text:
Newcastle ace, Kelly, takes on former IBF World ruler on Joshua Parker bill
Posted on March 9, 2018 by Michael Morgan
Josh Kelly will face former IBF World champion Carlos Molina for the WBA International Welterweight title at Principality Stadium in Cardiff on March 31, live on Sky Sports Box Office.
Kelly fights for his first title in just his sixth pro outing and first stadium appearance since turning over in April. The Team GB Rio Olympian has not needed to go beyond six rounds so far, with his last appearance an impressive sixth round stoppage win in his first scheduled ten rounder in London in December.
Kelly, who turned 24 today, now faces a huge step-up in class as he targets landing his first belt in the shape of former Super-Welterweight king Molina. The Mexican won his World crown in September 2013 beating Ishe Smith in his Las Vegas backyard and will provide Kelly with a stiff examination of the promise he's shown in his first year in the paid ranks – but Kelly says that trainer Adam Booth is ready to let him off the leash.
"Adam said it's time for me to step up, so let's go," said Kelly. "I feel this is a real test against a proven fighter. He's a former World Champion so you can't deny this is my biggest fight to date. I can't wait to step between those ropes in Cardiff and propel myself into the mix.
"He's never been stopped so it says a lot. Just look at the names on his record that he's been in with, I know how tough he will be. The motivation is to win against any fighter is enough for me, I want to perform on the big stage – I'm very excited to say the least.
"I'm ready for this. I feel physically, mentally and emotionally ready to take this step up and have been for a fight or two. I've learnt a lot in the fights and in training over the past year and also got to know myself even more. The learning process never stops, there's improvements to be made with each fight and every step up starting March 31."
Kelly's clash with Molina is part of a huge night of action in Cardiff as Anthony Joshua MBE and Joseph Parker meet in a World Heavyweight unification blockbuster.
Ryan Burnett defends his WBA World Bantamweight crown against Yonfrez Parejo and Alexander Povetkin puts his WBA Inter-Continental and WBO International titles against Liverpool favourite David Price and Welshman Joe Cordina fights for his first pro title with the WBA International Lightweight belt on the line against Andy Townend.
Limited tickets remain on sale for the event priced at £40, £60, £80, £300 and £600 from www.stubhub.co.uk


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text:
\section{Introduction.}\label{sec0}
In this note, we investigate a conjecture
of Aspinwall, Plesser
and Wang \cite{APW}.
Our calculations offer supporting evidence at the
level of toric $K$--theory and clarify some aspects
related to multiplicities of discriminants which play an
important implicit role in the
statement of the conjecture.

\medskip

In the projective Calabi--Yau case, homological mirror symmetry \cite{kont1} is stated
as an equivalence of the bounded
derived category
of coherent sheaves $D^b(X)$ on a smooth projective Calabi--Yau variety $X$ and
the derived Fukaya category $DFuk(Y)$ of the mirror Calabi--Yau variety $Y.$
Mirror symmetry and string theory considerations about $D$--brane moduli spaces
also predict an identification between
the global structure of locally trivial families of categories $D^b(X)$ and $DFuk(Y)$
over the K\"ahler parameters of $X$ and
the complex parameters of $Y,$ respectively.
As a by--product of his general conjecture, Kontsevich \cite{kont2}
conjectured that the action on cohomology of the group of
self--equivalences of $D^b(X)$
matches the monodromy action on the cohomology
of the mirror Calabi--Yau variety $Y.$  In particular,
for a smooth projective Calabi-Yau variety $X,$
the action of of the spherical twist induced by the structure
sheaf of $X$ is ''mirrored" by the monodromy action
around a distinguished ''primary" component in the
moduli space of complex structures of $Y.$ While such a
statement is implicit in the existing proofs of homological
mirror symmetry as well as in the statement of the Strominger--Yau--Zaslow
conjecture \cite{SYZ} as identifications at large complex/radius limits,
our work is concerned with a similar global identification in
the moduli spaces.

\smallskip

In the present work, we consider
the simplified case of
toric quasi--projective Calabi-Yau
Deligne-Mumford stacks.
The proposal by Aspinwall, Plesser and Wang
generalizes Kontsevich's
identification to all the irreducible components of the principal
$A$-determinant $E_A.$ The polynomial $E_A$
has integer coefficients and provides a remarkable
generalization of the classical discriminant and was introduced and studied by
Gelfand, Kapranov and Zelevinsky \cite{GKZbook}.
The APW conjecture predicts the spherical functors that correspond to the
various components of the principal $A$-determinant. Our result (Theorem \ref{thmm})
lends further support for the conjecture by checking its consistency
in terms of the ranks of the $K$-theory for the various toric Deligne-Mumford
stacks that enter the geometrical picture, and local multiplicity of discriminants
and their intersecting components.
The importance of these multiplicities has been recognized in the
original work of Aspinwall, Plesser and Wang \cite{APW}, and their
relevance is clearly explained
in the more recent paper of Kite and Segal \cite{KS}. Such multiplicities are also
featured in earlier works \cite{AHK}, \cite{HLS}.
In the purely categorical sense, they are implicit in the theory of ''windows"
that began with the string theoretical work of Herbst, Hori and Page \cite{HHP},
and was made rigorous by Ballard, Favero, Katzarkov \cite{BFK} and Halpern-Leistner \cite{HL}.
Our approach to determining multiplicities is elementary algebraic
(see Definition \ref{def:n}) and it uses the powerful GKZ toolbox
for studying discriminants. There are well known subtleties related to
the local topological picture of intersecting discriminants. A
categorification of
the classical braid factorization technique should offer
a truly conceptual picture of the expected semiorthogonal decompositions
beyond the $K$--theoretic point of view. For a glimpse of how such a procedure
might work, see \cite{AHK}.

\

In this note, we do not perform any mirror monodromy calculations
for the associated GKZ $D$--module. A true
check of the APW conjecture would require such computations,
but we leave this discussion for future work. However,
by simply combining the results of this work with the
analytic
monodromy calculations from \cite{hor}, \cite{BH2},
we can check the APW
predictions in some cases under certain transversality assumptions,
see Corollary \ref{cor:tr}.
It is important to note that the principal $A$-determinant
is determined by the characteristic cycle of the GKZ $D$--module.
We expect that the ''stacky" nature of the toric geometrical context
will require the use of the better behaved GKZ system
\cite{BH} and an appropriately adapted version of the topological mirror map.

\section{Review of toric geometry and $A$-discriminants}\label{sec1}

\subsection{Toric Geometry}
We briefly review the definitions of
reduced toric Deligne-Mumford stacks, as developed in \cite{BCS}
and \cite{Jiang}. A \emph{stacky fan} ${\bf {\bf \Sigma}}$ is defined as
the data $(\Sigma,\{v_i\})$
where $\Sigma$ is a simplicial fan in $\bar{N}= N \otimes_{\ZZ} \QQ,$
and $\{v_i\}$ $1\leq i \leq n,$
is a collection of elements in the
finitely generated group $N.$ We assume that the rays of the
simplicial fan $\Sigma$ are generated by
the possibly non-minimal non-zero integral elements $\bar{v}_i.$

The set $\{v_i\}$ defines a map
$
\alpha:\ZZ^n\to N
$
with finite cokernel. If we dualize, we get
an exact sequence
$$
0 \to N^\dual \to (\ZZ^n)^\dual \to N^{\prime} \to K \to 0.
$$
defining the Gale dual of $\alpha$ (see \cite{BCS}),
with $K$ a finite abelian group.
We apply the functor ${\rm Hom} (\cdot, \CC^{\times})$
to get the exact sequence
$$
0 \to K \to G \to (\CC^{\times})^n \to (\CC^{\times})^d \to 1,
$$
where $G$ is the algebraic group
$
{\rm Hom}(N^{\prime},\CC^{\times.}).
$

\medskip

Consider the subset $Z$ of $\CC^n$ that consists of all the
points ${\bf z}=(z_1,\ldots,z_n)$ such that the set of $v_i$ for
the zero coordinates of ${\bf z}$ is contained in a cone of $\Sigma$.
Then the toric DM stack $X_{\bf{\bf \Sigma}}$
that corresponds to the stacky fan
${{\bf \Sigma}}$ is defined as the stack quotient
$[Z/G]$ where $Z$ and $G$ are
endowed with the natural reduced
scheme structures.
It has been shown in \cite{BCS} that $X_{{\bf \Sigma}}$
is a Deligne-Mumford stack whose moduli space is
the simplicial toric variety $X_\Sigma$.
The category of coherent sheaves on $[Z/G]$
is equivalent to that of $G$-linearized coherent sheaves on
$Z$, see \cite[Example 7.21]{Vistoli}.

\medskip

To a cone $\sigma$ in $\Sigma,$ one can associate a closed substack
$X_{\sigma}$ of
$X_{{\bf \Sigma}}$ by looking at the quotient of $N$ by the subgroup
spanned by $v_i$ with $\bar{v}_i \in C$. The new fan $\Sigma/\sigma$
is defined as the image of the
link of $C$ in $\Sigma$, and the new $v_i$ are the images
of the old $v_i$ from the link. The quotient of $N$ may
have torsion, even in the case of a torsionfree $N.$ Therefore,
the closed substack $X_{\sigma}$ may not be reduced even
when $X_{\bf \Sigma}$ is.

\medskip

\subsection{Combinatorics of $A$-sets}\label{section:A}
In this section, we review some concepts discussed in chapter 7 of the
book \cite{GKZbook}.
A finite subset $A= \{ v_1, \ldots, v_n \}$ of
the integral lattice $N=\ZZ^d$ is called an {\bf $A$--set} if
the set $A$ generates the lattice $N$ as an abelian group, and
there exists a linear function
$h : N \to \ZZ$ such that $h(v_i)=1,$ for all
$i, 1 \leq i \leq n.$

For the rest of the paper, we will assume that the set
$$A= \{ v_1, \ldots, v_n \} \subset N=\ZZ^d$$
is an $A$--set. However, see Remark \ref{rm1}
below.

We will use the notations
\begin{equation*}
\begin{split}
Q&:= {\rm conv} (A) \subset \RR^{d-1},\\
K&:= \sum_{1 \leq i \leq n} \RR_{\geq 0} v_i= \RR_{\geq 0} Q \subset N \otimes \RR=\RR^{d},\\
S&:= \sum_{1 \leq i \leq n} \ZZ_{\geq 0} v_i= \ZZ_{\geq 0} A\subset N=\ZZ^{d}.
\end{split}
\end{equation*}
The rational polyhedral cone $K$
defines an affine toric variety
$Y={\rm Spec}\,\CC[K^\dual\cap N^\dual]$. The hyperplane condition on the
set $A$ implies that
this variety has Gorenstein singularities.
Moreover, any regular (i.e. projective) triangulation
of the polytope $Q$ with vertices among the elements of $A$ induces
a stacky fan structure ${\bf \Sigma}$ supported on the cone $K.$
The notion of a regular triangulation will be explained below.
There is a natural crepant birational morphism
$
\pi:X_{\bf \Sigma}\to Y
$
from the induced smooth toric Deligne-Mumford stack $X_{\bf \Sigma}$
to the toric affine Gorenstein singularity $Y.$

\medskip

A {\em triangulation} of the set $A= \{v_1, \ldots, v_n \}$ is a triangulation of the
polytope $Q$ such that all
vertices are among the the points $v_1, \ldots, v_n.$
A marked polytope $(P,B)$ consists of the convex polytope $P$
and a finite subset $B$ of $P$ such that $P=\hbox{Conv}(B).$

\begin{definition}
\cite[Definition 7.2.1]{GKZbook}
\label{def:polysub}
Given the set $A \subset N = \ZZ^d,$
a {\it polyhedral subdivision} $\Pa$ of $(Q,A)$ is a family
of $(d-1)$-dimensional marked polytopes
$(Q_i,A_i), i=1, \ldots, l,$ $A_i \subset A,$ such that
any intersection $Q_i \cap Q_j$ is a face (possible empty) of both
$Q_i$ and $Q_j,$
$$
A_i \cap (Q_i \cap Q_j) =
A_j \cap (Q_i \cap Q_j),
$$
and the union of all $Q_i$ coincides with $Q.$
\end{definition}

A function $f : Q \to \RR$ is said to be
{\it $\Pa$--linear} for the polyhedral subdivision
$\Pa=(Q_i,A_i)$ if $f$ is continuous and the
restriction of $f$ to each $Q_i$ is affine-linear.
A function $\psi : A \to \RR$ is said to be
{\it $\Pa$--linearizable}
if there exists a {\it $\Pa$--affine}
function $g_\psi : Q \to \RR$ such that
$g_\psi (v)=\psi(v)$ for any $v \in \cup A_i.$
A continuous function $g : Q \to \RR$ is
{\it concave}, if for any $x,y \in Q,$
we have $g(tx + (1-t) y) \geq tg(x) + (1-t) g(y),$ $0 \leq t \leq 1.$
For any polyhedral subdivision
$\Pa=(Q_i,A_i),$ let $C(\Pa)$ denote the cone
of $\Pa$--linearizable functions
$\psi : A \to \RR$ such that the associated
function $g_\psi : Q \to \RR$ is concave
and $g_\psi (v) \geq \psi(v)$ for all
$v \in A \setminus \cup A_i.$
We say that the subdivision $\Pa$ is {\it regular}
if the cone $C(\Pa)$ has relative
non-empty interior inside the linear
space of $\Pa$--linearizable functions.

\medskip

For two polyhedral subdivisions $\Pa=(Q_i,A_i)$ and $\Pa'=(Q'_j,A'_j),$ we
shall say that $\Pa$ is a {\it refinement} $\Pa'$ if, for each $j,$ the collection
of $(Q_i,A_i)$ such that $ Q_i \subset Q'_j,$ forms
a polyhedral subdivision of $(Q'_j,A'_j).$ The set of polyhedral subdivisions is then
naturally a partially ordered set (poset) with respect to refinement.
Triangulations are minimal elements of this poset. The maximal element
is the polyhedral subdivision $(Q=\hbox{conv}(A),A).$

\medskip

The cones $C(\Ta)$ for all the regular triangulations of $(Q,A)$ together
with all the faces of these cones form a complete fan $F(A)$
in $\RR^A= \RR^n$
called the {\it secondary fan}. We choose a translation invariant
volume form vol in $\RR^{n-1}$ (or in $\RR^n$) such that elementary simplex
in the lattice $N = \ZZ^n \subset \RR^n$ has volume $1.$ The characteristic
function of the triangulation $\Ta$ is the function $\phi_\Ta : A \to \RR$
defined by
$$
\phi_\Ta (v):=\sum_{v \in {\rm Vert}(\sigma)} {\rm vol} (\sigma),$$
where the summation is over all the maximal simplices of the triangulations
$\Ta$ for which $v$ is vertex. We set $\phi (v)=0$ if $v$ is not a vertex
of simplex in $\Ta.$ The {\it secondary polytope} $S(A)$ is the convex
hull in $\RR^A=\RR^n$ of the functions $\phi_\Ta$ for all the triangulations
$\Ta$ of $(Q,A).$
Theorem 7.1.7 in \cite{GKZbook} shows that
the secondary polytope is $n-d$ dimensional and its vertices
are the characteristics functions $\phi_\Ta$ for all triangulations
$\Ta$ of $(Q,A).$ Moreover, the secondary fan $F(A)$ is the normal
fan of the secondary polytope $S(A).$ We identify $\RR^A= \RR^n$
with its dual in the canonical way.

\medskip

Consider two regular triangulations
such that the corresponding vertices in the secondary polytope
are joined by an edge.
The two triangulations differ by a
{\it modification} along a circuit.
A {\it circuit} in $A$ is a minimal dependent subset $ \{ v_i, i \in I\}$
with $I \subset \{1,\ldots, N \}.$
In particular any circuit determines a relation
of the form
\begin{equation}\label{circuit}
\sum_{i \in I_+ } l_i v_i + \sum_{i \in I_-} l_i v_i =0,
\end{equation}
with $I= I_+ \cup I_-,$
where the two subsets $I_+:= \{ i: l_i > 0 \}$ and  $I_-:= \{ i: l_i < 0 \}$
are uniquely defined by the circuit up to replacing $I_+$ by $I_-$.
We will assume that the relation (\ref{circuit}) is {\it primitive},
i.e. that the integers $l_i$ have no common prime factor. Given a
circuit,
one can write a (possibly) non-primitve relation
by choosing
\begin{equation*}
|l_i|:= {\rm vol} ({\rm conv} (v_i, i \in I \setminus {i})).
\end{equation*}
When there is no danger of confusion, we may call the index subset $I$ a circuit.
The minimizing condition in the definition of a circuit implies that the
sets conv($v_i, i \in I_+$) and conv($v_i, i \in I_-$) intersect in their common interior point.
The polytope determined by $v_i,$ $i \in I$
admits exactly two triangulations $\Ta_+(I)$
and $\Ta_-(I)$ defined by the simplices conv($\{v_j, j \in I \setminus i$), for
$i \in I_+$, respectively $i \in I_-$.

\medskip

Suppose that the regular triangulations $\Ta$ and $\Ta^\prime$
of $A$ are obtained from each other by a modification
along a circuit $I.$ We say that a
subset $J \subset A \setminus I,$ is {\it separating}
for $\Ta$ and $\Ta^\prime$ if, for some $i \in I,$ the set
of $v_i,$ with $i \in (I \setminus i) \cup J$ is the set of vertices of a
simplex (of maximal dimension) of $\Ta_1.$ It turns out
that a separating set $J$ has the property that
$(I \setminus i) \cup J,$ for $i \in I_+,$ determine simplices
in $\Ta$ and $(I \setminus i) \cup J,$ for $i \in I_-,$
determine simplices in $\Ta^{\prime}.$

\medskip

\begin{proposition}\cite[Prop 7.2.12]{GKZbook} \label{prop:subd}
The polyhedral subdivision
$\Fa=\Fa(\Ta, \Ta^\prime)$
corresponding to the edge of the secondary polytope
joining the vertices determined by $\Ta$ and $\Ta'$ consists
of the simplices $(conv(K),K)$ which $\Ta$ and $\Ta'$
have in common and the polyhedra $(conv(I \cup J), I\cup J)$
for all separating subsets $J \subset A \setminus I.$
\end{proposition}

\subsection{Principal $A$-Determinants}
Define $\nabla_{A}$ as the Zariski closure in $\CC^n$
of the set of polynomials $f=\sum_{1 \leq j \leq n}
a_i x^{v_i}$ in $\CC[x_1, \ldots, x_d]$
such that there exists some
$y \in (\CC^{\times})^n$ with the property that
$f=0$ is singular at $y.$
By definition, the discriminant $\Delta_{A} \in \ZZ[a_1, \ldots,
a_n]$ is the irreducible polynomial (defined up to a sign)
whose zero set is given by the union of the irreducible
codimension $1$ components of $\nabla_{A}.$ For the case
codim$\nabla_{\cA} >1,$ one sets  $\nabla_{\cA}=1.$

\medskip

For any non-empty face $\Gamma$ of the polytope $Q,$
let $i(\Gamma)$ denote the index of the sublattice
$\ZZ( A \cap \Gamma)$ inside the lattice
$N \cap \RR \Gamma,$
\begin{equation}
i(\Gamma):= [N \cap \RR \Gamma: \ZZ (A \cap \Gamma)].
\end{equation}
Recall that
$S=\ZZ_{\geq 0} A$ is the semigroup generated by $A.$
Furthermore, let $S/\Gamma$ denote the image semigroup
of $S$ in the quotient free group $N_\RR/ \RR \Gamma,$
with $N_\RR= N \otimes_\ZZ \RR.$ The semigroup
$S/\Gamma$ generates a pointed cone in
$N_\RR/ \RR \Gamma,$ so we can define
\begin{equation}\label{def:u}
u(\Gamma):= {\rm vol} ( {\rm conv} \big(S/\Gamma) \setminus
{\rm conv} (S/\Gamma \setminus \{0\}) \big),
\end{equation}
where the volume form on $N_\RR/ \RR \Gamma$
is induced from the standard volume form
on $N.$

\begin{definition}\cite[Theorem 10.1.2]{GKZbook}\label{def:ea}
The {\it principal $A$-determinant} $E_A$ is the
polynomial in $\ZZ[a_1, \ldots, a_n]$
defined as
\begin{equation}
E_A:= \prod_{\Gamma} (\Delta_{A \cap \Gamma})^{u(\Gamma) \cdot i(\Gamma)},
\end{equation}
where the product is taken over all the non-empty faces of the
polytope $Q={\rm conv} (A).$
\end{definition}
For the faces $\Gamma$ such that
codim$\nabla_{\cA} >1,$ the corresponding factor
in the product is equal to $1.$
Hence the product is taken over the vertices and some the faces $\Gamma$
such that $A \cap \Gamma$ is not simplicial.

\medskip

The principal $A$--determinant has the following remarkable property.
\begin{theorem}\cite[Theorem 10.1.4]{GKZbook}
For a given set $A,$ the Newton polytope of $E_A$ coincides with
the secondary polytope $S(A).$ In particular, the vertices of the
Newton polytope of $E_A$ are given by the characteristic functions
$\phi_\Ta$ for all the regular triangulations $\Ta$ of $(Q,A).$
\end{theorem}

\section{K-theory and discriminant intersection multiplicities}

We will now compute the dimensions of the Grothendieck rings
of some toric stacks involved in the main result and conjecture discussed
in this paper.

Let $\Gamma$ be a non-empty face of the polytope $Q={\rm conv}(A).$
We will consider a stacky fan ${\bf \Sigma}_\Gamma=
(\Sigma_\Gamma, \{ \pi (v_i) \}_{i \in {\mathcal I}})$ where
$\pi : N \to N / \ZZ(A \cap \Gamma)$ is the canonical map,
and $\Sigma_\Gamma$ is a fan in the real linear space
$(N / \ZZ(A \cap \Gamma) \otimes \RR= N_\RR/\RR\Gamma$
with rays generated by the images of the
vectors $v_i$ for $i \in \Ia.$
The set $\mathcal I$ consists of all indices $i$
such that the image of $v_i$ in $N_\RR/\RR\Gamma$
is contained in the closure of the set
$$
{\rm conv} \big(S/\Gamma) \setminus
{\rm conv} (S/\Gamma \setminus \{0\}),
$$
where $S/\Gamma$ is the image semigroup
of $S=\ZZ A$ in $N_\RR/ \RR \Gamma.$
It is important to note that the set $A$
and the nonempty face $\Gamma$ uniquely determine
the vectors $v_i, i \in \Ia,$ but the cones
of the fan $\Sigma_\Gamma$ are not uniquely determined.
However, the following statement is true.

\begin{proposition}
For any two choices of stacky fans ${\bf \Sigma}_\Gamma$
and ${\bf \Sigma}_\Gamma^\prime$ as above,
the bounded derived categories of coherent categories $D^b(X_{{\bf \Sigma}_\Gamma})$
and $D^b(X_{{\bf \Sigma}_\Gamma^\prime})$ are equivalent.
\end{proposition}

\begin{proof} The stacky fans ${\bf \Sigma}_\Gamma$
and ${\bf \Sigma}_\Gamma^\prime$ are connected by a finite sequence of
toric birational flops. Each of them induces an equivalence of
the bounded derived categories of coherent sheaves
on the corresponding DM stacks \cite{BO}, \cite{Kaw}, so the result follows.
\end{proof}

The rank of the Grothendieck group of the
DM stack $Z_\Gamma$ is given by the following result.

\begin{proposition}\label{prop:kgamma}
The rank of the Grothendieck group
of the DM stack $Z_\Gamma$
is equal to the product $u(\Gamma) \cdot i(\Gamma).$
\end{proposition}

\begin{proof}
Since $N$ is a lattice,
the quotient group $N/ N \cap \RR \Gamma$ is torsion free.
This implies that the torsion
part of the quotient $N / \ZZ(A \cap \Gamma)$ is isomorphic
to the torsion part of the quotient
$N \cap \RR \Gamma/ \ZZ (A \cap \Gamma).$ The order
of the torsion part is given by the index $i(\Gamma).$
The discussion in section 6 of \cite{BH-K}, as well as
the work of Jiang-Tseng \cite{Jiang}, show that the rank of
$K_0(Z_\Gamma)$ is equal to $K_0(Z_{\Sigma_{\Gamma}}) \cdot i(\Gamma).$

\medskip

In order to finish the proof, we have to show that the dimension
$K_0(Z_{\Sigma_\Gamma})$ is equal to $u(\Gamma).$ Definition \ref{def:u}
shows that $u(\Gamma)$ is equal to the sum of the volumes of
the (simplicial) maximal dimensional cones in the fan $\Sigma_\Gamma.$
The result follows from
\cite[Proposition 3.20]{DKK} which shows that the desired rank
is equal to the sum of the volumes maximal dimensional cones in the fan
$\Sigma_\Gamma.$
\end{proof}

\medskip

We now consider two triangulations $\Ta$ and $\Ta^\prime$
corresponding to vertices of the secondary polytope
which are joined by an edge $F$. According to Proposition \ref{prop:subd},
there exists a circuit $I=\{ v_i \}, i \in I,$ associated to this
edge. Let $\ZZ I: = \sum _{i \in I} \ZZ v_i$ be the sublattice
generated in $N$ by the vectors of the circuit.
We
now define a stacky fan ${\bf \Sigma}_F=
(\Sigma_F, \{ \pi (v_i) \}_{i \in K})$ where
$\pi : N \to N / \ZZ I$ is the canonical map,
and $\Sigma_F$ is the fan in the real linear space
$N_\RR/ \RR I$ whose maximal cones with rays generated
by the images in $N_\RR/ \RR I$ of $v_i, i \in J,$ for
all the separating sets $J$ for the circuit $I$
and the pair of triangulations $\Ta$ and $\Ta^{\prime}.$
The set $\{ \pi (v_i) \}_{i \in K})$
consists of images of vectors $v_i$ which are not in $I$ but
are a vertex for one of the separating sets $J.$ Note that
all the separating sets $J$ are simplicial.
Denote by
$Z_F$ the toric DM stack defined by the stacky fan
${\bf \Sigma}_F.$ For the general theory of spherical functors,
see \cite{AL}.

\begin{proposition}\label{prop:ez}
There exists a spherical functor $D^b (Z_F) \to D^b (X)$
where $X$ is the toric DM stack induced by either one
of the triangulations $\Ta$ or $\Ta^\prime.$
\end{proposition}

\begin{proof} The $EZ$-twist argument in \cite{EZ} is mostly formal
and can be adapted to the toric DM stacky context. The result
also follows from the work of Halpern-Leistner-Shipman \cite{HLS}.
In the physics literature, the spherical twist induced
by this spherical functor is known as the ``wall monodromy".
\end{proof}

\begin{proposition}\label{prop:ki}
The rank of the Grothendieck group
of the DM stack $Z_F$
is equal to the sum
$$\sum_{J-{\rm sep}} [N: \ZZ (I \cup J)]$$
over all the separating sets $J.$
\end{proposition}

\begin{proof}
Since $I \cup J$ is a simplex, and
$\ZZ I \cap \ZZ J= (0),$ we can write a non-canonical
splitting for the torsion group
$N/ \ZZ(I \cup J)$ as a direct sum of finite torsion groups
$\ZZ^k/ \ZZ I \oplus \ZZ^{d-k}/\ZZ J.$ Note that
the index $\ZZ^{d-k}/\ZZ J$ is the volume of the
cone generated by the separating set $J$ in the
fan $\Sigma_I$ in $N_\RR / \RR I$ associated to the
toric DM stack $Z_F.$ As in the proof of Proposition \ref{prop:kgamma},
we conclude that the rank of the Grothendieck group
of the toric DM stack is indeed the sum $\sum_J [N: \ZZ (I \cup J)]$
over all the separating sets $J.$
\end{proof}

The main result of this note is the following theorem.

\begin{theorem} \label{thmm}
For any edge $F=[\phi_{\Ta}, \phi_{\Ta^\prime}]$ of the secondary polytope,
between the triangulations $\Ta$ and $\Ta^\prime.$
the following equality holds:
\begin{equation}\label{formula:k0}
\rk (K_0 (D^b(Z_F))) = \sum_{\Gamma \subset Q} n_{\Gamma,F} \cdot \rk (D^b (K_0(Z_\Gamma))),
\end{equation}
for some non-negative integers $n_{\Gamma,F}.$
\end{theorem}

The precise combinatorial definition of the non-negative
integers $n_{\Gamma,F}$
will be given below in Definition \ref{def:n}. It
arises naturally as a by-product of the proof of the theorem
and it is close in spirit to the string theoretical analysis in
\cite{AGM}. Geometrically,
the index $n_{\Gamma,F}$ coincides in most cases with
the intersection
multiplicity of the discriminant component $\nabla_{\Gamma,F}$
with the rational curve induced by edge $F$ in the
DM toric stack defined by the secondary fan. This point of view
been discussed in \cite{HLS}, \cite{KS}.

\begin{proof}
We will prove the result by
giving two interpretations to the coefficient
restriction
of the principal $A$-determinant $E_A$
to the edge $F$ of the secondary polytope.
By \cite[Prop 6.1.3]{GKZbook},
the coefficient
restriction of any Laurent polynomial
$P(x_1, \ldots, x_n)$ to any face $F$
of its Newton polytope
has the leading term of
the polynomial in $t$
$$t \mapsto P(t^{\psi_1} x_1,
\ldots , t^{\psi_n} x_n)$$
equal to
$t^{\psi(F)} P_F (x_1, \ldots, x_n),$
where $\psi$ is a lineat support function for the
face $F,$ and $P_F$ is the coefficient
restriction of $P$ to the face $F.$

Let $\Pa=(Q_i,A_i)$ be a polyhedral subdivision of the marked polytope $(Q,A).$
Theorem
1.12' in \cite{GKZbook} shows that, up to multiplication by a constant,
the coefficient
restriction of $E_A$ to the edge $F(\Pa)$ of the secondary polytope equals
$$
\prod_i (E_{A_i})^{[N: \ZZ A_i]} \ .
$$
In particular, if $\Fa$ is the polyhedral subdivision corresponding to the
circuit $I$ and the edge $[\phi_\Ta, \phi_{\Ta^\prime}]$ for
two triangulations $\Ta,$ $\Ta^\prime$
(see Proposition \ref{prop:subd}), then, up to multiplication by a
constant, $E_A$ is equal to
$$
\prod_{i \in K} (E_{A_i})^{[N: \ZZ A_i]} \
\cdot
\prod_{J-{\rm sep}} (E_{I \cup J})^{[N: \ZZ (I \cup J)]} \ ,
$$
where $({\rm conv}(A_i), A_i)_{i \in K}$ is the family of simplices
that $\Ta$ and $\Ta^\prime$ have in common,
and the second product is taken over all the separating sets $J$
such that $J \cup I$ has maximal dimension. Note that
each such $J \cup I$ contains as a unique circuit the circuit $I.$
In the terminology of \cite[pg 309]{GKZbook}, the set
$J \cup I$ is {\it weakly dependent}. As such, the argument
on \cite[pg 309]{GKZbook} shows that, up to a multiplication
by a constant, $E_{I \cup J}$ is in fact equal to the
the discriminant $\Delta_{I \cup J},$ which by \cite[Prop 9.1.8]{GKZbook}
is a non-zero scalar multiple of the polynomial
\begin{equation}\label{di}
\Delta_I:=(\prod_{i \in I_+} l_i^{l_i}) \prod_{i \in I_-} a_i^{-l_i}
-
(\prod_{i \in I_-} l_i^{-l_i}) \prod_{i \in I_+} a_i^{l_i},
\end{equation}
where $\sum_{i \in I_+ \cup I_-} l_i v_i=0$
is a primitive integer relation (unique up to sign)
associated to the circuit $I.$ Note \cite[Prop 9.1.8]{GKZbook}
assumes that the circuit generates the ambient lattice so
the choice
$
|l_i|:= {\rm vol} ({\rm conv} (v_i, i \in I \setminus {i})),
$
would work in that case. We do not make that assumption.

\medskip

Consider a linear form $\psi :\RR^A \to \RR$
lying in the
interior of cone $C(\Fa)$ (the normal cone
to the edge $F$) of the secondary fan as
defined in section \ref{section:A}.
This means that the edge $F$ is the
supporting face of $\psi.$
This means that
$\psi$ is equal to a constant $\psi(F)$ along $F,$
and $\psi(v_i) < \psi(F)$ when $v_i$ is not on $F.$
It follows that the coefficient of
the leading term in $t$ of the Laurent polynomial
$t \mapsto E_A (\sum_i t^{\psi_i} a_i x^{v_i})$ is, up to
multiplication by a constant and a monomial in the variables $a_i,$ the product
\begin{equation}\label{dom}
\prod_{J-{\rm sep}} (E_{I \cup J})^{[N: \ZZ (I \cup J)]}
=(\Delta_I)^{\sum_J [N: \ZZ (I \cup J)]}
\end{equation}
where the sum and product in the formula above are over all the
separating sets $J,$ a the discriminant polynomial
$\Delta_I$ is defined by (\ref{di}).

\medskip

On the other hand, according to Definition \ref{def:ea}
the principal $A$--determinant $E_A$ is the product
$$
E_A:= \prod_{\Gamma} (\Delta_{A \cap \Gamma})^{u(\Gamma) \cdot i(\Gamma)},
$$
where the product is taken over all the non-empty faces of the
polytope $Q={\rm conv} (A).$ Due to the properties of Minkovski sums for Newton polytopes,
the linear form $\Psi$ attains its maximum value
on the Newton polytope of $\Delta_{A \cap \Gamma}$ at either a vertex
or an edge parallel to $F.$
This means that, up to
multiplication by a constant and a monomial in in the variables $a_i$,
the product
the coefficient of the
leading term in $t$ of the Laurent polynomial
$t \mapsto \Delta_{A \cap \Gamma} (\sum_i t^{\psi_i} a_i x^{v_i})$ is equal to
$
(\Delta_I)^{n_{\Gamma,F}}.
$

\begin{definition}\label{def:n}
The multiplicity $n_{\Gamma,F}$ is
the non-negative integer defined by the property that, up to
multiplication by a constant and a monomial in the variables $a_i$,
the coefficient of the
leading term in $t$ of the Laurent polynomial
$t \mapsto \Delta_{A \cap \Gamma} (\sum_i t^{\psi_i} a_i x^{v_i})$ is equal
to $(\Delta_I)^{n_{\Gamma,F}}.$
\end{definition}
The theorem follows as a direct consequence \label{eq:n}
of the
results of Propositions \ref{prop:kgamma} and \ref{prop:ki}.
\end{proof}

\begin{remark}
Note that $n_{\Gamma,F}=0,$
when $\Gamma$ $\Gamma$ is a vertex of $Q.$
Furthermore, the definition shows that, if the circuit $I$ determined by the
edge $F$ is not contained in the face $\Gamma,$ then $n_{\Gamma,F}=0.$
\end{remark}

\begin{remark}
In practice, calculating the multiplicities
$n_{\Gamma,F}$ can be computationally daunting.
We will provide some examples in the next section.
Note that our definition does not use the
Horn uniformization of the $A$--discriminant (proved
in this case by Kapranov \cite{KapHorn})
which is a useful tool but only provides a birational parametrization
of the discriminant.
\end{remark}

The previous theorem adds to the body of evidence
supporting the conjecture of
Aspnwall-Plesser-Wang \cite{APW}. The importance
of multiplicities and other aspects
of this conjecture have been discussed by
Kite--Segal \cite{KS}.

\begin{conjecture}\label{conj:apw}

\

$1)$ For any edge $F$ of the secondary polytope, the category $D^b(Z_F)$
admits a semiorthogonal decompositon consisting of
$n_{\Gamma,F}$ components $D^b (Z_\Gamma)$ for each face $\Gamma$
of the polytope $Q$.

$2)$ For each face $\Gamma,$ there exist spherical functors
$D^b (Z_\Gamma) \to D^b (X)$ for any toric DM stack $X$
determined by a triangulation corresponding to a vertex of the secondary
polytope.
\end{conjecture}
The second part is a direct consequence of the first:
the wall monodromy functors $D^b (Z_F) \to D^b (X)$
are spherical, so result of Kuznetsov (unpublished)
and Halpern-Leristner-Shipman \cite{HLS}
implies that each component
of the semiorthogonal decomposition determines
a spherical functor.

\begin{remark}\label{rm1}
One of the hypotheses in the definition of an $A$-set is that
the set $A$ generates the lattice $N=\ZZ^d.$ In particular
this guarantees that the multiplicity of the ''primary"
component of the discriminant,associated to the face
$\Gamma=Q,$ has $u(\Gamma)= i(\Gamma)=1$ and
$Z_\Gamma$ is a point ${\rm Spec} \, \CC.$
This assumption is likely not essential.
We could simply ask
that $A$ linearly
generates the vector space $N_\RR=\RR^d.$ The adapted results of this note
should continue to hold, although the multiplicities involved gain the
extra factor $N: \ZZ A.$ On the other hand, the technical details
in \cite{GKZbook} are worked out in the book for $A$-sets,
so one would have to work out and adapt the calculations to the more
general situation.
\end{remark}

\section{Applications}
\label{section:exmples}
\example\label{ex1}
Recall that for a general set $A$ in the lattice $N=\ZZ^d,$
the rational polyhedral cone $K$ generated by $A$
defines a Gorenstein affine toric variety
$Y={\rm Spec}\,\CC[K^\dual\cap N^\dual]$.
For any stacky resolution
$\pi:X_{\bf \Sigma}\to Y$ induced by the a regular
trangulation of the polytope $Q,$
the structure of the exceptional set is
quite complicated. It may contain irreducible components
of different dimensions. Consider an edge
of the secondary polytope and
let $I=I_+ \cup I_-$
be the associated circuit in $A$. We assume
that $I$ is maximal dimension, $\dim I = \dim C.$
Let $X_+$ and $X_-$ denote the cooresponding toric
stacky resolutions of $Y$ that differ by a flop.
In this case $Z_F$ is the stacky point $[\rm{Spec} \, \CC/\ZZ_n]$
where $n$ is the index of the sublattice generated
by the circuit $I$ in $N, n=[N: \ZZ I].$
The sets $I_+$ and $I_-$ generate cones that determine
the exceptional loci $E_+ \subset X_+,$ and $E_- \subset X_-,$
and, by Proposition \ref{prop:ez}, we have spherical
functors $D^b({\rm Spec} \, \CC/\ZZ_n]) \to D^b(E_\pm) \hookrightarrow D^b(X_\pm).$
The category $D^b([\rm{Spec} \, \CC/\ZZ_n])$ splits into $n$
orthogonal copies of $D^b (\rm{Spec} \, \CC),$ so we get the spherical
functors predicted by Kontsevich's conjecture described in
the introduction. Theorem \ref{thmm} implies that
$n_{F,Q}=n,$ where $F$ is the edge of the secondary
polytope associated to the flop. This multiplicity
statement is not obvious if one attempts a direct calculation.
In the transversal case $n_{F,Q}=1,$ the
analytic
monodromy calculations from \cite{hor}, \cite{BH2}
for the classical GKZ module are valid, so we get that:

\begin{corollary}\label{cor:tr}
The wall monodromy spherical functor induced
by the edge $F$ is the spherical functor
$D^b {(\rm Spec} \, \CC) \to D^b(E_F) \to D^b(X),$
where $E_F \subset X$  is the exceptional locus
corresponding to the circuit $I.$ The $K$--theory
action of this functor matches the analytical continuation
functor along a correponding loop $\gamma_F$
(see \cite{BH2} )
\end{corollary}

A similar argument expands the range of cases
where Kontsevich's conjecture is true beyond the
the cases considered in  \cite{hor}. We hope to return
to this issue in future work.

\medskip

\example \label{exa2}
Let $X$ be the resolution of the $A_3$ singularity,
with $v_0=(1,0), v_1=(1,1), v_2=(1,2), v_3=(1,3),
v_4=(1,4).$ As it is well known, the secondary polytope
is combinatorially equivalent to a cube in $\RR^3.$
The principal $A$--determinant is
\begin{equation*}
\begin{split}
E_A&= a_0 a_4 \ \Delta_Q= a_0 a_4
 (256 a_0^3 a_4^3-192 a_0^2 a_1 a_3 a_4^2-128 a_0^2 a_2^2 a_4^2\\&+144 a_0^2 a_2 a_3^2 a_4
-27 a_0^2 a_4^4
+144 a_0 a_1^2 a_2 a_4^2-6 a_0 a_1^2 a_3^2 a_4-80 a_0 a_1 a_2^2 a_3 a_4\\
&+18 a_0 a_1 a_2 a_3^3
+16 a_0 a_2^4 a_4-4 a_0 a_2^3 a_3^2-27 a_1^4 a_4^2\\&+18 a_1^3 a_2 a_3 a_4
-4 a_1^3 a_3^3-4 a_1^2 a_2^3 a_4+ a_1^2 a_2^2 a_3^2).
\end{split}
\end{equation*}

\medskip

It is clear that in this case $Z_Q$ is the point
${\rm Spec} \, \CC.$ Let $X$ denote the DM stack $[\CC^2/\ZZ_4]$ whose
stacky fan has one cone and the rays generated by the
vectors $v_0, v_4.$ Consider the edge $F_1$ corresponding to the
birational transformation $X \leftrightarrow X_1,$ where $X_1$ is the
toric DM stack with cones determined by the pairs $v_0,v_1$ and $v_1,v_4.$
The associated polyhedral subdivision is $({\rm conv} \{0,4\}, \{ 0,1,4\}),$
and the circuit relation $I$ is $3v_0 - 4 v_1 + v_4=0.$ The discriminant
$\Delta_I$ is $256a_0^3 a_4 - 27a_1^4,$ and the leading term with respect
to the edge $F_1$ in the quartic discriminant $\Delta_Q$ is
$$
256 a_0^3 a_4^3 - 27 a_1^4 a_4^2= a_4^2 \cdot \Delta_I.$$
This means that $n_{Q,F_1}=1,$ which is consistent with the fact
that $Z_{F_1}= {\rm Spec} \, \CC.$ The associated spherical functor is
$$D^b({\rm Spec} \, \CC) \to D^b ([\CC^2/\ZZ_4]).$$

\medskip

Let $F_2$ denote the edge corresponding to the birational transformation
$X \leftrightarrow X_2,$ where $X_2$ is the
toric DM stack with cones determined by the pairs $v_0,v_2$ and $v_2,v_4.$
The associated polyhedral subdivision is $({\rm conv} \{0,4\}, \{ 0,2,4\}),$
and the circuit relation $I$ is $v_0 - 2 v_2 + v_4=0.$ The discriminant
$\Delta_I$ is $a_0 a_4 - 4a_2^2,$ and the leading term with respect
to the edge $F_2$ in the quartic discriminant $\Delta_Q$ is
$$
256a_0^3 a_4^3
-128 a_0^2 a_2^2 a_4^2 + 16 a_0 a_2^4 a_4
= 16 a_0 a_4 \cdot \Delta_I^2.$$
This means that $n_{Q,F_2}=2$ which is consistent with the fact
that $Z_{F_2}$ is the stacky point $[{\rm Spec} \, \CC/\ZZ_2].$ The derived category
$D^b([{\rm Spec} \, \CC/\ZZ_2])$ splits into two copies of
$D^b({\rm Spec} \, \CC),$ the associated
wall monodromy spherical functor is
$$D^b([{\rm Spec} \, \CC/\ZZ_2])
\to D^b ([\CC^2/\ZZ_4]).$$

\medskip

\example \label{ex3}
Let $X_1$ be the well known quasi--projective Calabi-Yau
toric variety defined as the total space of the canonical bundle of $\PP^2.$
The toric structure is given by the vectors $v_0=(0,0,1), v_1=(1,0,1),
v_2=(0,1,1)$ and $v_3=(-1,-1,1).$ The lattice of linear relations is one dimensional and
generated by the vector $(-3,1,1,1),$ and the secondary polytope
is the segment $F=[(3,2,2,2),(0,3,3,3)].$
There are two toric birational models in this case, namely
$X_1$ and the stacky quotient $X_2 = [\CC^3/\ZZ^3].$
The principal $A$--determinant is $E_A= a_1^2 a_2^2 a_3^2 (a_0^3 -27 a_1 a_2 a_3).$
The multiplicity of the discriminant components corresponding
to $v_1,v_2,v_3$ are all equal to $2,$ and $n_{F,Q}=1.$ The spherical
twist of $D^b(X_1)$ is determined by the spherical object $\Oa_{\PP2} \subset X_1.$

\example \label{ex4}
Let's consider the case the Calabi-Yau
toric variety $X_1$ is the total space of the canonical bundle of the Hirzebruch surface $F_2.$
The toric structure is given by the vectors $v_0=(0,0,1), v_1=(1,0,1),
v_2=(0,1,1), v_3=(-1,2,1)$ and $v_3=(0,-1,1)$ with the obvious cones.
The lattice of relations is two dimensional and
generated by the vectors $(-2,0,1,0,1)$ and $(0,1,-2,1,0).$ There are four stacky
toric birational models for $X_1.$

\medskip

\begin{figure}
\begin{center}
\includegraphics[height=6cm,width=8cm]{secf2.jpg}
\caption{The $F_2$ example.}
\label{fig:f2}
\end{center}
\end{figure}

The discriminant picture is obtained by studying the singularities
of the Laurent polynomial $\sum a_i x^{v_i}.$ Besides the vertices
corresponding to $v_1, v_3$ and $v_4$ there are two other
faces that contain circuits. The discriminant component
determined by the face $\Gamma$ generated by the
vectors $v_1, v_2$ and $v_3$ is
$$
f_{\Gamma}= a_2^2 - 4 a_1 a_3.$$
For the maximal dimensional face $Q$ we obtain the discriminant
$$
f_{Q}= a_0^4 -8a_0^2 a_2 a_4 +16  a_2^2 a_4^2 -64 a_1 a_3 a_4^2.$$
The principal $A$-determinant is
$$E_A=a_1^2 a_2^2 a_4^2 \cdot f_\Gamma \cdot f_Q.$$
If we choose the coordinates adapted to the large complex
structure point
$$
x:= \frac{a_2 a_4}{a_0^2}, \quad y= \frac{a_1 a_3}{a_2^2}
$$
the we obtain that the two relevant discriminant components are given by
$$y=\frac{1}{4} (1-\frac{1}{4x})^2 \quad \text{and} \quad
y= \frac{1}{4}.$$
Figure 1 is suggestive and intuitively very useful and
appeared for the first time in the paper by Dave Morrison \cite{DRM} on the $3$-fold octic
in $\PP^4(1,1,1,2,2)$ which is a higher dimensional generalization
of our example.

\medskip

The analysis of the neighborhood of the point $A$ is completely analogous to the
work \cite{AHK}, so we will not discuss here.
By analzying the moduli space picture,
we see that a rational curve $x={\rm constant}$ infinitesimally close to the
$y$-axis corresponds to a transition from $X_1$,
the total space of the canonical bundle of $F_2$
to the stacky version $X_4$ of the weighted projective space
$\PP(2,1,1).$ This curve intersects $E_A=0$ in the neighborhood
of the point $B.$
The circuit $I$ associated to the
corresponding edge of the secondary quadrilateral is
$v_1 -2v_2 + v_3=0.$ One of the associated EZ-spherical functor
is induced by the diagram
\begin{equation*}
\begin{split}
\xymatrix{
E={\mathbb A}^1 \times {\mathbb P}^1 \; \ar@{^{(}->}[r] \ar[d]_q & X_1\\
Z={\mathbb A}^1&
}
\end{split}
\end{equation*}

\begin{figure}
\begin{center}
\includegraphics[height=6cm,width=8cm]{2par.png}
\caption{The ''mirror" complex discriminant moduli space.}
\label{fig:2par}
\end{center}
\end{figure}

The birational transformation
corresponding a rational curve $x={\rm constant}$
infinitesimally close to the
$y=\infty$ corresponds to a transition from $X_2$, a
different birational model of $X_1$ to $X_4$ which
is the stacky quotient $[\CC^3 / \ZZ_4].$
This curve intersects $E_A=0$ in the neighborhood
of the point $C.$
One of the
associated spherical functors is
is induced by the diagram
\begin{equation*}
\begin{split}
\xymatrix{
E=[{\mathbb A}^1/\ZZ_2] \times {\mathbb P}^1 \; \ar@{^{(}->}[r] \ar[d]_q & X_4=[\CC^3 / \ZZ_4]\\
Z=[{\mathbb A}^1 / \ZZ_2]&
}
\end{split}
\end{equation*}
It is interesting to note that
the circuit $I$ associated to this transition and determined by
corresponding edge of the secondary polytope is also
$v_1 -2v_2 + v_3=0.$

\medskip

Recall that in determining multiplicities, the circuit discriminant
$\Delta_I= 4a_1 a_3 - a_2^2$ played a crucial role. For the transition
$X_1 \leftrightarrow X_4,$ we can consider the linear form $\psi$
in the cone $\Ca(\Fa_{14})$ whose
supporting face is the edge $F_{14}$ given by
$$
\psi(v_0)= u, \psi(v_1)=  \psi(v_2)=  \psi(v_3)= 1, \psi(v_4)=0,
$$
with $u < 1/2$. A similar linear form $\psi$ will work for the transition
$X_1 \leftrightarrow X_4,$ with $u> 1/2$ in that case. The monomials in $f_Q$
$a_0^4, a_0^2 a_2 a_4, a_2^2 a_4^2, a_1 a_3 a_4^2$ have the weights
$4u, 2u+1, 2, 2,$ respectively. The leading term for the edge $F_{14}$
is $a_0^4.$ The highest weight terms for the edge $F_{23}$ are
$a_2^2 a_4^2$ and $a_1 a_3 a_4^2,$ so in this case the leading term
for $f_Q$ is
$$
16a_2^2 a_4^2 -64 a_1 a_3 a_4^2= - 16 a_4^2 \cdot \Delta_I.$$
We conclude that $n_{Q,F_{1,4}} =0$ and $n_{Q,F_{2,3}}=1,$ as
Figure \ref{fig:2par} indicates. In this simple example, we do have
the true parametrizations of the components, so the weight calculation is not needed.
However, only the birational Horn parametrizations are available in the general case.

\medskip

\section{Final comments and some speculations}

The proposed constructions of spherical functors appearing
in this paper seem to be related of the notion
of co-sheaf over the topological
space of the cone $K = \sum \RR v_i,$ in the
sense of Bressler-Lunts \cite{BL},
section 6.7.1.

Since the defined spherical
functors can be thought as
categorical maps corresponding
to the Bressler-Lunts strata, it is
a very intriguing question whether
the whole structure can be expressed
in the language of schobers introduced
by Kapranov and Schetchman \cite{KaSc}.
Such an abstract realization would be consistent
with the fact that our $EZ$--spherical functors
map "stalks'' between various strata of
a Whitney stratification of the characteristic
cycle of the GKZ hypergeometric system.
Such a picture would implicitly subsume our multiplicity result.
The stalks
in this case would be bounded derived categories
of coherent sheaves of the form $D(Z_\Gamma),$
indexed by the faces $\Gamma.$ In particular,
if $\Gamma= \emptyset,$ the "stalk'' is
the bounded derived category of coherent sheaves on the
ambient toric DM stack. Donovan \cite{Do} studied
certain wall crossing phenomena in terms of spherical pairs and
schobers. In a combinatorial context similar to ours,
\v{S}penko and Van den Bergh \cite{SvB}
produced a categorification of the GKZ system in the quasi-symmetric toric case.
More work is needed in order to make our speculation
rigorous.

\bigskip

redpajama_set_name: RedPajamaArXiv


text:
moving forward – GRB's MOVEMEDIA in Seattle
By Victoria Farr Brown|2017-02-07T02:20:48-08:00October 14th, 2015|
Last week Grand Rapids Ballet—whose artistic director is former Pacific Northwest Ballet star ballerina Patricia Barker—had its début in Seattle with a double bill.
After dancing the Seattle premiere of Whim W'Him artistic director/choreographer Olivier Wevers's witty evening-long Midsummer Night's Dream for 3 nights, GRB switched gears on the weekend to MOVEMEDIA, a lively mixed repertory program of contemporary pieces by 4 choreographers: Annabelle Lopez Ochoa, a favorite with Whim W'Him audiences, for whom she has made several pieces and will create a new one in the 2015-16 season; Penny Saunders, whose exploration for Whim W'Him of the works of American painter Edward Hopper garnered much praise; Mário Radačovský, with whose National Theatre Brno (Národní divadlo Brno) Olivier is now working in the Czech Republic; and David Parsons.
I spoke with Michael Auer, GRB creative director (and Patricia's husband) about the genesis of MOVEMEDIA. When the couple first went to Grand Rapids, the major emphasis there was classical. Some contemporary works were being done, but separately, in a "workshop, black box way." On the one hand, Michael says, if you really wanted to make contemporary dance meaningful at GRB, "you needed to put on more full-fledged productions and charge more than $5," while on the other hand, you would also need to avoid "confusion on the customer's part." Contemporary dance was not necessarily what the company's long-established audiences came to see. The solution:
MOVEMEDIA
DANCE/MEDIA/INNOVATION
A contemporary dance series featuring dance works merging visual elements of sets, props, costumes and lighting, framed by technology and digital media to enhance the body in space by providing internationally recognized choreographers a creative canvas.
As Michael puts it, GRB created "an identifiable product by slapping a name on it.
A funding opportunity arose, and Patricia took it." Resources were mainly contributed by the Rosemary and David Good Family Foundation, for a couple of years at first, but now the two part program is going into its 5th season. Whim W'Him's Olivier Wevers was the first recipient of the two-part opportunity: first, to set an already created piece on GRB; and then, to create a new work for the company. The Sofa, choreographed for that initial MOVEMEDIA season, has now been performed not just by the Grand Rapids company, but by Whim W'Him in Seattle and at the Joyce Theater in New York City, and will open later this month in Brno, Czech Republic, where Olivier is currently rehearsing.
Over these 4 years, MOVEMEDIA has produced 36 ballets—by local, emerging and established choreographic talent, the full spectrum. By now, however, MOVEMEDIA is fully integrated into the repertory. And rather different choices are being made: more works by well-known choreographers, less total pieces, more time and resources to be devoted to each, more light designer time, etc.
Annabelle Lopez Ochoa's haunting 'Written & Forgotten'
There is always limited money. The program can't put the choreographers up in fancy hotels. And it's interesting, says Michael, to give local choreographers a platform.
So especially in the the beginning, rather as with Whim W'Him's Choreographic Shindig, MOVEMEDIA afforded, among other things, an opportunity for young people (such as dancer Yuka Oba) to experience and experiment in other aspects of dance than performing. "There was a less pressured environment for choreographers." With some of the newer choreographers, Michael says he has taken an active role—so they can bounce things off of him, help shape their concept. "What do you want?" he would ask. "I've got strong ideas on how you might—I don't say should!— proceed." He might help the shaping of ideas in the choreographer's own mind. They have had conversations on choice of music, development of vocabulary, lighting.
MOVEMEDIA dancers in PNB rehearsal hall
Says Michael, "I say, 'Think about what the audience will walk away with.' Personally, I like to walk away from a piece and it's done something to me. Story, visual interest, something. I'm okay with pretty dancing, but do it on purpose. Show something—anger, frustration, relationships. They need to show up at the table." It's what he calls the "Alice in Wonderland Syndrome," paraphrasing the Cheshire Cat: "If you don't know where you're going, then it doesn't matter which way you go." He adds that there's a way in which Balanchine did a disservice to young choreographers, when he said, "I just go into the room and create." For of course Mr. B. entered the studio already knowing the music he would use in great detail, along with possessing a great fund of ideas, experience and technique to inform what he did there.
'Slight,' an evocative meditation on light by Penny Saunders
I asked "But if the MOVEMEDIA emphasis—as in the program that came to Seattle—is now on more professional and polished work, what about the notion of bringing young choreographers along?" Michael says that GRB is still attending to them, sending out feelers, working on collaborations with companies to pool resources. "Choreographers need space to work, we have space, and so on." The point, he notes, isn't necessarily to produce a single great choreographer or great work, but to continue finding ways to provide a forum for exploration, for career purposes and for broadening out the experience and depth of the work, as well as to expose audiences to a whole new array of talent and approaches of dance.
'The Envelope' by David Parsons
"We are positioned at Grand Rapids so we can do really good contemporary dance," he continues. "Our dancers learn fast and are flexible. GRB is larger than Aspen-Santa Fe, say, or Whim W'Him. It's a middle-sized company. Annabelle's Dangerous Liaisons
[whose US premiere will take place at GRB in February 2016] will have 20-22 dancers—a big stage, more people. It's nice to see larger possibilities in the numbers of dancers."
'Beethoven Excerpt' of Mário Radačovský
As the only professional ballet company in Michigan, GRB needs to continue covering the territory, as it still does every year in one or two productions of "old classics, done the traditional way." But at least partly under the broadening influences of MOVEMEDIA, even the classics have been getting a new look in Grand Rapids. Michigan reviewer Jeffrey Kaczmarczyk wrote of Mário Radačovský's Black Swan White Swan, "Great art is great, in part, because, whether it's Shakespeare's plays or Beethoven's symphonies, it can be created new again." In new works, such works as Black Swan, Annabelle's Liaisons, and Olivier's Midsummer, stories are told in a series of almost cinematic scenes. "People are comfortable with story," notes Michael, "and these stories are told in 4-6 minute vignettes. It works well for the audience and lets choreographers show different qualities, temperaments, moods." At Grand Rapids Ballet, as old stories are reinvented and contemporary sensibilities investigated, a very old art form is being reinvigorated.
Victoria Farr Brown is a former historian, a writer, now mostly of fiction, and a long-time lover of dance. She has written about Whim W'Him from its first steps in 2009.

redpajama_set_name: RedPajamaCommonCrawl


The only Neon Orange is one of our Special Collection. Because of its neon color, it is more noticeable and looks very cool. The sole of Neon Orange is made of jute rope and recycled textile fibers from Barcelona.


text:
Q: cxf-codegen-plugin in Gradle classpath in jax-ws-catalog.xml Firstly idea was to generate java classes without internet connection, so I decided use jax-ws-catalog.xml file to rewrite uri on resource/local directory.
I defined location of xsd files by using jax-ws-catalog.xml with classpath.
But every compilation is wrong and catching error:
WARN org.apache.cxf.resource.URIResolver - Resource classpath:/local/xml.xsd was not found in the classloaders.
WSDLToJava Error: org.apache.ws.commons.schema.XmlSchemaException: Unable to locate imported document at 'http://www.w3.org/2001/xml.xsd', relative to ...{my file}...

build.gradle
plugins {
    id 'java'
    id 'java-library'
    id "io.mateo.cxf-codegen" version "1.0.2"
}
import io.mateo.cxf.codegen.wsdl2java.Wsdl2Java

ext {
    artifactId = 'soap-adapter'
}

jar {
    enabled = true
}

bootJar {
    enabled = false
}

dependencies {
    implementation "javax.xml.ws:jaxws-api:${jaxwsApiVersion}"
    implementation "javax.jws:javax.jws-api:${javaxJwsApiVersion}"

    implementation "com.sun.xml.messaging.saaj:saaj-impl:${sunSaajImplVersion}"

    implementation "org.apache.cxf:cxf-rt-frontend-jaxws:${apacheCxfVersion}"
    implementation "org.apache.cxf:cxf-rt-transports-http:${apacheCxfVersion}"
    implementation "org.apache.cxf:cxf-rt-ws-security:${apacheCxfVersion}"

    cxfCodegen "jakarta.xml.ws:jakarta.xml.ws-api"
    cxfCodegen "jakarta.annotation:jakarta.annotation-api"
    cxfCodegen "ch.qos.logback:logback-classic"

    testImplementation 'org.springframework.boot:spring-boot-starter-test'
}

// Common properties of java generation from wsdl.
tasks.withType(Wsdl2Java).configureEach {
    toolOptions {
        markGenerated.set(true)
        outputDir.set(file("${buildDir}/generated"))
        addToMainSourceSet.set(true)

        wsdl.set(file("${projectDir}/src/main/resources/wsdl/myWsdl.wsdl"))
        catalog.set("${projectDir}/src/main/resources/META-INF/jax-ws-catalog.xml")
    }
}

compileJava.dependsOn wsdl2java

jax-ws-catalog.xml
<?xml version="1.0"?>
<catalog xmlns="urn:oasis:names:tc:entity:xmlns:xml:catalog">
    <rewriteURI uriStartString="http://www.w3.org/2001/xml.xsd" rewritePrefix="classpath:/local/xml.xsd"/>
</catalog>

I think that classpath is determined wrong.

A: I discovered the next:

*

*At the programm compilation jax-ws-catalog.xml using only relative paths from itself.

*At the runtime you can use classpath.

*And jax-ws library using resurce/META-INF/jax-ws-catalog.xml by default.

So I just duplicated this jax-ws-catalog.xml into next directories with different properties:
resource/jax-ws-catalog.xml
<?xml version="1.0"?>
<catalog xmlns="urn:oasis:names:tc:entity:xmlns:xml:catalog" prefer="system">
  <rewriteURI uriStartString="http://" rewritePrefix="local/"/>
  <rewriteURI uriStartString="https://" rewritePrefix="local/"/>
</catalog>

resource/META-INF/jax-ws-catalog.xml
<?xml version="1.0"?>
<catalog xmlns="urn:oasis:names:tc:entity:xmlns:xml:catalog" prefer="system">
  <rewriteSystem systemIdStartString="http://www.w3.org/2006/03/addressing" rewritePrefix="classpath:local/www.w3.org/2006/03/addressing"/>
  <rewriteSystem systemIdStartString="http://www.w3.org/2005/08/addressing" rewritePrefix="classpath:local/www.w3.org/2005/08/addressing"/>
</catalog>

In build.gradle I changed the next
tasks.withType(Wsdl2Java).configureEach {
    toolOptions {
        markGenerated.set(true)
        outputDir.set(file("${buildDir}/generated"))
        addToMainSourceSet.set(true)

        wsdl.set(file("${projectDir}/src/main/resources/wsdl/myWsdl.wsdl"))
        catalog.set("${projectDir}/src/main/resources/jax-ws-catalog.xml")
    }
}

redpajama_set_name: RedPajamaStackExchange


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text:
Drivers line up to be first across new Champlain Bridge
The new bridge opened officially to traffic just before 5 a.m.
Jason Magder, Montreal Gazette
Suzanne Bienvenu watched the new Samuel De Champlain Bridge take form during the last four years from the window of her home; on Monday, she was one of the first to drive on it.
"It was amazing," Bienvenu said. "I was able to see it being built and kept saying to myself, 'one day, I'm going to be on it.' I made history (today). It was great just to be there."
Bienvenu convinced her neighbour to drive her over the bridge so she could take pictures as she crossed. She woke up at 4 a.m. and waited in a line of cars for about a half-hour so she could be one of the first across.
Cars cross over the new Samuel De Champlain Bridge from the South Shore toward Montreal hours after it was opened to traffic in on Monday, June 24, 2019. The super-structure of the old Champlain Bridge is at left. John Mahoney / Montreal Gazette
Including engineers working on design, more than 2,000 people logged nearly 8.5 million hours to build the bridge. At peak on the work site, there were 1,600 workers as the project was done in a fast-track process, which involved designing the bridge as it was being built.
One of the first people to conceive of the new bridge was Guy Mailhot, chief engineer for Infrastructure Canada, which oversaw the project. He was involved in its first conceptions back in 2012, three years before the project broke ground.
"When you go back seven years, it seems like so long away to be able to drive across it and get to the end result," Mailhot said. "It's nice to be able to see the end result."
Cars cross over the new Samuel De Champlain Bridge from the South Shore toward Montreal hours after it was opened to traffic on Monday, June 24, 2019. John Mahoney / Montreal Gazette
The new bridge — one of the widest in the world — is built to last 125 years. It has three lanes in each direction for car traffic and a shoulder on the right side, which is now being used as a reserved lane for buses until 2021, when the REM — an electric light-rail train — is slated to start service over the bridge's middle span. When the train is running, that lane will be reserved for cars with mechanical problems.
The bridge is being opened in stages. Monday was for Montreal-bound traffic. It will open to South Shore-bound drivers on Monday, July 1. Until then, anyone leaving from Montreal will take the old Champlain Bridge to get across, and the new one to return. A cyclist and pedestrian pathway will open in September or October.
The old bridge — built in 1962 — will be closed on July 1 and demolition on part of it will begin soon after because crews need to free up space to finish work on the new bridge. The complete demolition of the bridge won't begin for several months, and the process will take at least four years.
François-Philippe Champagne, the minister for infrastructure said the federal government wanted the bridge — the busiest in Canada with 50 million annual crossings — to stand out. He believes builders succeeded in making the cable-stayed bridge an iconic monument for the city.
"We talk about the Golden Gate Bridge and the Brooklyn Bridge, and I think this will be a historic landmark for Montreal," Champagne said.
Drivers will notice there are still plenty of construction workers on site and scaffolding on the main tower. Most of the work left on the bridge is painting and touch-ups, while the road network around the bridge is being completed by the end of the year. The jetties — consisting of one million tonnes of stones — to serve as work sites for the project will have to be removed and the river returned to its former state by that time.
The $4.239 billion of taxpayer money covers the construction of the bridge and its maintenance for 30 years, but the final bill will have to be worked out by lawyers for both the federal government and Signature on the St. Lawrence, the consortium of companies including SNC-Lavalin that built the new structure, as the original deadline for the project was December 2018, and penalty fees may be applied to the consortium.
Builders promise a smooth ride on new Champlain Bridge
Montrealers invited to weigh in on demolition of old Champlain Bridge
jmagder@postmedia.com
twitter.com/jasonmagder
facebook.com/jasonmagderjournalist
 Watch: Ride across the new Champlain Bridge on opening day While you were sleeping: Holiday road closings, four-alarm fire


In 2003, the NMWA purchased "St. Thomas" by Georges de La Tour (1593-1652). To introduce this new acquisition to the general public, the NMWA presents "Georges de La Tour: World of Light and Shadow", featuring approximately 20 of the artist's works.
Known for his mastery in depicting the effects of light, La Tour's paintings are rendered with an almost modern feel. His skillful portrayal of candlelight in particular, gave birth to his considerable fame. Forgotten almost immediately after his death, it was not until the 20th century that his works were rediscovered by the art world. Today, only about 40 authentic works by La Tour remain.
This exhibition will have a third of the artist's extant paintings on display, alongside copies and related works.


The National Democratic Congress (NDC) has ordered President Nana Addo Dankwa Akufo-Addo to publish the full report of the Emile Short Commission of Enquiry.
According to the party, the failure of government to publish the report is an affront on transparency and accountability.
"We recall that the Short Commission presented its report to the President on 15th March, 2019, and yet till date – almost a month, the President for some unknown reason, is sitting on the said report. This is a blatant disregard for the principles of transparency and accountability that the President swore to uphold. In spite of the fact that the NDC boycotted the proceedings of the Commission, we believe that the Ayawaso West Wuogon issue is a matter of intense national and international interest, and that the people of Ghana deserve to know the findings and recommendation of the Commission," the party said in a statement.
The statement added: "What makes this demand imperative is the fact that till date, the perpetrators of the violence at the bye-election who were captured live on videos, are still free and walking the streets of Ghana. What is even more disturbing is that we still do not know the status of the police investigation into the said matter, despite assurances to the country".


text:
HomeCountyUnemployment rate continues to drop in Rio Blanco County
Unemployment rate continues to drop in Rio Blanco County
August 6, 2014 Sean McMahon County 0
RBC I For the fourth straight month, the June unemployment rate is down slightly in Rio Blanco County, using seasonally non-adjusted figures. Unemployment numbers are down more than 1.8 percent from June 2013.
Unemployment as of June 2013 was 6.0 percent, using the same seasonally non-adjusted figures. Since early in 2014, the jobless numbers in Rio Blanco have continued to fall, however slowly, to 4.2 percent in June 2014.
That drop to 4.2 percent is a drop of 0.1 percent from 4.3 percent in May.
For June, there were 4,308 in the available county workforce, and, of those, 4,129 were employed while 179 were without jobs.
In May, there were 4,216 workers in the available county work pool. Of those, 4,033 were employed, and 183 were unemployed.
In June 2013, there were 4,347 workers available in the county workforce. Of those, 4,087 were gainfully employed while 260 potential workers were jobless.
Statewide, the Colorado unemployment figures continue to drop.
In Colorado, the unemployment rate in June 2013 was 7.3 percent, and that had fallen to 5.5 percent in May 2014 and stood at 5.4 percent for June 2014.
The Colorado county with the highest unemployment rate in June was Castillo County (county seat San Luis), the only county in the state that had an unemployment rate over 10 percent during June and May. In June, the rate had dropped to 10.1 percent, in May it stood at 11.6 percent. One year ago, while there were several counties with unemployment in excess of 10 percent, Castillo was still the highest with 12.5 percent.
While no other counties reported unemployment in excess of 10 percent for the past two months, the June 2013 statistics showed six counties above 10 percent: Castillo at 12.5 percent; Conejos (county seat Conejos) at 11.5 percent; Saguache (Saguache) and Huerfano (Walsenburg) at 11.4 percent; Las Animas (Trinidad) at 10.5 percent and Pueblo (Pueblo) at 10.1 percent. By June this year, things were looking up for Pueblo County, where the rate had dropped to 7.8 percent.
The county with the lowest unemployment rate in the state during June was Hinsdale County (Lake City) with a 2.3 percent jobless rate. The other five counties with unemployment rates below 3 percent were: Baca (Springfield) and Cheyenne (Cheyenne Wells) at 2.5 percent; Yuma (Wray) and Mineral (Creede) counties with a 2.8 percent rate; and Jackson County (Walden) with a 2.9 percent rate.
Cowboys preparing for upcoming football season
Women-only Cast, Blast & Twang set in Steamboat
RBC hires emergency and natural resources manager
March 24, 2021 Lucas Turner 0
RBC | Rio Blanco County has a new employee in a brand new position heading into spring. Edward (Eddie) Smercina has been on the job for about one month with the newly created title of […]
Men of the Outdoors: Trey Morris
November 6, 2011 Special to the Herald Times 0
MHS senior Trey Morris bagged this 28 inch mule deer in unit 21 near Rangely with the help of his dad, Todd. They harvested the massive buck after school let out Wednesday, Oct. 26.
Canadian wins Classic
September 14, 2019 Herald Times Staff 0
This years 2019 Meeker Sheep dog trials bared the storm on the championship run day. Despite the storm the final top twelve competitors took center field at the award ceremony Sunday evening. The bleachers were […]


text:
Homemade Bomb Found, Two Arrested
Written by Jessie Williams
Published: Tuesday, October 11,2016
Two Westmoreland men were arrested on Tuesday, October 4, after a City of Lafayette police officer discovered a homemade bomb in their vehicle during a routine traffic stop.
According to patrolman Derrick Gann's report, 27-year-old Zack Ward and 36-year-old Robert Carter were pulled over near Sunshine Market on Highway 52 for expired tags.
"I immediately observed the passenger having his arm setting on a shotgun," the report states. "At this time, for my safety, I had both subjects exit the vehicle, and requested backup."
Suspecting the duo may have more firearms, officer Gann asked for consent to search the vehicle. After Mr. Ward gave his consent, Officer Gann began administering a field sobriety test due to suspicion of possible impairment, while Deputy Tim Freeman began a search of the vehicle.
Officer Gann paused testing at one point to assist Deputy Freeman with the search.
The report reads, "While searching, Deputy Freeman and I came upon a suspicious item that was wrapped in duct tape. As we began to inspect the item, we found that the duct tape was wrapped up over a 45 caliber round, detonator cord and several nails. At this time, Sget. Looper walked over to assist and advised us that we had found an explosive device, and to get away from the vehicle."
Ward and Carter were then detained and Sergeant Matt Looper advised dispatch that an explosive device had been located and to contact the Tennessee Highway Patrol Explosive Ordinance Disposal Unit.
When asked about their intentions for the device, Ward and Carter stated that it was for killing coyotes and that the detonator cord had been found on his property.
Both Ward and Carter were transported to the Macon County Jail, and the EOD unit was escorted to Winding Stairs Park where the device was safely detonated.
Zack Ward was charged with weapon prohibited, possession of a firearm during a violent felony, possession of explosive components, driving under the influence and violation of registration law. His bond is set at $7,150.
Robert Carter was charged with weapon prohibited, possession of firearm during a violent felony, possession of explosive components and public intoxication. His bond was set at $5,750.
Both men are scheduled to appear in General Sessions Court on November 9, 2016.
The investigation is ongoing and additional charges could be brought forth.


This Winter Greens Salad with Citrus Vinaigrette recipe is sponsored by Juice Products Association and the Healthy Aperture Blogger Network.
YES YES YES!!! I'm all about this Winter Greens Salad with Citrus Vinaigrette and it's exactly what you need on your holiday table.
I love getting in a bowl full of green goodness all the time, but especially around the holidays when we are bombarded with sweets. This salad bowl is so nourishing and hearty thanks to the winter greens and hearty grains. They fill you up and make you feel so freaking awesome eating it.
That can't-stop-won't-stop-all-day-errr-day energy that we most definitely NEED for all the holiday shopping and getting the house ready for friends and family to visit.
This Winter Greens Salad with Citrus Vinaigrette has all the good stuff in it.
WINNING! Add grilled chicken or garbanzo beans to make this salad a meal, or serve it alongside your holiday spread.
This Tangy Citrus Vinaigrette is made with 100% orange juice! 100% juice is filled with the same important vitamins and nutrients as the whole fruit it is squeezed from and made with no added sugar.
And since 80% of Americans (yes, that's pretty much everyone) do not eat the recommended amount of fruit and vegetables in their diet, juice can be an easy way to increase your intake! If you have kiddos, the Academy of Pediatrics suggests serving sizes of 4 ounces for young kids 1-6 and 8 ounces for older children ages 7+.
This Winter Greens Salad with Citrus Vinaigrette tho. It's exactly what should be on your holiday table— warm acorn squash, apple slaw, winter greens, hearty grains and a tangy citrus vinaigrette! Add grilled chicken or garbanzo beans to make this salad a meal, or serve as a salad in your holiday spread.
Slice acorn squash in half lengthwise, scoop out seeds and thinly slice the squash. Arrange squash on baking sheet and toss with coconut oil and cinnamon. Bake for 15-20 minutes until golden and skins are soft.
In small bowl or mason jar prepare Citrus Vinaigrette. Whisk together orange juice, apple cider vinegar, chili garlic sauce, olive oil, and maple syrup. Season with salt to taste.
In medium bowl, toss together apple, onion, cabbage, grapes, and 2-3 tablespoons of dressing.
Prepare salad by tossing mixed greens with salad dressing. Top with hearty grains, apple slaw and acorn squash.
What a lovely salad! Squash, apples, cabbage… I am in love. Seriously. Making it next week!
Mmmmm what a lovely salad for this time of year! I should be eating more salads, but I'm all about the carbs right now. Need to get into gear and start eating more healthily again!
I love everything about this. And I could TOTALLY use some extra veggies right now. Pinning to try!


text:
Last edited by Shaktisida
2 edition of West of Plymouth found in the catalog.
West of Plymouth
Virginia Cole Trenholm
by Virginia Cole Trenholm
Published 1978 by s.n.] in [s.l .
Cole family.,
Selecman family.,
Bedford family.
Statement by Virginia Cole Trenholm.
LC Classifications CS71.C69 1978
Visit your local bookshop in Plymouth - Drake Circus today for a great selection of books and gifts. Our friendly booksellers are here to help you find your next favourite book. Lookup People, Phone Numbers, Addresses & More in Plymouth, MI. Whitepages is the largest and most trusted online phone book and directory.
Plymouth Meeting - Plymouth Meeting, PA. Plymouth Meeting The Metroplex Chemical Road Plymouth Meeting, PA Store Hours: Call for Hours. For the safety of our communities and booksellers, this location is temporarily closed. » Add this store to my Address Book» Add to My B&N Stores Connect with us. Book your tickets online for the top things to do in Plymouth, Devon on Tripadvisor: traveler reviews and photos of Plymouth tourist attractions. Find what to do today, this weekend, or in June. We have reviews of the best places to see in Plymouth. Visit top-rated & must-see attractions.
Guest Book for Edward West of Plymouth, MA. Share your condolences, send flowers, find funeral home information. Guestbook provided by Old Colony Memorial. Researchers can search grantor books or grantee books at no charge. Plymouth County deeds can now be viewed online. Plymouth Probate and Family Court 52 Obery Street Plymouth, MA Plymouth County probate files and records can now be viewed online at Fax: Hours: Mon. - Fri. a.m. to p.m.
Loving Geordie
Tuberculosis and how to combat it
Bulletin No. 6
Fundamental musicianship for the elementary classroom teacher.
Structure plan for south east Dorset
Submission of the National Association of Women and the Law to the Labour Canada Committee on benefits for part-time workers
room with a view overseas
Logistics in service industries
chemical sensitivity of Lakes to acidic deposition and the risk to fish populations.
Money-Order Division of the Sixth Auditors Office. Letter from the Acting Secretary of the Treasury, requesting an appropriation for the Sixth Auditors Office for the months of May and June, 1891, and for the fiscal year ending June 30, 1892.
A letter from George Nicholas, of Kentucky, to his friend, in Virginia.
Post-secondary art education.
Ethics4Everyone
Marco Polo (Groundbreakers Explorers)
Oriental immigrant and Canadas Protestant clergy, 1858-1925
Medical Microbiology Illustrated
Frames of mind
Johnson: prose & poetry
West of Plymouth by Virginia Cole Trenholm Download PDF EPUB FB2
The Landmark History of the American People, Volume 1: From Plymouth to the West [Daniel J. Boorstin, Ruth F. Boorstin] on *FREE* shipping on qualifying offers. The Landmark History of the American People, Volume 1: From Plymouth to the West/5(4). Plymouth, Ma: A Scavenger Hunt (The Look Book: Family Fun for Everyone!) by Barbara Tibbetts | Nov 5, out of 5 stars 1.
Paperback $ $ 9. Get it as soon as Thu, May FREE Shipping on orders over $25 shipped by Amazon. Only 1 left in stock (more on the way). More. Return to USGenWeb Plymouth. Plymouth Books Online This page is still under West of Plymouth book - many other books are listed on the Lookups page.
These listings are for books that are in the public domain, and of which page images or transcrptions are available West of Plymouth book one or more sites. The second volume in our series explores post-WWII Plymouth through Like its predecessor, this volume is an oversized page paperback book with illustrations.
A compelling look at our recent past. Publication of the Plymouth Public Library Corporation: print editions are also available to borrow and to purchase. Plymouth, MA - Plymouth, Massachusetts Map & Directions. Adventure Babies Plymouth, Plymouth. likes. Story and sensory adventures bring brilliant books to life to support vital early communication for babies from birth up to our pre-school classes.5/5.
Meet @Work, coworking space in Dar es salaam. See reviews, photos, amenities, prices, availability, and book online with on: Zanaki Street, Dar es salaam, Tanzania.
Book Stop opened in with two floors of books packed into one of Tavistock's oldest Tudor buildings. The current owners, Simon and Natasha Church, met while working for Author: Guardian Staff.
Welcome to West Wood Village. On FebruWest Wood Village, an active 55+ community, became a resident owned community located in Plymouth, Massachusetts. This close knit 66 home community received 5 out of 5 stars from an independent appraisal firm.
For more information about West Wood Village, check out the rest of our site. Dar es-Salaam (/ ˌ d ɑːr ɛ s s ə ˈ l ɑː m /; from Arabic: دار السلام , romanized: Dār as-Salām, meaning: Place of Peace) is the largest city and former capital of is the largest city in East Africa and the seventh-largest in Africa, with a population of 6, page 2 Located on the Swahili coast, Dar es-Salaam is an important economic centre and one of Districts: List, Ilala, Kigamboni, Kinondoni, Ubungo.
Access property records, Access real properties. Loading. Plymouth Live, Plymouth. likes 37, talking about this. Providing Plymouth news, sport and entertainment to you 24/7Followers: K. The Plymouth Public Schools does not discriminate on the basis of race, color, religious creed, national origin, sex, gender identity, sexual orientation, homelessness, genetic information, ancestry, status as a veteran or U.S.
uniformed military service member, disability, or age in admission to, treatment in, or employment in its services, programs, and activities. Need help with Book 2, Chapter 25 in William Bradford's Of Plymouth Plantation. Check out our revolutionary side-by-side summary and analysis. Thomas Weston is an English merchant, investor, and arguably one of the main antagonists of the book.
Initially he helps the English separatists obtain and negotiate a charter with the Virginia Company, whereby they'll sail to the Plymouth plantation and slowly pay off their debts. The second of two lavishly illustrated titles from acknowledged railway expert Colin G. Maggs, presenting the story of Devon's branch lines.
Shop Plymouth State University Apparel, Textbooks, Merchandise and Gifts at the Panthers Bookstore. Flat-Rate Shipping. Search the world's most comprehensive index of full-text books. My library. Town of Plymouth, Massachusetts 26 Court St., Plymouth, MA PH: FAX: Town Hall Hours: Mondays, Wednesdays, and Thursdays, a.m.
The History of Plymouth in Devon, England, extends back to the Bronze Age, when the first settlement began at Mount Batten a peninsula in Plymouth Sound facing onto the English continued as both a fishing and continental tin trading port through the late Iron Age into the Early Medieval period, until the more prosperous Saxon settlement of Sutton, later renamed Plymouth, surpassed it.
Originally published as The Landmark History of the American People from Plymouth to Appomattox, which came out in 22 chapters fromthis edition features new text alongside updated and revised text and new, colorful illustrations, timelines, and maps.
The first in a 2-volume series, this book covers the plantations and earliest settlements from Massachusetts to Virginia, the path Pages: The last section of the book is the vital records that were entered on the Colony records from to that pertain to Plymouth.
These were previously published by Nathaniel B. Shurtleff, but these are known to have transcription errors, so the corrections that appeared in the Mayflower Descendant were used for this book.Looking for plymouth brethren books? hosts over of the finest online book stores and booksellers, including specialists in plymouth brethren.
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Senator Harry Reid (D-NV), the Senate Majority Leader, has introduced the Senate version of the "health care" bill. The bill is over 2,000 pages in length and the senators were given 36 hours to read through it. Then there will be a vote Saturday evening. What is it with these Saturday votes? If Senator Reid and Representative Pelosi really want to remedy our health care problems and craft vital legislation, why are they ramming through these monstrosities on weekend nights? Why are they afraid of the light of the day? And, why won't they give our legislators sufficient time to review these unwieldy propositions?
I think the answer is clear... they don't want analysis, they want a political victory and they are not concerned about the human cost.
Tom Peters at The American Papist blog highlights a speech by Sen. Mike Johanns (R-Neb) who states that all it will take to defeat the public funding of abortion is for one pro-life Democratic senator to stand up and vote no. Who will that senator be? Let's hope and pray that a pro-life Democrat has the courage and the conviction to stand up for human life.