text:
\section{Introduction}
\label{sec:int}
The theory of {\em linear preservers} (linear operators preserving certain
families of matrices or polynomials) is a widely developed and active area of
mathematics (see, e.g., \cite{SU} and the references therein).
Linear preservers have found applications in many
areas such as approximation theory, probability theory, and
statistics (see, e.g., \cite{karlin}), and have even been
used to give interesting reformulations of the Riemann Hypothesis
\cite{csordas}. One of the most classical instances of the theory of linear
preservers occurs in the setting of real-rooted polynomials, initiated in the
late 19th century by Laguerre and Hermite.

Given a sequence of real numbers  $\gamma=\{\gamma_{j}\}^\infty_{j=0}$
consider the linear operator $T_{\gamma}: \mathbb{R}[x]\to \mathbb{R}[x]$ acting on
each $x^j$ by multiplication by $\gamma_j$. We refer to such a $T_\gamma$ as
the {\em diagonal operator} corresponding to $\gamma$. Let
$R\!R\subset\!\mathbb{R}[x]$ denote the collection of polynomials all of whose
complex roots are real, i.e., {\em \underline{r}eal-\underline{r}ooted}
polynomials. Following
\cite{PS} we
call $\gamma$ a {\em multiplier sequence} ({\em``Faktorenfolge"})
{\em of the first kind} if $T_\gamma(R\!R)\!\subseteq\!R\!R$. Similarly,
let $S\!S$ denote the subset of $R\!R$ consisting of polynomials $p$ whose
nonzero roots (all real, by assumption) are all of the \underline{s}ame
\underline{s}ign.
A {\em multiplier sequence of the 2nd kind} is then a $\gamma$ with
$T_\gamma(S\!S)\!\subseteq\!R\!R$.

The following result of Pol\'ya and Schur is fundamental.
\begin{thma}\label{th:PS} \cite{PS}
Let $\gamma=\{\gamma_j\}^\infty_{j=0}$ be a sequence of real numbers
and $T_\gamma : \mathbb{R}[x] \to  \mathbb{R}[x]$ the corresponding diagonal  operator.
Then:
\begin{enumerate}
\item[(i)]{$\gamma$ is a multiplier sequence of the $1$st kind
(i.e., $T_\gamma(R\!R)\!\subseteq\!R\!R$)
iff for all $n\!\in\!\mathbb{N}$ we have $T_\gamma((1+x)^n)\!\in\!S\!S$.}
\item[(ii)]{$\gamma$ is a multiplier sequence of the $2$nd kind
(i.e., $T_\gamma(S\!S)\!\subseteq\!R\!R$)
iff for all $n\!\in\!\mathbb{N}$ we have $T_\gamma((1+x)^n)\!\in\!R\!R$. \qed}
\end{enumerate}
\end{thma}
\begin{remark}
Pol\'ya and Schur also obtained a transcendental
characterization in terms of the generating function
$\Phi_\gamma(x)=\sum\limits^\infty_{k=0} \frac{\gamma_k}{k!} x^k$.
\end{remark}

There exist obvious versions of these notions for polynomials of bounded
degree. In particular, a sequence $\gamma=(\gamma_0,\gamma_1,...,\gamma_k)$  will be
referred to as a {\em multiplier sequence of length $k+1$} or simply a
{\em finite multiplier sequence} if it has the above mentioned properties when
acting on the linear space $\mathbb{R}_k[x]$ of real polynomials of degree at most
$k$. In particular, we define $R\!R_k:=R\!R\cap\mathbb{R}_k[x]$ and
$S\!S_k:=S\!S\cap R\!R_k$.

Craven and Csordas proved 60 years later that for a finite length
multiplier\linebreak
\scalebox{.93}[1]{sequence $\gamma$, checking whether $\gamma$ is of first or second
kind can be reduced to checking the}\linebreak
\scalebox{.95}[1]{image of just one polynomial under $T_\gamma$
(see \cite[Thm.\ 3.7]{CC2} and \cite[Thm.\ 3.1]{CC1}).}
\begin{thmb}\label{th:MS}
Let $\gamma=(\gamma_0,\ldots,\gamma_k)$ and $T_\gamma$ the corresponding diagonal
operator. Then for all $k\!\in\!\mathbb{N}$, we have:
\begin{enumerate}
\item[(i)]{$T_\gamma(R\!R_k)\!\subseteq\!R\!R_k$ iff $T_\gamma\!\left((1+x)^k\right)\in
S\!S$.}
\item[(ii)]{$T_\gamma\!\left(S\!S_k\right)\!\subseteq\!R\!R_k$ iff
$T_\gamma\!\left((1+x)^k\right)\in R\!R$.}
\end{enumerate}
\end{thmb}
\begin{remark}
While Assertion (i) is merely a rewording of \cite[Thm.\ 3.7]{CC2},\linebreak
Assertion (ii) appears to be new and follows upon a closer examination
of Section 3 of \cite{CC2}.
\end{remark}

Letting $q(x)\!:=\!x^m(1+x)^2$, note that $q\inS\!S\subsetneqqR\!R$ and $q$
has $-1$ as a root of multiplicity $2$.
It then follows that if one decreases the coefficient of $x^{m+1}$
in $q$ (and leaves the coefficients of $x^m$ and $x^{m+2}$ fixed)
then the resulting polynomial has non-real roots. With a little more work one
then easily concludes that any multiplier sequence $\gamma=(\gamma_0,\gamma_1,
\ldots)$ of first or second kind must satisfy {\em Tur\'an's Inequalities}
(see, e.g., \cite{CVV} and  \cite[Problem 4.8]{CC3}):
$\gamma^2_j\geq\gamma_{j-1}\gamma_{j+1}$ for all
$j\geq 2$. Since we can naturally identify any finite
multiplier sequence $(\gamma_0,\ldots,\gamma_k)$ with the
infinite sequence $(\gamma_0,\ldots,\gamma_k,0,0,\ldots)$ the
Tur\'an Inequalities clearly hold for finite length multiplier sequences
(of first or second kind) as well. The converse fails, however, as
can be easily seen by perturbing the nonzero coefficients of $x^m(1+x)$
instead. The occurence of roots of multiplicity $>\!1$ here is one reason
it is natural to start thinking of discriminants (see also Figures 1 and 2
below).

\begin{remark}
Positive sequences satisfying Tur\'an's inequalities are called
{\em log-concave} and find frequent applications in combinatorics.
An analoguous notion with the coefficients weighted by binomial
coefficients is known as {\em ultra log-concavity} \cite{Li,KSh}.
\end{remark}

We will return to $x^m(1+x)$ momentarily but observe now that the polynomial
$x^m(1+x)^2$ has the following special property: all polynomials
obtained by arbitrary sign flips of its coefficients also belong to $R\!R$.
\begin{dfn}
A real polynomial $p$ is called {\em sign-independently
real-rooted} if $p$ is real-rooted and all polynomials obtained by
arbitrary sign flips of the coefficients of $p$ are real-rooted as well.
We let $\si$ denote the set of all \underline{s}ign-\underline{i}ndependently
real-rooted polynomials and $S\!I^\geq$ denote the subset of $\si$
consisting of polynomials with all coefficients nonnegative.
Finally, we call $\gamma$ a {\em multiplier sequence of the
$3$rd kind} if $T_\gamma\!\left(S\!I^\geq\right)\!\subseteq\!R\!R$.
\end{dfn}

Clearly, $S\!I^\geq\!\subsetneqq\!\si\!\subsetneqq\!S\!S\!\subsetneqq\!R\!R$.
Another simple example of a sign-independently real-rooted polynomial is
$x^m(1+x)$ and less trivial examples can be found in Section \ref{sub:real}.
Similar to our earlier development we define $\si_k\!:=\!\si\cap \mathbb{R}_k[x]$
and $S\!I^\geq_k\!:=\!S\!I^\geq\cap\mathbb{R}_k[x]$. The sets $S\!I^\geq_3$, $S\!S_3$, and $R\!R_3$ are
illustrated in Figure 2 below.

\medskip
Our main results are summarized by the following $2$ theorems and
a corollary.
\begin{thm}\label{th:main} $\gamma$ is a
multiplier sequence of the third kind (finite or infinite)
iff it is log-concave, i.e., $T_\gamma\!\left(x^n(1+x)^2\right)\inR\!R$ for all
$n\!\in\!\mathbb{N}$. Moreover, any such $\gamma$ satisfies
$T_\gamma\!\left(S\!I^\geq\right)\!\subseteq\!S\!I^\geq$.
\end{thm}

\begin{corollary}
If $p(x)=a_0+a_1x+\cdots+a_kx^k\inS\!I^\geq_k$ then
$a^2_\nu\geq 4a_{\nu-1}a_{\nu+1}$ for all $\nu\in\{1,\ldots,k-1\}$,
and any truncated polynomial $a_mx^m+a_{m+1}x^{m+1}+\cdots+a_nx^n$
(for $0\leq m<n\leq k$) has all its nonzero roots negative.
\end{corollary}

\noindent
\begin{picture}(200,220)(0,-37)
\put(150,0){\epsfig{file=regions.eps,height=2.6in}}
\put(-40,0){\epsfig{file=cubicdisc.eps,height=2.6in}}
\put(13,176){{\small $b$}}
\put(30,60){{\small {\bf 3 real, with}}}
\put(30,50){{\small {\bf different signs}}}
\put(80,110){{\small {\bf only 1 real}}}
\put(144,150){\begin{rotate}{45}
{\tiny {\bf 3 negative}} \end{rotate}}
\put(167,20){$a$}
\end{picture}

\vspace{-1.8cm}
\noindent
\begin{minipage}[t]{2.4in}
{\scriptsize {\sc Figure 1}: The discriminant variety of the
family\linebreak

\vspace{-.3cm}
$1+ax+bx^2+x^3$ separates the coefficient space\linebreak

\vspace{-.45cm}
into regions according to the number of real roots.}
\end{minipage}
\hspace{.3in}
\begin{minipage}[t]{2.2in}
{\scriptsize {\sc Figure 2}: Corresponding slices of $S\!I^\geq_3$,
$S\!S_3$,\linebreak

\vspace{-.3cm}
and $R\!R_3$: $S\!I^\geq_3$ is in black, $S\!I^\geq_3\!\subsetneqq\!
S\!S_3\!\subsetneqq\!R\!R_3$,\linebreak

\vspace{-.45cm}
and the complement of $R\!R_3$ is white. }
\end{minipage}

\medskip
\noindent
Davenport and Pol\'ya observed earlier \cite{DaPo} that
log-concave positive sequences
form a semigroup with respect to the {\em Hadamard product}
$(\gamma_0,\gamma_1,\ldots)\cdot (\gamma'_0,\gamma'_1,\ldots)\!:=\!
(\gamma_0\gamma'_0,\gamma_1\gamma'_1,\ldots)$. In particular, it will be fruitful
to observe later that the image of such sequences under coordinate-wise
logarithm forms a cone.

More to the point, via {\em $A$-discriminant theory}
\cite{GKZ}, we can reinterpret the sets $S\!I^\geq_k$, $S\!S_k$, and $R\!R_k$
in terms of the complement of an important hypersurface associated
to $k$. This point of view yields yet another new family of multiplier
sequences, in some sense dual to $S\!I^\geq$. \\
{\bf Definition 2.} {\em
We define $I\!I^\geq_k$ to be the set of those polynomials
$p(x)=a_0+a_1x+\cdots+a_kx^k$ such that (i) $a_j\!\geq\!0$ for all
$j$, (ii) $a_0,a_k\!>\!0$, (iii) $p$ has exactly $1$ or $0$
real roots according as $k$ is odd or even, (iv) for any polynomial $p^*$
obtained from $p$ by multiplying any subset of the $a_i$ with $i\!\in\!\{1,
\ldots,k-1\}$ by $-1$, $p^*$ also has maximally many
\underline{i}maginary roots in the sense of Condition (iii).}

\noindent
Note that for $k$ even, any polynomial $p\!\in\!I\!I^\geq_k$ is positive
on all of $\mathbb{R}$, and any $p^*$ obtained from $p$ (as in Condition (iv) above)
is also positive on all of $\mathbb{R}$. \\
{\bf Theorem 2.}
{\em A positive sequence $\gamma\!:=\!(\gamma_0,\ldots,\gamma_k)$ satisfies
$T_\gamma\!\left(I\!I^\geq_k\right)\!\subseteq\!I\!I^\geq_k$ iff

\vspace{-.25cm}
\noindent
$\gamma^k_j\!\leq\!
\left(\frac{\gamma_k}{\gamma_0}\right)^j$ for all $j\!\in\!\{1,\ldots,k-1\}$.}

\noindent
Within the next section,
we will see how $\si^\geq_k$ and $\ii^\geq_k$ correspond naturally
to opposite connected components of a particular {\em amoeba} complement.

\section{Background on Discriminants and Amoebae}
The first ingredient to proving our main results is the
following construction:\linebreak
\scalebox{.94}[1]{Consider the map
$\mathrm{Log}|\cdot|: (\C^*)^{k+1}\to \mathbb{R}^{k+1}$ sending
${\bf a}\mapsto\bigl(\log|a_0|,\log|a_1|,\ldots,
\log|a_k|\bigr)$,}\linebreak
where ${\bf a}=(a_0,a_1,...,a_k)\in
(\C^*)^{k+1}$. Notice that $\mathrm{Log}|\cdot|$ maps $\mathbb{R}_+^{k+1}$
diffeomorphically onto $\mathbb{R}^{k+1}$ where $\mathbb{R}_+$ is the set of all positive
real numbers.

\addtocounter{footnote}{1}
For any polynomial $q\in \mathbb{C}[a_0,...,a_k]$ one defines its
{\em amoeba} $\mathrm{Amoeba}(q)$ as the image of
the complex algebraic hypersurface\\
\mbox{}\hfill
$H_q:= \left\{{\bf a}=(\left. a_0,\ldots,a_k)\in(\C^*)^{k+1}\; \right| \;
q({\bf a})=0\right\}$ \hfill\mbox{}\\
under $\mathrm{Log}|\cdot|$.
Recall also that the {\em Newton polytope} of $q(x):=\sum_{{\bf \alpha}\in A}
c_{\bf \alpha} x^{\alpha}$, written $\mathrm{Newt}(q)$, is the convex hull of$^1$ of
$\{\alpha\in\mathbb{Z}^{k+1} \; | \; c_\alpha\!\neq\!0\}$, where the notation
$x^{\alpha}:=x^{\alpha_0}_1 \cdots x^{\alpha_k}_k$
is understood.\footnotetext{i.e., smallest convex set containing...}
There is a natural $1$-$1$ correspondence between unbounded
connected components of the complement $\mathbb{R}^{k+1}\setminus \mathrm{Amoeba}(q)$ and the
vertices of $\mathrm{Newt}(q)$.
\begin{lemma}
\label{lem:gkz}
\cite[Prop.\ 1.7 \& Cor.\ 1.8, pp.\ 195--196]{GKZ}
Suppose a polynomial $f \in \mathbb{C}[x_1,\ldots,x_n]$ has
Newton polytope $P$ and $v$ is a vertex of $P$. Also let
$C$ denote the closure of the cone of inner normals to $v$. Then there
is a unique unbounded
connected component $\Gamma$ of the complement to $\mathrm{Amoeba}(f)$
containing a translate of the cone $C$. \qed
\end{lemma}

\noindent
The cone $C$ above is also called the {\em recession cone} of $\Gamma$, since
it consists of all translations $y\!\in\!\R^n$ with $y+\Gamma\subseteq\Gamma$.

Let $\Delta_k$ denote the discriminant of the family of polynomials
$a_0+\cdots+a_kx^k$, i.e., $\Delta_k\!\in\!\mathbb{Z}[a_0,\ldots,a_k]$ is the
unique (up to sign) irreducible polynomial such that
$a_0+\cdots+a_kx^k$ has a root of multiplicity $>\!1$ implies
that $\Delta_k(a_0,\ldots,a_k)\!=\!0$. For instance,
$\Delta_3:=-27a^2_0a^2_3+18a_0a_1a_2a_3+a^2_1a^2_2-4a_0a^3_2-4a^3_1a_3$.
More generally, $\Delta_k$ can be computed using a number of arithmetic
operations polynomial in $k$ (via a\linebreak
\scalebox{.95}[1]{standard formula involving a $(2k-1)\times (2k-1)$
determinant), and is the special case}\linebreak
\scalebox{.94}[1]{$A\!=\!\{0,\ldots,k\}$ of an {\em $A$-discriminant}
(see \cite[Ch.\ 9 \& 12]{GKZ} for further background).}

\vbox{
Amoebae of $A$-discriminants have a more refined structure.
For example, the boundary of $\mathrm{Amoeba}(\Delta_k)$ is contained in the image of
the real part $H^\mathbb{R}_{\Delta_k}$ of the complex algebraic hypersurface
$H_{\Delta_k}$ under $\mathrm{Log}|\cdot|$ (see Figure 3 below).
The latter fact  motivates the following
definition.
\addtocounter{dfn}{1}
\begin{dfn}
For a complex algebraic hypersurface
$H_q\subset \mathbb{C}^{k+1}$ given by\linebreak
$q(a_0,a_1,...,a_k)=0$ we define its {\em
complete reflection}
$H^\dagger_q$ as the union of the $2^{k+1}$ hypersurfaces given by $q(\pm a_o,
\pm a_1,..., \pm a_k)=0$ for all $2^{k+1}$ possible choices of signs of
coordinates (see, e.g., Figure 4 below).
\end{dfn}
\begin{minipage}[t]{2.4in}
\mbox{}\hspace{-.5cm}\epsfig{file=cubicdiscamoeba.eps,height=2in}

\vspace{-.4cm}
{\scriptsize {\sc Figure 3:} The amoeba of the specialized cubic\linebreak

\vspace{-.3cm}
discriminant $\Delta_3(1,a,b,1)$ (in yellow), and the\linebreak

\vspace{-.45cm}
image of $H^\mathbb{R}_{\Delta_3(1,a,b,1)}$ under $\mathrm{Log}|\cdot|$ (in blue).}
\end{minipage} \hspace{.5cm}
\begin{minipage}[t]{2.2in}
\epsfig{file=refcubic.eps,height=2in}

\vspace{-.4cm}
{\scriptsize {\sc Figure 4:} The real part of the discriminant\linebreak

\vspace{-.3cm}
variety of the family $1+ax+bx^2+x^3$ (bold)\linebreak

\vspace{-.45cm}
and its sign flips, i.e., $H^\dagger_{\Delta_3(1,a,b,1)}$.}
\end{minipage}}

Consider the restriction of the real part
$(H^\dagger_q)^\mathbb{R}$ of $H^\dagger_q$ to $\mathbb{R}_+^{k+1}$. Notice that
by the above remark each connected component of $\mathbb{R}_+^{k+1}\setminus
(H^\dagger_q)^\mathbb{R} $ is mapped  by $\mathrm{Log}|\cdot|$ diffeomorphically either
onto a connected component of the complement $\mathbb{R}^{k+1}\setminus \mathrm{Amoeba}(q)$
or onto $\mathrm{Amoeba}(q)$ itself. One thus sees that $\mathrm{Amoeba}(q)$ is the union of
the images of some number of the latter connected components.

Returning to $\Delta_k$, it is well known (see, e.g., \cite[pg.\ 271]{GKZ})
that $\Delta_k$ has the two homogeneities:\\
\mbox{}\hfill $\Delta_k(\lambda a_0,\lambda a_1,\lambda a_2, \ldots,\lambda
a_k)=\lambda^{2(k-1)}\Delta_k({\bf a})$\hfill\mbox{}\\
and\\
\mbox{}\hfill $\Delta_k(a_0,\lambda a_1,\lambda^2 a_2, \ldots, \lambda^k
a_k)=\lambda^{k(k-1)}\Delta_k({\bf a})$.\hfill \mbox{}\\
This immediately implies that $\mathrm{Newt}(\Delta_k)$
has codimension at least $2$. In fact, the codimension is exactly $2$,
and it is then easy to see that $\mathrm{Amoeba}(\Delta_k)$ is an $\mathbb{R}^2$-bundle
over a base that is an amoeba of smaller dimension. In particular, one
can take the base to be the amoeba of $\Delta_k(1,a_1,\ldots,a_{k-1},1)$,
thus explaining why our illustrations for $k\!=\!3$ are in the plane, as
opposed to $\mathbb{R}^4$.

There is also a combinatorial formula for the monomials in $\Delta_k$
with exponents corresponding to vertices of $\mathrm{Newt}(\Delta_k)$ (see
\cite[pgs.\ 300 \& 302]{GKZ}). Namely, each such
vertex monomial corresponds to
a unique subdivision of the line segment $[0,k]$ into a collection of
segments $\{[0,k_1],[k_1,k_2],\ldots,[k_m,k]\}$, with
integers $0<k_1<k_2<\ldots< k_m<k$. In particular, the finest subdivision
$\{[0,1],\ldots,[k-1,k]\}$ of $[0,k]$ into unit intervals is
associated with the monomial\\
\mbox{}\hfill
$\pm a_1^2a_2^2\cdots a_{k-1}^2=\pm(a_1a_2\cdots a_{k-1})^2$,\hfill (1)
\hfill\mbox{}\\
whereas the second finest subdivisions, having one segment $[l-1,l+1]$ of
length two and all other
segments of unit length, correspond to the monomials\\
\mbox{}\hfill$\pm 4\,a_{l-1}a_l^{-2}a_{l+1}(a_1a_2\cdots a_{k-1})^2\,,\quad
l\in\{1,\ldots,k-1\}$.\hfill (2)\hfill\mbox{}\\
Moreover, thanks to Lemma \ref{lem:gkz}, we obtain a trinity of associations
(see also Figure 5 below): \\
\begin{picture}(200,160)(0,-50)
\put(30,40){vertex monomials of $\Delta_k$}
\put(75,55){\begin{rotate}{45}\scalebox{2}[1]{$\pmb{\longleftrightarrow}$}
\end{rotate}}
\put(100,100){certain unbounded connected}
\put(230,80){\begin{rotate}{-45}\scalebox{2}[1]
{$\pmb{\longleftrightarrow}$}\end{rotate}}
\put(90,90){components of the complement of $\mathrm{Amoeba}(\Delta_k)$}
\put(200,40){triangulations of $\{0,\ldots,k\}$}
\put(140,40){\scalebox{2}[1]{$\pmb{\longleftrightarrow}$}}
\end{picture}

\vspace{-1.1in}
Combinatorially $\mathrm{Newt}(\Delta_k)$ is a cube of
dimension $k-1$ and the monomials (2) represent the vertices
$v_0+e_{l-1}-2e_l+e_{l+1}$ neighboring the vertex $v_0=(0,2,2,\ldots,2,0)$
corresponding to the monomial (1).

\subsection{Archimedean Newton Polygons}
An arguably more direct association between polynomials of degree $k$
and subdivisions of the point set $\{0,\ldots,k\}$ can be
obtained via the {\em Archimedean Newton polygon}, which dates
back to work of Ostrowski in the 1940s \cite[pp.\ 106 \& 132]{ostrowski}.
This particular kind of Newton polygon further
elucidates the connection between Theorems 1 and 2, and
we use the appelation ``Archimedean''
to complement the {\em non-Archimedean} Newton
polygons coming from number theory and tropical geometry.
\begin{dfn}
Given any polynomial $f(x)\!=\!a_0+a_1x+\cdots+a_kx^k$, its
{\em Archimedean Newton polygon}, written $\mathrm{ArchNewt}(f)$, is the
convex hull of the finite point set $\{(i,-\log|a_i|)\; | \;
i\in\{0,\ldots,k\}\}$. We also call any edge of $\mathrm{ArchNewt}(f)$ a
{\em lower} edge if it has an inner normal with positive last coordinate.
\end{dfn}

One can observe experimentally that there is a deep correlation between the
slopes of the lower edges of $\mathrm{ArchNewt}(f)$ and the absolute values of the
roots of $f$.\linebreak
\scalebox{.98}[1]{In particular, paraphrasing in more modern language,
Ostrowski proved remarkable}\linebreak
\scalebox{.93}[1]{explicit bounds revealing how the slopes of the lower edges
of $\mathrm{ArchNewt}(f)$ approximate}\linebreak
\scalebox{.95}[1]{the negatives of the logs of the norms of the
roots of $f$ \cite[pp.\ 106 \& 132]{ostrowski}.}

Even more directly, one notes that the lower hull of $\mathrm{ArchNewt}(f)$
naturally associates, via orthogonal projection onto the first coordinate,
a triangulation of $\{0,\ldots,k\}$ to $f$.
In particular, it is easy to derive that the strict
log-concavity\footnote{Strict
log-concavity for $(\gamma_0,\ldots,\gamma_k)$ simply means that
$\gamma^2_j>\gamma_{j-1}\gamma_{j+1}$ for all $j\in\{1,\ldots,k-1\}$.}
of the sequence of coefficients of $f$ is nothing more than the condition that
$\mathrm{ArchNewt}(f)$ have exactly $k-1$ lower edges. However, unless $\mathrm{ArchNewt}(f)$
is sufficiently ``bowed'', a degree $k$ polynomial $f$ having
$\mathrm{ArchNewt}(f)$ with $k-1$ edges need {\em not} correspond to a point in
the corresponding component $\Gamma$ of the complement of $\mathrm{Amoeba}(\Delta_k)$.

\vspace{.15cm}
\noindent
\begin{picture}(200,220)(0,-37)
\put(-40,0){\epsfig{file=cubicdiscamoeba.eps,height=2.6in}}
\put(140,160){\epsfig{file=0123.eps,height=.25in}}
\put(50,140){\epsfig{file=023.eps,height=.3in}}
\put(120,70){\epsfig{file=013.eps,height=.3in}}
\put(30,60){\epsfig{file=03.eps,height=.3in}}
\put(200,70){\epsfig{file=bowed.eps,height=.5in}}
\end{picture}

\vspace{-1.8cm}
\noindent
\begin{minipage}[t]{2.4in}
{\scriptsize {\sc Figure 5}: Lower hulls of $\mathrm{ArchNewt}(f)$,\linebreak

\vspace{-.3cm}
and associated subdivisions of $\{0,1,2,3\}$,\linebreak

\vspace{-.3cm}
corresponding to the unbounded components\linebreak

\vspace{-.45cm}
of the complement of $\mathrm{Amoeba}(\Delta_3(1,a,b,1))$.}
\end{minipage}
\hspace{.3in}
\begin{minipage}[t]{2.2in}
{\scriptsize {\sc Figure 6}: \textcolor{red}{$1+2.9x+2.9x^2+x^3$} does not
lie\linebreak

\vspace{-.3cm}
in the upper right component of Figure 5,\linebreak

\vspace{-.3cm}
but $\pmb{1+9x+9x^2+x^3}$ (with a more ``bowed''\linebreak

\vspace{-.45cm}
lower hull for its $\mathrm{ArchNewt}$) does... }
\end{minipage}

\noindent
For instance, $1+2.9x+2.9x^2+x^3$ has only $1$ real root, but
$1+9x+9x^2+x^3$ has $3$ real roots.

As we will see in Lemma \ref{lem:push} of the next section, multiplier
sequences can be used to make the lower hull of an $\mathrm{ArchNewt}(f)$ more
bowed. Similarly, the sequences
highlighted in Theorem 2 can clearly be identified with those $f$
having $\mathrm{ArchNewt}(f)$ with exactly $1$ lower edge.
Thus, Theorem 1 (resp.\ Theorem 2) appears to relate
maximal (resp.\ minimal) triangulations with polynomials having maximally
(resp.\ minimally) many real roots.

\subsection{Supporting Results on Real-Rooted Polynomials}
\label{sub:real}
Using the notation $x_l=\log|a_l|$ we see that $\mathrm{Amoeba}(\Delta_k)$ is the set
of vectors $(x_0,...,x_k)\in\mathbb{R}^{k+1}$ such that
the torus $|a_0|=e^{x_0},\ldots, |a_k|=e^{x_k}$ intersects the
discriminant locus $H_{\Delta_k}$.

\begin{prop}\label{pr:monom}The map $\mathrm{Log}|\cdot|$ is a
diffeomorphism from $S\!I^\geq_k$ to the connected component of the complement of
$\mathrm{Amoeba}(\Delta_k)$ corresponding to the monomial (1).
\end{prop}

\noindent
The proof of this proposition is based on several additional statements.
Along the way, we will also see some more examples of sign-independently
real-rooted polynomials.

First consider the vector $s\in\mathbb{N}^{k-1}$ given by\\
\mbox{}\hfill $s_j=\left(\left|\frac{k}{2}-j\right|+1\right) +
\left(\left|\frac{k}{2}-j\right|+2\right) + \cdots + \frac{k}{2},
\quad  j\in\{1,\ldots,k-1\}$,\hfill\mbox{}\\
for $k$ even, and by\\
\mbox{}\hfill
$s_j=\left(j-\frac{k-1}{2}\right) + \left(j+1-\frac{k-1}{2}\right) + \cdots
+ \frac{k-1}{2}, \quad j\in\{1,\ldots, k-1\}$,\hfill\mbox{}\\
for $k$ odd.

The first few instances of $s$ are $(1)$ for $k=2$; $(1,1)$ for $k=3$;
$(2,3,2)$ for $k=4$; $(2,3,3,2)$ for $k=5$; $(3,5,6,5,3)$ for $k=6$;
$(3,5,6,6,5,3)$ for $k=7$;\linebreak
\scalebox{.93}[1]{$(4,7,9,10,9,7,4)$ for $k=8$;
$(4,7,9,10,10,9,7,4)$ for $k=9$;  and  $(5,9,12,14,15,14,12,9,5)$}\linebreak
for $k=10$.

\begin{lemma}
\label{lem:push}
The polynomial\\
\mbox{}\hfill $p_k(x) = 1 + \lambda^{s_1}x + \lambda^{s_2}x^2 + \cdots +
\lambda^{s_{k-1}}x^{k-1} + x^k$\hfill\mbox{}\\
of degree $k$ is sign-independently real-rooted for any sufficiently large
value of the positive real parameter $\lambda$.
\end{lemma}

\begin{proof}

This follows from the fact that for large $\lambda$ the polynomial $p_k$ has
coefficients approaching the polynomial $q_k$ given by:\\
\mbox{}\hfill
$q_k(x) = (x+\lambda^{-k/2})(x+\lambda^{1-k/2})\cdots(x+\lambda^{k/2})$
\hfill\mbox{}\\
if $k$ is even, and by\\
\mbox{}\hfill $ q_k(x) =
(x+\lambda^{-(k-1)/2})(x+\lambda^{1-(k-1)/2})\cdots(x+\lambda^{(k-1)/2})$
\hfill\mbox{}\\
if $k$ is odd.

Indeed, in order to see that $p_k$ is real-rooted for large positive
$\lambda$, one observes that the
roots of $q_k$ are all real, and since they are given by distinct powers
of $\lambda$, there are $k$ of different magnitude. Hence, under the small
change of real coefficients that is needed to deform $q_k$ to the original
polynomial $p_k$, the roots remain well apart, and hence cannot form any
conjugate pair of complex roots. Now,  one can easily check that for
sufficiently large $\lambda$ changing arbitrarily signs of roots of $q_k$ one
obtains $2^k$ polynomials close to $2^k$ polynomials obtained from $q_k$ by
arbitrary sign changes of its coefficients. Thus, any change  of signs of
some of the
coefficients of $p_k$ just corresponds to an appropriate sign change in
some of the roots of $q_k$, and the preceding argument again shows that
the polynomials are still real-rooted.
\end{proof}

\begin{lemma}
\label{lem:cont}
The set $S\!I^\geq_k$ is fibered over $S\!I^\geq_{k-1}$ with
contractible $1$-dimensional fibers.
\end{lemma}

\begin{proof} Notice that the restriction of $S\!I^\geq_k$ to the hyperplane
$a_0=0$ is in obvious $1$-$1$ correspondence with $S\!I^\geq_{k-1}$ obtained
by dividing a polynomial $p(x)=a_1x+\cdots+a_kx^k$ from the former set  by the
variable $x$. To finish the proof we show that for any $p(x)=a_0+a_1x+\cdots
+a_kx^k$ belonging to $S\!I^\geq_k$ the family of polynomials
$p_{\tau}=p-a_0\tau,\;\tau\in[0,1]$ belong to $S\!I^\geq_k$ thus forming
the required fiber of the projection in question.  Indeed, consider  for any
real rooted polynomial
$p(x)=a_0+a_1x+\cdots+a_k x^k$ the family $p_\varepsilon(x)=p(x)+\varepsilon$ where
$\varepsilon\in \mathbb{R}$. It is obvious that
$p_\varepsilon(x)$ is real-rooted if and only if $\varepsilon\in [v_{min}, V_{max}]$ where
$v_{min}$ is the maximal local minimum of $p(x)$ and $V_{max}$ is its minimal
local maximum. Now take $p\in S\!I^\geq_k$ and consider its family
$p_\varepsilon(x)$. Since all the $a_i$ are now nonnegative,
consider $p_{-}(x)=-a_0+a_1x+\cdots+a_kx^k$ which must also be real-rooted.
Thus at least for $\varepsilon$ in the interval $[-2a_0,0]$ one has that $p_\varepsilon(x)$
is real-rooted. Exactly  the same argument works for all $p_\pm$ obtained
from $p$ by  arbitrary sign changes of its coefficients proving that the
family $p-a_0\tau,\; \tau\in [0,1]$ sits inside $S\!I^\geq_k$.
\end{proof}

\subsection{Finding Recession Cones}
Denote by $\Gamma_k$ the connected component of\linebreak $\mathbb{R}^{k+1}\setminus
\mathrm{Amoeba}(\Delta_k)$ corresponding to the monomial (1), and let $C_k$ denote the
recession cone of $\Gamma_k$. We now prove the following crucial result.
\begin{lemma} \label{lem:cone}
The cone $C_k$ is given by the inequalities $2x_l\ge x_{l-1}+x_{l+1}$, for
$l\in\{1,\ldots,k-1\}$.
\end{lemma}

\begin{proof}
Recall that for a polynomial $p({\bf z})$ in $n$ complex variables
${\bf z}=(z_1,...,z_n)$, one defines its {\em Ronkin function} $N_p(x)$,
in $n$ real variables $\bar{x}=(x_1,\ldots,x_n)$, by the formula
$$\frac{1}{(2\pi i)^n}\int_{{\mathrm{Log}}^{-1}(\bf x)}\log|p({\bf z})|
\frac{dz_1}{z_1}\wedge \cdots \wedge \frac{dz_n}{z_n},$$
where ${\bf x}=(x_1,\ldots,x_n)$. It is known that the Ronkin function is
convex, and it is affine on each connected component of the complement of the
amoeba $\mathrm{Amoeba}(p)$. Equivalently, $N_p$ is given by the
integral $$N_p({\bf x})=\frac{1}{(2\pi)^n}\int_{[0,2\pi]^n}\log|p({\bf z})|
d\theta_1 \cdots d\theta_n,$$
where $${\bf z}=\left(e^{x_1+i\theta_1},\ldots, e^{x_n+i\theta_n}
\right)$$ \cite{PT}. As an example, the Ronkin function of a monomial
$p({\bf z})=az_1^{l_1}\cdots z_n^{l_n}, \; a\neq 0$
is given by $$N_p({\bf x})=\log|a|+l_1x_1+\cdots+l_nx_n.$$
From general results proved in
\cite{PR} one knows that the Ronkin function of $\Delta_k$ is equal to
$\log|c_v|+\langle
v,x\rangle$ in the component corresponding to a vertex monomial $c_vx^v.$

In particular, in the components of the special vertex monomials (1) and
(2), the Ronkin function coincides with  the affine linear functions\\
\mbox{}\hfill $2x_1+\cdots +2x_{k-1}=2\,(x_1+\cdots+x_{l-1})$\hfill
\mbox{}\\
and\\
\mbox{}\hfill $x_{l-1}-2x_l+x_{l+1}+2\,(x_1+\cdots+x_{k-1})$\hfill\mbox{}\\
respectively. Now, by \cite{PST} one knows that the amoeba of
$\Delta_k(1,a_1,\ldots,a_{k-1},1)$ does
not have any other unbounded connected components to its complement
other than those corresponding to the vertices of
$\mathrm{Newt}(\Delta_k(1,a_1,\ldots,a_{k-1},1))$.
Now let $S_{\Delta_k}$ denote the spine (see \cite{PR} for its
definition) and let $S_\infty$ denote a sufficiently small neighborhood
of $S_{\Delta_k}$ about infinity. It then follows that $S_\infty$ is
exactly a neighborhood about infinity of the corner locus
of the pieceswise linear convex function (or tropical polynomial)\\
\mbox{}\hfill $\max_v\bigl(\log|c_v|+\langle v,x\rangle\bigr)$,\hfill
\mbox{}\\
where $v$ ranges over the vertices of the Newton polytope of $\Delta_k$.
The unbounded connected components of the complement of the spine
$S_{\Delta_k}$
are convex polyhedral cones where one of the affine linear functions
dominates all the others, and the closure of such a cone is the recession
cone of the unbounded connected component of the complement to
$\mathrm{Amoeba}(\Delta_k)$.
For the special vertex monomial (1)
we obtain in this way that the recession cone $C_k$ of $\Gamma_k$ is given
by the inequalities: $$ 2\,(x_1+\cdots+x_{k-1}) \ge
x_{l-1}-2x_l+x_{l+1}+2\,(x_1+\cdots+x_{k-1}),\,
l\in\{1,\ldots,k-1\}\,,$$
or, equivalently, $2x_l\ge  x_{l-1}+x_{l+1}$, for $l\in\{1,\ldots,k-1\}$.
\end{proof}

We will later need the following refinement of Lemma \ref{lem:cone}
that characterizes the unique translate $C^s_k$ of $C_k$ {\em supporting}
$\Gamma_k$.
\begin{lemma}
\label{lem:sup}
The cone $C^s_k$ defined by the inequalities
$2x_l\geq x_{l-1}+x_{l+1}+\log 4$ for all $l\!\in\!\{1,\ldots,k-1\}$
contains $\Gamma_k$, but $y+C^s_k$ does not contain $\Gamma_k$ for
any $y$ in the interior of $C_k$.
\end{lemma}

\noindent
{\em Proof.}
First note that each polynomials $x^m(1+x)^2$, for
$m\!\in\!\{0,\ldots,k-2\}$, lies on a unique facet of the
cone $C^s_k$, and that this cone has exactly $k-1$ facets.
So to conclude, we need only show that each such polynomial
lies on the boundary of $\Gamma_k$. However, the last statement
was already observed in the introduction, during our discussion
of perturbing middle coefficients. \qed

\medskip
\noindent
{\em Proof of Proposition 1.} From our earlier discussion, we know that
the set $S\!I^\geq_k$ (if non-empty) consists of some number of connected
components of the complement $\mathbb{R}^{k+1}\setminus \Delta^\dagger_k$ where
$\Delta^\dagger_k$ is the reflected
discriminant of $\Delta_k$ (see, e.g., Figure 4). Indeed, $S\!I^\geq_k$ is
the intersection of the set of all degree $k$ real-rooted polynomials
having only simple zeros with all similar sets
obtained by all possible sign changes of the coefficients. By Lemmata
\ref{lem:push} and \ref{lem:cont}
the set $S\!I^\geq_k$ is non-empty and connected, so
$S\!I^\geq_k$ coincides with a unique
connected component of $\mathbb{R}^{k+1}\setminus \Delta^\dagger_k$.

To conclude, we have to show that the
image of $S\!I^\geq_k$ under $\mathrm{Log}|\cdot|$ coincides with the component of the
complement to $\mathrm{Amoeba}(\Delta_k)$ corresponding to the monomial (1). We show
that the vector $s\in\mathbb{N}^{k-1}$ from Lemma \ref{lem:push} is an interior
point in the recession cone of the unbounded connected component $\Gamma_k$
of the complement of the discriminant amoeba corresponding to the
finest subdivision of $\{0,\ldots,k\}$. Indeed, this recession cone is
defined by the inequalities $2x_j \ge  x_{j-1} + x_{j+1}$,
$j\!\in\!\{1,\ldots,k-1\}$ with the dehomogenizing convention
$x_0=x_k=0$, thanks to Lemma \ref{lem:cone}. This
means that the coefficients $\lambda^{s_j}$ of the polynomial $p_k$ from
Lemma \ref{lem:push}, for large enough $\lambda$, represent a point in
$\Gamma_k$. But the polynomial $p_k$ was seen
to be sign-independently real-rooted for large $\lambda$, and this concludes
the proof.  \qed

\section{The Proofs of our Main Results}

\subsection{Theorem 1}
The proof of the ``only if'' direction is easy, as outlined in the
introduction: If $T_\gamma(S\!I^\geq)\!\subseteq\!R\!R$ then we must
certainly have $T_\gamma(x^m(1+x)^2)\!\in\!R\!R$ for all $m$, since
$x^m(1+x)^2\!\in\!S\!I^\geq$ for all $m$. Thus, $\gamma$ must be log-concave.

The proof of the ``if'' direction is more intricate but now follows
easily from our preceding development: By Proposition 1 and
Lemma \ref{lem:cone}, $\mathrm{Log}|\cdot|$ of the set of log-concave
$\gamma=(\gamma_0,\ldots,\gamma_k)$ is precisely the recession cone $C_k$ of
$\Gamma_k$, and\linebreak
$\mathrm{Log}|\cdot| : S\!I^\geq_k\longrightarrow \Gamma_k$ is a diffeomorphism.
So any such $\gamma$ satisfies $T_\gamma\!\left(S\!I^\geq_k\right)\!\subseteq\!S\!I^\geq_k$,
and we are done. \qed

\subsection{Corollary 1}
The first part of the Corollary follows immediately from
Lemma \ref{lem:sup}. The second part follows easily by
applying Lemma \ref{lem:cont} inductively. \qed

\subsection{Theorem 2}
Our proof here will be completely parallel to that of
Theorem 1, so let us start with some analogues of
$\Gamma_k$ and $C_k$: First, let us denote by $\Gamma'_k$ the connected
component of $\mathbb{R}^{k+1}\setminus
\mathrm{Amoeba}(\Delta_k)$ corresponding to the trivial (single-celled) subdivision
of $\{0,\ldots,k\}$. Also let $C'_k$ denote the recession cone of $\Gamma'_k$.
\begin{lemma}
\label{lem:back}
The cone $C'_k$ is given by the inequalities $kx_j\leq j(x_k-x_0)$, for
$j\in\{1,\ldots,k-1\}$. \qed
\end{lemma}

\noindent
Lemma \ref{lem:back} follows easily from the
development of \cite{GKZ,PT} just like Lemma
\ref{lem:cone}, so we proceed to an analogue of Proposition 1:
\begin{prop} The map
$\mathrm{Log}|\cdot| : I\!I^\geq_k \longrightarrow \Gamma'_k$ is a diffeomorphism. \qed
\end{prop}

\noindent
Proposition 2 is proved in exactly the same way as Proposition 1,
save that one uses a different deformation argument along the way:
Lemma \ref{lem:push} is replaced by the observation that
(a) $q_k(x):=1+\lambda^{-1}x+\cdots+\lambda^{-1}x^{k-1}+x^k\!\in\!I\!I^\geq_k$
for all sufficiently large $\lambda$, and (b) the roots of
$q_k$ approach those of $x^k+1$ as $|\lambda|\rightarrow \infty$.

\medskip
We are now ready to prove Theorem 2: \\
{\em Proof of Theorem 2:}
The ``only if'' direction can be proved as follows:
For any $j\!\in\!\{1,\ldots,k-1\}$, consider the
polynomial $p_j(x):=(k-j)-kx^j+jx^k$.
It is then easily checked that (a) $p_j$ has a unique degenerate
real root, (b) $p_j$ has exactly $1$ or $2$ real roots according
as $k$ is even or odd, (c) $p^-_{j,\varepsilon}(x):=(k-j)-k(1-\varepsilon)x^j+jx^k\in
I\!I^\geq_k$ for all $\varepsilon\!\in\!(0,1]$, and
(d) $p^+_{j,\varepsilon}(x):=(k-j)-k(1+\varepsilon)x^j+jx^k\not\inI\!I^\geq_k$
for all $\varepsilon\!>\!0$ (To prove (a)--(d) one can simply
apply Descartes' Rule of Signs and a clever formula for the
discriminant of a trinomial from \cite[Prop.\ 1.2, pg.\ 217]{GKZ}.)
Thus, should the stated inequalities involving $(\gamma_0,\gamma_j,\gamma_k)$
fail to hold, we can easily find an $\varepsilon\!>\!0$ such that
$T_\gamma(p^-_{j,\varepsilon})\!\not\in\!I\!I^\geq_k$ (with $\gamma:=(\gamma_0,
\underset{j-1}{\underbrace{1,\ldots,1}},\gamma_j,
\underset{k-j-1}{\underbrace{1,\ldots,1}},\gamma_k)$) and obtain a \linebreak

\vspace{-.6cm}
\noindent
contradiction.

The proof of the ``if'' direction is more intricate, but follows
easily from our development: By Proposition 2 and Lemma
\ref{lem:back}, $\mathrm{Log}|\cdot|$ of the set of
$\gamma=(\gamma_0,\ldots,\gamma_k)$ satisfying the stated inequalities is precisely
the recession cone $C'_k$ of $\Gamma'_k$,
and\linebreak
$\mathrm{Log}|\cdot| : I\!I^\geq_k\longrightarrow \Gamma'_k$ is a diffeomorphism.
So any such $\gamma$ satisfies $T_\gamma\!\left(I\!I^\geq_k\right)\!\subseteq\!I\!I^\geq_k$,
and we are done. \qed

\section{Future Directions}

\noindent
{\bf Problem 1.} {\em How does one count connected components of the
complement to the reflected discriminant of a given discriminant? In
particular, is it true
that the number of connected components of the complement to the reflected
discriminant of univariate polynomials of degree $k$ restricted to $\mathbb{R}_+^k$
equals $2^{k}$? }

\medskip
\noindent
{\bf Problem 2.} {\em Find an elementary proof of Theorem~\ref{th:main}
avoiding the use of discriminant amoebae.}

\medskip
\noindent
Regarding the last problem, we observe that we first derived our
characterization of the recession cone relevant to $S\!I^\geq_k$ via some quick,
informal calculations using the {\em Horn-Kapranov Uniformization}
\cite{kapranov,PT}. (The Horn-Kapranov Uniformization is a remarkably useful
rational parametrization of the $A$-discriminant variety. An intriguing fact
is that the resulting parametric formula for $H_{\Delta_k}$ has size polynomial
in $k$, while $\Delta_k$ has a number of monomials (and coefficient
bit-sizes) exceeding $2^{k-1}$ \cite{bhpr}.) It is likely that the
Horn-Kapranov Uniformization can yield an alternative proof of
Theorem \ref{th:main} without Ronkin functions. This could be seen
as a step toward solving Problem 2.

\section*{Acknowledgements}
The second author thanks the Wenner Gren Foundation for the support of
his visit to Stockholm University during the start of this project.
The second author was also partially supported by
NSF CAREER grant DMS-0349309 and Sandia National Laboratories. We are also
sincerely grateful to P.\ Br\"anden for finding a mistake in the initial
version of the paper (and pointing out a number of relevant references), and
to Jan-Erik Bj\"ork for pointing out the reference \cite{ostrowski}.

redpajama_set_name: RedPajamaArXiv


text:
2018 BMW 230i xDrive Coupe: A DriveWays Review…
With automobiles, especially an expensive sport coupe like the 2018 BMW 230i xDrive, it's all about the return on investment.
It is the same in business as well, but there the focus is on profits measured in dollars. With cars, it's about the tangibles and intangibles they deliver.
Buy a minivan and you get practicality for family vacations. Buy a BMW 230i xDrive and practicality flies out the window. The return on that investment comes in driving enjoyment, preferably with two people on board.
This compact two-door coupe, though it delivers strong performance, capable handling and decent fuel economy, has a cramped back seat. And because the front seatbacks move minimally forward, it requires athletic ability to access.
Offsetting that is a large trunk of almost 14 feet that can swallow a couple's luggage for a week. If the trip is longer or you have a lot of stuff, the rear seatbacks fold to expand the cargo-carrying capability to 53 cubic feet.
Arguably, the 230i xDrive, which is the lowest-price sedan in the BMW lineup, is the lineage successor to the 1967 BMW1600-2, which had a bigger and more accommodating back seat. Car and Driver Magazine trumpeted it as "the World's Best $2,500 Car." It was a boxy two-door with a cavernous trunk that in this writer's family was big enough to conceal all of Santa Claus's gifts for four children.
It had an 84-horsepower, 1.6-liter four-cylinder engine that drove the rear wheels through a four-speed manual gearbox. Its independent suspension system and rack-and-pinion steering delivered great handling and the capability to hammer steadily over railroad tracks at 100 mph, all while delivering fuel economy in the mid-20s.
Contrast the 1600-2 (later joined by the more powerful 2002) with the tested 2018 230i xDrive and you see a great deal of price creep. The writer's 1600-2, with options, had a $2,850 price tag, which has inflated over the years to $20,920 in 2017. The 2018 230i—the lowest-priced sedan in BMW's expensive lineup—starts at $34,145, much of the difference because of modern emissions, safety and convenience requirements.
The tested xDrive, which is BMW-speak for all-wheel drive, started at $37,795. With $12,520 in options, it ended up with a $50,315 price tag.
Whether that investment delivers a substantial return depends on the individual owner's delight and involvement, and whether he or she concludes that it merits the "ultimate driving machine" label. But the 230i xDrive is a sweet piece of machinery.
Its tidy size—three inches shy of 15 feet long—and quick steering makes for confident moves in city and freeway traffic. Need to make a quick lane change or dodge a clueless and careless driver intruding into your lane? A flick of steering and tap on the throttle and the troubles are gone.
Open highway cruising is relaxing. A supple ride, supportive sport seats with good seatback bolstering and a quiet interior means you can put on many miles without fatigue.
The engine is a 248-horsepower, 2.0-liter turbocharged four-cylinder engine that makes 258 pound-feet of torque, or twisting force. Power gets to all four wheels through an eight-speed automatic transmission that can be shifted manually with paddles on the steering wheel. For traditionalists, the 230i xDrive can be ordered with a six-speed manual gearbox.
Though the engine slurps premium gasoline, city/highway/combined fuel consumption with the automatic works out to 24/33/27 mpg.
The test car was well equipped with optional leather upholstery (leatherette is standard), navigation system, rear-view camera, adaptive cruise control, motorized glass sunroof, power front seats with memory settings, SXM satellite radio, wireless smart phone charging, WiFi hot spot and Apple CarPlay. A $2,300 track handling package included an adaptive suspension system, sport brakes and variable steering.
Still, there were shortcomings. On the automatic settings, the air conditioning could not keep up on a hot day. Fortunately, BMW included a knob that delivered a manual maximum air conditioning blast. However, with a capable system it should not be needed.
In addition, the infotainment system is needlessly complicated, sun visors did not slide on their support rods to fully block sunlight from the side, and there was no blind spot warning, though it is not needed if the outside mirrors are properly adjusted.
For those drivers who value a car for visceral entertainment as opposed to pedestrian competence, the 230i xDrive returns a bonus on the investment.
Model: 2018 BMW 230i xDrive two-door coupe.
Engine:0-liter four-cylinder, turbocharged, 248 hp, 258 lb-ft torque.
Transmission: Eight-speed automatic with manual shift mode.
EPA city/highway/combined fuel consumption: 24/33/27 mpg. Premium fuel required.
Disclaimer: This test drive was based on a loan of the vehicle from the manufacturer. It was driven by the author in circumstances similar to everyday driving by consumers.
Photos (c) BMW.
BMW, Car Reviews, Frank A. Aukofer, Uncategorized
2018 BMW 230i xDrive, German Cars, Luxury, new for 2018, Performance Coupes
Scroll to the bottom of the page, and you'll discover links to stories, reviews and rants from the writers who bring you The Review Garage:

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Braves By The Numbers: Game 1 Recap: Dbacks 6/28/13 "Chipper Jones #10 Retirement"
Game 1 Recap: Dbacks 6/28/13 "Chipper Jones #10 Retirement"
Amazing, amazing night at Turner Field. I was blessed enough to be in attendance for the ceremony, including the unveiling of Chipper's number along the ring of honor at the stadium.
A picture says a thousand words, so I'll just post that from last night. Our seats were down the 3rd base line right in prime position to see the new plaque. But even after the ceremony, and even after Martin Prado got a standing ovation when Chipper recognized him in his speech, the Braves had a game to win. And win they did, 3-0 on a dominant pitching performance by Julio Teheran.
At the beginning of the season in spring training I said, "If he (Teheran) can carry even 50% of his spring potential into the season, he and Medlen may be in a duel for the hot young phenom honors." While Kris hasn't received the win/loss record he'd like, he's currently in the top 20 of NL ERA for starters, along with two other Braves: Mike Minor and Julio Teheran. Julio showed off his prowess last night with a 6 inning, 4 hit, no run, and 10 K performance. He was blowing away Dback hitters with a combination of power and finesse. Add in 3 more innings by the bullpen: Avilan to Walden to Kimbrel, yer out! Braves win! Nobody was going to touch up the Braves staff on Chipper Jones night in the Ted.
The hitting wasn't spectacular, but it didn't have to be with Teheran dealing. The Braves posted 11 hits and 5 walks, but only went 2-10 with RISP, and the other run was a solo homer by Simmons. The Braves continue to live on a knife's edge game to game, which drives us all a little crazy. However, they did get the lead early and they did get a home run in the same game. If they combine those two things, and the game is at home? We're practically undefeated on the season. Also, take a look at Dan Uggla's average. Is it? Could it be? YES! He's above the Mendoza line! Let there be a grand clubhouse feast! Bring out the finest meats and cheeses in all the land! Now only BJ remains, sitting at .180 after his two doubles last night (one was a very generous call since the Dback CF completely lost the ball in the sky).
Heyward, Uggla, and BJ all had multi-hit games. Justin, Simmons, and Reed Johnson picked up RBIs. The Reed Johnson PH was the thing of beauty. With Uggla's leadoff triple in the 8th, Reed came to the plate with a unique plan. He decided to lay down the perfect safety squeeze to score Uggla. That move caught the Dback defense off guard, and allowed the 3rd run to cross the plate. A little extra insurance never hurts, and it shows that the Braves can actually get to the plate late in the game with a strategy that doesn't include just hacking at the pitch.
The CPA correctly predicted a win, and moves to 39-29 on the season. Can the Braves carry the momentum of alumni weekend into game 2?
This could be a tighter game unless the Braves get a handle on Kennedy early. He'll give up some hits, and he'll give up some walks, but the Braves have to string those together with RISP to make it count. Also, Hudson has been on the receiving end of some terrible Braves hitting, so you would hope they wake up in time to get him a win.


Lexmark E460dw Driver Download The Lexmark E460dw printing performance is absolutely its shining factor in our testimonial. This Lexmark cordless printer is without a doubt one of the most reputable printer for high-volume jobs as a result of its 80,000 regular monthly duty pattern (52,000 pages/month anticipated outcome) at a print rate of 40 web pages each minute.
the Lexmark E460dw printing performance is its absence of different colors printing capacities. While this isn't a major issue for most legal or technical printing uses, it is troublesome if you need to print the occasional bar chart in different colors. and also the bottom-loading input drawer extends out to accept as much as 250 sheets of 20-pound paper, a conventional amount for any monochrome printer. All 3 need you to link the printer to a host computer system at first with a USB 2.0 cable.
During our testing, we presented the Lexmark E460dw cordless sensitivity to be enough for home offices as well as smaller, single-floor businesses. We designated the Lexmark E460dw a rating of 80 out of a possible 100, Lexmark E460dw it lost factors due to some problem we had attaining a convenient signal on the ink-jet printer. Essentially this indicates that you could have a little difficulty finding a signal originally, yet you'll be fine when you look after the preliminary setup and configuration.
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"My alcoholic father used to beat me every night. I slept for less than two hours at times," said 16-year-old Hamsa Raj, a child rights representative. "People ask us to dream. We don't even get to sleep," he said.
Raj, along with many other children, spoke about the daily violation of their rights and dignity at a programme organised to release an alternative report, prepared by child representatives across Tamil Nadu, on the implementation of the United Nations Convention on the Rights of Children (UNCRC) here on Wednesday.


Attorneys for Purdue University say the school has settled a federal lawsuit over the forced retirement of Indiana-Purdue Fort Wayne's former chancellor.
The school's attorneys said in court filing Monday that Michael Wartell's lawsuit "has now been resolved," but that filing included no details of the settlement.
Wartell sued Purdue in federal and Tippecanoe County court in 2013, claiming discrimination and harassment by the university under then-Purdue President France Cordova.
His federal suit states that in late 2010 or early 2011, Cordova announced in a meeting that, before her term as president was over, she wanted to increase the number of women in the school's administration.
The settlement ends years of wrangling over a report prepared by an attorney who investigated Wartell's removal, The Journal Gazette reported.
The newspaper reported that records released Tuesday by Purdue show it spent about $19,000 to prepare its report on Michael Wartell, but it then cost the school more than $153,000 over 18 months fighting to keep it from the public, the media and Wartell's attorneys.
Purdue released the documents nearly six months after The Journal Gazette requested information on how much Purdue spent fighting the report's disclosure.


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Fertility Journey Update: Anna Victoria
The fitness influencer and Instagram star continues to give the latest news on her progress
Anna Victoria has been very open about her infertility struggle, and has made it a point to keep her Instagram and YouTube page updated with her progress, in the hopes that it can help other women on the same path. In April we did an article that highlighted what the fitness star was doing as she began her infertility journey-- from learning to understand why she could not conceive, to changes she made to her diet and her exercise routines in the hopes of becoming pregnant. Since then, Anna Victoria has undergone a couple of infertility treatments, including IVF. In her latest video, she shares how her and her husband have persevered throughout this journey and how hopeful they are that this procedure will work for them.
In this article we're going to discuss:
What changes Anna Victoria has made to her lifestyle to promote fertility
Her IVF journey
What the future has in store
Sometimes you might need to make changes in your diet or lifestyle in order to get your body ready for a pregnancy.
Making changes to exercise and diet could increase the chance of pregnancy
Anna and her husband had been trying to conceive naturally for most of 2018. Each time she thought she was pregnant, it turned out that she wasn't. Although it can sometimes take couples over a year to conceive, she considered that maybe there was something else at play here. She was eventually diagnosed with unexplained infertility-- something she said she could have never dreamed of in a million years. At first she was convinced she was doing something wrong-- that there was something about her lifestyle that kept preventing her from becoming pregnant. She realized that this was just another way women tend to still blame themselves for their infertility-- and in no way is that true. She took it upon herself to understand her menstrual cycle and pinpoint the days where conception was most likely.
After this still did not yield the results she wanted, her doctors asked her about her fitness and eating routines. As a personal trainer and fitness guru, working out and eating healthy is a very important part of Anna Victoria's life. Although a large part of her life, she also knew she had to do everything in her power to get pregnant. After reducing her high impact workouts, adding a variety of foods to her meals, and increasing her overall calories, Anna Victoria still struggled through her infertility. It was then that her and her husband made the decision to try IVF.
Anna has been very open about her IVF treatments, and has given women hope that this procedure might be right for them as well.
IVF Process
Everyone starts somewhere
Anna Victoria's latest entry on her YouTube channel shows her starting her IVF process with daily injections to stimulate her follicles in the hopes of creating eggs that are viable for retrieval. Her husband helped with her fertility injections in the days leading up to the retrieval procedure-- and this is done to help mature multiple eggs. Multiple eggs are preferred as pregnancy rates tend to be higher when more eggs are able to be retrieved.
Anna worked closely with her doctor to develop a plan for egg stimulation and monitoring. After going in for an ultrasound to check on the status, her doctor was able to determine that the follicles were growing and had reached sufficient number. This meant that it is now time for her to receive the trigger injection, before the eggs have a chance to ovulate. The next step in Anna Victoria's IVF journey meant that the eggs were ready for retrieval, and she and her doctor set up a time for this procedure. Her eggs were removed-- 25 in all-- by utilizing a gentle suction that allows the eggs to come out with the follicular fluid. Her eggs will then be monitored for quality and then prepared for fertilization.
Anna had a successful egg retrieval, but there is still some waiting to do in order to decide if a viable embryo has been created and can eventually be implanted.
What's In Store For Anna's Future?
No one can predict the future, but after a successful egg retrieval, Anna and her husband will hopefully be able to eventually implant a fertilized embryo. There is more waiting after this process, to confirm that the pregnancy is in fact viable and the embryo has implanted in the uterus.
Anna Victoria has been extremely open and up front about her struggles with infertility, and has given many other women hope that just because they cannot conceive naturally, doesn't mean they cannot conceive at all! By documenting her IVF process, she has taken on a lot of taboos that still surround IVF and other fertility procedures, and instead makes them seem very normal and relatable. We hope to give another update in the future on Anna Victoria's progress, so stay tuned to the eIVF blog for more updates.


Being an owner of a startup or small business, you might not have enough budget to spend on expensive advertising as your biggest competitors do. In order to compete with your rivals on the same level with a limited budget, you need to become a smart marketer.
In order to get local traffic through digital marketing, you need to have unique search strategy focused on users searching for your products or services in the same geographical area. Keep the design and layout of your website clean. You can also add an address map on your website that will build a trust and help you convert website visitors. Besides, your website must have information related to your services and products that match the keywords your target audience search for.
Publicity helps you build awareness of your online business. If you do it correctly, it may help you be active in your community with ease. You can also sponsor a community activity which is somehow connected to your industry. When something new happens in your business like the launch of a new product or service, you must float a press release. It's always better to go with the online press releases that may publish your news for free.
Social media is one of the most popular and effective marketing channels that can really help you target potential customers with ease. But, you need to share some valuable content on social media that doesn't seem promotional and engage the audience. Moreover, you need to choose the social media platforms very carefully. Research and find out which will be the most relevant to your online business. Remember, social media is free, but you need to think about the value of your time before signing up at any channel.
You must add your company's address on local directories such as Google, Yelp, Manta, Yahoo, etc. Don't forget to leverage your presence on these local business directories with well-written content about your business. After the approval, you will get full control on your profiles. Most local business directories may charge a fee, so make sure you are going with the ones that will be of value to your online business.
The most important part of your digital marketing strategy is finding and choosing the right keywords. Make sure your chosen keywords are relevant and logical with your targeted location. Targeting audience in another location with your keywords will only waste your funds. So, make sure you are going to target your industry with relevant keywords.
Online Advertising is another popular way to make your small website found for target keywords. You can use Google Adwords (pay-per-click), Facebook ads, and Twitter ads to promote your business instantly. At the initial level, you should only target local customer with particular keywords related to your products or services.
Your small business needs to provide every bit of contact details that can convert a visitor into a customer. Make it easy for your visitors to communicate with you without filling out too much details. You can add an active phone number, email, social media pages and other methods to make your customers contact you effortlessly. It will also help you build trust with possible customers.
If you have a significant amount of budget, then Search Engine Marketing (SEM) can be very effective. For SEM, you need to use some popular adverting platforms such as Google Adwords, and Bing/Yahoo Ads that will provide you with the options to target local customers for your product or services effectively. In order to achieve some positive leads, you need to create a well-researched local keyword strategy to target a specific audience.
Update the content of your site on a regular basis. Make sure every information on your website is accurate and up to date. Besides, you need to monitor the customer feedback on social media channels such as Facebook, Instagram, Google local, Yahoo local, and others. You must answer to every query suggestions provided by your customers.
In this highly competitive online market, your website must be optimized for all kind of mobile screens. If you haven't updated your business website, then you need to do it today. Make sure your website is fully responsive and works flawlessly on all kind of mobile devices and operating systems.
You might have already heard about all these strategies mentioned above. But, if you are not applying them in your digital marketing plans, then roll up your sleeves and try these cost-effective digital marketing strategies for your small business today.


Not sure which butterfly it is but it was intense heat out there. Very fast moving butterfly though.
What a beautifully colored flower!
I like the color! Gorgeous.
The color of the zinnia is so warm and rich!
Heat helps them grow big, right?
Then I need some of that.
Oh a beautiful bright flower! Awesome!
Great Shot…I have a few potted zinnia's myself,they are very hardy!
It's very difficult to capture butterflies, you've done a great job here. There are many little critters out and about today. Great shots.
Zinnias are really pretty and I like the bug in here too super macro shot!
That's a very warm colour, isn't it! And a pretty butterfly as a bonus!
I love zinnias, and a butterfly visitor is such a nice treat!
What a good shot, I can never catch wildlife on camera – if only it would stay still! Your zinnias are a beautiful colour.


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Canadian Prime Minister, Justin Trudeau, speaks during a news conference at U.N. headquarters during the General Assembly of the United Nations in Manhattan, New York, U.S., September 26, 2018. (Amr Alfiky/REUTERS)
Trudeau criticized for tweet to TV personality pledging money for education
By Levon Sevunts |
english@rcinet.ca
Posted: Monday, December 3, 2018 13:30
Prime Minister Justin Trudeau's tweet pledging $50 million for an international NGO has caused a huge backlash on social media with thousands of people arguing the money would be better spent on programs for Canadians in need.
On Sunday, Trudeau tweeted to South African comedian and host of "The Daily Show" Trevor Noah that Canada would pledge $50 million to Education Cannot Wait, an organization that funds education for children affected by conflicts, natural disasters and other crises.
Trudeau thanked Noah for celebrating Nelson Mandela's legacy at the Global Citizen festival, an initiative aimed at ending extreme poverty by 2030, and said he was sorry he couldn't attend the festival in person.
"But how about Canada pledges $50M to @EduCannotWait to support education for women & girls around the world? Work for you? Let's do it," Trudeau tweeted.
Hey @Trevornoah – thanks for everything you're doing to celebrate Nelson Mandela's legacy at the @GlblCtzn festival. Sorry I can't be with you – but how about Canada pledges $50M to @EduCannotWait to support education for women & girls around the world? Work for you? Let's do it.
— Justin Trudeau (@JustinTrudeau) December 2, 2018
Reaction on Twitter was swift.
Conservative Leader Andrew Scheer accused Trudeau of pledging $50 in a tweet to impress a TV personality.
"Taxpayers need a defender not somebody who throws their money around to be popular with celebrities," Sheer tweeted. "This is how deficits become massive and permanent."
Pledging $50 million in a tweet to impress a TV personality?
Taxpayers need a defender not somebody who throws their money around to be popular with celebrities.
This is how deficits become massive and permanent. https://t.co/lBdAtzH0eX
— Andrew Scheer (@AndrewScheer) December 3, 2018
More than twenty thousand people chimed in, saying the money would be better spent on a range of needs in Canada – from supporting Canadian war veterans, job creation programs for oil industry workers laid off in Alberta to paying down Canada's ballooning debt.
Thousands of others supported Trudeau, saying the money will go to a very worthy cause that the majority of Canadians support.
With files from The Canadian Press
Categories: International, Politics
Tags: international aid, Justin Trudeau, Twitter
Do you want to report an error or a typo?	Click here!
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Texas Sheriff: We Don't Wait, We Engage Active Shooters and Save Lives
A North Texas sheriff left no doubt about his policy on engaging active shooters. The sheriff made it clear his deputies will not wait for SWAT or other agencies. "We go into engage and stop the shooter and save lives," the sheriff wrote in a memorandum on Wednesday.
Denton County Sheriff Tracy Murphree sent a memo to all employees of his office on Wednesday making a clear policy statement on active shooters. The statement comes on stark contrast to the apparent lack of action taken by Broward County sheriff's deputies in Florida.
Sheriff Murphree wrote:
With the recent tragedy in Florida, I wanted to make clear my policy on responding to an active shooter. All commissioned Deputies, if you repond to an active shooter you are expected to take immediate action. We do not stage and wait for SWAT, we do not take cover in a parking lot, and we do not wait for another agency. We go in and do our duty. We go in to engage and stop the shooter and save lives.
The sheriff closed the letter by asking any deputy who felt they could not follow the directive to notify their supervisor and request a re-assignment.
Four Broward County Sheriff's Office deputies waited outside the Marjory Stoneman Douglas High School while the attack on the school continued inside, Breitbart News' AWR Hawkins reported on Friday. Initial reports indicated that one deputy remained outside the school. A CNN report later expanded the number of deputies waiting outside to four.
Coral Springs police officers were reportedly stunned and upset that the four deputies remained outside the school — even after the officers entered the building.
After questions arose regarding the actions of the deputies, Broward County Sheriff Scott Israel refused to accept responsibility for his deputies' lack of action.
"I gave him a gun. I gave him a badge. I gave him the training," Sheriff Israel told an NBC6 South Florida reporter in a video interview tweeted by Erika Glover. "If he didn't have the heart to go in, that's not my responsibility."
Sheriff Murphree served as a Texas Ranger before running for office as a county sheriff. Texas Rangers are known to follow the legendary quote, "One Riot, One Ranger." The Texas Department of Public Safety reports that quote is more of a paraphrased statement of the sentiments of Ranger Captain W.J. McDonald. Nevertheless, the legend lives on in the attitude of Sheriff Murphree.
If the Broward County deputies were confused about the policies of their boss, Denton County, Texas, deputies have no such doubts.
In the very clear words of Sheriff Murphree, "We go in to engage and stop the shooter and save lives."
Bob Price serves as associate editor and senior political news contributor for Breitbart Texas. He is a founding member of the Breitbart Texas team. Follow him on Twitter @BobPriceBBTX, Gab, and Facebook.
Border / Cartel ChroniclesCrimeActive ShooterBroward County Sheriff Scott IsraelBroward County Sheriff's OfficeFlorida school shootingMarjorie Stoneman Douglas High School
Dershowitz: 'Articles of Impeachment Are Two Noncriminal Actions'


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Netaji's INA Trials in Red Fort Led British Panic and Leave India Claims Book by Bhuvan Lall
TOPICS:Bhuvan LallBookIndian National ArmyNetajiSubhash Chandra BoseThe Man India Missed the Most
Posted By: manoshi sinha
Here is Bhuvan Lall's version of a man of extraordinary will, who had embarked on death defying journeys for India's freedom across the treacherous mountains of Afghanistan, threatening waves of Atlantic and Indian Oceans, tortuous jungles of Burma – pursued by assassins and the spectre of death. Faced with impossible odds, he was determined to triumph over an undefined future with courage, dignity and honor. He became the stuff of legends in his own lifetime. He was 'The Man India Missed the Most'. He was Netaji Subhash Chandra Bose.
Relive the life lived by Netaji Subhash Chandra Bose, reading word by word, paragraph after paragraph, page after page, until you reach the 'finish' part of The Man India Missed the Most, a novel (BOOK) by Bhuvan Lall.
The truth about Netaji Bose and the sacrifices he made for India have been wiped out from History. Bhuvan Lall's The Man India Missed the Most taps into the single greatest emotional truth of India's freedom movement – the Indian National Army trial at the Red Fort that affected the withdrawal of the monumental British Raj from India. It was Netaji who facilitated India's freedom. His impact led the British panic and leave India.
To quote Bhuvan Lall, "Firmly holding the tricolor, the Indian National Army advanced with the war cry 'Delhi Chalo' (Onwards to Delhi). Their military objective was India's independence. And their leader was Subhash Chandra Bose. With shells blasting, guns blazing and blood flowing, this was the defining moment of his life. He was the solitary Indian nationalist to confront the mighty British Empire on the battlefield."
Bhuvan Lall's The Man India Missed the Most was launched on THE 120th Birth Anniversary of Netaji on January 23rd this year amid the presence of Maj Gen (Dr.) Gagandeep Bakshi, Subhash Ghai, Film Director and other eminent personalities. Here is an excerpt of an interview with Bhuvan Lall on his book on Netaji, taken by Manoshi Sinha, Editor of My India My Glory Blogazine.
Please tell us a few lines about your book The Man India Missed the Most?
My book The Man India Missed the Most chronicles the impact of the iconic leader Subhash Chandra Bose on India's freedom movement in a never-before style. It is a tale of heroism and nationalism, of adventure and sacrifice and of victories and catastrophic defeat. His story illustrates how life can often transcend works of literature in terms of idealistic spirit, dramatic episodes and tragedy.
The Man India Missed the Most is also an account of the ferocious will of a nationalist who refused to give up during the war for India's Independence. The hard power approach he favored for gaining India's freedom from colonialism often paralleled and sometimes overshadowed the non-violent civil disobedience movement. In effect, Bose's contribution has not been adequately celebrated in Independent India.
What was your objective behind writing a book on Netaji?
The heroism and patriotism of Netaji Subhash Chandra Bose should be shared and remembered. This was my objective behind writing a book on Netaji. The Man India Missed the Most is also a heartfelt tribute to the sacrifices, struggles, daring and compassion of all the men and women who fought for India's freedom.
The Man India Missed the Most is aimed for all age groups, more especially for the younger generation of India growing 7 decades after independence.
What about reaching a wide audience across India and the world so that they uncover the truth about Netaji Bose?
Cinema is the best medium to reach the widest audience irrespective of regions. Making of a Hollywood movie and a TV series based on my book is in the agenda.
How many years did it take to research on Netaji?
I have spent almost 20 years researching and documenting the life and times of the foremost Indian nationalist Netaji Subhash Chandra Bose.
Any detailed documentation on Netaji on how he gave shape to his Indian National Army outside India?
The Man India Missed the Most spans three continents, taking the reader across the expanse of European imperialism in India from the moment Vasco da Gama sighted the Indian coast in the summer of 1498 to the liberation of Goa in December 1961.
My book also focuses on the global footprint of India's freedom movement ranging from the West Coast of North America in San Francisco and Vancouver to the capital cities of European superpowers in London and Berlin and finally to the Asian nations of Japan, Singapore and Burma.
About Bhuvan Lall
Bhuvan Lall is an award winning filmmaker, international journalist and creative entrepreneur. He is the Executive Chairman of LALL Entertainment Private Limited that deals in animation, broadcasting, and cinema.
Buy this book from Flipkart.
manoshi sinha
Manoshi Sinha is a writer, history researcher, avid heritage traveler; Author of 8 books including 'The Eighth Avatar', 'Blue Vanquisher', 'Saffron Swords'.
Latest posts by manoshi sinha (see all)
Nagarjuna: Wizard in Indian Alchemy and Metallurgy; Introduced Gold-Shine Mechanism - January 20, 2020
Lt Rama Mehta: Joined Netaji's INA at 17, Tourist Guide Post Independence; An Interview - January 20, 2020
Lt. PN Oak: His Role in Netaji's INA, His Expose of Taj Mahal Truth - January 20, 2020
Netaji Bose and Azad Hind: INA Trials in Red Fort Led British Panic and Leave India Netaji, not Gandhi got India Freedom, Claims GD Bakshi's Book Archival Data and Attlee's Statement Prove Netaji and INA Trials Forced British to Leave India: GD Bakshi Royal Indian Navy Mutiny and INA Trials: The Trigger for End of British Raj Did British Fear Another 1857-Type Revolt after INA Trials? Fact File and INA Museum Gandhi once Decided to Leave India and Settle Down in Pakistan!


The hair transplant method is alluded to as moving or moving the hair follicles from the protected zone of the contributor to the beneficiary one's scalp, where it is expected to embed the follicles to cover the uncovered segment and additionally moving the hair follicles in such a way, to the point that the tasteful outcomes could be accomplished as far as normal hairline configuration and the best possible thick join example of the bare segment as imperative. The procedure of the hair transplant is picking up its pertinence and significance step by step because of its economical outcomes for hair regrowth and in addition characteristic hairline outline. It is a vital part of worry about the system of the hair transplant with the goal that the higher number of hair follicles could be moved in a solitary session. The second thing which comes in this viewpoint is about the cost of the hair transplant for the specific strategy that is being picked regarding reaping most extreme joins and getting more regular outcomes.
To keep away from the disadvantages amid the arranging of hair reclamation surgery, it is essential to have an intensive learning of both the methods that choose the appropriateness for a specific one in view of your state of sparseness and physical or anatomical strata.
There are two principle sorts of hair transplant systems named; FUT (Follicular unit transplant) and FUE (Follicular unit extraction).
FUT (Follicular Unit Transplant): The FUT strategy is a capable and a honest to goodness decision if there is the need to extricate the higher number of follicular units to cover the higher NW class and in FUT, it is conceivable to accomplish the most noteworthy number of join gathering and embedding in a solitary session. The FUT procedure of hair transplant is more predominant among those patients, having the most elevated review of sparseness and it requires an exceptional arrangement of abilities from the specialist's side to accomplish a tasteful outcome for the same. In a FUT hair transplant it needs a tasteful ability from the specialist's side that how productively they remove the strip and it's every one of the an inquiry pretty much their mastery and flawlessness in workmanship. It is required to trim the strip territory with the sharp edge having less profundity so the hair roots will be suitable and in the wake of trimming them, we have to hold with the guide of the snare so the shallow line gets the hair establishes in the most reasonable way.
Previously mentioned are some vital inquiries previously picking the FUT surgery from the specific center and every one of the appropriate responses that you are looking for ought to be in positive from your specialist's point of view.
FUE (Follicular Unit Extraction): The follicular unit extraction strategy of the hair transplant is in itself a precise procedure of single follicular unit extraction, in which specialists remove the follicular units one by one straightforwardly from the focused on spot by utilizing the mechanized or manual punch of DIA 0.75 to 1 mm. Here the odds of harm rate will be higher on the grounds that the strategy requires the exceptional arrangement of aptitudes of flawlessness and ability by which a specialist chooses in punching profundity and it's about the theories and forecasts that the specialist applies amid the extraction of follicular units. The punching and boring of reaping the hair roots ought to be at the best possible point and bearing so the specialist productively separates the follicular units and subsequently the exchange/harm rate of follicular units are higher there and the results are likewise of lesser esteem. FUE is reasonable for those hopefuls who are experiencing the lesser review of hair sparseness with least NW class.
The FUE method sets aside much time for gathering the unions and the specialist reaps the unions over and again until the point that enough unite is gathered for the arranged hair rebuilding. The FUE system takes an entire day to finish the strategy, however there might be numerous sessions as the unite prerequisites to cover the bare segment and the recuperation time fluctuates according to the sessions are increased and the required mending time over the course is of 5-7 days. The injury after the surgical system of FUE will show up as modest white scars in estimated 1-mm in measure. These scars can be effortlessly covered in the back and sides of the scalp containing the hair.
Presently it has been effortlessly confirmed that the specific cost for a specific method of hair transplant according to the usable strategy and scopes of follicular units that are collected in a specific methodology shifts in like manner.
Connected system chooses the impediments of unions gathering and specifically decides the cost of hair reclamation strategy as the cost of per joins goes between RS/30-50INR.
The Degree of NW class that chooses the required number of unions to cover the bare segment has a noteworthy part in cost assurance.
Topographical limits additionally assume a part in cost assurance for hair reclamation administrations in light of the fact that the cost for the hair transplant system in other created countries is far higher if contrasted with India.
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She would've been 55 today.
Lord knows we miss Whitney Houston!
Today would have been her 55th birthday, but sadly, Whitney is no longer here. However, we plan to celebrate her life and incredible music forever!
In honor of the vocal goddess' Earth strong, check out these beautiful, funny, real moments that will certainly have you missing Whitney.


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Home Tech News Satellite images suggest Iran preparing for space rocket launch
Satellite images suggest Iran preparing for space rocket launch
Iran appeared to be readying for a space launch as satellite images showed a rocket on a rural desert launch pad, just as tensions remain high over Tehran's nuclear program.
By: AP
| Updated on: Jun 15 2022, 17:48 IST
Satellite image from Maxar Technologies shows a rocket erected at a launch pad in Iran. (AP)
The images from Maxar Technologies showed a launch pad at Imam Khomeini Spaceport in Iran's rural Semnan province, the site of frequent recent failed attempts to put a satellite into orbit.
One set of images showed a rocket on a transporter, preparing to be lifted and put on a launch tower. A later image Tuesday afternoon showed the rocket apparently on the tower.
Iran did not acknowledge a forthcoming launch at the spaceport and its mission to the United Nations in New York did not immediately respond to a request for comment.
However, its state-run IRNA news agency in May said that Iran likely would have seven homemade satellites ready for launch by the end of the Persian calendar year in March 2023. A Defense Ministry official also recently suggested Iran soon could test its new solid-fueled, satellite-carrying rocket called the Zuljanah.
It wasn't clear when the launch would take place, though erecting a rocket typically means a launch is imminent. NASA fire satellites, which detect flashes of light from space, did not immediately see any activity over the site late Tuesday night.
Asked about the preparations, State Department spokesman Ned Price told reporters in Washington that the U.S. urges Iran to de-escalate the situation.
"Iran has consistently chosen to escalate tensions. It is Iran that has consistently chosen to take provocative actions," Price said.
A Pentagon spokesman, U.S. Army Maj. Rob Lodewick, said the American military "will continue to closely monitor Iran's pursuit of viable space launch technology and how it may relate to advancements in its overall ballistic missile program."
"Iranian aggression, to include the demonstrated threat posed by its various missile programs, continues to be a top concern for our forces in the region," Lodewick said.
Over the past decade, Iran has sent several short-lived satellites into orbit and in 2013 launched a monkey into space. The program has seen recent troubles, however. There have been five failed launches in a row for the Simorgh program, a type of satellite-carrying rocket. A fire at the Imam Khomeini Spaceport in February 2019 also killed three researchers, authorities said at the time.
The launch pad used in Tuesday's preparations remains scarred from an explosion in August 2019 that even drew the attention of then-President Donald Trump. He later tweeted what appeared to be a classified surveillance image of the launch failure. Satellite images from February suggested a failed Zuljanah launch earlier this year, though Iran did not acknowledge it.
The successive failures raised suspicion of outside interference in Iran's program, something Trump himself hinted at by tweeting at the time that the U.S. "was not involved in the catastrophic accident." There's been no evidence offered, however, to show foul play in any of the failures, and space launches remain challenging even for the world's most successful programs.
Meanwhile, Iran's paramilitary Revolutionary Guard in April 2020 revealed its own secret space program by successfully launching a satellite into orbit. The Guard launched another satellite this March at another site in Semnan province, just east of the Iranian capital of Tehran.
Judging from the launch pad used, Iran likely is preparing for the Zuljanah test launch, said John Krzyzaniak, a research associate at the International Institute for Strategic Studies. Krzyzaniak earlier this week suggested a launch was imminent based on activity at the site.
The rocket's name, Zuljanah, comes from the horse of Imam Hussein, the grandson of the Prophet Muhammad. Iranian state television aired footage of a successful Zuljanah launch in February 2021.
The launch preparations also come as the Guard reportedly saw one of its soldiers "martyred" in Semnan province under unclear circumstances over the weekend. Iran's Defense and Armed Forces Logistics Ministry, however, later claimed the man worked for it.
The United States has alleged that Iran's satellite launches defy a U.N. Security Council resolution and has called on Tehran to undertake no activity related to ballistic missiles capable of delivering nuclear weapons. The U.S. intelligence community's 2022 threat assessment, published in March, claims such a satellite launch vehicle "shortens the timeline" to an intercontinental ballistic missile for Iran as it uses "similar technologies."
Iran, which has long said it does not seek nuclear weapons, previously maintained that its satellite launches and rocket tests do not have a military component. U.S. intelligence agencies and the International Atomic Energy Agency say Iran abandoned an organized military nuclear program in 2003.
However, Iran's likely preparations for a launch come as tensions have been heightened in recent days over Tehran's nuclear program. Iran now says it will remove 27 IAEA surveillance cameras from its nuclear sites as it now enriches uranium closer than ever to weapons-grade levels.
Both Iran and the U.S. insist they are willing to re-enter Tehran's 2015 nuclear deal with world powers, which saw the Islamic Republic drastically curb its enrichment in exchange for the lifting of economic sanctions. Trump unilaterally withdrew America from the accord in 2018, setting in motion a series of attacks and confrontations beginning in 2019 that continue today into the administration of President Joe Biden.
Building a nuclear bomb would still take Iran more time if it pursued a weapon, analysts say, though they warn Tehran's advances make the program more dangerous. Israel has threatened in the past that it would carry out a preemptive strike to stop Iran — and already is suspected in a series of recent killings targeting Iranian officials.
First Published Date: 15 Jun, 17:48 IST
https://tech.hindustantimes.com/tech/news/satellite-images-suggest-iran-preparing-for-space-rocket-launch-71655284214404.html
https://tech.hindustantimes.com/tech/news/spacex-faces-nasa-hurdle-for-starship-backup-launch-pad-as-international-space-station-at-risk-71655284225830.html
https://tech.hindustantimes.com/tech/news/5g-auction-in-india-approved-kolkata-delhi-hyderabad-and-10-more-cities-getting-it-first-71655290109076.html
https://tech.hindustantimes.com/tech/news/5g-in-india-soon-to-be-10-times-faster-than-4g-71655284396970.html


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The science of moving dirt
AMES, Iowa -- It can compact, grind, mix, shake and otherwise analyze the soil and earthworks that support roadways and construction projects.
And it can do all that testing right next to the heavy equipment doing all the dirty work.
Iowa State University's new Geotechnical Mobile Lab is a $350,000, one-of-a-kind laboratory-on-wheels that allows engineers to study construction sites and come up with immediate answers. Researchers say those answers could lead to more efficient building projects, reduced construction costs, better use of taxpayer investments and better performing roads.
"This really allows us to do cutting-edge research and be a national leader in doing geotechnical research," said Tom Cackler, the director of the Partnership for Geotechnical Advancement at Iowa State University's Center for Transportation Research and Education. "It's pretty exciting."
Dave White, an Iowa State assistant professor of civil, construction and environmental engineering, recently took the lab to a Minnesota Department of Transportation construction site along U.S. Highway 14 near Janesville, Minn. Engineers used it to do soil compaction tests. They compared the results of their tests with the results from new technology built into a rolling machine.
He said the mobile lab provided results twice as fast and allowed all the testing equipment to be right on the job site.
The lab's 44-foot aluminum trailer includes a diesel generator, a conference area, a satellite Internet system, a laptop computer, a weather station, soil compactors, a soil shaker, a soil grinder, a soil mixer, ovens to dry soil, a vibrating table to test soil density, global positioning technology and lots more.
"This provides us the opportunity to conduct research in the field and come back with real-time results," White said. "It also provides the opportunity to train future engineers with state-of-the-art test equipment."
Vern Schaefer, an Iowa State professor of civil, construction and environmental engineering, said the lab will help advance what's known as intelligent construction, the use of computer, laser and other technologies to improve the efficiency and quality of construction.
Schaefer said the lab will also be useful in testing and verifying new construction equipment and technologies. Departments of transportation often require new ideas to be proven before they'll use them on a job site.
Heath Gieselman, a research technician for the Center for Transportation Research and Education, recently led a tour of the mobile lab. Everything was in its place. And everything was spotless.
That's the way researchers want to keep it, he said. It makes for the best science. It's also a necessity because there's not another lab like it. That means it attracts visitors every time it rolls onto a job site.
The lab is sponsored by the McAninch Corp. of West Des Moines.
Tom Cackler, Center for Transportation Research and Education, (515) 294-3230
Vern Schaefer, Civil, Construction and Environmental Engineering, (515) 294-9540
Dave White, Civil, Construction and Environmental Engineering, (515) 294-1463
Heath Gieselman, Center for Transportation Research and Education, (515) 294-7720
Mark Anderson-Wilk, Center for Transportation Research and Education, (515) 294-7124
Mike Krapfl, News Service, (515) 294-4917
Iowa State University's new Geotechnical Mobile Lab is a $350,000, one-of-a-kind laboratory-on-wheels. It will allow engineers to study construction sites right next to the heavy equipment doing all the dirty work. And it could help build better roads and reduce construction costs.
"This really allows us to do cutting-edge research and be a national leader in doing geotechnical research."
Tom Cackler, the director of the Partnership for Geotechnical Advancement at Iowa State University's Center for Transportation Research and Education


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The original Sound Blaster Live!
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I have used both this card and an audigy 2. I don't need any amazing functionality, etc.
Include Add a Protection Plan: The Effect algorithms were creative labs sound blaster live 5.1 digital by a development system that integrated into Microsoft Developer Studio. Published on November 22, Image Unavailable Image not blaxter for Color: One person found this helpful. Learn more about Amazon Prime. However, the rate-conversion was only applied when the audio signal was passed through the effects engine. Show More Show Less. It is really an excellent and outstanding card.
The card installed easily and many features I probably will never use. Amazon Inspire Digital Educational Resources. The item may have some signs of cosmetic wear, but is fully operational and functions as intended.
PCI Sound Card 6 product ratings 4. August saw the introduction of the EMU10K1 audio processor. Product number databases are incomplete, have errors, and CreativeLabs web site isn't helpfull in identifying them either.
Get to Know Us. Card works good most of the time but every so often I need to reboot to stop distortion. Was very easy to install, worked the first time i creative labs sound blaster live 5.1 digital it. See all 13 pre-owned listings. Skip to main content.
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World Breastfeeding Week is a global event between 1 – 7th August that aims to celebrate, protect, and promote the act of breastfeeding. Breast milk is a wonderful and really quite miraculous first form of nourishment and protection for babies, with profound health benefits for both baby and mother. However, it's not always easy (or even possible) for some women to breastfeed. Many women need extra guidance from their midwives, consultations with lactation specialists and for some, additional nutritional support via supplements or herbal medicine.
Before I go on to outline some natural herbs and supplements that can support breastfeeding, please remember that you and your baby are unique. You may be struggling big-time with breastfeeding for a variety of reasons, or perhaps you aren't even able to breastfeed full-stop. Throughout this week (and beyond) don't beat yourself up and load on the Mummy-guilt if it hasn't worked out! Chat with a natural health professional to find out ways to enhance your alternative feeding choices in a way that best suits you and your babies unique needs.
If you are breastfeeding, a good healthy diet for yourself is essential during this time. You can read my guide to a healthy breastfeeding diet here.
Fennel (Foeniculum vulgare): Fennel supports the stimulation of breast milk production. It has the added benefit of being a carminative plant. This means that the volatile oils in fennel can help to calm the digestive tract, and once ingested by Mum, these properties can help reduce wind for baby.
Aniseed (Pimpinella anisum): Aniseed can help to support the normal flow of breast milk. Again, via the breastmilk, it also helps to reduce and ease griping and wind issues in baby.
Nettle (Urtica dioica): Nettle can help to support milk production and also increases the nutrient quality of breast milk. It has a replenishing tonic effect for Mum and contains iron which is a vital nutrient postpartum.
Blessed Thistle (Cnicus benedictus): Blessed thistle can help to increase milk production, so it's useful if you have a low milk supply. It seems to work best when taken in conjunction with fenugreek.
Probiotics: Taking a probiotic supplement is an excellent way to ensure that you have a healthy gut. Good gut health is vital to a strong immune system.
Lecithin: Lecithin can be used by nursing mothers to help prevent blockages in the milk ducts. The lecithin can help decrease the viscosity of the milk, making it easier to pass through the ducts. Soy lecithin can be purchased as a supplement. You can also find good levels of lecithin in eggs, dairy products, beef, and peanuts, as well as various fruits and vegetables.
Vitamin C: If you are eating a diet high in fresh fruits and vegetables, you should be getting enough vitamin C for you and baby. However, if you are finding you are getting sick often or have recurrent bouts of mastitis, it may pay to consider a vitamin C supplement.
The following herbs should be avoided when breastfeeding, as they can contribute to a decrease in milk supply; black walnut, sage, yarrow, and peppermint.
While a good diet is vital to both the health of the breastfeeding mother and the quality of breastmilk for baby, sometimes a little more nutritional support is needed to ensure that all bases are covered. This is when a breastfeeding multivitamin can be beneficial. Speak to your health professional or chemist about what vitamin may be right for you.
Important Note: Before introducing any herbal medicines into your diet, please speak to your midwife or a qualified health professional for advice as to what is best suited to you and your specific needs.


The Carnival of Barcelona is welcoming the party mood to the city this week! The first written reference to the Carnival in Barcelona dates back to 1333, when the battle of oranges was banned (today it is performed as la "Taronjada") and the use of masks was regulated. If you are planning to come to Barcelona, get the most unique experience with BE Apartment in the best luxury apartments in Barcelona and other cities, for your security, pleasure and comfort. The Carnival 2018 is celebrated between the 8th and the 14th of February. The King Carnestoltesá debauchery will arrive soon: the Carnival parades will tour the city on Saturday, February 10 and the party will be extended from Fat Thursday until Ash Wednesday.
According to the tradition, after several days of "sins", the King Carnestoltes is judged on Shrove Tuesday. On Ash Wednesday, the final act of the Carnival of Barcelona is organized, called "The Burial of the Sardine", which serves as a farewell to the King Carnestoltes. Moreover, it traditinally gives a start to the Lent, the period before Easter. Popular events are organized in various neighborhoods of the city and, of course, there the typical sardines will not be lacking! Do not miss this opportunity to enjoy Barcelona and get to know the city better with BE Apartment. If you´re visitng the city, take a look at our luxury apartments that we have throughout the city.


Active Immunotherapy: Could It Benefit Men at An Earlier Stage of Prostate Cancer?
In this video, Dr. Paul Schellhammer discusses recent research that suggests that active immunotherapy could be used to treat earlier stages of prostate cancer and shares his perspective about what he believes this means for men with less advanced disease. Dr. Schellhammer also explains how this research changes the landscape of treatment and what he feels are developing areas of hope for men with men with advanced and/or metastatic disease.