.IOException;
import java.io.PrintWriter;
import java.util.List;
import java.util.Set;

import jakarta.enterprise.context.Conversation;
import jakarta.inject.Inject;
import jakarta.servlet.ServletException;
import jakarta.servlet.annotation.WebServlet;
import jakarta.servlet.http.HttpServlet;
import jakarta.servlet.http.HttpServletRequest;
import jakarta.servlet.http.HttpServletResponse;

@SuppressWarnings("serial")
@WebServlet("/servlet/*")
public class Servlet extends HttpServlet {

    @Inject
    private Message message;

    @Inject
    private Conversation conversation;

    @Inject
    private DestroyedConversationObserver observer;

    @Override
    protected void doGet(HttpServletRequest req, HttpServletResponse resp) throws ServletException, IOException {
        String uri = req.getRequestURI();
        if (uri.endsWith("/display")) {
            printInfo(resp.getWriter());
        } else if (uri.endsWith("/redirect")) {
            resp.sendRedirect("servlet/display"); // do a redirect, the cid is not propagated;
        } else if (uri.endsWith("/begin")) {
            conversation.begin();
            message.setCid(conversation.getId());
            printInfo(resp.getWriter());
        } else if (uri.endsWith("/end")) {
            conversation.end();
            printInfo(resp.getWriter());
        } else if (uri.endsWith("/set")) {
            setMessage(req);
            printInfo(resp.getWriter());
        } else if (uri.endsWith("/invalidateSession")) {
            observer.reset();
            req.getSession().invalidate();
            printInfo(resp.getWriter());
        } else if (uri.endsWith("/listDestroyedMessages")) {
            PrintWriter writer = resp.getWriter();
            writer.append("DestroyedMessages: ");
            printMessages(writer, observer.getDestroyedMessages());
        } else if (uri.endsWith("/listConversationsDestroyedWhileBeingAssociated")) {
            PrintWriter writer = resp.getWriter();
            writer.append("ConversationsDestroyedWhileBeingAssociated: ");
            printSessionIds(writer, observer.getAssociatedConversationIds());
        } else if (uri.endsWith("/listConversationsDestroyedWhileBeingDisassociated")) {
            PrintWriter writer = resp.getWriter();
            writer.append("ConversationsDestroyedWhileBeingDisassociated: ");
            printSessionIds(writer, observer.getDisassociatedConversationIds());
        } else {
            resp.setStatus(404);
        }
        resp.setContentType("text/plain");
    }

    @Override
    protected void doPost(HttpServletRequest req, HttpServletResponse resp) throws ServletException, IOException {
        String uri = req.getRequestURI();
        if (uri.endsWith("/set")) {
            setMessage(req);
            printInfo(resp.getWriter());
        } else {
            resp.setStatus(404);
        }
        resp.setContentType("text/plain");
    }

    private void printInfo(PrintWriter writer) {
        writer.append("message: ").append(message.getValue());
        writer.append('\n');
        writer.append("cid: [").append(conversation.getId());
        writer.append(']');
        writer.append('\n');
        writer.append("transient: ").append(Boolean.toString(conversation.isTransient()));
        writer.append('\n');
    }

    private void printSessionIds(PrintWriter writer, Set<String> ids) {
        for (String id : ids) {
            writer.append('<').append(id).append('>');
        }
    }

    private void printMessages(PrintWriter writer, List<Message> messages) {
        for (Message message : messages) {
            writer.append("<M:").append(message.getCid()).append('>');
        }
    }

    private void setMessage(HttpServletRequest request) {
        String value = request.getParameter("message");
        if (value == null) {
            throw new IllegalArgumentException("message must be specified");
        }
        message.setValue(value);
    }
}

redpajama_set_name: RedPajamaGithub


In this article we'll share some places that are unknown to most tourists, little-hidden gems that you will be able to admire while you buzz about following ''The Musts tour'' and its trail of attractions. Traditional restaurants, wonderful shops, fantastic gelato parlours leading you to discover Rome "off the beaten track", that only a few guides care to mention.
Thanks to the iPad and GPS navigator provided with our cute Buzz buggies and our iconic Vespas, Buzz4tours has provided pre-set itineraries around Rome for some time now. A Fun, quick, easy and ecological way to visit the marvels of the Eternal City.
Passetto di Borgo: a secret elevated passage between Vatican and St. Angel's Castle in Rome.
The first stop on our ''The Musts'' itinerary is the mighty St Angel's Castle, but how well do you know Borgo Pio? Located just around the corner from our office, ebbed inside the XIII century walls of the Passetto, the secret passage way that runs from the Vatican to the nearby castle, it is vibrant with restaurants and artisan's shops strewn along its narrow cobbled streets, enough to charm the most hardened of travellers. Now, are you a foody…? Good, a nice way to start the day is to taste one of the delicious cakes made by the Pasticceria il Farinone (Vicolo del Farinone 19) or try the Roman's favourite breakfast (cappuccino + cornetto) at the La Latteria (Borgo Pio 48).
Trapizzino is a triangular pocket of pizza dough filled with classic Roman recipes.
Back on board Buzz or Vespa let us continue our journey to Trastevere, the place to hang-out for the locals and not. If you are feeling like something more than a snack you should try Trapizzino (Piazza Trilussa, 46). You can sink your teeth into a Trapizzino which is a triangular shaped wedge of white pizza which is filled with typical Roman food classics. The piazza can also host you for a sundowner before you move off for dinner at Enzo's (Via dei Vascellari, 29), and have their excellent Carbonara.
Trastevere is one of the Capital's most popular districts. Discovering its alleyways is an experience not to be missed by all those who wish to plunge into Roman habits and breath the air of a Rome gone by. Right here there is a gem known only to a few.
A visit to the ancient Spezieria (literally Spices shop) amongst its ampoules, fine vases, a liquor lab and XVII century furniture is definitely a "dive into the past" giving you a peak of everyday life in those days. You will find it in Piazza della Scala 23.
The beautiful Municipal Rose Garden is a public garden in Rome, located on Aventine Hill in Rome.
Buzz will probably feel the added weight following your stop but just click on "The Magic Keyhole" on the pre-set GPS and drive off over Tiber Island to take a peek through the keyhole before a short drive back down to none other than the Circus Maximus. Not many know that, just nearby, you can admire the wonderfully beautiful Rose Garden (Via di Valle Murcia, 6). Be amazed by over 1100 different varieties of roses from all over the world. This is an absolute MUST if you are in Rome in the spring.
The next few stops do not need an introduction, the Roman Forum, Capitoline Hill, and the Colosseum. Our nippy buzz buggies will get you there in no time as will our Vespa, as you pretend to be Gregory Peck or Audrey Hepburn (Roman Holidays). Having carried on to the Pantheon and feeling slightly stunned by all this ancient beauty, you may decide a gelato (ice-cream) would go down really well. "Said and done", you happen to be near two of the better known gelaterias in the whole of Rome that stand out a mile from the many touristy places you may have happened on before. Giolitti (Via degli Uffici del Vicario, 40) and the more modern Grom (Via della Maddalena, 30a).
Next stop Piazza Navona and the nearby Campo de' Fiori – here you'll find two very different places that will satisfy both your love of all things sweet and that for culture. These are ZUM (Piazza del Teatro di Pompeo, 20), who specialize in the well known Tiramisù, the typical italian dessert made with Mascarpone (creamy white cheese) and coffee on a bed of biscuits, possibly not very slimming but delicious… The other, is the Biblioteca Angelica (Piazza di S. Agostino, 8), a wonderful library designed by Luigi Vanvitelli that hosts some 200,000 books both modern and ancient.
The Dolls' Hospital a place where antique dolls are given new life. Photo credit: © Gary Jones.
So, you may have visited the Trevi Fountain, or you have perhaps gone mad with your credit card on the four floors of the nearby Rinascente in via del Tritone, with over 800 brands from international fashion houses, or perhaps you're sitting on the one of the famous Spanish Steps before descending on Piazza del Popolo in the very heart of Rome. Fine, now walk down Via di Ripetta one of the three roads off the piazza, and proceed to #29. Here you will find a really strange place where time seems to have stopped, this is the Dolls' Hospital run by able craftsmen who repair dolls, puppets and vintage toys. The atmosphere is truly unreal: not to be missed!
In 1904 the Palazzo was bought by Enrichetta Hertz, who refurbished it and then made it the home of a wonderful Paintings Collection and a very rich library specializing in Art literature, known today as the Biblioteca Hertziana (Hertzian Library).
Art and deformity meet again in a building just off Piazza di Spagna. The place is called l'Anamorfosi del Convento del Sacro Cuore" right next to the Church of Trinità dei Monti.
Anamorfistic is said of an image in which the subject may only be perceived by looking at it from a specific angle. To admire it, you should enter one of the upper galleries of the cloister where you will find a large painting of St.Francis of Paola. As you will walk alongside the painting and reach its middle, it will transform itself magically revealing a bucolic scene.
Your journey will end in St.Peter's Square amongst Papal vestiges and works of art that have made history. There is much more to Rome of course and other great itineraries await.


text:
Archive : 24th AIFF 2018 : Bergman 100
Bergman 100: A Century of Cinema
On the occasion of the celebration of one hundred years after the great Swedish director's birth, the 24th AIFF participates in the events of the global cinema community paying a modest homage to a colossal work that is worth being remembered forever.
The audience will have the opportunity to explore a marvelous installation dedicated to unpublicized material from Bergman's career, that is presented in Greece for the first time, Jane Magnusson's much-debated documentary about the unseen aspects of a film career and a whole life, as well as the digitally remastered copy of «Shame»: one of the iconic director's most important films, which is going to be presented in an anniversary screening, 50 years after its release.
With the support of the Embassy of Sweden in Athens
Legendary Swedish director, writer, and producer who worked in film, television, theatre and radio. Bergman directed over sixty films and documentaries for cinematic release and for television, most of which he wrote. A nine-time Oscar nominee, Bergman also directed over 170 plays.
1982 Fanny and Alexander
1974 Scenes From a Marriage
1972 Cries & Whispers
1968 Shame
1966 Persona
1963 The Silence
1960 The Virgin Spring
1957 Wild Strawberries
1957 The Seventh Seal
More films in : Bergman 100
22/9/18, 17:45, Danaos 1
28/9/18, 22:30, Astor
Bergman: A Year in a Life
23/9/18, 19:15, Odeon Opera 1


Have you heard about makeover? No! Don't you want to rejuvenate your personality? Yes, Great! Black Hair Salon Review in Northcliff there are numerous beauty salons established in Miami to take care of individuals' appearances. Best Hair Salons For Black Hair some of these are sophisticated and high-quality. Beauty salons are now going out of their ways to focus on customized solutions to enhance individuals' looks.


text:
Aceros Camesa S.A. de C.V.
Margarita Maza De Juarez/154//Nueva Industrial Vallejo//Cien Metros Y AV Central 7700 Iztapalapa Distrito Federal MX
Where does Aceros Camesa S.A. de C.V. buy?
At Zapp Precision Wire, Inc.
475 International Cir, Summerville, SC 29483, US
318 suppliersexplore
To whom does Aceros Camesa S.A. de C.V. sell?
To Wireco Worldgroup Inc.
West 75 TH St. Prairie Village US
Which products does Aceros Camesa S.A. de C.V. buy?
Stranded wire, ropes and cables, of iron or steel (excl. electrically insulated products and twisted fencing wire and barbed wire)
392 purchased productsexplore
Which products does Aceros Camesa S.A. de C.V. sell?
From which countries does Aceros Camesa S.A. de C.V. buy?
in United States of America
19,779.4 t
38 import countriesexplore
To which countries does Aceros Camesa S.A. de C.V. sell?
119,633 t
On this page you will get insights into the market activities of Aceros Camesa S.A. de C.V. which have been individually prepared by our research team.
Who are the contact persons in the purchasing department of Aceros Camesa S.A. de C.V.?
ABRAMS world trade wiki offers the search for current contact persons in the purchasing department of Aceros Camesa S.A. de C.V. as a free service.
Who are the contact persons in the sales department of Aceros Camesa S.A. de C.V.?
ABRAMS world trade wiki offers the search for current contacts in the sales department of Aceros Camesa S.A. de C.V. as a free service.
Where can I find information about compliance at Aceros Camesa S.A. de C.V.?
ABRAMS world trade wiki offers the search for latest compliance news from all over the world about Aceros Camesa S.A. de C.V. as a free service.
What are the latest press releases about Aceros Camesa S.A. de C.V.?
ABRAMS world trade wiki offers to search for the latest press releases from around the world about Aceros Camesa S.A. de C.V. as a free service.
Who are the suppliers of Aceros Camesa S.A. de C.V.?
The performance of a company depends heavily on its suppliers. Therefore, a reliable network of partners (suppliers, dealers and manufacturers) is elementary. In bills of lading, a total of 318 international suppliers of Aceros Camesa S.A. de C.V. could be identified on which the company is relying. One of the suppliers is, for example, the company Zapp Precision Wire, Inc., which delivered 69 shipments.
Among 38 identified supplying countries, an important source to purchase products from is: United States of America. 1,827 shipments of goods could be identified from this country.
Get an overview of Aceros Camesa S.A. de C.V.'s international procurement and its 318 suppliers with Company Transparency
What does the supply chain of Aceros Camesa S.A. de C.V. look like?
Generally you can differentiate the supply chain or supply network of Aceros Camesa S.A. de C.V. by two perspectives:
The supplier side (a so-called Tier-1-supplier is a direct supplier, Tier-2 is a supplier of the Tier-1 supplier, Tier-n is a supplier at any level in the Aceros Camesa S.A. de C.V. supply network).
The buyer side (a so-called Tier-1 buyer is a direct buyer, Tier-2 is a buyer of the Tier-1 buyer, Tier-n is a buyer at any level in the Aceros Camesa S.A. de C.V. distribution network).
To ensure business stability and continuity, a supply chain or supply network must be constructed in such a way that external influences can cause as little damage as possible to Aceros Camesa S.A. de C.V.. This applies both in the area of procurement (the loss of a supplier must be covered by an alternative supplier) as in the area of sales (the loss of a buyer must be covered by an alternative buyer).
Get an overview of the international supply chain of Aceros Camesa S.A. de C.V. (upstream/procurement and downstream/sales) from Tier-1 to Tier-n
E.G.: Tier 1 supplier Zapp Precision Wire, Inc.Tier-1 Buyer Wireco Worldgroup Inc.
Is the supply chain of Aceros Camesa S.A. de C.V. stable?
Each tier in the supply chain or supply network on the supplier side of Aceros Camesa S.A. de C.V. involves a single risk. The more tiers, the more difficult it is to analyze and monitor the business partners. Suppliers of suppliers are generally unknown, so there is already an increasing risk at the second (Tier 2) level.
Analyze the potential of shortening and therefore stabilizing the supply chain of Aceros Camesa S.A. de C.V.
Which products does Aceros Camesa S.A. de C.V. purchase?
Based on the analysis of international bills of lading (bill of lading/ bills of lading/ BOL), a total of 392 different product categories that have been purchased by Aceros Camesa S.A. de C.V., can be identified.
"Stranded wire, ropes and cables, of iron or steel (excl. electrically insulated products and twisted fencing wire and barbed wire)". This HS code was sighted 576 times in total.
From the product descriptions of the analyzed bills of lading (bill of lading/ bills of lading/ BOL), the following purchased products from various suppliers can be identified as examples:
CABLE DE ACERO GALVANIZADO CABLE DE ACERO GALVANIZADO CABLE DE ACERO GALVANIZADO
ALAMBRE PARA BOBINAR
ALAMBRES DE ACERO INOXIDABLE
ALAMBRE DE ACERO SIN ALEAR ELECTROGALVANIZADO PARA LA FABRICACION DE GRAPAS
CABLE DE ACERO GALVANIZADO CABLE DE ACERO GALVANIZADO 28 MM
ALAMBRE DE COBRE PARA BOBINAR ALAMBRE DE COBRE PARA BOBINAR
TARIMAS DE ACERO CARRETES DE ACERO
CARRETES DE ACERO TARIMAS DE ACERO
ALAMBRE DE ACERO INOXIDABLE DE SECCION TRANSVERSAL CIRCULAR ALAMBRE DE ACERO INOXIDABLE DE SECCION TRANSVERSAL CIRCULAR ALAMBRE DE ACERO INOXIDABLE DE SECCION TRANSVERSAL CIRCULAR
CABLE DE ACERO GALVANIZADO 10 MM
Get an overview of internationally purchased products by Aceros Camesa S.A. de C.V. with Company Transparency
Is Aceros Camesa S.A. de C.V. able to reduce costs in the upstream supply chain?
Each tier in the supply chain or supply network of Aceros Camesa S.A. de C.V. is adding value and therefore costs. The more tiers, the more expensive the product gets. Supply chains are normally not very transparent, trade secrets are part of the survival tactics in each stage of the value chain, especially for trading companies.
If e.g. a Tier-1 supplier from Aceros Camesa S.A. de C.V. that primarily trades a product and doesn't add any value (only fulfills the distributor function) it is possible to shorten the supply chain and reduce purchasing costs. A previous Tier-2 supplier now turns into a Tier-1 supplier, which we call "go-direct". In our tool Supply Chain Intelligence as far as international suppliers are concerned, this analysis can be done with one click:
Analyze the potential of shortening the supply chain of Aceros Camesa S.A. de C.V. and saving costs on the procurement side.
Who are the buyers of Aceros Camesa S.A. de C.V.?
The performance of a company depends fundamentally on its buyers. Therefore, a largest possible base of stable buyers is crucial. A total of 233 international buyers of Aceros Camesa S.A. de C.V. could be identified in bills of lading. For example, one of the buyers is the company Wireco Worldgroup Inc., which has received 167 shipments.
Among 35 identified buying countries, an important target country to sell products to is: United States of America. 6,372 shipments of goods were shipped to this country in the period covered.
Get an overview of the international sales and its 233 buyers of Aceros Camesa S.A. de C.V. with Company Transparency
Based on the analysis of international bills of lading (bill of lading/ bills of lading/ BOL), a total of 63 different product categories sold by Aceros Camesa S.A. de C.V., can be identified.
"Stranded wire, ropes and cables, of iron or steel (excl. electrically insulated products and twisted fencing wire and barbed wire)". This HS code was sighted 2,267 times in total.
CABLE ELECTROMECANICO CABLE ELECTROMECANICO CABLE ELECTROMECANICO
CABLE DE ACERO NEGRO CABLE DE ACERO NEGRO CABLE DE ACERO NEGRO
CABLE DE ACERO GALVANIZADO ALAMBRE DE ACERO SIN ALEAR GALVANIZADO CABLE DE ACERO GALVANIZADO
CABLE DE ACERO NEGRO ALAMBRE DE ACERO SIN ALEAR GALVANIZADO CABLE DE ACERO GALVANIZADO
CABLE DE ACERO GALVANIZADO CABLE DE ACERO NEGRO CABLE DE ACERO NEGRO
CABLE DE ACERO NEGRO CABLE DE ACERO GALVANIZADO CABLE DE ACERO NEGRO
CABLE DE ACERO CABLE DE ACERO NEGRO CABLE DE ACERO NEGRO
ALAMBRE DE ACERO GALVANIZADO (GALFAN)
ALAMBRE DE ACERO NEGRO ALAMBRE DE ACERO GALVANIZADO
Get an overview of internationally sold products by Aceros Camesa S.A. de C.V. with Company Transparency
Is Aceros Camesa S.A. de C.V. able to optimize its profits on the buyer side of the supply chain? (downstream)
Each tier in the buyer side of the supply chain or supply network of Aceros Camesa S.A. de C.V. is adding value and therefore costs. The more tiers, the more expensive the product gets. Supply chains are normally not very transparent, trade secrets are part of the survival tactics in each stage of the value chain, especially for trading companies.
If e.g. a Tier-1 buyer from Aceros Camesa S.A. de C.V. that primarily trades products and doesn't add any value (only fulfills the distributor function) it is possible to shorten the supply chain downstream giving a potential to optimize profit margins for Aceros Camesa S.A. de C.V.. A previous Tier-2 buyer now turns into a Tier-1 buyer, which we call "go-direct". In our tool Supply chain Intelligence as far as international buyers are concerned, this analysis can be done with one click:
Analyze the potential to shorten the supply chain and optimize the profit margins on the sales side of Aceros Camesa S.A. de C.V.
Is Aceros Camesa S.A. de C.V. a reliable supplier?
Each company contributes significantly minimizing risks for its buyers, through its own stability. Internal risks (endogenous factors) can be e.g. financially (financial ratios, free cash flow, balance sheets) or operationally (production, lead times). Since 2021, external risks have increased due to worldwide shortages and are an important factor as to whether a supplier such as Aceros Camesa S.A. de C.V. can itself reliably deliver due to external dependencies (exogenous factors).
How is the supply chain or supply network structured on the supplier side of Aceros Camesa S.A. de C.V.? Are there suppliers or products in the supply chain that are reported as critical, e.g. in international media?
Which buyers does Aceros Camesa S.A. de C.V. consistently sell to over a long period of time that can be identified as a good reference?
Does Aceros Camesa S.A. de C.V. have problems supplying at the moment?
The more constant outgoing shipments of a company's, the easier it is to assess potential problems for a supplier like Aceros Camesa S.A. de C.V.
Since 2021, global supply shortages have increasingly added external factors as to whether a supplier like Aceros Camesa S.A. de C.V. can deliver reliably itself through dependencies.
Here you can see an analysis of shipments by Aceros Camesa S.A. de C.V. to international buyers in the last few months with Company Transparency
Does Aceros Camesa S.A. de C.V. have procurement problems at the moment?
The more constant incoming shipments, the better one can determine possible procurement problems of a company like Aceros Camesa S.A. de C.V.
Here you can see an analysis of shipments from international suppliers to Aceros Camesa S.A. de C.V. in the last few months with Company Transparency
What is the CO₂ footprint and sustainability of Aceros Camesa S.A. de C.V. products?
Sourcing products as regionally as possible with short shipping routes and the use of sustainable materials define the CO₂ footprint and sustainability of the products of any company, including Aceros Camesa S.A. de C.V.
Setting and meeting sustainability goals for own products of Aceros Camesa S.A. de C.V.
Based on the development of current shipments (in terms of number of shipments and quantity of goods) and changes in international suppliers and routes, it is possible to analyze the extent to which Aceros Camesa S.A. de C.V. has already made efforts in recent months and years, to focus on environmentally relevant aspects, like carbon footprint and sustainability in its business activities.
Get an overview of the international shipments that have an impact on the CO₂ footprint, as well as the sustainability of Aceros Camesa S.A. de C.V.
Who are the competitors of Aceros Camesa S.A. de C.V.?
Generally you can differentiate the competition of Aceros Camesa S.A. de C.V. by the purchase and sales perspective.
Who are the competitors of Aceros Camesa S.A. de C.V. on the purchase side?
Competitors of Aceros Camesa S.A. de C.V. on the purchase side (procurement of raw materials and components for production or distribution) are exposed on ABRAMs wiki under Competitive Intelligence, purchased products, classified by HS-Codes (e.g. 731210, 732690) or by top product terms.
Who are the competitors of Aceros Camesa S.A. de C.V. on the sales side (selling similar products)?
Competitors on the sales side of Aceros Camesa S.A. de C.V. (selling raw materials or components for production or distribution) are exposed on ABRAMS wiki under Competitive Intelligence, sold products classified by HS-Codes (e.g. 731210, 721710) or by top product terms.
from Aceros Camesa S.A. de C.V.
Which companies buy similar products like Aceros Camesa S.A. de C.V.?
Kiswire Cord Ltd.
Cty Tnhh Kone Việt Nam
Which companies sell similar products like Aceros Camesa S.A. de C.V.?
Kos Wire Ltd.
Vessel Co., Inc.
Metal One Steel Products Corporatio


text:
Berkshire Organics in Pittsfield, Massachusetts, were one of the first businesses to receive a loan from the Berkshire Columbia Investment Network. From left, front row: Ellen Russo, Aleisha Gibbons, Robin Zeamer, Signe Schaefer, Natalie Shiras. Back row, l to r: Peter Schulte, David Curtis, Brian Gibbons, Richard Evans, Christopher Schaefer, Rich Lumma.
Berkshire Columbia Investment Network: A helping hand to small businesses
By David Scribner Sunday, Nov 13, 2016 Trade and Commerce
Great Barrington – Two and half years ago, Robin Zeamer and Christopher Schaefer and his wife Signe were having dinner with their friend Rich Lumma at Radio Mexican Restaurant in Hudson, N.Y. They were mulling over the creation of a Local Investor Opportunities Network (LION) through which individuals could provide low-interest financing to local enterprises.
Such informal investment associations had been popping up across the country, as alternatives to conventional small business financing arrangements, preferring a personal connection between business and investor that institutional loans did not have. LION associations have been created in Washington, California, Wisconsin, Oregon and New York City.
Since that session, Zeamer, Lumma, and Schaefer — along with Monte Wasch of New Lebanon, New York – have put together a loose affiliation of investors called the Berkshire Columbia Investor Network (BCIN) whose 12 members have now provided more than $200,000 in loans – at between 5 and 8 percent interest – to local entrepreneurs. The annual BCIN membership fee is $15. Among BCIN's beneficiaries, so far, are Laura Meister at Farm Girl Farm in Sheffield, Berkshire Organics in Pittsfield, the Stanton Home life-sharing foster care homes, and Ooma Tesaro's Marinara Sauce in Windsor.
"We were four friends who got together because we were tired of no return on our investments, and we were locally engaged," Schaefer explained, during an interview in his snug living room on Hillside Avenue. "We are locally engaged, and we wondered what we could do with money that's useful to the local economy and to helping people through hard times."
They got their answer from a similar investor network in Port Townsend, Washington, in East Jefferson County, that became the model for the Berkshire Columbia Investor Opportunities Network. Established in 2006, the Port Townsend LION now has 60 members, and has issued $3 million in investments, primarily low-interest loans.
BCIN is now seeking to expand its membership and opportunities for small businesses, and will be holding an information forum on Monday, November 14, from 5 to 6:15 p.m. at the South Berkshire Friends Meetinghouse, 280 State Rd, in Great Barrington. Guest speakers will include Laura Meister, Peter Stanton, and Dan Esko, general manager of the Berkshire Co-op Market.
"We need another 20 members," explained Zeamer. "What we do is broker a conversation between investor and entrepreneur, and develop a wonderful kind of relationship between them, thereby using investment in a different way. We're not a corporation. We are not an impersonal system. But we are recognized by Keith Girard at the Small Business Administration office in Pittsfield, who sends people to us, as does David Curtis at 1Berkshire. And we've established a relationship with Berkshire Grown."
Most LIONs rely on pre-existing relationships between investors and business owners, thus avoiding federal securities laws, and the LION doesn't invest as a group, but allows eah member to make his or her decision and negotiate terms.
"We don't do start-ups, and we require that businesses be at a break-even point," Schaefer explained. "We do offer bridge loans."
For example, Laura Meister at Farm Girl Farm did not receive her USDA loan for spring planting in April, so she was successful in getting a loan from BCIN.
"We try to educate people about how they may be investing their disposable income," Schaefer said. "But our loans to entrepreneurs have been successful. The monthly checks from our investments are rolling in. So far, everybody's paying."
For information on BCIN, contact Zeamer, Schaefer or Wasch at bcin@outlook.com.
The media. It sounds like a convention of spiritualists.
— Tom Stoppard
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